Equilibrium structural parameters (Vibrational Spectra and Structure; 24)

Citation preview

EDITORIALBOARD Dr. Lester Andrews University of Virginia Charlottesville, Virginia USA

Dr. J.A. Koningstein Carleton University Ottawa, Ontario CANADA

Dr. John E. Bertie University of Alberta Edmonton, Alberta CANADA

Dr. George E. Leroi Michigan State University East Lansing, Michigan USA

Dr. A.R.H. Cole University of Western Australia Nedlands WESTERN AUSTRALIA

Dr. S.S. Mitra University of Rhode Island Kingston, Rhode Island USA

Dr. William G. Fateley Kansas State University Manhattan, Kansas USA

Dr. A. MiJller Universit~it Bielefeld Bielefeld WEST GERMANY

Dr. H. Hs. G~inthard Eidg. Technische Hochschule Zurich SWITZERLAND

Dr. Mitsuo Tasumi University of Tokyo Tokyo JAPAN

Dr. P.J.Hendra University of Southampton Southampton ENGLAND

Dr. Herbert L. Strauss University of California Berkeley, California USA

PREFACE TO THE SERIES

It appears that one of the greatest needs of science today is for competent people to critically review the recent literature in conveniently small areas and to evaluate the real progress that has been made, as well as to suggest fruitful avenues for future work. It is even more important that such reviewers clearly indicate the areas where little progress is being made and where the chances of a significant contribution are minuscule either because of faulty theory, inadequate experimentation, or just because the area is steeped in unprovable yet irrefutable hypotheses. Thus, it is hoped that these volumes will contain critical summaries of recent work, as well as review the fields of current interest. Vibrational spectroscopy has been used to make significant contributions in many areas of chemistry and physics as well as in other areas of science. However, the main applications can be characterized as the study of intramolecular forces acting between the atoms of a molecule; the intermolecular forces or degree of association in condensed phases; the determination of molecular symmetries; molecular dynamics; the identification of functional groups, or compound identification; the nature of the chemical bond; and the calculation of thermodynamic properties. Current plans are for the reviews to vary, from the application of vibrational spectroscopy to a specific set of compounds, to more general topics, such as force-constant calculations. It is hoped that many of the articles will be sufficiently general to be of interest to other scientists as well as to the vibrational spectroscopist. As the series has progressed, we have provided more volumes on topical issues and, in some cases, single author(s) volumes. This flexibility has made it possible for us to diversify the series. Therefore, the course of the series has been dictated by the workers in the field. The editor not only welcomes suggestions from the readers, but also eagerlv solicits your advice and contributions.

James R. Durig Kansas City, Missouri

PREFACE TO VOLUME 24

The current volume in the series, Vibrational Spectra and Structure, is a single topic volume on gas phase structural parameters. The title of the volume, Equilibrium Structural Parameters, covers the two most common techniques for obtaining gas phase structural parameters: microwave spectroscopy and the electron diffraction technique. Since the quantum chemical method provides equilibrium geometries, the volume is an attempt to provide a connection between the experimental and theoretical parameters. In chapter 1, Professor Harmony has provided a review on molecular structure determinations from spectroscopic data using scaled moments of inertia. He has pointed out the limited number of molecules for which equilibrium parameters have been obtained and the requirement of a large amount of microwave data needed to obtain the equilibrium structural parameters. In chapter 2, Dr. Mastryukov reviews the electron diffraction technique. He also describes how the technique can incorporate structural information from microwave spectroscopy, vibrational spectroscopy, or theoretical calculations to improve the determination of the structural parameters by electron diffraction studies. In chapter 3, Dr. Groner has reviewed the theory and methods of microwave spectroscopy, describing in some detail r 0 and r s structures as well as r m structures and corrections based on ab initio calculations. The final chapter by Professor Bell reviews in some detail the accuracy of the molecular geometry predictions by quantum chemical methods. Professor Bell has recently calculated many of the predicted structural parameters and has presented the data in graphic form rather than in tabular form. The graphic presentation makes it possible to readily note the difference in the parameters predicted at the various levels of quantum mechanical calculations. Therefore, it is believed that the four authors have provided a rather coherent description of the various structural parameters obtained experimentally along with treatments needed to extract equilibrium bond distances and angles. The final chapter provides the necessary data for the reader to ascertain the expected quality of the ab initio predicted parameters based upon the level of the calculation. The editor thanks his administrative assistant, Mrs. Linda Smitka, for providing the articles in camera ready copy form and enduring some of the various tasks associated with the completion of the volume. He also thanks Mr. Richard Hester for typing the subject index and his wife, Marlene, for preparation of the pages with figures as well as some t f the proofreading.

James R. Durig Kansas City, Missouri vii

CONTRIBUTORS TO VOLUME 24

STEPHEN BELL Department of Chemistry, University of Dundee, Dundee DD 1 4HN, Scotland, U.K. PETER GRONER Department of Chemistry University of Missouri-Kansas City Kansas City, Missouri 64113, USA MARLIN D. HARMONY Department of Chemistry University of Kansas Lawrence, Kansas 66045, USA VLADIMIR S. MASTRYUKOV Department of Physics University of Texas Austin, Texas 78712, USA

ix

CONTENTS OF OTHER VOLUMES

V O L U M E 10

VIBRATIONAL SPECTROSCOPY USING TUNABLE LASERS, Robin S. McDowell INFRARED AND RAMAN VIBRATIONAL OPTICAL ACTIVITY, L. A. Nafie RAMAN MICROPROBE SPECTROSCOPIC ANALYSIS, John J. Blaha THE LOCAL MODE MODEL, Bryan R. Henry VIBRONIC SPECTRA AND STRUCTURE ASSOCIATED WITH JAHN-TELLER INTERACTIONS IN THE SOLID STATE, M.C.M. O'Brien SUM RULES FOR VIBRATION-ROTATION INTERACTION COEFFICIENTS, L. Nemes

V O L U M E 11

INELASTIC ELECTRON TUNNELING SPECTROSCOPY OF HOMOGENEOUS CLUSTER COMPOUNDS, W. Henry Weinberg

SUPPORTED

VIBRATIONAL SPECTRA OF GASEOUS HYDROGEN-BONDED COMPOUNDS, J. C. Lassegues and J. Lascombe VIBRATIONAL SPECTRA OF SANDWICH COMPLEXES, V. T. Aleksanyan APPLICATION OF VIBRATIONAL SPECTRA TO ENVIRONMENTAL PROBLEMS, Patricia F. Lynch and Chris W. Brown TIME RESOLVED INFRARED INTERFEROMETRY, Part 1, D. E. Honigs, R. M. Hammaker, W. G. Fateley, and J. L. Koenig VIBRATIONAL SPECTROSCOPY OF MOLECULAR SOLIDS-CURRENT TRENDS AND FUTURE DIRECTIONS, Elliot R. Bemstein

xv

xvi

CONTENTS OF OTHER VOLUMES

VOLUME 12 HIGH RESOLUTION INFRARED STUDIES OF SITE STRUCTURE AND DYNAMICS FOR MATRIX ISOLATED MOLECULES, B. I. Swanson and L. H. Jones FORCE FIELDS FOR LARGE MOLECULES, Hiroatsu Matsuura and Mitsuo Tasumi SOME PROBLEMS ON THE STRUCTURE OF MOLECULES IN THE ELECTRONIC EXCITED STATES AS STUDIED BY RESONANCE RAMAN SPECTROSCOPY, Akiko Y. Hirakawa and Masamichi Tsuboi VIBRATIONAL SPECTRA AND CONFORMATIONAL ANALYSIS OF SUBSTITUJED THREE MEMBERED RING COMPOUNDS, Charles J. Wurrey, Jiu E. DeWitt, and Victor F. Kalasinsky VIBRATIONAL SPECTRA OF SMALL MATRIX ISOLATED MOLECULES, Richard L. Redington RAMAN DIFFERENCE SPECTROSCOPY, J. Laane

VOLUME 13 VIBRATIONAL SPECTRA OF ELECTRONICALLY EXCITED STATES, Mark B. Mitchell and William A. Guillory OPTICAL CONSTANTS, INTERNAL FIELDS, AND MOLECULAR PARAMETERS IN CRYSTALS, Roger Frech RECENT ADVANCES IN MODEL CALCULATIONS OF VIBRATIONAL OPTICAL ACTIVITY, P. L. Polavarapu VIBRATIONAL EFFECTS IN SPECTROSCOPIC GEOMETR/ES, L. Nemes APPLICATIONS OF DAVYDOV SPLITTING FOR STUDIES PROPERTIES, G. N. Zhizhin and A. F. Goncharov

OF CRYSTAL

RAMAN SPECTROSCOPY ON MATRIX ISOLATED SPECIES, H. J. Jodl

CONTENTS OF OTHER VOLUMES

xvii

V O L U M E 14 HIGH RESOLUTION LASER SPECTROSCOPY OF SMALL MOLECULES, Eizi Hirota ELECTRONIC SPECTRA OF POLYATOMIC FREE RADICALS, D. A. Ramsay AB INITIO CALCULATION OF FORCE FIELDS AND VIBRATIONAL SPECTRA,

G~za Fogarasi and Peter Pulay FOUR/ER TRANSFORM INFRARED SPECTROSCOPY, John E. Bertie NEW TRENDS IN THE THEORY OF INTENSITIES IN INFRARED SPECTRA, V. T. Aleksanyan and S. Kh. Samvelyan VIBRATIONAL SPECTROSCOPY OF LAYERED MATERIALS, S. Nakashima, M. Hangyo, and A. Mitsuishi

V O L U M E 15 ELECTRONIC SPECTRA IN A SUPERSONIC JET AS A MEANS OF SOLVING VIBRATIONAL PROBLEMS, Mitsuo Ito BAND SHAPES AND DYNAMICS IN LIQUIDS, Waker G. Rothschild RAMAN SPECTROSCOPY IN ENERGY CHEMISTRY, Ralph P. Cooney DYNAMICS OF LAYER CRYSTALS, Pradip N. Ghosh THIOMETALLATO COMPLEXES" VIBRATIONAL SPECTRA AND STRUCTURAL CHEMISTRY, Achim Mtiller ASYMMETRIC TOP INFRARED VAPOR PHASE CONTOURS AND CONFORMATIONAL ANALYSIS, B. J. van der Veken WHAT IS HADAMARD TRANSFORM SPECTROSCOPY?, R. M. Hammaker, J. A. Graham, D. C. Tilotta, and W. G. Fateley

xviii

CONTENTS OF OTHER VOLUMES

VOLUME 16 SPECTRA AND STRUCTURE OF POLYPEPTIDES, Samuel Krimm STRUCTURES OF ION-PAIR SOLVATES FROM MATRIX-ISOLATION/SOLVATION SPECTROSCOPY, J. Paul Devlin LOW FREQUENCY VIBRATIONAL SPECTROSCOPY OF MOLECULAR COMPLEXES, Erich Knozinger and Otto Schrems TRANSIENT AND TIME-RESOLVED RAMAN SPECTROSCOPY OF SHORT-LIVED INTERMEDIATE SPECIES, Hiro-o Hamaguchi INFRARED SPECTRA OF CYCLIC DIMERS OF CARBOXYLIC ACIDS: THE MECHANICS OF H-BONDS AND RELATED PROBLEMS, Yves Marechal VIBRATIONAL SPECTROSCOPY UNDER HIGH PRESSURE, P. T. T. Wong

V O L U M E 17A SOLID STATE APPLICATIONS, R. A. Cowley; M. L. Bansal; Y. S. Jain and P. K. Baipai; M. Couzi; A. L. Verma; A. Jayaraman; V. Chandrasekharan; T. S. Misra; H. D. Bist, B. Darshan and P. K. Khulbe; P. V. Huong, P. Bezdicka and J. C. Grenier SEMICONDUCTOR SUPERLATTICES, M. V. Klein; A. Pinczuk and J. P. Valladares; A. P. Roy; K. P. Jain and R. K. Soni; S. C. Abbi, A. Compaan, H. D. Yao and A. Bhat; A. K. Sood TIME-RESOLVED RAMAN STUDIES, A. Deffontaine; S. S. Jha; R. E. Hester RESONANCE RAMAN AND SURFACE ENHANCED RAMAN SCATTERING, B. Hudson and R. J. Sension; H. Yamada; R. J. H. Clark; K. Machida BIOLOGICAL APPLICATIONS, P. Hildebrandt and M. Stockburger; W. L. Peticolas; A. T. Tu and S. Zheng; P. V. Huong and S. R. Plouvier; B. D. Bhattacharyya; E. Taillandier, J. Liquier, J.-P. Ridoux and M. Ghomi

CONTENTS OF OTHER VOLUMES

xix

V O L U M E 17B STIMULATED AND COHERENT ANTI-STOKES RAMAN SCATTERING, H. W. SchrStter and J. P. Boquillon; G. S. Agarwal; L. A. Rahn and R. L. Farrow; D. Robert; K. A. Nelson; C. M. Bowden and J. C. Englund; J. C. Wright, R. J. Carlson, M. T. Riebe, J. K. Steehler, D. C. N guyen, S. H. Lee, B. B. Price and G. B. Hurst; M. M. Sushchinsky; V. F. Kalasinsky, E. J. Beiting, W. S. Shepard and R. L. Cook RAMAN SOURCES AND RAMAN LASERS, S. Leach; G. C. Baldwin; N. G. Basov, A. Z. Grasiuk and I. G. Zubarev; A. I. Sokolovskaya, G. L. Brekhovskikh and A. D. Kudryavtseva OTHER APPLICATIONS, P. L. Polavarapu; L. D. Barron; M. Kobayashi and T. Ishioka; S. R. Ahmad; S. Singh and M. 1. S. Sastry; K. Kamogawa and T. Kitagawa; V. S. Gorelik; T. Kushida and S. Kinoshita; S. K. Sharma; J. R. Durig, J. F. Sullivan and T. S. Little

V O L U M E 18 ENVIRONMENTAL APPLICATIONS OF GAS CHROMATOGRAPHY/FOURIER TRANSFORM INFRARED SPECTROSCOPY (GC/FT-IR), Charles J. Wurrey and Donald F. Gurka DATA TREATMENT IN PHOTOACOUSTIC Michaelian

FT-IR SPECTROSCOPY,

K. H~

RECENT DEVELOPMENTS IN DEPTH PROFILING FROM SURFACES USING FTIR SPECTROSCOPY, Marek W. Urban and Jack L. Koenig FOURIER TRANSFORM INFRARED SPECTROSCOPY OF MATRIX ISOLATED SPECIES, Lester Andrews VIBRATION AND ROTATION IN SILANE, GERMANE AND STANNANE AND THEIR MONOHALOGEN DERIVATIVES, Hans Biirger and Annette Rahner FAR INFRARED SPECTRA OF GASES, T. S. Little and J. R. Durig

xx

CONTENTS OF OTHER VOLUMES

VOLUME 19 ADVANCES IN INSTRUMENTATION FOR VIBRATIONAL OPTICAL ACTIVITY, M. Diem

THE

OBSERVATION

OF

SURFACE ENHANCED RAMAN SPECTROSCOPY, Ricardo Aroca and Gregory J. Kovacs DETERM/NATION OF METAL IONS AS COMPLEXES I MICELLAR MEDIA BY UV-VIS SPECTROPHOTOMETRY AND FLUORIMETRY, F. Fernandez Lucena, M. L. Marina Alegre and A. R. Rodriguez Fernandez AB INITIO CALCULATIONS OF VIBRATIONAL BAND ORIGINS, Debra J. Searles

and Ellak I. von Nagy-Felsobuki APPLICATION OF INFRARED AND RAMAN SPECTROSCOPY TO THE STUDY OF SURFACE CHEMISTRY, Tohru Takenaka and Junzo Umemura INFRARED SPECTROSCOPY OF SOLUTIONS IN LIQUIFIED SIMPLE GASES, Ya. M. Kimel'ferd VIBRATIONAL SPECTRA AND STRUCTURE OF CONJUGATED CONDUCTING POLYMERS, Issei Harada and Yukio Furukawa

AND

V O L U M E 20 APPLICATIONS OF MATRIX INFRARED SPECTROSCOPY TO MAPPING OF BIMOLECULAR REACTION PATHS, Heinz Frei VIBRATIONAL LINE PROFILE AND FREQUENCY SHIFT STUDIES BY RAMAN SPECTROSCOPY, B. P. Asthana and W. Kiefer MICROWAVE FOUR/ER TRANSFORM SPECTROSCOPY, Alfred Bauder AB INITIO QUALITY OF SCMEH-MO CALCULATIONS OF COMPLEX INORGANIC

SYSTEMS, Edward A. Boudreaux CALCULATED AND EXPERIMENTAL VIBRATIONAL SPECTRA AND FORCE FIELDS OF ISOLATED PYRIMIDINE BASES, Willis B. Person and Krystyna Szczepaniak

CONTENTS OF OTHER VOLUMES

xxi

V O L U M E 21 OPTICAL SPECTRA AND LATTICE DYNAMICS OF MOLECULAR CRYSTALS; % N. Zhizhin and E. I, Mukhtarov

VOLUME 22 VIBRATIONAL INTENSITIES, B. Galabov and T. Dudev

VOLUME 23 MOLECULAR APPROACH TO SOLIDS, A. N. Lazarev

CHAPTER MOLECULAR SPECTROSCOPIC

STRUCTURE DATA USING

1

DETERMINATION SCALED

MOMENTS

FROM OF INERTIA

Marlin D. H a r m o n y Departmem of Chemistry University o f K a n s a s L a w r e n c e , KS 66045, U S A

I.

INTRODUCTION

II.

REVIEW

..................................................................................................... 2

O F B A S I C S .............................................................................................. 6

III. M O L E C U L A R

STRUCTURE

METHODS

......................................................... 14

A.

r 0 and r e Structures ...................................................................................................... .14

B.

r s Structures .................................................................................................................. 24

C.

r m Structures ................................................ ................................................................ 31

D.

Miscellaneous Methods ............................................................................................... 40

E.

rPm Structures .............................................................................................................. 46 1. Equilibrium structures ofpolyatomic organic molecules ...................................... 46 2. Some pivotal observations ..................................................................................... 48 3. rPm Heavy-atom structures .................................................................................... 53 4. rPm Structures of hydrogen-containing molecules ................................................. 66 5. rPm Applications .................................................................................................... 75

IV.

CONCLUSION

........................................................................................................ 77

ACKNOWLEDGMENTS

REFERENCES

............................................................................................... 79

................................................................................................................ 80

2

I.

HARMONY

INTRODUCTION A principal end result of the high-resolution spectroscopic study of gasphase molecules has been the determination of molecular structures. The concept of structure, or geometrical arrangement of atoms or atomic nuclei in molecules, has been a natural one at the heart of molecular sciences throughout the 20 th century. Its theoretical origin traces to the early work in quantum mechanics and is embodied most clearly in the Bom-Oppenheimer (B-O) approximation [ 1]. This work leads to the concept of the Bom-Oppenheimer potential surface, which maps out the energy of the N atoms (or nuclei) in a molecule as they undergo their vibrational motions in a 3N-6 (or 5) dimensional space. The molecular structure defined by the positions of the nuclei at the global minimum of the 3N--6 (or 5) dimensional surface is commonly known as the e q u i l i b r i u m structure and is usually denoted as an r e structure. A molecule usually has distinctly different B-O potential surfaces (and, hence, r e structures) in its ground and excited electronic states [2]. The discussion in this work will focus attention implicitly upon the ground state potential surface and molecular structure. Based upon the Born-Oppenheimer separation of electronic and nuclear energies, the B-O potential surface should be isotopically invariant and the r e structure of a molecule should also be isotopically invariant. Most of the tightly bound molecules of interest to chemists conform very well to the B-O approximation, although all molecules deviate when examined at sufficient precision. Weakly bound hydrogen-bonded species or van der Waals molecules such as Ar'..HCI are exceptions that require special considerations [3 ].

MOLECULAR STRUCTURE DETERMINATION

3

While the r e structure represents the most well-defined molecular geometry, it is not, unfortunately, one that exists in nature.

Real molecules exist in the

quantum states of the 3N-6 (or 5) vibrational states with quantum numbers (v l, v2,...V3N-6 (or 5)), vi = 0, 1, 2 . . . . .

Even in the lowest (ground) (0,0...0)

vibrational state, the N atoms of the molecule undergo their zero point vibrational motions, oscillating about the equilibrium positions defined by the B-O potential energy surface. It is necessary then to speak of some type of average or effective structures, and to account for the vibrational motions, which vary with vibrational state and isotopic composition.

In spectroscopy, a molecule's

structural

information is carried most straightforwardly by its molecular moments of inertia (or their inverses, the rotational constants), which are determined by analysis of the pure rotational spectrum or the resolved rotational structure of vibrationrotation bonds. Thus, the spectroscopic determination of molecular structure boils down to how one uses the rotational constants of a molecule

Av lV2- 9 9V3N-6 (or 5) Bv lV2 9 9 9V3N-6 (or 5)

(1)

CVlV2 9 9 9V3N-6 (or 5)

in one or more specific states (Vl . . . .

V3N-6 (or 5)) to extract the geometry. Of

particular importance will be the ground state rotational constants (symbolized simply by A 0, B 0, CO), which are those most easily accessible to experiment. What emerges then is that there is a plethora of structural types, depending upon precisely what data are used and precisely how they are used.

Using

spectroscopically determined rotational constants (and, thus, moments of inertia),

4

HARMONY

one can compute and distinguish several types of structures, including re, r 0, (rv), rs, rm, rz, rgi, and map (see Table 1). We will review the characteristics of these various structural types or methods, concentrating eventually our principal attention upon the rm o structure [4-10], which has recently been developed as an excellent approximation to the r e structure of relatively large polyatomic molecules. There is, of course, another important structural method for gas-phase molecules, viz., electron diffraction. Here one does not probe the properties of particular quantum states by spectroscopy; rather, the scattering of electrons from the atoms of a thermal distribution of molecules is studied in a fashion analogous to x-ray diffraction of crystalline solids. This leads to entirely different types of structures, for example, rg. Vibrational motions are, of course, once again the complicating factor. With appropriate corrections, electron diffraction data have led to r e structures of high accuracy. The abundant literature should be consulted for the methods of electron diffraction [ 11 ]. We shall see shortly that it is really quite easy to obtain the r e structures of diatomic molecules to very high accuracy. Also, for a relatively minor number of small polyatomic molecules (such as OCS, SO2, CO 2, CH4), it has been possible to determine r e structures. Certainly, when possible, this is the desired result. As we shall shortly describe, the task is formidable once the molecule reaches 4 or 5 atoms or more, in general. Indeed, for an important prototype organic molecule such

as

ethane

(C2H6) ,

a

true

re

structure

(as

described

below)

MOLECULAR STRUCTURE DETERMINATION

5

T A B L E 1. Types of structures

Symbol

Name

Procedure

ro

Effective

Structure obtained by fitting directly ground state moments of inertia

re

Equilibrium

Structure obtained by fitting equilibrium moments of inertia

rs

Substitution

Structure obtained with coordinates computed from Kraitchman's equations with ground state moments of inertia of several isotopomers

rm

rOm

rz

Mass-dependence

Structure obtained using Im values computed from

or Watson' s

isotopic I0 values by Watson's equations

Scaled moment of

Structure computed from I Pm values obtained by

inertia

scaling I 0 values

Average

Structure computed from moments of inertia of the molecule when atoms are in their average positions Approximate corrections for vibration-rotation

re,I

contribution to I0; resulting structure closely equivalent to r s rg

Thermal average

Thermal average of distances and angles from electron diffraction data

has never been determined. Our interest was, in fact, to attack the problem of r e structures for polyatomic molecules such as ethane, propane and formic acid. This led to the development of the rOm structural method, which provides an excellent approximation to the r e structure for molecules larger even than ethane.

6

HARMONY

Over the past decade or so, the capabilities of ab initio quantum theory have advanced remarkably due to improvements in computer speeds. It is now possible to compute r e structures by ab initio methods for relatively large molecules at the Hartree-Fock level using reasonably sophisticated basis sets [12]. With the inclusion of electron correlation by CI or perturbation methods [13], computed ab initio bond distances are now becoming competitive in quality with experimental r e values. However, since experimental r e values have been largely unavailable for organic molecules of even modest size, it has not been possible to do careful benchmarking. With the addition of the rm ~ (scaled moment of inertia) method, it is now possible to make legitimate comparisons. As an example, we recently determined the near-equilibrium (rm~ ) CC distance in ethane to be 1.522 A from available spectroscopic data [8], providing a firm benchmark for this fundamental prototype CC single bond. Recent ab initio calculations [14] at the HF level with a 6-31G** basis set yield a value of 1.527 A, while addition of MP2 electron correlation gave 1.524 A.

This is really quite excellent quality and

demonstrates the convergence of experiment and theory.

II.

REVIEW OF BASICS In the principal axis system (PAS) of a general polyatomic molecule with the origin of the molecule-fixed axes at the center of mass (COM), the principal moments of inertia are related to the coordinates of the atoms of mass m i by

I a = Iaa = Z m i (b 2 + c 2) i

(2a)

M O L E C U L A R STRUCTURE D E T E R M I N A T I O N

I b =Ibb = ~ m i ( a 2 + c 2)

7

(2b)

i

2 I c = Ice = E m i ( a 2 + b i ) i

(2c)

In the PAS centered at the COM, the off-diagonal components of the inertial tensor I vanish:

lab = - ~ miaib i = 0 i

Iac = - ~ m i a i c

i=0

(3)

i

Ibc = - ~ m i b i c

i= 0

i

Also, the COM condition leads to

~mig i =0 (4)

i

gi = a i , b i , c i From spectroscopy, the rotational constants A, B, C are actually measured"

A = h/87t2I a

(5a)

B = h/8zr2I b

(5b)

C = h/8zr2I c

(5c)

where, for practical purposes h/87t 2 = 505379 amu A 2 MHz. By convention, we choose

A>_B_>C

or

Ia < Ib < Ic

For prolate symmetric rotor molecules,

(6)

8

HARMONY

A>B=C

(7a)

while for oblate rotors

A=B>C

(7b)

In the linear molecule case we have I a = 0 and

I=I b-I c

orB=C

(7c)

Finally, a spherical rotor is defined by

A=B=C

(7d)

In all cases, by virtue of Eqs. (2), it is clear that the molecular moments of inertia of a particular set of atomic isotopes is entirely determined by the atomic coordinates, i.e., the structure. It is also obvious that the moments are isotopomerdependent, since they depend upon atomic mass. It is always possible (in principle) to express Ig (g = a, b, c) in terms of internal parameters, such as bond lengths, bond angles and dihedral angles. These

will be no more than 3N--6 (or 5 for linear molecules) in number, depending upon symmetry restrictions.

Thus, for linear X-X, X-Y, Y-X-Y, X - Y - Z and non-

linear Y-X-Y, X-Y-Z, the number of internal parameters is 1, 1, 1, 2, 2, 3,

respectively. For larger organic molecules, the number of independent parameters grows rapidly; thus, for formic acid (HCOOH, the smallest organic acid) in either of its two planar conformations, there are seven internal structural parameters.

MOLECULAR STRUCTURE DETERMINATION

9

In a few simple cases, the formulas in terms of internal structural parameters can be written easily. Thus, for a diatomic molecule with interatomic distance r and atomic masses m l and m 2,

Ib = i = ~2

where

(8)

(9)

mlm2 ml m2

while for an X - Y - Z linear molecule (such as OCS) with masses m x, my, m z (total mass M) and distances r(XY) and r(YZ),

1 {mxmyr(Xy)2 + mymzr(Zy)2 + mxmz[r(XY ) + r(YZ)]2 } (10) I = I b = ~-

In most cases, analytical expressions are not conveniently written, but in all cases the problem is easily tractable on a computer. The key point to make here is that moments of inertia and, hence, rotational constants (via Eqs. (5)) contain all the information about structure. It should be stressed here also that all the moment of inertia equations above deal with a collection of point masses with fixed positions.

They also

specify properties of a vibrating molecule (a collection of point mass atoms) at some instant in time, or at some point (perhaps fictitious) in 3N-6 (or 5) dimensional space at which the atoms are motionless, such as the equilibrium position. There is a set of very important equations which deal with the solution of Eqs. (2) under the assumption that the atomic positions do not change under isotopic substitution.

If we substitute a particular atom with an isotope that

10

HARMONY

changes the atomic mass by Am, the total mass of the new molecule will change from M to M + Am and the moments of inertia will, in general, change by amounts AIg (g = a, b, c). Then, as shown by Kraitchman [15], the magnitude of the a coordinate of the substituted atom in the PAS of the original (unsubstituted or

parent) molecule is given by

I APa (1

lal- / ~t'

and [b[ and

+

APb Pb - Pa

)(1 +

APc Pc - Pa

)

] 1/2

(11)

[el are obtained by cyclic permutations of a, b and c. The Pg are the

planar secondmoments, for example,

and

1 Pa = 2 (Ib + Ie - Ia)'

(12)

~t, = Mam M+Am

(13)

The other two principal planar moments and expressions for

Ibl and Icl are given

by cyclic permutations of a, b, c. Equations of the form of Eq. (11) are known as the "Kraitchman" equations. symmetry elements.

Special cases arise for molecules with various

For substitution of an atom in a linear molecule, the

magnitude of the a coordinate of the substituted atom is

lal = (AIb/~') ~

(14)

More complete discussions of Kraitchman's equations can be found in standard treatises [16]. In addition, analogous equations are available for multiple isotopic substitution of equivalent atoms, e.g., H20 to D20 [ 17].

MOLECULAR STRUCTURE DETERMINATION

11

The details of spectral analysis need not concem us here; standard sources should be consulted [ 16,18]. All we need to know for our purposes is that analysis of rotational spectra or vibration-rotation bands leads to what are called the

effective

rotational constants as symbolized in Eqs. (1). For simplicity we shall

often write simply A v, B v and Cv, but it is imperative to remember that the simple subscript signifies all the vibrational quantum numbers. The constants are related to the

equilibrium rotational

effective

rotational

constants by

di

Av = A e - ~( V t~A ii

+ -~-) ...+

(15a)

Be- )-"( VOtiBi

di + -~--)...+

(15b)

di Cv --- Ce - )-"i~c (v i + -~-) + ...

(15c)

Bv=

in which the sums are over all the unique vibrations of degeneracy d i and the vibration-rotation constants ai are, in general, different for the three axes and the various modes.

For many purposes, the neglected higher-order terms are

negligible. From measured A v, Bv, Cv values, effective moments of inertia Iav, Ibv, Icv are computed from Eqs. (5). If the ot's are also determined experimentally (see below), then Ae, B e and C e can be obtained, which then leads to the equilibrium moments Iae, Ibe and Ice. Sometimes it is more convenient to define the vibration-rotation contributions in terms of the moments of inertia.

For

example, Igv and Ige (g = a, b, c) are related by

di Igv = Ige + EiI~g(vi + y ) ...+

If all higher terms in Eqs. (15) and (16) are neglected, then

(6)

12

HARMONY

(17)

~ig ~ (-~-~)I2e (zgi

Note that for a diatomic molecule (with I b - I and gb = ec = ~;), Eq. (16) becomes

I v = I e + e(v + 89

(18)

I 0 = I e + 89 e

(19)

or

in the ground vibrational state. Thus, it is seen clearly that even the ground state m o m e n t o f inertia suffers a vibration-rotation defect so that it differs from the equilibrium value. It is worth noting the experimental task involved in evaluating equilibrium rotational constants. For the diatomic molecule, it is necessary to measure B in a m i n i m u m o f two states, e.g., v = 0 and v = 1. Then Eq. (15) yields B0 = Be

mc 2

B1 = Be

3me 2

The experimental values o f B 0 and B 1 yield the desired B e and me values.

(Here

we note for the diatomic molecule that mb = mc = m is usually written me to identify it as an equilibrium property.

W e have avoided the subscript e in Eqs. (15) and

(17) to simplify notation.) For the linear XYZ molecule, with two non-degenerate and one doubly degenerate vibrations, one B e value and three ot values must be determined: B v = B e - m1(v 1 + 89 - m2(v2 + 1) - m3(v 3 + 89

M O L E C U L A R STRUCTURE DETERMINATION

13

Thus, if the (000), (100), (010) and (001) states are studied, the following four equations can be solved to obtain Be, orl, or2, and ot3" B(000) = B e - 89 ct1 - c t 2 - 89 ot3 B(100) = B e - 3/2tXl - c t 2 - 89 ot3 B(010) = B e - 89 ~ 1 - 2ct2 - 89 t~3 B(001) = B e - 89 ct 1 - a2 - 3/2 a3 Non-linear polyatomic molecules are treated similarly except now there are three rotational constants in general.

Thus, for non-linear XYZ (such as C1NO)

molecules, experimental values of A v, B v and C v in the non-degenerate states (000), (100), (010), (001) lead to values of Ae, B e and C e and the nine agvalues. It should be clear then that for even modest-sized polyatomic molecules, the determination of me, B e and C e requires a formidable amount of data from excited vibrational states. Specifically, for an N-atom polyatomic asymmetric rotor (all di values = 1), a total of 3N-6 vibration modes exist and thus 3N-6 excited vibrational states must be probed in addition to the ground state. It turns out then that equilibrium rotational constants (and equilibrium moments of inertia) are not available for very many non-linear polyatomic molecules. It should be mentioned here finally that it is assumed that the rotational spectral analysis has properly accounted for centrifugal distortions in all cases and for other possible factors such as Coriolis interactions, Fermi resonances, internal

rotation, etc., so that the rotational constants A v, B v and C v of Eqs. (15) have these influences removed [16,18]. Moreover, special additional care is needed for

14

HARMONY

those relatively few gas-phase molecules having either electronic spin or orbital angular momentum. With this very brief summary, it is possible now to move on to a discussion of structure. Although our principal aim is to describe and utilize a very recently developed procedure for obtaining what we call rm ~ structures, it is useful and important to review briefly the most common spectroscopic structural procedures.

III.

MOLECULAR STRUCTURE METHODS

A.

r 0 and r e Structures The simplest structural procedure is to use ground state (v = O) effective moments via equations of the form of Eqs. (2), (8) or (10). Such structures are commonly known as r 0 or effective structures since they utilize only A 0. B0, C O values.

For most polyatomic molecules it is customarily

assumed that the interatomic distances are isotopically invariant, even though this is true only for a rigid, non-vibrating or equilibrium molecule. Therefore, data from several isotopic species can be invoked. Thus, for OCS, ifB 0 is determined for each of the isotopes 16-12-32 and 16-13-32, the two B 0 values permit determination of the C-O and C-S distances from two equations with the form of Eq. (10).

Indeed, various isotopomer

combinations are possible, as summarized in Table 2. It is seen that the r 0 structure parameters vary rather widely (far outside experimental error), depending upon which pair is selected for the calculation.

This is a clear

example of the deleterious effects of uncorrected zero-point vibration terms (a or e), along with the assumption of isotopic invariance.

M O L E C U L A R STRUCTURE DETERMINATION

15

T A B L E 2. Effective (r0) structure of OCS a

Isotopomers used

C - O (A)

C-S (A)

16-12-32, 16-12-34

1.1647

1.5576

16-12-32, 16-13-32

1.1629

1.5591

16-12-34, 16-13-34

1.1625

1.5594

16-12-32, 18-12-32

1.1552

1.5653

Mean

1.1614

1.5604

Range

0.0095

0.0077

aSee Ref. [ 19].

Consider the simplest diatomic case. Here it is possible to obtainB v and I v in several excited states v as briefly summarized in Table 3 for the molecules CsC1 and HC1. Then from Eq. (8) we obtain rv:

rv = (Iv/~t)l/2

(20)

Note the regular variation in r v as v changes; indeed, note that r shrinks as the vibrational quantum number decreases. It is the well-known non-harmonic (anharmonic), potential curve that leads to this characteristic variation. Indeed, quantum mechanics shows to a high approximation that r v is a welldefined average

rv= v 1/2

(21)

16

HARMONY

T A B L E 3. Rotational constants and bond lengths for 133Cs35C1 and H35Cla.

v

Bv (MHz)

rv (A)

CsCI 3

2125.955

2.9303

2

2136.013

2.9234

1

2146.092

2.9165

0

2156.191

2.9097

B e = 2161.189

r e = 2.9063

HCI 3

285847.5

1.3434

2

294835.9

1.3228

1

303876.4

1.3030

0

312990.9

1.2839

B e = 317582.6

r e = 1.2746

aSee Ref. [20] for more extensive data.

On the other hand, for a polyatomic molecule, no well-defined average

distance exists.

For example, for the general polyatomic molecule, the

experimental Igv value is given by an average of I corrected for Coriolis terms [21 ]:

1 = ( 1 ) v + Coriolis terms Igv lg

(22)

It is clear that no simple relation can be written for the distances even if the Coriolis contributions are evaluated. (Imagine averaging the inverse of Eq. (10)).

MOLECULAR STRUCTURE DETERMINATION

17

Now, for diatomic molecules such as CsCI or HC1, the B e, I e and, hence, r e values are easily measured because B 0, B 1, B2 9 9 9 data are usually readily obtained. (Table 3 contains merely a sample of measured B v values [20].) Using any two B v values (such as B 0 and B 1) or doing a least-squares fit, we obtained directly the B e and ~e values and thus r e is obtained as

re = (Ie/~t)l/2

(23)

Table 3 summarizes these results for the two example molecules. Note that r e is obtained alternatively by extrapolating the r v values to v = - 8 9 the hypothetical vibrationless state. Data for the isotopomers of HC1 provide insight into typical mass-dependent structural properties.

From the

isotopomer data of Table 4, it is immediately evident that the r e bond distance of HC1 is invariant (within 1/104) to isotopic substitution, in accord with the expectations of the B-O approximation. It is clear also from the HCI, DCI, TC1 data that the effective bond distance shrinks substantially as the H-atom is substituted by a heavier isotope. It is seen, in contrast, that substitution of a heavy atom (37C1) for a light atom (35C1) has no effect (for HC1, at least) upon the bond length to within 10-4 A. Generally, heavy atom substitutions lead to changes on the order of 0.0001 A, with the trend in the direction identical to the H-atom case. Thus, the r 0 value of 13CO is 0.0001 A shorter than for 12CO [20]. Qualitatively, this mass dependence is easily understood as arising from the heavier isotopomers falling more deeply into the potential

18

HARMONY

TABLE 4. Structural isotope effects for HC1a. r0

re

HC1

1.2839 A

1.2746 A

DCI

1.2812

1.2746

TCI

1.2801

1.2746

H37C1

1.2839

1.2746

a35c1 isotope unless noted. See Ref. [20] for more extensive data.

well, which always leads to shrinkage. It is significant to note that the bond length mass dependence is an order of magnitude greater for the H-atom than for the heavy atoms.

In principle, the r e distance of HC1 might also be

considered to be the value that would be obtained by extrapolating the HC1, DC1, TCI r 0 data to the point at which the hydrogen atom mass goes to infinity. In summary then, diatomic molecule r e structural data are relatively readily available because only one vibrational mode exists and, hence, only one vibration-rotation constant (a) needs to be determined.

Table 5

summarizes data for just a few of the many diatomics that have been investigated.

Note that the experimental precision is sufficiently high in

general to permit the r e length to be specified with an accuracy better than 1 • 10-4 A. Extensive data are available in the literature [20,22].

MOLECULAR STRUCTURE DETERMINATION

TABLE 5.

19

Equilibrium rotational and structural parameters of diatomic molecules a.

HC1

Be

me

re

(MHz)

(MHz)

(A)

317582.7

9209

1.2746

KC1

3856.38

23.68

2.6667

BaO

9371.93

41.73

1.9397

SnO

10664.19

64.24

1.8325

CO

57898.35

OH

566932

524.6 2.171 • 104

1.1283 0.9697

NO

50123.8

513

1.1507

CN

56953

520.7

1.1718

H2

1.8243 • 106

9.180 x 104

0.7414

N2

59905.8

519.2

1.0977

02

43337.9

474.8

1.2075

C2

54557

529

1.2425

C12

7314.6

44.7

1.9879

aAll species are 12g except for OH, NO, CN, and 02, which are 2II, 21I, 2s

and

3 Xg, respectively. More extensive tabulations are given by Huber and Herzberg [22] and by Lovas and Tiemann [20]. Data refer to the most abundant isotopic species.

Returning to the linear triatomic molecule case, we note now that the unsatisfactory situation summarized in Table 2 can be remedied for many triatomics, since the existence of only three (at most) ot values for each principal axis makes the r e structure determination still relatively tractable. Table 6 summarizes some of the key parameters for several triatomic

20

HARMONY

T A B L E 6. Equilibrium rotational and structural parameters for triatomic molecules a ota

ctb

ore

Rotational

Equilibrium

(MHz)

(MHz)

(MHz)

constant (MHz)

structure

Mode

SO 2

1

33.60

50.42

42.75

A e = 60502.05

r e = 1.431 A

(C2v)

2

-1127.41

-2.39

15.98

B e = 10359.24

0 e = 119.3 ~

3

612.44

34.24

32.03

C e = 8845.57

F20

1

-430.95

72.08

39.32

A e = 58744.90

r e = 1.405 A

(C2v)

2

--699.02

42.36

53.28

B e = 10985.28

0 e = 103.1 ~

3

585.01

69.56

115.25

C e - 9255.27

H20

1

0.750*

0.238*

0.2018"

A 0 = 835833

r e = 0.959 A

(C2v)

2

-2.941"

-0.160"

0.1392"

B0=435094

0 e = 103.9 ~

3

1.253"

0.078*

0.1445"

C o = 278372

OCS

1

18.13

(Coov)

2

-10.59

3

36.43

CO 2

1

0.00126*

(Dooh)

2

0.00076*

3

0.00309*

(CO)e = 1.154 A B e = 6099.22

(CS)e = 1.563/~

B e = 0.391625*

r e = 1.160 A

aAsterisks indicate units of cm -1. For more extensive data, see Lovas [23], Herzberg [2,18], Harmony et al. [24], and Landolt-Bomstein [25].

Data refer to the most abundant isotopic

species.

molecules for illustrative purposes.

For the triatomics, the r e precisions are

usually in the range 1 x 10 -4 - 1 • 10-3 A, so we list the distances r o u n d e d to 1 • 10-3 A. Angles are listed with comparable relative precision. M o r e generally, for large polyatomic molecules, it is often possible to obtain sufficient data for an r o structure but not for an r e structure. A simple e x a m p l e can illustrate the problem. Consider ethylene oxide (C2v s y m m e t r y ,

MOLECULAR STRUCTURE DETERMINATION

21

see Fig. 1), which has five independent structural parameters. By studying the microwave spectra of simply the normal isotopomer and one additional species (such as 13C in natural abundance, or perhaps even 180 in natural abundance, or the D 4 species by isotopic enrichment via the compound synthesis), six independent rotational constants are obtained in the ground state (A0, B0, C O for two isotopomers).

Thus, these data, along with the

assumption of isotopic invariance, lead via a least-squares fit to a conventional r 0 structure.

Just as was observed for OCS in Table 2, the

derived parameters will vary, depending upon which isotopic data are utilized. Berry and Harmony [7] have reported an r 0 structure for ethylene oxide using a least-squares fit of the 12 10 values from four isotopomers. On the other hand, obtaining a true r e structure for ethylene oxide requires that 3N--6 = 15 ct values must be determined for each axis; i.e., 45 otgparameters must be evaluated to obtain a set of Ae, Be, Ce values for a single isotopomer. This requires complete analysis of the ground vibrational state and 15 excited vibrational states, a total of 16 complete rotational spectral analyses,

Then, the process must be repeated for an additional

isotopomer in order to obtain a sufficient number of equilibrium rotational constants to compute the five structural parameters.

Thus, the task of

obtaining an r e structure is truly formidable for even a relatively small polyatomic molecule. Of course, for a few polyatomics with high symmetry, r e structures

22

HARMONY

O

,/

H" HI

0

FIG. 1. Structural sketch of ethylene oxide.

have been obtained. Methane (CH4) is a prime example. With T d symmetry, there exist only four unique vibrational modes (with degeneracies d 1 = 1, d 2 = 2, and d 3 = d 4 = 3) and one structural parameter, which has been determined to be re(CH ) = 1.0858 A [26]. A second example is acetylene (Dooh) with five modes (three with d i = 1, two with d i = 2) and two parameters whose r e values are r e (C-C) = 1.2026 A and re(C-H ) = 1.0622 A [27]. Finally, we close this discussion of r e and r 0 structures by presenting in Table 7 the r 0 structure of the bona fide organic polyatomic molecule propane (C3H8) , with a C2v shape, as shown in Fig. 2. The nine independent structural parameters were determined by a least-squares fit of the groundstate moments of inertia of six isotopomers, or a total of 18 independent I 0 values [9]. (As usual, the structure is assumed to be isotopically invariant.)

MOLECULAR STRUCTURE DETERMINATION

23

TABLE 7. Molecular structure of propane a

rgm

ro

rs

CC

1.5209 (9)

1.5337 (10)

1.5279

CH

1.0929 (20)

1.0952 (23)

1.0944

CH s

1.0877 (35)

1.0911 (41)

1.0864

CH a

1.0907 (19)

1.0945 (22)

1.0937

LCCC

112.35 (11)

112.14 (13)

112.23

ZHCH

106.13 (32)

106.54 (37)

106.31

ZHaCH a

107.04 (28)

107.50 (32)

107.32

LCCH s

111.60 (31)

111.30 (35)

111.89

LCCH a

110.62 (10)

110.50 (12)

110.63

aAll distances in ,~ and angles in degrees. Subscripts s and a identify methyl group hydrogens lying in and out of the molecular symmetry plane, respectively. Unlabeled hydrogens occupy the methylene group. See Ref. [28] for original data. Reprinted from Ref. [9] with permission.

H

/

H

/ /

H CJ C c/H \

/

H

H

FIG. 2. Structural sketch of propane.

"H

24

HARMONY

Note that as few as three isotopomers might have sufficed (nine I 0 values), so that a whole range of r 0 structures (as for OCS in Table 2) might have been calculated. The use of all the available data leads to the best overall average r 0 structure. Treating the deviations as being merely statistical, we list the standard deviation (a) in parentheses as some kind of representation of the quality of the fit.

The other structures in the table (rgm and rs) will be

discussed shortly. We note that no true, complete, spectroscopic r e structure is available or even ever likely to become available. For each of a minimum of three isotopomers, a total of 27 excited states (3N--6) plus the ground state would need to be fully spectrally analyzed to obtain the necessary nine Ie values!

B~

r s structures The most extensively used spectroscopic structural method for polyatomic molecules is known as the substitution (rs) method and was proposed by Costain [29] some 40 years ago. In this method, the equations of Kraitchman (Eq. (11)) are used with ground state (I0) moments of inertia for a parent (or normal) molecule and an appropriate collection of isotopomers. It has been generally supposed that the resulting r s coordinates and structural parameters are better approximations to the equilibrium (re) parameters than the simpler r 0 effective parameters. This idea arises from the following simple argument.

Consider a diatomic parent molecule with

moment I 0 and one of the isotopomers with moment I0'. From the diatomic

M O L E C U L A R STRUCTURE D E T E R M I N A T I O N

25

molecule Kraitchman equation (14), calling the coordinate z s and writing I 0 for the Ib0, the square of the coordinate is

z 2 = ~ . I ~ I~

(24)

s

If now Eq. (19) is inserted

2 - " ( I . ' - I ) +e 8 9

e)

Z$

(25)

Then it is clear that if e is a slowly varying function of atomic mass, the vibration-rotation contributions to the moments of inertia will at least partially cancel out, leaving a result closer to the equilibrium result. Indeed, if c z E', then

2 ~ -I e'-Ie , ~ =

Z s

Z

2

(26)

e

so z s ~-, Ze, which shows that the resulting interatomic distance will satisfy r s -~ r e. Costain showed that for the diatomic molecule [29]

r s ~ 89 (r o + re)

to a first-approximation,

(27)

so that the r s distance is, indeed,

a better

approximation to r e than r 0. Because r e < r 0 as described earlier,

re < rs < ro

will always be true for diatomics.

(28)

Moreover, according to Eq. (27), the r s

procedure is expected to correct for about 50% of the r 0 distance error. As an

26

HARMONY

example, if the I 0 values of H35C1 (the parent), D35C1 and H37C1 are used with Eq. (24) to obtain the H and 35C1 z-coordinates, the H-CI distance is found to be r s = 1.2783 A. Comparing this value to the r 0 and r e values in Table 3, it is clear that Eq. (28) is satisfied. Moreover, the r s procedure has removed about 60% of the r 0 discrepancy. As the latter example has illustrated, for the diatomic molecule more data are required to obtain an r s structure than are needed

for an re

structure.

Thus, there is no advantage to be gained generally in obtaining an r s structure for a diatomic molecule, especially since the result is still inferior to the r e value

because

of the

residual

uncancelled

vibration-rotation

terms.

Consequently, r s values of diatomic molecules are seldom computed or tabulated. For polyatomic molecules, even the simplest linear ones, there is no proof that the r s structure should be closer to equilibrium, but it has nevertheless been assumed to be the case. Because the Kraitchman equations rely most heavily on AI values upon isotopic substitution, it has seemed reasonable to suppose that even an incomplete cancellation of vibrationrotation terms would lead to a structure that was closer to equilibrium than the r 0 structures.

For this reason, and because r e structures are simply not

possible, the r s method has been used extensively to obtain structures for polyatomic molecules.

Unfortunately, although the early work of Costain

[29] suggested that the r s structure would be a fairly reliable representation of

MOLECULAR STRUCTURE DETERMINATION

27

the equilibrium structure, the passage of time has abundantly illustrated [410] that this was an overly optimistic view. There are a few obvious and not-so-obvious problems with the r s method. First, it is easy to show that small coordinates are determined very poorly (i.e., with large uncertainties) by Kraitchman's equations [23j. Indeed, an extreme case of this occurs when the value of the quantity in square brackets on the right side of Eq. (11) becomes negative, which frequently occurs due to the residual vibration-rotation terms exceeding in magnitude (but with opposite sign) the contribution from the equilibrium moments of inertia. What results then is an imaginary coordinate, which is, of course, physically impossible. In some cases, for example when the atom in question lies in a symmetry plane (say, the ab plane), it is evident that the value of the c coordinate of the atom can be physically set equal to zero. (We should mention that special symmetry-specific forms of Kraitchman's equations can be used for molecules with planes of symmetry that invoke the zero coordinates at the outset.

As one simple example, planar molecules

have c i = 0 for all atoms i, which results also in I a + I b - I e = 0 for any rigid planar molecule or instantaneously planar molecule.) On the other hand, the imaginary coordinate may arise accidentally (not symmetry related) simply because the atom lies near to a principal axis.

In this case, some other

procedure or assumption will obviously be needed to use the coordinate in obtaining a bond length, which, incidentally, is generally obtained from the coordinates of two atoms m and n by

r(mn) = {(am - an)2 + (b m - bn) 2 + (c m - Cn)2} 89

(29)

28

HARMONY

The most common procedure to solve the small coordinate problem, when only one small coordinate exists along a given axis, is to use the firstmoment (or COM) Equation (4).

Then if the problem lies with the c

coordinate of atom j, its value is obtained from

-~mic i

cj =

~ mj

(30)

where the sum omits atom j but includes all other atoms whose r s coordinates c i are determined from Kraitchman's equations. In this regard, it should be noted that the first-moment equations are not satisfied perfectly by complete sets of r s coordinates [29].

For example, in the PAS of the 16012C32S

molecule, the value o f E miz i is 0.0330 amu A using the r s coordinates [19]. This discrepancy is relatively small but nevertheless shows that the zeropoint vibration effects have not been entirely eliminated. It is evident that the r s method requires strictly that I 0 be obtained for all necessary isotopomers, that is, one for each symmetry-unique isotopic substitution.

This means that either the isotopomers must be observed in

natural abundance (frequently possible for 13C or 15N and sometimes possible for 180 and other less abundant isotopes) or they must be obtained by enrichment through chemical synthesis. In either case, the labor may be non-trivial, but has been accomplished for many molecules [24]. Of course, atoms without more than one stable isotope, such as fluorine (~9F) are excluded from the substitution method in its strict application.

However,

when only a single F atom is present, as in HC-CF, its coordinate can be

MOLECULAR STRUCTURE DETERMINATION

29

found from the first moment or COM relation as described above for the ccoordinate case (Eq. (30)). We will present numerous r s structural results later, but it is useful to discuss the r s structure of propane as a typical example. Table 7 presents the results along with those from the r 0 structure described previously and the rm 9 structure to be described shortly. It was mentioned that the r 0 structure was obtained from the 18 I 0 values of six isotopomers. These include the parent (or normal) molecule with all 12C and 1H isotopes and five more isotopomers for which each unique atom has been replaced by a heavier isotope (2H =- D). The six molecules are 1.

CH3CH2CH 3

2.

13CH3CH2CH 3

3.

CH313CH2CH 3

4.

CDsH2CH2CH 3

5.

CHsHaDaCH2CH 3

6.

CH3CHDCH3,

where for the methyl group s represents an in-plane H or D while a represents an out-of-plane H or D atom.

These six isotopomers and their

rotational constants (or moments of inertia, I0a, I0b, I0c) constitute what we call a substitution data set (SDS). For any molecule, an SDS consists of the minimal set of isotopic species needed to compute a complete r s structure. The data set outlined here was, in fact, used to obtain the r s coordinates and, hence, the r s structure parameters of Table 7. It might be noted at this point

30

HARMONY

that different SDSs are frequently possible for any given molecule.

For

example, in the case of propane, in place of molecule 6 the CH3CD2CH 3 doubly-substituted molecule would be suitable along with the equations of Chutjian [17].

(In fact, in this case, no other data are available, but the

particular example mentioned is not unusual because double-deuterium methylene substitutions are synthetically not uncommon.) Turning attention now to the r 0 and r s distances in Table 7, it is seen that the r s value for the CC distance is 0.008 A smaller, while the r s values for the CH distances are smaller by as much as 0.005 A. The magnitudes of these variations are typical, although it is easy to find examples for which the r 0 - r s differences are either larger or smaller.

Moreover, although in this

case the parameters are in accord with the diatomic result of Eq. (28), viz., r s < r 0, it turns out that when data are reviewed for many molecules, it is about as likely for r s to exceed r 0 as the converse. And there appears to be no way of knowing how the r s values compare to the theoretically well-defined r e value. In our extensive review [4-7] of polyatomic molecules whose r e and r s structures are both known (ignoring X - H distances, X = C, N, O, etc.), r s r e tends mostly to be greater than zero, with r s - r e _--_0.006 being the largest difference. On the other hand, about one-third of the parameters exhibited r s - r e < 0, with relatively small magnitude deviations of___- 0.002 A. The point we are making here is that there is really no simple a priori way of knowing how close (and in which direction) the r s structure is to the r e structure for polyatomic molecules. For this reason, although rs structures continued to be

M O L E C U L A R STRUCTURE DETERMINATION

31

the standard for polyatomic molecules for many years [24], it was clear that r s results were not going to provide the necessary precision and certainty in the future. Thus, a number of efforts began in the 1970s to find more reliable spectroscopic structural methods that might lead to distance parameters within 0.001 A of the unknown but desired r e value.

C.

r m structures In 1973, J. K. G. Watson [30] presented a new theory based upon isotopic mass effects to obtain a structure (known as rm) that was potentially a good approximation to r e. Consider for simplicity a linear molecule, and compute the s u b s t i t u t i o n

coordinate z i of each of the N atoms using an

appropriate SDS as previous described. Eq. (24) for the diatomic molecule applies for each of the N atoms of any linear molecule:

2

AI

Zsi = ~ gi

i- 1 ...N

(31)

Then define the moment of inertia I s for the molecule whose atoms have the coordinates Zsi:

I s = Zmiz2i

i = 1...N

(32)

Note that I s is not a directly measured quantity, but is perfectly well-defined and computable.

For simplicity also, define a vibration-rotation parameter

g g, which relates to the e g of Eq. (16) by (for v i = 0)

g = '~ t;g(di / 2) + .. i

(33)

32

HARMONY

Then for the linear molecule, with only a single axis of concern,

I0 = Ie + g = I0 _ Ie

or (34)

Then g measures the entire discrepancy (or error) between the measured ground state moment of inertia and the equilibrium moment. Now, Watson showed that if a new moment I m is defined by

I m = 2 1s - I 0

(35)

then I m is related to I e by

1 Ie = I m - ~ -

02(M~) miam i 0m 2 +...

Im 8

i =I...N

(36)

That is, I m is the zeroth-order approximation to I e and the sum (divided by M) represents the error to the first-order in the isotopic mass changes Am i. If this quantity and the higher order terms (all of which we define as 5) are sufficiently small, then I m will be a good approximation to I e and might then be expected to yield a structure close to r e. As usual, the first test of the theory used a diatomic molecule, CO [30]. For this simple molecule, I 0 data are available [31 ] for four SDSs as follows: 1.

12-16, 13-16, 12-18

2.

13-16, 12-16, 13-18

3.

12-18, 13-18, 12-16

4.

13-18, 12-18, 13-16

MOLECULAR STRUCTURE DETERMINATION

33

where the parent is underlined in each case. For each of these parents, the I s values can be computed using Eqs. (31) and (32), and then I m is obtained from Eq. (35). These various moments are summarized in Table 8 along with the experimental I e values [32]. From Table 8 it is apparent that I m is a very good approximation to I e in all four cases, especially compared to either the I 0 or I s values. In particular, for the

12C160 parent, we see for illustrative

purposes that = I 0 - I e = 0.0397 amu A 2 and

5 = I m - I e = - 0 . 0 0 3 6 amu A 2

Thus, whereas I 0 is in error by 0.45%, the magnitude of the error in I m is only 0.04 %.

It is interesting to note for CO that Watson's method has

overcorrected the original I 0 data, that is, 8 < 0. Nevertheless, the small I m error of 0.04% guarantees that the r m distances will be very close to r e. Table 9 summarizes the r0, r m and r e distances computed from the Table 8 values. It is evident that r m is, indeed, for this diatomic molecule, a very good, even an excellent, approximation to r e .

For our purposes, the number of

significant figures given in Table 9 is entirely suitable.

Watson [30, 32]

should be consulted for finer details involved with still smaller B-O corrections. The result for CO is essentially typical for diatomic molecules, i.e., the r m distance is expected to be within approximately 0.0001 A of the r e distance (except in the case of H-atom-containing molecules).

Note, of

34

HARMONY

T A B L E 8. Carbon monoxide moments of inertia a. Parent molecule

I0

Is

Im

Ie

12C16 0

8.76846

8.74680

8.72514

8.72873

13C160

9.17186

9.14999

9.12811

9.13122

12C180

9.20688

9.18492

9.16295

9.16617

13C180

9.65274

9.63059

9.60843

9.61105

aAll values in units of arnu A 2. See Ref. [31 ] for extensive ground state data and Refs. [30] and [32] for data used for the Is, Im, and Ie values.

T A B L E 9. Carbon monoxide interatomic distances a. Parent molecule

r0

rm

re

12C160

1.1309

1.1281

1.1283

13C 160

1.1308

1.1281

1.1283

12C180

1.1308

1.1281

1.1283

13C18 0

1.1308

1.1282

1.1283

aAll distances in A. Computed from data in Table 8.

course, that the I m structure, like the r s structure, requires more moment of inertia data than the r e structure, so once again the I m method has no advantage in general for diatomic molecules. The question then is how does the r m method work for polyatomic molecules?

The general answer is that although the procedure always

MOLECULAR STRUCTURE DETERMINATION

35

produces I m values that are relatively close to equilibrium, the resulting_ structures are often of uncertain quality. Why that occurs can be seen by a few examples. Consider SO 2 (C2v symmetry), for which extensive data are available. Using the most common SDS (32S1602, 34S1602, 32S180160), Im can be evaluated for both the a and b axes (and via the inertia defect the c axis as well [30]), yielding Ima and Imb values that differ from Iea and Ieb by 0.055% and 0.0040%, respectively.

Then, since SO 2 has only two structural

parameters, the two I m moment of inertia values permit rm(SO ) and 0m(OSO ) to be computed, yielding r m = 1.4307 A and 0 m = 119.33 ~ which are nearly identical to the equilibrium values = 1.4308 A and 0 e = 119.33. Similarly, excellent I m results will occur for 0 3 (C2v symmetry) and CO 2 (Dooh symmetry) or any molecule whose structure can be evaluated using a single SDS, and which does not contain hydrogen. The problem with hydrogen is that the substitution H--+D (or H---~T, D---~H, etc.) leads to 5 values (see Eq. (36)), which are too large to be neglected in comparison to the leading term I m in the expansion in powers of Ami. When data from more than one SDS are required, slight distortions in the I m values may be introduced by the neglected terms in 5 that fluctuate markedly in magnitude and sign due to the finite changes in Ami (see Eq. (36)). Two examples from Watson [30] and one from Kuchitsu [33] as reported by Harmony et al. [4] are presented in Table 10.

36

HARMONY

TABLE 10. Some r m structures a. re

rm N20b NN

1.1281 (8)

1.1284 (3)

NO

1.1842 (8)

1.1841 (3) OCS b

CO

1.1587 (13)

1.1543 (2)

CS

1.5593 (10)

1.5628 (4) COC12e

CO

1.1808 (62)

1.1756 (23)

CC1

1.7351 (28)

1.7381 (19)

zCICCI

112.08 (28)

111.79 (24)

aAll distances in A. bReported in Ref. [30]. eReported in Ref. [4] with data from Ref. [33].

For linear molecule N20, a minimum of two independent I m values and thus two SDSs with unique parents are needed. In fact, eight uniqueparent I m values were available, so the complete set of data was used in a least-squares fit to give the tabulated values.

In this case, the agreement

between r m and r e values is very good, so apparently the effects of the neglected terms lead to no significant distortions. N20 is an interesting case (as mentioned earlier), because it has a very small coordinate for the central N atom. Indeed, when 15N14N160 is used as the parent, the substitution of 15N for 14N leads to AI < 0, which means the coordinate is imaginary by Eq.

MOLECULAR STRUCTURE DETERMINATION

37

(31). In Watson's treatment, these small or imaginary coordinates are not dropped, set equal to zero, or recomputed by some other method (such as COM relation); in fact, the values from Kraitchman's equations (Eq. (31) for the linear molecule) provide the essential mass-dependent vibration-rotation contributions to the moments of inertia. Now, while the N20 results are very good, those for the entirely analogous OCS case (using again eight unique parents) are really rather poor, with CO and CS variations of +0.0044 and-0.0035 A, respectively. Here, the small distortions in the I m values produced by the neglected terms in 5 lead to substantial structural distortions, even though all I m values are within 0.02% of I e. This discouraging state of affairs was investigated theoretically by Smith and Watson [34], who showed by model force-field calculations that the trouble was an unavoidable consequence of the neglected finite masschange terms Am i in 5. Indeed, if one extrapolated to Ami = 0 (by using two different substitution patterns for each atom, i.e., 160 ~ 170 and 180, 12C 13C and 14C and 32S ~ 33S and 34S), so that 5 ~ 0, then I m ~ I e and the r m structure approaches very closely the r e structure! Thus, Smith and Watson [34] concluded that one could not be confident in general that r m parameters would be even within 0.001 A of the equilibrium values unless the results could be obtained entirely from the I m value(s) of a single SDS. As a final example of a still more complicated molecule, Table 10 shows the results for COC12. Here there is much data available, so that eight parent-molecule SDSs are fully available, yielding

38

HARMONY

eight unique sets of Ima, Imb and Imc values (of which only two of the three are considered independent for the planar molecule). Again, a least-squares fit yields the r m values of the two distances and the bond angle. While a complete spectroscopic determination of r e is not available, Kuchitsu [33 ] has obtained a reliable structure (although with rather large uncertainties) by combining spectroscopic and electron diffraction data. Note that the r m and r e distance values differ in the range 0.003-0.005 A.

The deviations are

within the listed errors (least-squares a for rm), of course, but the point is that the rm structure has such large uncertainties that it is not useful as an estimate o f r e. When the r m procedure is applied to molecules involving hydrogen, the H ~ D substitutions that are required may lead to rather large changes in the mass-dependent contributions and, hence, the neglected terms ~5 may be especially significant. Even for the diatomic molecule HCI, the r m distance computed for the H35C1 parent is ~0.0015 A smaller than r e, a deviation ten times greater than for the CO example described earlier. Of course, for Hcontaining polyatomics the problem is doubly troubled if more than one SDS is necessary. Thus for HCN, Watson [30] found that the r m values of the CH and CN distances were in error (compared to re) by -0.0051 and +0.0012 A. Thus, although the r m method had great promise, it led, unfortunately, to no real advances in distance determinations, especially for polyatomic molecules.

Still, because the method undeniably yields Im values that are

much nearer to I e than either I 0 or Is, the method continued to attract attention

MOLECULAR STRUCTURE DETERMINATION

39

and one wondered whether somehow the method might be "fixed up" in some way. In closing this section, it is perhaps worthwhile to note that the isotopic data requirements of the r m method are rather extreme, so much so that the method would probably not be considered for most polyatomic molecules, even if the resulting structure were reliable. Consider chloroacetylene, H C=C-C1.

For this species, with three structural parameters, three unique-

parent SDSs would then be required to obtain the necessary three I m values. This would require, in tum, that the spectra of a total of ten isotopomers be analyzed to obtain their B 0 and, hence, I 0 values. Suitable SDSs would be, for example: 1.

1-12-12-35, 1-13-12-35, 1-12-13-35, 1-12-12-37, 2-12-12-35

2.

2-12-12-35, 2-13-12-35, 2-12-13-35, 2-12-12-37, 1-12-12-35

3.

1-12-12-37, 1-13-12-37, 1-12-13-37, 1-12-12-35, 2-12-12-37,

where the unique isotopomers are underlined. Note that each SDS yields an I s value and thus an I m value for the parent molecule (always the first listed). If the r m structure were desired for the most stable planar conformer of formic acid (Fig. 3), the simplest organic carboxylic acid with seven independent structural parameters, a total of four unique-parent SDSs would be necessary in order to yield four pairs of (Ima, Imb) values. If four suitable SDSs are written down, one finds that a total of 16 isotopomers must be fully rotationally analyzed in the ground vibrational state. The impracticability of this procedure is only exceeded by the task of obtaining an r e structure,

40

HARMONY

O

H

/

O

H FIG. 3. Structural sketch of most stable conformation of formic acid.

namely, the ground and nine excited vibrational states must be fully rotationally analyzed for four isotopomers, a total of 40 rotational spectral analyses! D.

Miscellaneous Methods The next section will present the major emphasis of the chapter, viz., the scaled moment of inertia or rm 0 structure method. We should mention just a few additional spectroscopic structure procedures that have been used in the past.

First, the double-substitution method of Pierce [35] was

developed to handle the small-coordinate problem of the r s method. In this procedure, essentially, the small coordinate is determined in two different axis systems using two different pairs of isotopomers.

Then, upon

subtracting a second time (to yield AAI-type terms) and relating the two

MOLECULAR STRUCTURE DETERMINATION

41

independent axis systems, the small coordinate is obtained with reduced influence of the vibration-rotation terms.

The method leads to much

improved values for the troublesome small r s coordinates at the expense of more experimental work (assignment and high-precision analysis of more isotopomer spectral data). It does not change the overall quality of the r s method, but certainly provides a way of solving the small coordinate problem. Often, especially for polyatomic organic molecules, it is not feasible or practical to obtain a complete set of substitution structure data, or perhaps there are atoms such as fluorine that cannot be substituted. In these cases, one might nevertheless have data for several isotopomers.

For a general

asymmetric rotor, if the number of independent moments of inertia exceed by at least three the number of independent internal structure parameters, then a least-squares fit of AIg data leads to a structure that has the qualities of the r s structure. Such a method, which might be labeled as rAl , has been proposed and described by N/3sberger et al. [36] and other workers [37]. For example, consider ethylene oxide of Fig. 1.

(C2H40), the C2D40 isotopomer

I f the spectra of the normal species and the

12C13CH40 isotopomer

are

analyzed, a total of nine independent moments of inertia are obtained. Then six AIg values can be obtained (where AIg = Ig (isotopomers) - Ig (parent)), which may be used in a least-squares fit to obtain the "best" values of the five independent structure parameters.

Because only AI values are used, it is

reasonable to expect that the vibration-rotation terms will cancel out to first order as in the r s method. In fact, if a complete SDS is used, the rAI method

42

HARMONY

will lead to a structure very similar (but not identical) to the r s structure. Rudolph [38] has discussed this and a related method in great detail. In the related case, he suggests taking the ~g of Eq. (33) to be isotope independent (this is not true), so that for a given molecule only three values (~ a, g b, ~ c) exist, no matter how many isotopic Ig0 values exist. Then Rudolph proposes to do a least-squares fit of the experimental Ig0 values to obtain the best values of the independent structural parameters plus the three ~ g values. Thus, for ethylene oxide again, the same set of data as discussed in the previous paragraph (nine Ig0 values) could be used to obtain the best values of the five structure parameters and the three ~ parameters. This structure method, which might be called re, I , is intrinsically appealing because it explicitly includes (even if incorrectly) the offending vibration-rotation terms as fitting parameters.

Interestingly, as described by Rudolph [38], the re,I

method leads to structural results that are identical to those of the rAi method for the same input set of data. Neither Pierce's double-substitution method nor the rAI or re,I procedures provide results that are particularly good approximations of the equilibrium structure, basically being variations of the r s procedure. Nakata et al. [39] have pointed out the importance of using complementary data sets when applying Watson's r m procedure. Here, it was noted (actually, Watson [34] made the same observation) that with proper selection of data the effects of the neglected 5 term in Eq. (36) could be minimized.

Consider OCS as an example.

Then the I m value for

MOLECULAR STRUCTURE DETERMINATION

43

16012C32S has zxmi terms in 5 of +2, +1, +2. Now consider the I m value for 18013C34S. In this case, the Ami terms of 5 have the values - 2 , - 1 , -2. Now the other factors in 8, in particular the second derivative, should not change much from one isotopomer to another. Therefore, it is expected that ~5(16012C32S) ~ - 5(18013C34S) These two isotopomers are "complementary" pairs.

Similarly, 16013C32S

and 18012C34S are complementary pairs. Nakata et al. [39] propose using sets of complementary-pair data to evaluate the results, which in effect leads to a cancellation of the 5 values to first order. The procedure, known as the r e structure method, has very limited applicability because of the increased data demands, but has been found to be useful in a few cases such as C120 [39] and COCI 2 [33]. However, the still higher-order terms in ;5 are potentially troublesome and even in simple cases are apparently important. In a very thorough analysis of OCSe, Le Guennec et al. [40] have shown that the r c structure is not of high quality (i.e., is not very near to re) unless additional procedures are utilized to cancel out the higher order terms. Finally, we should mention a procedure whose aim is not to approximate the equilibrium structure but, rather, to evaluate the average structure rz in the ground vibrational state by correcting the measured I0s. It has been shown, correct to terms of order (v + 89 that the moment of inertia about the g-axis of a molecule with all atoms in their average ground state positions, I'g, is given by [41-44]

44

HARMONY

and

I*g = Ig0 - ~ g (harm)

(37)

I*g = Ige + ~ g (anharm)

(38)

where

g = ~ g (harm) + g g (anharm)

(39)

is defined in the v = 0 state by Eqs. (33) and (16). Here the key feature is that the vibration-rotation terms ~ can be broken up into a portion containing only the harmonic contribution to the vibrational potential function and a part containing everything else (the anharmonic part). Moreover, Eq. (37) shows that only the harmonic part of g must be known to correct the experimental effective ground state moments of inertia to give the moments of the average configuration.

While not trivial, these harmonic correction terms (which

include the Coriolis terms) can be made in a straightforward manner from the usually well-known harmonic force field (e.g., Wilson's F matrix [45]). Then the moments I*g can be used to obtain the atomic coordinates in the average configuration and, hence, the average values of the structural parameters designated r z. For the diatomic molecule, it is perhaps useful to summarize the definitions [ 16, 41 ]: I 0 = lxro2 ~, ~t 1.

More generally, for a non-linear molecule, the particular axes must be specified. For the a-axis,

MOLECULAR STRUCTURE DETERMINATION

69

8ZD

g

.=____ =,...._

(a)

8.bD

c3aD ~CD

(b)

FIG. 4.

Bond elongation vector for X-D bond in (a) a general linear polyatomic molecule and (b) a general non-linear polyatomic molecule.

Components along principal axes are shown in

latter case. Reprinted from Ref. [7] with permission.

n

(iPma )corr D - (IPma)D = + 2 mD ~ ( b i S b i + ciSci)

(47)

i

gives the correction generally when n symmetry-equivalent H atoms are substituted by D atoms, b i and atoms and 6bi,

6C i a r e

ci

are the coordinates of the D

components of 5~D , which is always along the

C-D bond direction [7, 8]. Using data for several well-characterized

70

HARMONY

molecules, we have selected 15~D] = 0.0028 A [7] as a reliable average value for X-H bonds. One expects in principle that the value IS~DI might depend upon the heavy atom X. However, it appears that a single average value is suitable in practice. [Please note that Eq. (12) in Ref. 7 is not a general expression, applying for n = 2 (for example, H20 --~ D20 substitutions) but not for n = 3. The more general result given above as Eq. (47) was first presented in Ref. 8.] We will look at results shortly, but it is worth mentioning that the empirical procedure involving a shift of the D atom along the X-D bond by 0.0028 .A is intuitively pleasing because the shitt is about the amount that occurs upon H ~ D substitution. (See r 0 data in Table 4 and r 0 and rz data in Table 11.) Note, however, that the sign of the shit~ (i.e., the direction of 5f D) depends upon whether pg is greater or less than unity. When pg < 1, which is most common, the X-D bond is elongated as depicted in Fig. 4, while if pg > 1, the X-D bond is shortened by the same amount. It is best to avoid giving too much physical interpretation (or reality) to the shift correction. Rather, Eqs. (46) and (47) should be simply looked upon as being the empirical procedure needed to correct the I0m values of those isotopomers that involve replacement of H by D. Finally, the results for the b- and caxes are written from Eq. (47) by cyclic permutations of a, b, c and, if the SDS involves D ~ H substitution, m D in Eq. (47) is replaced by m H and 5~H is used in place of 5~D with a value ofSr H =-0.0056 A.

MOLECULAR STRUCTURE DETERMINATION

71

For more detailed discussions of the form of Eqs. (46) and (47), see Ref. 7 (especially the Appendix) and Ref. 8. In Table 18 we present r0m structure computations that utilize the corrections given by Eqs. (46) and (47).

In all cases reasonably

reliable r e structures are available for comparison. It is first interesting to compare the HCN and HCCH results of Table 18 with those given in Table 17, where no corrections to the basic scaling procedure were made. Note that in both cases the heavy-atom distances (C-N and C-C, respectively) have improved remarkably, with nearly exact agreement with the r e values.

Naturally, the C-H distances have

improved also, but it is clear that the correction procedure still does not yield highly accurate H-atom distances. Next it is worthwhile looking at the entire set of distances in the table.

The first conclusion we can make is that, for all seven

molecules, the r s distances are inferior to the r Om distances. Second, it is clear that the heavy-atom distances are all of high quality, with only two cases (HNC and CH2C12) leading to deviations greater than 0.001 A. If we compute the average deviations, we obtain X-H distances:

(Irmp - r e 1 ) = 0 . 0 0 1 7 A ( I r s - r e 1) = 0.0045 A

heavy-atom distances:

(Irmp -re I)=0.0005 h ( [ r s - r e 1) = 0.0023 A

72

HARMONY

TABLE 18. rOm structures of hydrogen-containing molecules with corrections for H --~ D substitutions a.

rs HCN

HNC

HN2 +

HCCH

HCCCI

H2CO

CH2C12

re

rm p

Ref. b,c

CH

1.0631

1.0655 (2)

1.0668 (2)

CN

1.1553

1.1532 (0)

1.1531 (1)

NH

0.9862

0.9970

0.9923 (3)

NC

1.1719

1.1684

1.1701 (1)

NH

1.0319

1.0336 (4)

1.0347 (1)

NN

1.0950

1.0928 (1)

1.0929 (0)

CH

1.0586

1.0622 (2)

1.0631 (0)

C=C

1.2058

1.2026 (1)

1.2026 (0)

CH

1.0550

1.0605

1.0599 (4)

C-C

1.2036

1.2030

1.2032 (8)

C-CI

1.6369

1.6353

1.6357 (6)

CH

1.1042

1.1005 (20)

1.1012 (2)

C=O

1.2049

1.2033 (10)

1.2031 (1)

LHCH

116.55

116.30 (25)

116.25 (4)

CH

1.0836

1.0874

1.0851 (11) c

C-C1

1.7687

1.7648

1.7636

LHCH

112.26

111.51

111.90 (17)

LCICC1

112.06

112.03

112.25 (3)

b,c

b,c

c

c,d

c

(3)

aDistances in A, angles in degrees. bSee Ref. [6] in text. cSee Ref. [7] in text. dEquilibrimn structure obtained by high quality ab initio calculations of ~. See Ref. [57].

Thus, it is evident that the heavy-atom distances for H-containing molecules are of the same high quality as for the strictly heavy-atom

MOLECULAR STRUCTURE DETERMINATION

73

species when the corrected (I0m)D values are used. It is important to stress that if the H-atom corrections are not made, the deficient (I 0rn)D values cause the heavy-atom structural parameters to deteriorate markedly. The poorest heavy-atom result of Table 18 is the N - C distance in HNC. This species has been looked at in some detail by model forcefield calculations, which show that the low-frequency bending mode (032 = 477 cm -1) is probably the origin of the somewhat large

discrepancy [6].

We have experienced only one r0m case that

apparently leads to an unacceptably large discrepancy compared to r e . For ethylene, we reported [7] r0m (C=C) = 1.3297 (5) A, while the reported r e value [58] is r e (C=C) = 1.334 (2). Even accounting for the rather low quality re structure, it appears that the r0m value is somewhat low.

Based upon detailed considerations [9], and by

analogy to HNC, it appears that this may be caused by the relatively low-frequency out-of-plane wagging mode of the CH 2 groups. We have proposed that the true equilibrium C=C distance in ethylene is most probably 1.332 + 0.002 A [9]. Now it should be noted that the r0m procedure has not been applied to molecules such as H20, NH 3 and CH 4. For these species the changes in vibration-rotation contributions to the moments of inertia are dominated by the H ~ D substitutions and, thus, the scaling

74

HARMONY

procedure proposed here will be inadequate even with the H ~ D correction term.

Fortunately, species of the type XH n represent a

negligible fraction of organic polyatomic molecules and, therefore, failure of the method here is of little consequence. To summarize where matters now stand, the results presented in Sections E.2 to E.4, plus other examples and supporting theoretical work [4-10], lead us to the following conclusions concerning r0m structures:

a)

Heavy-atom distances are expected to be within 0.001-0.002 A of r e if all necessary H ~ D corrections are made via Eqs. (46) or (47).

b)

Hydrogen atom distances are not given with high reliability by the procedure even after making the H ~ D corrections. Still, it seems likely that X-H distances will be within 0.002-0.003 A of r e .

c)

The r0m structure is a much more reliable representation of the r e structure than is the r s structure, especially for heavy atoms. A word about precision is perhaps in order. We typically list the least-squares o value that results from the least-squares fit in our tabulated data to show the degree to which the scaled moments of inertia model the presumed rigid, equilibrium molecule. Of course, experimental uncertainty in the I0's will also contribute to o.

It is not our intention that one should

MOLECULAR STRUCTURE DETERMINATION

75

automatically use the o value as a bona fide measure of the uncertainty in the parameter. Note from Table 18 that the r~m o values are commonly (but not always) smaller than the deviation from r e. Our conservative view is that when the uncertainty o is less than 0.001 A (or 0.002 A), the parameter uncertainty should be stated as 0.001 A (or still more conservatively as •

A).

When the parameter uncertainty o exceeds 0.001 A (or 0.002 A), then the parameter uncertainty is best stated as •

Finally,

we add one more conclusion (or caveat) to the above list of 3:

d)

Structural distortions may occur that cause the accuracy to decline from the normal 0.001-0.002 A range if especially lowfrequency bending vibrations of hydrogen atoms exist.

5.

rOm applications Here we mention just a few rOm structures of organic polyatomic molecules of particular interest.

First, our rgm structure of ethane

H3C-CH 3 [8] provided the best available spectroscopic estimate of the r e value of the prototypical C-C single-bond length.

The reported

value of rOm (C-C) = 1.522 + 0.002 A is in very good agreement with the estimated r e value from electron diffraction work (1.524 + 0.003 A) [59]. It is interesting next to look at the results for propane (CH 3CH2-CH3) that were presented in Table 7. Note for propane that the rOm (C-C) value is 1.521 A, nearly identical to the ethane value as

76

HARMONY

expected for these very-low polarity alkane molecules. Note further that if one were to select either the r 0 or r s value for the propane C-C distance, one would really be in a different realm since the deviations from rm p are quite substantial. The propane results for C-C (and three out of four of the C-H distances) also illustrate a common, but not absolute observation, viz.,

r0 > rs > rPm

(48)

Recall that for diatomic molecules, Eq. (28) ordered the distances as r 0 > r s > r e. There is no proof of Eq. (28) for polyatomic molecules, but our observation in Eq. (48) for polyatomic molecules is equivalent to Eq. (28) if rPm ~ r e, which is our hypothesis. The rPm structures of symmetric rotors such as the methyl halides are troublesome unless the A rotational constant is available. Using A values from IR data, Le Guennec et al [60] have reported the r o 9

m

structures of CH3CI and CH3Br. For CH3C1, the rPm values of the CC1 and CH distances deviate from the r e values by +0.0017 a n d 0.0019 A, respectively.

For CH3Br, the deviations for the CBr and

CH distances are +0.0013 and -0.0007 A, respectively. The results are satisfactorily in accord with our expectations based upon nonsymmetric rotor molecules. The principal impetus for developing the rPm structure method was the desire to establish near-r e structures for polyatomic organic

MOLECULAR STRUCTURE DETERMINATION

77

molecules. Table 19 presents a summary of results for a variety of small molecules treated by the rOm procedure (including all H --~ D corrections).

In all cases the accuracy (with respect to the true r e

value) is expected to be within +0.001 A or +0.002 A. In a few cases, bona fide experimental r e structures exist (such as for C2H 2 , H2CO and CH3C1), but we list our rPm values for consistency. (In any case, the r gin and r e values are in agreement for these cases as previously discussed.) The various molecules provide interesting comparisons of some key bonds in various bonding situations, including the two cyclic molecules ethylene oxide and cyclopropene. data

provide

reliable

benchmarks

for

ab

In addition, the initio

theoretical

computations.

IV.

CONCLUSION

The scaled moment of inertia (rOm) structure method has been demonstrated to yield an excellent approximation to the theoretically significant equilibrium (re) structure. A practical variant of Watson's r m procedure [30], the method requires only the spectroscopic data necessary to perform a complete ground state substitution (rs) structure determination.

The resulting rOm structure appears to be invariably a more

reliable estimate of the true r e structure than is the r s structure.

For polyatomic

molecules, the heavy-atom distance parameters are expected to be within 0.001-0.002 A of the r e value in most cases. For hydrogen-containing molecules, a special empirical

78

HARMONY

TABLE 19. Selected values of near-equilibrium bond distances (in A). C-C

C=C

C=C

C - - O C=O

C--C1 Ref.

Acetylene

HCCH

1.203

[7]

Chloroacetylene

C1CCH

1.203

1.636 [7]

Formaldehyde

H2CO

Methyl chloride

CH3C1

1.778 [60]

Methylene

CH2C12

1.764

1.203

[7]

[7]

chloride Formic acid

HCO2H

Ethylene oxide

C2H40

Ethylene

H2CCH 2

Ethane

H3CCH 3

1.522

Ethyl chloride

CH2C1CH3

1.510

Vinyl chloride

C2H3C1

Propene

CH3CHCH 2

1 . 4 9 6 1.334

[9]

Cyclopropene

C3H4

1.505 1.293

[7]

Propane

CH3CH2CH 3

1.522

[9]

1.459

1.340 1.196

[9]

1.425

[7]

1.332

[7] [8] 1.789

1.328

[9]

1.726 [61 ]

correction is needed to handle the data for the hydrogen isotopomers. This correction ensures that the quality of the heavy-atom parameters remains high, and leads to H-atom parameters of reduced, but still perhaps useful, accuracy (0.002-0.004 A). The new procedure permits for the first time the practical estimation of reliable near-r e structures for polyatomic organic molecules from gas-phase spectroscopic ground state rotational constants. Along with gas electron diffraction and modem high-quality

MOLECULAR STRUCTURE DETERMINATION

79

ab initio computations, the rOm methodology should provide a reliable new tool for the

high-precision structure determination of modest-sized (6-8 heavy atoms or so) polyatomic organic molecules. ACKNOWLEDGMENTS Over the many years of this structural research, the support of the National Science Foundation, the Petroleum Research fund, and the Research Corporation has been greatly appreciated.

This particular manuscript would not have been possible

without the technical assistance and encouragement of Nancy M. Harmony.

80

HARMONY

REFERENCES 1. ,

M. Born and J. R. Oppenheimer, Ann. Phys., 84, 457 (1927). G. Herzberg, Electronic Spectra of Polyatomic Molecules, Van Nostrand, New York, 1966.

3.

J.M. Hutson and B. J, Howard, Mol. Phys., 45, 791 (1982).

4.

M.D. Harmony and W. H. Taylor, J. Mol. Speetrose., 118, 163 (1986).

5.

M.D. Harmony, R. J. Berry and W. H. Taylor, J. Moi. Speetrose., 127, 324 (1988).

6.

R.J. Berry and M. D. Harmony, J. Mol. Speetrose., 128, 176 (1988).

7.

R.J. Berry and M. D. Harmony, Struet. Chem., I, 49 (1990).

8.

M.D. Harmony, J. Chem. Phys., 93, 7522 (1990).

9.

H.S. Tam, J.-I. Choe and M. D. Harmony, J. Phys. Chem., 95, 9267 (1991).

10. M.D. Harmony, Aeets. Chem. Res., 25, 321 (1992). 11. K. Kuchitsu and S. J. Cyvin, Molecular Structure and Vibrations (S. J. Cyvin, ed.), Elsevier, Amsterdam, 1972, p. 183. 12. W. J. Hehre, L. Radom, P. v. R. Schleyer and J. A. Pople, Ab Initio Molecular Orbital Theory, Wiley, New York, 1986. 13.

S. Saebo and P. Pulay, Ann. Rev. Phys. Chem., 44, 213 (1993).

14. K.B. Wiberg, private communication. 15. J. Kraitchman, Am. J. Phys., 21, 17 (1953). 16. W. Gordy and R. L. Cook, Microwave Molecular Spectra, Wiley-Interscience, New York, 1984. 17. A. Chutjian, J. Mol. Spectrose., 14, 361 (1964). 18. G. Herzberg, Spectra of Diatomic Molecules, Van Nostrand, Princeton, NJ, 1950.

MOLECULAR STRUCTURE DETERMINATION

81

19. C.H. Townes, A. N. Holden and F. R. Merritt, Phys. Rev., 74, 1113 (1948). 20.

F.J. Lovas and E. Tiemann, J. Phys. Chem. Ref. Data, 3, 609 (1974).

21.

V. W. Laurie, Critical Evaluation of Chemical and Physical Structural Information (D. R. Lide and M. A. Paul, eds.), National Academy of Sciences, Washington DC, 1974, p. 67.

22.

K. P. Huber and G. Herzberg, Constants of Diatomic Molecules, Van Nostrand Reinholt, New York, 1979.

23.

F.J. Lovas, J. Phys. Chem. Ref. Data, 7, 1445 (1978).

24.

M. D. Harmony, V. W. Laurie, R. L. Kuczkowski, R. H. Schwendeman, D. A. Ramsay, F. J. Lovas, W. J. Lafferty and A. G. Maki, J. Phys. Chem. Ref. Data, 8, 619(1979).

25. Landolt-Bornstein Numerical Data and Functional Relationships in Science and Technology, New Series, Group II, Springer, Berlin, Vol. 6, 1972. 26.

D.L. Gray and A. G. Robiette, Mol. Phys., 37, 1901 (1979).

27.

A. Baldacci, S. Ghersetti, S. C. Hurlock and K. N. Rao, J. Mol. Speetrosc., 59, 116 (1976).

28.

D.R. Lide, J. Chem. Phys., 33, 1514 (1960).

29.

C.C. Costain, J. Chem. Phys., 29, 864 (1958).

30.

J.K.G. Watson, J. Mol. Spectrosc., 48, 479 (1973).

31.

B. Rosenblum, A. H. Nethercot, Jr. and C. H. Townes, Phys. Rev., 109, 400 (1958).

32.

J.K.G. Watson, J. Mol. Spectrosc., 45, 99 (1973).

33.

M. Nakata, T. Fukuyama and K. Kuchitsu, J. Mol. Spectrose., 83, 118 (1980).

34.

J.G. Smith and J. K. G. Watson, J. Mol. Speetrose., 69, 47 (1978).

35.

L. Pierce, J. Mol. Spectrosc., 3, 575 (1959).

36.

P. N6sberger, A. Bauder and H. Giinthard, Chem. Phys., 1, 418 (1973).

82

HARMONY

37. R. H. Schwendeman, Critical Evaluation of Chemical and Physical Information (D. R. Lide and M. A. Paul, Eds.), National Academy of Sciences, Washington, DC, 1974, p. 94. 38.

H.D. Rudolph, Struet. Chem., 2, 581 (1991).

39.

M. Nakata, M. Sugie, H. Takeo, C. Matsumura, T. Fukuyama and K. Kuchitsu, J. Mol. Speetrose., 86, 241 (1981).

40.

M. Le Guennec, G. Wlodarczak, and J. Demaison, J. Mol. Speetrosc., 157, 419 (1993).

41.

D.R. Herschbach and V. W. Laurie, J. Chem. Phys., 37, 1668 (1962).

42.

V.W. Laurie and D. R. Herschbach, J. Chem. Phys., 37, 1687 (1962).

43.

T. Oka, J. Phys. Soe. Japan, 15, 2274 (1960).

44.

T. Oka and Y. Morino, J. Mol. Speetrose., 8, 300 (1962).

45.

E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw-Hill, New York. 1955.

46.

P. Botschwina, N. Oswald, J. Fltigge, A. Heyl and R. Oswald, Chem. Phys. Letters, 209, 117 (1993).

47.

T.J. Balle and W. H. Flygare, Rev. Sci. Instru., 52, 33 (1981).

48.

D.J. Clouthier and J. Karolczak, J. Chem. Phys., 94, 1 (1991).

49.

R.J. Saykally, Ace. Chem. Res., 22., 295 (1989).

50. A.C. Legon, Ann. Rev. Phys. Chem., 34, 275 (1983). 51.

T.A. Miller, Ann. Rev. Phys. Chem., 33, 257 (1982).

52. M. Le Guennec, G. Wlodarczak, W. D. Chen, R. Bocquet and J. Demaison, J. Mol. Speetrose., 153, 117 (1992). 53.

G. Cazzoli, C. D. Esposti, P. Palmieri and S. Simeone, J. Mol. Spectrose., 97, 165 (1983).

MOLECULAR STRUCTURE DETERMINATION

54.

83

W.H. Taylor, Ph.D. Dissertation, University of Kansas, Lawrence, KS, 1983.

55. H.H. Nielsen, Rev. Mol. Phys., 23, 90 (1951). 56.

I. M. Mills, in Molecular Spectroscopy: Modern Research (K. N. Rao and C. W. Mathews, Eds.), Vol. 1., Academic Press, New York, 1986, pp. 115-140.

57.

P. Botschwina, private communication.

58.

J.L. Duncan, Mol. Phys., 28, 1177 (1974).

59.

J.L. Duncan, D. C. McKean and A. J. Bruce, J. Mol. Spectrosc., 74, 361 (1979).

60.

M. Le Guennec, G. Wlodarczak, J. Burie and J. Demaison, J. Mol. Spectrosc., 154, 305 (1992).

61.

I. Merke, L. Poteau, G. Wlodarczak, A. Bouddou and J. Demaison, J. Mol. Speetrose., 177, 232 (1996).

CHAPTER ELECTRON

DIFFRACTION:

WITH OTHER

2 A COMBINATION

TECHNIQUES

Vladimir S. Mastryukov Department o f Physics University o f Texas Austin, T X 78712, U S A

I.

I N T R O D U C T I O N ...................................................................................................

86

II.

T H E E L E C T R O N D I F F R A C T I O N T E C H N I Q U E .... , ....................................... 88 A.

Experimental Equipment ............................................................................................. 88

B.

Basic Relations ............................................................................................................ 92

C.

Structure Analysis ........................................................................................................ 97

III. C O M B I N E D A P P L I C A T I O N O F E X P E R I M E N T A L T E C H N I Q U E S ......... 100 A.

Supersonic Nozzle ..................................................................................................... 101

B.

Mass Spectrometer ..................................................................................................... 102

C.

Laser .......................................................................................................................... 103

IV. T E C H N I Q U E S I N C O R P O R A T E D IN S T R U C T U R A L A N A L Y S I S ............. 109

V.

A.

Vibrational Spectroscopy ........................................................................................... 110

B.

Rotational Constants .................................................................................................. 115

C.

Theoretical Calculations ............................................................................................ 121

D.

Liquid Crystal NMR Spectroscopy ........................................................................... 134

CONCLUDING REMARKS AND ACKNOWLEDGEMENTS

...................... 137

R E F E R E N C E S ..............................................................................................................

85

141

86

MASTRYUKOV

"There is no more basic enterprise in chemistry than the determination of the geometrical structure of a molecule. Such a determination, when it is well done, ends all speculation as to the structure and provides us with the starting point for the understanding of every physical, chemical and biological property of the molecule" R. Hoffmann, 1983

I.

INTRODUCTION Gas-phase electron diffraction (GED) will be 70 years of age in the year 2000. It is about 20 years older than microwave spectroscopy, the other of just two methods currently in use to determine the geometry of free molecules. Other structural data come from studies of crystals done by X-ray and neutron diffraction. The purpose of this chapter is to briefly describe the major milestones of the evolution of GED during the seven decades of its history focusing

exclusively at

the interactions of this method with other techniques. In 1988, all

these individual interactions were reviewed by a large international group of electron diffractionists (see Part A of Ref. [ 1]) and, therefore, our goal is to briefly characterize them and to give a good representation of the literature that has appeared after 1988; comprehensive coverage is not intended. GED, like any other method, has its own inherent weaknesses as well as unique strengths.

Its data are otten insufficient to provide all the necessary

information for a full evaluation of the structure of any but the smallest molecule. Therefore, interactions with other methods have been found useful to increase the power of GED and extend its range of application. These interactions are of two distinctly different kinds. The first type of interaction occurs at the experimental

ELECTRON DIFFRACTION

87

level and implies some modifications of standard GED equipment; this will be characterized in Section III. The second type of interaction refers to the way the experimental GED data are treated and it implies the inclusion of some additional information from independent sources of both experimental and computational nature; this will be described in Section IV. These two sections are preceded by Section II, which serves as an introduction to both of them, describing the standard GED equipment and standard structure analysis. The first GED experiment was reported in 1930 by Herman Mark and Raimund Wierl in Germany. Many interesting details related to this event can be found in Mark's personal recollections (see Introduction in Ref. [1 ] and Ref. [2]). However, an argument can be made that Lawrence Brockway, more than anyone else may be said to be the father of the GED method [3 ]. Brockway, as a graduate student of Linus Pauling, initiated GED in the United States and as early as in 1936 he was already able to write a review article in Review of Modern Physics which contained structural information for 146 molecules. Currently, there are about 20 groups, in the world, doing GED and they are in the following countries: Belgium, Germany, Hungary, Japan, Norway, Russia, and the United States. Their scientific production reaches about 150 papers per year. With this publication rate it is important to keep a record of all relevant literature and The Sektion flir Spektren- und Strukturdokumentation at the Universit~it Ulm, Germany does this job. This group currently headed by Jiargen Vogt created a structural chemical database for gas-phase compounds [4] and molecular spectroscopy [5] and also participates in publication of LandoltB6mstein Tables, a unique source of structural information [6-10]. In 1973, The Chemical Society in Great Britain began to publish an annual series of

88

MASTRYUKOV

comprehensive reports on molecular structure by diffraction methods. A survey of earlier electron diffraction studies has been written by B. Beagley [11 ]. The last volume in this series was published in 1978 and in a contribution by Sch~ifer [12] one can find all the previous references. In recent years Rankin and Robertson have continued this work and we mention here only the latest contributions [ 13 ]. Finally, a useful source of structural information for selected classes of compounds was included in a book written by Vilkov, Mastryukov and Sadova* [ 14] and Part B of a book edited by Hargittai and Hargittai [ 1].

II.

THE ELECTRON DIFFRACTION TECHNIQUE Before we discuss a combination of GED with other techniques it is convenient to give a brief overview of the electron diffraction method itself. Therefore, in subsection A we describe the experimental equipment currently in use, followed by the basic equations of GED (B) and structure analysis (C). As a logical consequence, Section III describes the changes in running the GED experiment, while Section IV characterizes those modifications, which are used in a combined structure analysis. A.

Experimental Equipment The electron diffraction method is based on measuring the intensity of electrons scattered from a gas jet injected into a high vacuum.

A highly

simplified schematic arrangement of a conventional GED unit is shown in Fig. 1; it includes the following three sections:

(1) An electron optical

The epigraph chosen for this chapter is a quotation from a Foreword written by R. Hoffmann for this book.

ELECTRON DIFFRACTION

89

system producing a well-collimated electron beam; (2) A diffraction chamber equipped with the specimen inlet system (nozzle); and (3) The detector; photographic data recording is usually used in most GED units, although counting techniques (see below) have also been applied successfully. The scattering picture consists of a series of concentric rings superimposed upon a steeply descending background, falling off from the diffraction center towards higher scattering angles.

Since the scattering

intensity decreases so rapidly, a rotating sector made of metallic sheet is used to screen the photographic plate. The sector, a spiral- or heart-shaped cam, is placed immediately above the photographic plate with its axis of rotation coinciding with the incoming beam. Sectored pictures can easily be record zd by a microphotometer.

Typical experimental conditions are as follows:

electron accelerating voltage about 40KV, which corresponds to electron wavelength, E, about 0.06A, vacuum about 10-5 Torr and sample pressure at the container about 20 Torr. Most of the GED apparatuses used all over the world are laboratory built and their characteristics were reviewed by Hilderbrandt [15] and Oberhammer [16] in 1975 and 1976, respectively.

A more recent

contribution by Tremmel and Hargittai [ 17] contains many details about the GED unit used in Budapest, Hungary. Finally, the Antwerp GED unit has been described, in 1997 for the first time [ 18]. A very interesting and promising development of the registration technique is reported by Iijima, Suzuki and Yano in 1998 [19]. The authors

90

MASTRYUKOV

photograhic plate rotating sector

/I

electron gun

9

electron beam

nozzle~l I vacuum chamber

FIG. 1

scattered electrons

gas

Scheme of the gas electron diffraction experiment.

suggest the use of imaging plates (IP) instead of the photographic plates used so far. Due to the wide dynamic range of IP, data of similar quality were obtained both with and without a rotating sector.

Another improvement

consists of the use of a read-out system of a stand-alone type ivstead of a microdensitometer. All this allows the authors to conclude their paper by a very strong statement: "The experimental technique of gas-phase electron diffraction has been characterized by the sector and the microphotometer and is thus called the sector microphotometer method. Both of these are going to be obsolete by virtue of the wide dynamic range of the imaging plates." Although the photographic registration of the scattered electrons is undoubtedly the most widely used technique in conventional GED units, it is not the only one. There were several successful attempts to eliminate the photographic intermediary and sector device.

The first unit using counting

ELECTRON DIFFRACTION

91

technique was built in 1970 at Indiana University by Fink and Bonham [20,21 ]. Later, a similar apparatus has been build by Fink and coworkers at the University of Texas [22]. The high sensitivity of the counting technique allows one to study compounds of low volatility: a sample pressure of 10-2 Torr is sufficient to run an experiment instead of the conventional 20 Torr. In the same Austin group, a unique diffraction unit was built equipped with a M611enstedt analyzer, which allows the measurement of both elastic and inelastic cross sections. It was applied to a study of correlation effects in the Ne atom.

This study represents a major step forward in the details with

which calculated wave functions can be examined experimentally [23 ]. Another non-photographic instrument for GED studies was build by Sch~ifer and coworkers at the University of Arkansas in 1984 [24]. In this machine the scattered electrons are detected by a fluorescent screen, which is optically coupled to a custom multichannel analyzer.

Later several

improvements were introduced into the original design [25-27] and the possibility of using real-time GED as a detector in gas chromatography was demonstrated [28]. Some recent studies of SF 6 [29] and SeF 6 [30] show the application of this procedure. As a continuation of achievements in non-photographic techniques in the 1970s and 1980s, in the 1990s three more GED units were build resembling one of the two types mentioned above. In 1992 BSwering and coworkers at the University of Bielefeld, Germany reported their results [31 ] and a more detailed account was given later [32].

A similar advance

occurred with a machine built in 1992 by Zewail and coworkers at the

92

MASTRYUKOV

California Institute of Technology [33,34].

In this apparatus a newly

designed computerized two-dimensional charge-coupled device (CCD) was used as a detection system. This permitted immediate visualization of the diffraction pattern; a color image is shown in Ref. [33] while a black and white is reported in Ref. [34]. The authors called this technique Ultrafast Electron Diffraction (UED) [33,34].

Finally, in 1995 the experimental

apparatus was described by Geiser and Weber at Brown University, Rhode Island [35]. Therefore, we can conclude that the counting technique started in 1970 lead to significant progress in recent years which is related to progress in technology. As can be seen in later sections, most of these machines were designed for special purposes where the photographic technique is not applicable.

B.

Basic Relations The basis of the use of GED as a structural tool lies in the relationship between the intensity of scattered electrons as a function of scattering angle, 8, and P(r), the probability function which expresses the distribution in internuclear separation.

The molecular parameters determined by this

relationship are rij, the effective internuclear distance between atoms i and j, and lij, the amplitude of vibration (sometimes also denoted as uij). The total scattered intensity, I T, is usually expressed as a function of the variable s, instead of the scattering angle, 8, where s = (4n/~.) sin (0/2)

ELECTRON DIFFRACTION

93

and ~. is the electron wavelength.

When a smooth background, I B, is

subtracted from the total intensity, the molecular term, IM, is obtained. I M = I T - I B. The background itself does not contain information conceming the molecular structure; it is mainly determined by the charge distribution in the atoms of which the molecule it built, and by inelastic and extraneous scattering.

Theoretically, the molecular term can be expressed to an

approximation sufficient for most structure work as follows (for more details of theory see Refs. [ 14, 21, 36-40]): IM -

~ l f i [ I f j l cos (rli _ i~j

rlj)exp(-ll/j

s:)sin S(ra/ . -

9

(1)

The complex atomic scattering factor,

f i(s) = f i(s)] exp[i ~Ti(s)] is usually taken from existing tables. The most recent scattering factors were calculated by Ross, Hilderbrandt and Fink in 1992 [41].

The asymmetry

parameter, K, represents a slight deviation of the argument from a linear function of s and to a good approximation is equal to

al4/6

where a is the

Morse constant. The molecular term slM, or sometimes the ratio S(IM/IB) = sM(s), is analyzed by a least-squares method and the bond distances r a, bond angles and some of the amplitudes of vibration are determined (see more about this in the next section).

The r a distance, sometimes called an operational

parameter, is an ill-defined parameter since it refers to the maximum position of any peak of the

P(r)/r

function.

The better parameter, the rg distance,

94

MASTRYUKOV

which corresponds to the center of gravity of the P(r), peak and is the thermal average of internuclear distance. The relation between these two parameters and the definitions of other parameters, which are of special importance in GED, are given in Table 1. Normally, every GED paper reports the original intensities I(s) and sM(s) curve as recommended by the Commission on Electron Diffraction of the Intemational Union of Crystallography [42]. Such an example from the electron diffraction study of dichlorodimethylsilane [43] is shown in Fig. 2. The experimental background shown in the upper part of Fig. 2 can be either hand-drawn or can be generated computationally using a series of polynomials satisfying a number of criteria. The initial background can be improved in the course of structure analysis [44]. Another curve that is much used in GED study is the Fourier transform of the molecular intensity function, the radial distribution curve.

It is

customarily defined by: Smax

fE(r) =

sM E(s)

exp (-ks 2) sin (sr) ds.

(2)

Smi n

The integration limits Smin and Smax are limited by the experimentally available s range. The constant k is an artificial damping constant that helps to suppress error ripples in the radial distribution curve. The radial distribution curve has the advantage of being more understandable than the intensity curve since it has a peak for each intemuclear distance. The peak is rather narrow and is centered around every

ELECTRON DIFFRACTION

95

TABLE 1. The most important internuclear distance parameters.

ra

effective internuclear distance in the expression of the molecular contribution to electron scattering intensities, Eq. (1), equal of the center of gravity of the P(r)/r distribution function for the experimental temperature.

rg

average value of interatomic distance (for a particular temperature), equal to the center of gravity of the probability distribution P(r) for each pair of atoms in the molecule. It is related to r a by rg = r a -12/r.

rt~

distance between average nuclear positions in the thermal equilibrium at temperature T; calculated from rg or r a

ra = r g - K = r a -12/r- K.

K is the perpendicular amplitude correction (IV.A) K = ( + )/2r e. value of ra extrapolated to temperature of 0 K. rO

limT~ 0 (r~).

It is related to r a by

r T - r 0 = 3//22a I ( l 2) T_ (12) 0 ] + 5r T + K 0 _ (/2)T

/r0

a is the Morse anharmonicity constant, l an rms amplitude of vibration, 8r the change due to centrifugal distortion, K the perpendicular amplitude corrections. r0

effective

intemuclear

parameter

which

reproduces

ground-state

rotational constants A 0, B0, C O(section IV.B). rZ

distance between mean positions of atoms in the ground vibrational state; represents same physical quantity as rO but differs in origin since it is calculated from spectroscopic data.

re

distance between equilibrium positions related to rg by r e = r g - 3al2/2.

96

MASTRYUKOV

l(s)

(CH 3 )2 SiCI2

| r

sM(s)

IV

VVUVVVVV" 2A

. . . .

ilk

A

l

.

.

l

5 FIG. 2

.

.

I

15

_-----_ _ _ A

l

l

25

I

s,/~ -1

Electron-diffraction intensity curves. The two uppermost curves are the total intensities, I(s), obtained at two different "nozzle-toplate" distances.

Empirical background is shown.

Below is a

composite molecular intensity curve sM(s). The bottom curve is the difference between experiment and theory multiplied by a factor of two. distance r encountered in a molecule; it has an approximately Gaussian shape unless several imemuclear distances contribute to the same peak. The radial distribution curve for (CH3)2SiCI 2 [43] shown in Fig. 3 gives examples of

ELECTRON DIFFRACTION

97

f r)

f

C-H

l

Si-C Si-CI Si...H C...C C...CI CI...CI (]...H

A I 0

1 FIG. 3

3

r,A

Experimental radial distribution curve for dimethyldichlorosilane. All the peaks are assigned according to the molecular model shown in the upper right comer; the difference curve shows how well the final model corresponds to the experiment [43 ].

both of these possibilities: the first and the last peaks are very well separated and have almost Gaussian shapes while all the others are composite peaks having either two or three components. C.

Structure Analysis An electron-diffraction structure determination is nowadays based exclusively on comparing experimental, sME(s), and theoretical, sMT(s), molecular intensity curves. The structure parameters are adjusted until the

98

MASTRYUKOV

best fit is obtained and a least-square procedure is always used for this purpose. From the form of Eq. (1) it is clear that the least-squares procedure is nonlinear in the structural parameters. The usual linearizing method based on Taylor's series expansion is used so that the parameters actually adjusted are the shitts in the internal coordinates. The algorithm was formulated a long time ago by Hedberg and Iwasaki [45] and it was implemented in many programs like the one used by the Norwegian school [46]. Structure analysis begins with building a trial model [47]. The radial distribution curve can be very useful for this purpose. From the positions of the peaks one has trial values of rij and from their half widths one can estimate trial values of lij (see Fig. 3). Earlier practice was also to take the initial values for the amplitudes of vibration from similar molecules [48-52], however, at the present time the amplitudes are calculated from the available force fields (see more in Section IV.A.). When the trial model is completed one can calculate sMT(s) using Eq. (1) and compare it with its experimental counterpart, sME(s). This time the structure refinement begins [47]. The least-squares program changes all r's and l's until the difference

k=l reaches its minimum value (n is the number of observed points and Wk is a weight function).

The number of adjustable parameters depends on the

accuracy and on the amount of experimental data and on the complexity of molecule as well. The analysis often requires assumptions about molecular

ELECTRON DIFFRACTION

99

symmetry and those parameter values on which the molecular intensity depends only weakly. These "weak" parameters are constrained to calculated values or to values known from previous experience. These constraints of the amplitudes of vibration and the geometry parameters are discussed in more detail in Sections IV.A and IV.C, respectively. The quality of fit is characterized by the R factor, which is also used in X-ray crystallography:

1/2 m

_

_

k=l

n

. . . .

(4)

Note that the numerator in this expression is identical to Eq. (3). In our example of (CH3)2SiC12 molecule (see Figs. 2 and 3) an R factor was 7.6% [43]. It is not rare in the practice of GED that several models are found which have very similar R factors. It is important to recognize that all these models are solutions of the problem in the mathematical sense and to make a good decision between them is not at all easy. One such example is given by the study of thionlytetrafluoride, SOF4, by Gundersen and Hedberg [53]. These authors found four models in excellent agreement with experiment. Multiple solutions are a serious problem of the GED method and the most general way to solve this problem is by addition of information from independent sources.

In the case of SOF 4, this was done by using three

rotational constants from microwave spectroscopy as constraints [54]. Other examples will be considered in Section IV.

100

MASTRYUKOV

We conclude this Section by mentioning a very important modification of the common least-squares procedure suggested in 1975 by Bartell, Romenesko and Wong

[55] and called the "Method

of Predicate

Observation." In this method, a set of reasonable values of parameters based on similar measurements or experience is incorporated into the least-squares procedure as additional observations.

This addition helps to remove the

natural or accidental linear dependence among the parameters which interfere with their reliable determination. The important feature of this approach is that a guessed value used as a predicate observation does not rigidly constrain the optimized parameters to the a priori estimate. Therefore, the systematic error introduced could be much less critical than if the parameters are frozen.

This powerful method is perhaps the most objective way of

combining GED with molecular mechanics, ab initio calculations and X-ray analysis. A promising example of the application of this procedure known as SARACEN is discussed in Section IV.C.

III.

COMBINED APPLICATION OF EXPERIMENTAL TECHNIQUES In this section we describe those changes in the standard GED apparatus (see Section II.A and Fig. 1, in particular) which have been designed to expand the application of the method. The presentation in this section, as well as in the next one, follows the chronological development. In 1967, a supersonic nozzle replaced the regular one in a GED unit and, in Subsection A, we will see what kind of new information this alteration may produce.

ELECTRON DIFFRACTION

101

In 1975, a quadrupole mass spectrometer was coupled with the GED unit. This type of combination unit is sometimes referred to as ED + MS (see Subsection B). Finally, in 1979 a laser was used for the first time in the GED experiment and it is described in Subsection C. A.

Supersonic Nozzle As early as the late 1960s the Orsay electron diffraction group in France successfully used a supersonic nozzle for production of large cluster~ of rare gas atoms, carbon dioxide and water [56-61]. Bartell and his coworkers at the University of Michigan [62-65] continued this important research in the early 1980s. First, formation of benzene clusters was studied [63] followed by a similar study of n-butane [64]. In the course of the latter study, an interesting phenomenon called "conformational cooling" [65] was observed. A detailed description of these and other findings was given in a review paper written by Bartell in 1986 [66]. The interested reader will find this paper a good introduction to the field and can learn more about the supersonic nozzle system, statistical modelling of clusters, solid and liquid like clusters of polyatomic molecules and trends in nucleation and cluster growth. In the subsequent years large water clusters were studies in detail [67-70] together with nanocrystals of transition-metal hexafluorides, MoF 6 and WF 6 [71 ]. In 1992, after almost three decades of using the supersonic nozzles exclusively for cluster production, a group of German physicists at the University of Bielefeld found another exciting application for this inlet system.

It was suggested to use a supersonic molecular beam to study

102

MASTRYUKOV

Spatially Oriented Molecules [72-77]. A continuous beam of CH3C1 or CH3I molecules is oriented via the hexapole technique before their scattered intensities are measured in the GED unit described before [31,32].

The

molecules are oriented preferentially parallel or antiparallel to the electron beam and show oscillating deviations from the scattering pattern of unoriented molecules up to 4% as a function of momentum transfer, quite similar to previous calculations using the independent-atom model (IAM) [78,79]. In 1994, Fink and co-workers introduced a pulsed supersonic quartz nozzle, which might find a future application in GED [80]. B.

Mass Spectrometer Every GED study is done with the assumption that the composition of the gas phase is known. This assumption is good enough if we deal with stable compounds, which have no tendency to self-association like benzene, CCI4, NH 3, etc. On the other hand, there are cases when a researcher has doubts about what kind of species are present under the experimental conditions. Two typical examples include" 9 there is a dynamic equilibrium between monomeric (M) and dimeric (D) species 2M # D, 9 the compound in question is unstable and is prepared in the course of the GED experiment. Note that both processes are temperature dependent. Normally, when there are doubts about the vapor composition a common practice is to determine it from an independent mass spectrometric

ELECTRON DIFFRACTION

103

measurement. This approach is exemplified by a recent work by Konaka et al. [81] who studied CH3-N=CH2, a product of thermal decomposition of trimethylamine. In this case, the mass spectra were measured to determine the optimum heating conditions.

However, there are situations when the

structural problem can be solved only by simultaneous mass spectrometric and GED measurements. The first attempt for such a combination was made by Kohl and Kennerly at the University of Texas in a study of the thermal isomerization of cyclopropane.

Unfortunately, the results for this pioneer

study remained unpublished and are described only in the dissertation by Kennerly in 1975 [82]. In their apparatus, the quadrupole mass spectrometer was located outside the diffraction chamber on the flange opposite the nozzle.

It had its own high-vacuum system and was connected to the

diffraction chamber via a valve. Slightly later, in 1977, a similar combination was developed in Budapest, Hungary by Hargittai and co-workers [83]. For more details, see also Ref. [17].

Still later, another group in Ivanovo, Russia headed by

Girichev built analogous equipment.

Table 2 shows several molecules

studied in these two groups by a combined approach GED + MS. C.

Laser In 1979, Arvedson and Kohl were the first to use a laser to study vibrationally pumped molecules by GED [84]. The basis for the use of this combination lies in the ability of GED to measure the amplitudes of vibration. This is how the authors describe their reasoning:

104

MASTRYUKOV

TABLE 2. Molecules studied in a combined electron diffraction/ quadrupole mass spectrometer experiment. No.

Molecule

85

8

CI2Gee

84

A12Br6

85

9

Cl2Sid

89

3

BeCI 2

86

10

C13NbO

90

4

Be2CI 4

86

11

F3NbO

91

5

BeF2

87

12

GeI2 e

92

6

Br2Gea

88

13

C22H38CuO4

93

7

Br2Sib

89

No.

Molecule

1

AIBr 3

2

Ref.

Bis(dipivaloylmethanato)copper(II) 620~

aproduced by reaction Ge(s) + GeBr4 (v) bproduced by reaction Si(s) + SiBr4 (v)

1200~ 660~

eProduced by reaction Ge(s) + GeCI4 (v)

1200~

dproduced by reaction Si(s) + Si2CI6 (v) eproduced by reaction GeI4(g) + Ge(s) .

Ref.

653K

> 2 GeBr2 (v). >2 SiBr2 (v). > 2 GeCI2. >3 SiCI2 (v).

' 2 GeI 2 (g).

Since the mean amplitudes of vibration, the l/j's, are functions of the vibrational temperature of the molecule, their relative change with respect to a change in the molecular vibrational energy will depend in the distribution of energy among the modes.

Therefore, information

concerning intramolecular energy transfer among modes can be obtained from a study of the relative changes in the mean amplitudes of all atom pairs of a molecule as a result of the deposition of energy into a particular vibrational mode. [94] The authors studied the absorption by room temperature SF 6 of the 10~t radiation emitted by a CO 2 laser. An electron diffraction unit was modified

ELECTRON DIFFRACTION

105

to allow the intersection at right angles of an electron beam, a gas jet of SF6, and focused 40W c w C O 2 laser beam. The focal spot was positioned just off the gas inlet nozzle, which was equipped with a thermocouple, to monitor its temperature. They found that the relative rise in the vibrational amplitudes indicated that the distribution of absorbed energy was not thermal. Therefore, this work has shown that it is possible to observe the effects of laser excitation of molecules by GED. The same molecule was studied in more detail by Bartell and coworkers [95-98] soon after the first study by Averdson and Kohl. Over 200 diffraction plates were taken at SF 6 sample pressures ranging from 50 to 600 Torr, and at various laser powers and wavelengths. Conclusion reached by the authors can be summarized as follows: Vibrational excitation corresponding to the absorption of up to 3.6 photons/molecule was deduced from the increased amplitudes of vibration of the SF, FFci s, and FFtran s atom pairs and the lengthening of the SF bond.

At high

excitations, electron diffraction intensities were accounted for best if it was assumed that two subsets of molecules were produced, one much hotter than the other.

Vibration-

vibration relaxation from v 3 to the other stretching modes was too fast to be followed.

Relaxation of stretching to

bending could be monitored, crudely, at lower pressures where approximately 30 collisions were needed at the depressed temperatures in the jet. At higher pressures and excitations V-T/R relaxation was observed, corresponding to a transfer of perhaps one-tenth of the vibrational excitation in the course of 103 collisions. [98]

106

MASTRYUKOV

In the course of these studies of laser-pumped SF 6, a number of diffraction patterns were recorded of vibrationally hot molecules that had been excited by accidental irradiation of the nozzle tip. These patterns, when analyzed, were found to yield information about vibrational anharmonicity that is difficult to derive from spectroscopy.

This accident helped to

introduce a new method in which gas molecules can be heated from room temperature to well over 1500 K in the order of a microsecond. Using this procedure the vibrations and thermal expansions of SF 6, CF4, SiF4, and CF3CI have been investigated up to 1200-1700 K [99-102]. Various models proposed for the treatment of increases in bond length have been assessed, among which an anharmonic Urey-Bradley field accounted well for results [99l. In all GED studies described so far, a continuous electron beam was used. In 1983, Ischenko and co-workers at the University of Moscow, Russia introduced a new technique where exciting optical pulses of the laser were synchronized with the pulsed electron beam [103,104]. In this method, the GED intensities are recorded at a well-defined time interval following excitation of the scattering molecules, and repeated electron pulses all diffract from molecules that have the same age relative to the time of excitation.

Using this method it should be possible to characterize short-

lived species or states, to study time-resolved energy absorption, distribution and redistribution, and to measure structural changes.

Clearly, for kinetic

studies it is crucial to choose the time duration of the electronic pulse and the delay time between pumping and probing shorter than the life-times of the

ELECTRON DIFFRACTION

107

species or the duration of the phenomena under investigation. This technique was called Stroboscopic Gas Electron Diffraction. Using the experimental arrangel~ent described in Refs. [ 103 ] and [ 104] a study of the IR multiphoton dissociation of the CF3I molecule, according to the reaction CF3I

hv >CF 3 + I, was initiated with the hope of recording

diffraction pattems of the short-lived CF 3 free radical. However, the authors were successful only in demonstrating changes in scattering intensities of this molecule induced by irradiation; the data have not been analyzed, and it was impossible to judge the success of the experiment from the information given. Later, Mastryukov [105] critically analyzed the data and found that the conclusions reached by the authors [103,104] were too optimistic. This criticism was never answered since the equipment in the University of Moscow was dismantled. A quantitative study of the photodissociation of CF3I was reported by Zewail and co-workers in 1994 who used UED [34]. Here we can find all the documentation lacking in the original publication by Ischenko et al. [ 103,104]. Earlier experiments in Moscow were of limited scientific value because the requisite detection schemes did not exist at the time but they opened the way for similar studies. Later, these studies continued at the University of Arkansas using the on-line data recording techniques mentioned in Section II.A [106].

In the early 1990's Sch~ifer and co-workers studied two

interesting systems using laser irradiation coupled with their equipment. They called their method Time-Resolved Electron Diffraction (TRED) and

108

MASTRYUKOV

proposed it as a tool that can be used to investigate the evolution of internuclear distances in reacting molecular systems. First, the 193-nm photodissociation of isomeric 1,2-dichloroethens was studied [ 107]. The authors conclude: When the time delay between excitation and diagnostic electron scattering is 15 ns, only fragmentation and no cistrans isomerization is observed.

When the time delay is

several milliseconds, cis-trans isomerization is prevalent. This finding confirms the thesis that, because of faster intervening dissociative processes, cis-trans isomerization is not

a

primary

photochemically

driven

unimolecular

rearrangement of 193-nm irradiated dichloroethenes but hinges upon intermolecular collisions. [ 107] The second example was the 193-nm photodissociation of CS 2 [ 108]. This is how this study was characterized in the abstract to this paper: A novel data analysis procedure is described, based on a variational solution of the SchrSdinger equation, that can be used to analyze gas electron diffraction (GED) data obtained from

molecular

ensembles

in

nonequilibrium

(non-

Boltzmann) vibrational distributions. The method replaces the conventional expression used in GED studies, which is restricted to molecules with small-amplitude vibrations in equilibrium distributions, and is important in time-resolved (stroboscopic) GED, a new tool developed to study the nuclear dynamics of laser-excited molecules.

As an

example, the new formalism has been used to investigate the structural

and

vibrational

kinetics

of

C-S,

using

stroboscopic GED data recorded during the first 120 ns following the 193 nm photodissociation of CS 2. Temporal changes of vibrational population are observed, which can

ELECTRON DIFFRACTION

109

be rationalized by collision-induced electronic-to-vibrational energy transfer from excited S(1D) atoms to ground state C-S and CS 2. The time-evolution of the energy transfer is modeled by determining the vibrational distributions and mean internuclear distances (ra, g) of C-S as functions of delay time.

Inverted (non-Boltzmann) distributions are

observed, and the refined parameters of model distributions are presented. [ 108] We conclude our description of experiments involving the photon excitation by mentioning a single contribution by British researchers who combined GED with flash-photolysis [109].

These authors conducted

diffraction studies on the decomposition of chlorine dioxide and biacetyl using electron pulses. Study

of the

photodissociation

dynamics

and

nonequilibrium

vibrational distribution of laser excited species requires significant changes in the theory used earlier for structure determination and described in Section II.B. Therefore, it is not surprising that a number of theoretical papers were published by different groups in an attempt to give a better description of the phenomena involved [ 110-116].

IV.

T E C H N I Q U E S I N C O R P O R A T E D IN S T R U C T U R A L A N A L Y S I S In this section we come back to the use of our regular GED equipment (See Section II. A) and we will see how the usual structure analysis described in Section II. C can be modified in order to get more reliable and more precise structural information.

So far, four different and independent sources of

information were proved to be particularly useful for incorporation into the regular structure analysis. Curiously enough, these techniques were introduced to GED

110

MASTRYUKOV

during four consecutive decades showing rather accurately their chronological sequence. 9 Vibrational Spectra (symbolized as IR) 1950s Rotational Constants, mostly taken from microwave spectroscopy (MW) 1960s 9 Theoretical methods, (Molecular Mechanics, ab initio, DFT) 1970s 9 Liquid crystal NMR (LC NMR) 1980s. These combinations will be reviewed in this order in subsections A, B, C and D. Each of these subsections contains tables listing molecules studied by these particular combinations with GED.

However, there are many cases in which

several techniques were applied simultaneously, then the molecule is mentioned only once in the corresponding table giving the priority to the same sequence of the methods: IR, MW and Theory. The only exception is made for LC NMR, otherwise these cases would have been absorbed by the previous combinations. The ultimate goal of a combined structure analysis is to obtain a self-consistent molecular model by the process schematically depicted in Fig. 4.

This is a

modified version of Fig. 10-2 taken from a review written by Geise and Pyckhout [ 117]. Such a combination became possible when the physical significance of the structural parameters produced by any individual method was clarified and they were made compatible with electron diffraction. A.

Vibrational Spectroscopy Historically, vibrational spectroscopy was the first method which found a very useful application in the GED structure analysis. As early as in 1949, the Karles [ 118] were able to carry out an electron diffraction experiment of

ELECTRON DIFFRACTION

111

Theory Geometry, Energy and Force Field

1 ED

IR

MW

LC NMR

Self-Consistent Molecular Model

FIG. 4.

Components in the determination of a self-consistent molecular model.

sufficient precision to allow a determination not only of the interatomic distances, but also of mean amplitudes of vibrations. The studies reported in this paper were of CO 2 and CC14. The experimental amplitudes of vibration were compared with those calculated by the authors for CO 2 from formulas given by Debye [119] and with those calculated by James [120] for CC14. It had been shown by Debye and James that the amplitudes of vibration could be calculated from molecular vibration frequencies and atomic masses, together with assumptions about the form of the harmonic force field of the molecule. This is how J. and I. L. Karle describe this period of time in their recollections [ 121 ]: Confidence in the increased accuracy with which interatomic distances could be determined, as well as the possibility of

112

MASTRYUKOV

obtaining reliable values for root mean squared amplitudes of vibration, was enhanced by satisfactory comparisons with results from infrared and microwave spectroscopy.

An

incident that occurred soon after our first work using the new

analytical techniques

was

published provided

a

poignant example of the value of such comparisons. One of the first molecules investigated by the new analytical techniques was CO 2 (Karle and Karle, 1949). We reported a value of 0.040+0.007 A for the rms amplitude for the O...O distance. However we had calculated a value of 0.029 A from spectroscopic data which indicated a discrepancy somewhat larger than our reported limit of error for the electron diffraction experiment.

It was soon shown by

Yonezo Morino (1950) that the formula that we were using was in error and that atter correction the computed value for spectroscopic data was 0.041 A in very good agreement with the result from the electron diffraction experiment. [ 121 ] (The authors refer to their own paper, which we already mentioned [118], while the contribution by Morino can be found under Ref. [ 122]). In the early 1950s Morino and his co-workers in

Japan established a

systematic treatment for the calculated of amplitudes of vibration from harmonic force fields, based on the widely used Wilson GF matrix method [ 123], which is still the foundation for the majority of such calculations. The following quotation taken from Morino's personal recollection [ 124] describes the next milestone of this story: In 1959 I visited Trondheim and had a chance to talk with Otto Bastiansen about the future prospects of gas electron diffraction.

My presentation at a colloquium there on the

method of calculation of mean square amplitudes apparently

ELECTRON DIFFRACTION

113

stimulated Sven S. Cyvin, who later became an expert in the field (Cyvin, 1968). Bastiansen took me to a ski-house on the top of the Holmenkollen Olympic Stadium near Oslo, where he told me about his new discovery that distance between two nonbonded atoms in a linear molecule, such as allene or dimethylacetylene, deviates appreciably from the sum of their bond lengths (Almenningen, Bastiansen and MuntheKaas, 1956; Almenningen, Bastiansen and Traetteberg, 1959). An idea came to me that this systematic difference would be attributable to the bending vibrations of linear molecules.

Upon my return to Tokyo, we made a

calculation and found that the observed magnitudes of the 'shrinkages'

were what we expected from the force

constants obtained by spectroscopic measurements (Morino, Nakamura and Moore, 1962). [124] As we will see, the conversation in Holmenkollen had a very important conceptual impact on the GED method. This phenomenon is known as the "Bastiansen-Morino shrinkage effect" or more simply as the "shrinkage effect". Shrinkage effect is a direct consequence of the molecular vibrations and it shows that the bonded and non-bonded interatomic distances measured by GED are not self-consistent, i.e., they do not correspond to a set of distances calculated from a rigid geometrical model.

This phenomenon is

illustrated on a simplified diagram below for a linear triatomic molecule A B 2 (simplification in this case means that for this type of vibration while two B atoms move up, atom A moves down which is ignored in this diagram).

114

MASTRYUKOV

rg

-,~

r e

,~

The B...B distance observed by GED (rg) is shorter than that obtained by doubling the A-B distance as we expect for a linear system. This effect arises because at every instantaneous position in the bending mode, except precisely at the linear (r e equilibrium) configuration, the B...B distance must be less than twice the A-B distance. Usually the shrinkage effect is routinely included in electrondiffraction least-squares refinement.

In order to do so, it has been found

appropriate to introduce a third distance type ra defined as the distance between mean positions of atoms at a particular temperature [125,126].

If

the harmonic force field is known, ra may be calculated from r a according to

Eq. (5)" l2 ra = r a + - - - K r

whereK

2r

(Ax 2) and (Ay2) are the mean square perpendicular vibrational amplitudes.

(5)

ELECTRON DIFFRACTION

115

In Section IV. B will see that this new ra structure plays an important role to make the different methods compatible. Several programs were published to calculate amplitudes of vibration and shrinkage corrections [127-129]. In order to do so, one needs to know the force field of a molecule. There are typically three options for this: 1) An approximate force field is built by transferring the elements, from similar molecules (See, for example, [130, 131 ]). 2) The force field can be calculated theoretically using either molecular mechanics [132, 133] or ab initio calculations. 3) The force field is obtained from the normal coordinate analysis of the vibrational spectrum of a molecule. Data collected in Table 3 illustrates these options showing how it is practically done in different electron-diffraction groups [ 134-165].

B.

Rotational Constants According to Robiette [ 166], there are three major topics that comprise the main areas of interplay between spectroscopy and GED: the amplitudes of vibration, the shrinkage effect and the inter-relation of molecular structures obtained from spectroscopic and electron-diffraction data.

The

first two topics were discussed in the previous subsection while this subsection concentrates on the third topic. In the 1950s it was noticed that bond lengths (ro) derived from spectroscopic measurements of rotational constants (Bo) may differ from GED bond lengths (ra, rg) by an amount which is clearly outside

116

MASTRYUKOV

TABLE 3. Molecules studied by joint analysis of electron diffraction and vibrational spectroscopy.

No.

Molecular formula

Name

Other methods useda

Reference

BBr 3

Boron tribromide

134

B2GaH9

Hydridogallium bis(tetrahydroborate)

BiF 3

Bismuth trifluoride

136

BiI3

Bismuth triiodide

137

Br6H2Si3

Hexabromotrisilane

CdC12

Cadmium dichloride

139

CdI2

Cadmium diiodide

140

CeI 3

Cerium triiodide

141

CI2Mg

Magnesium dichloride

142

10.

C13Ga

Gallium trichloride

143

11.

C13In

Indium trichloride

143

12.

CI4Mg2

Magnesium dichloride dimer

142

13.

C14U

Uranium tetrachloride

14.

F3Sb

Antimony trifluoride

136

15.

F4U

Uranium tetrafluoride

145

16.

I3Sb

Antimony triiodide

137

17.

CFNO 5

Fluorocarbonyl peroxynitrate

Th

146

18.

CFN30

Fluorocarbonyl azide

Th

147

19.

CF202

Fluoroformyl hypofluorite

Th

148

20.

CH2C12

Dichloromethane

MW, Th

149

Th

Th

Th

135

138

144

ELECTRON DIFFRACTION

117

TABLE 3. (continued).

No.

Molecular formula

Name

Other methods useda

Reference

21.

C2FNO2

Fluorocarbonyl isocyanate

Th

147

22.

C2F202S 2

Bis(fluorocarbonyl) disulfane

Th

150

23.

C2F6S

Trifluoroethylidynesulfur Trifluoride

Th

151

24.

C2H203

Formic anhydride

MW, Th

152

25.

C2H6C12Ti Dimethyltitanium dichloride

Th

153

26.

C3H3CIO

2-Chloroacrolein

Th

154

27.

C3H403

Acetic formic anhydride

Th

155

28.

C3H6C12Si Dichloromethylvinylsilane

29.

C3H60 2

Ethyl formate

MW, Th

157

30.

C4H60

Methacrolein

MW, Th

158

31.

C4H60

N-Chloro-N-ethylethanamine

Th

159

32.

C4H100

Diethyl ether

MW, Th

160

33.

CsHloO 2

Isopropyl acetate

Th

161

34.

C6H140

tert-Butyl ethyl ether

Th

162

35.

CTHTCIO

Chloromethyl phenyl

Th

163

36.

C7H7CIO

Chloromethyl phenyl ether

Th

164

37.

C8H22Si2

1,2-Di-tert-butyldisilane

Th

165

156

aThe abbreviations have the following meaning: Th - theoretical calculations (Molecular Mechanics, ab initio and density functional theory); and MW- microwave spectroscopy

experimental error (for definitions of these parameters see Table 1). Critical examinations were made and it became clear that this arises because the spectroscopic and electron-diffraction bond lengths are derived from

118

MASTRYUKOV

observed quantities in which the effects of molecular vibrations are averaged in quite different ways.

In 1961, Bartell, Kuchitsu, and de Neui [167]

published a paper on mean and equilibrium structures of CH 4 and CD 4 where for the first time the precise relationship of electron-diffraction bond lengths to spectroscopic bond lengths was determined. Furthermore, the relation of both types of these structures to the equilibrium structure, r e, was found. This confirmed the essential equivalence of molecular information derived from two different methods if suitable corrections are applied to each. The seminal contribution by Bartell, Kuchitsu and de Neui [167] opened the gate for a stronger interaction between the two previously conflicting methods and Morino [124] stresses the importance of an open discussion" Probably the strongest evidence for the success of the collaboration between electron diffraction and spectroscopy may be found in the prosperity of the Austin Symposium on Gas-Phase Molecular Structure organized by James E. Boggs.

The central theme of the Symposium is the

discussion of the fundamental problems in molecular structural studies by electron diffraction, spectroscopy and other techniques including ab initio calculations.

The

symposium has been held every other year since 1966, with increasing numbers of participants and reports. According to Boggs

[168], the history of the interaction between

experimental structure determinations by microwave spectroscopy and by GED has clearly three different eras: "(1) competition and antagonism, (2) comparison and correction, and (3) integration of analysis" If we agree that

ELECTRON DIFFRACTION

119

the second period began in 1961 with the work by Bartell, Kuchitsu and de Neui [167] mentioned above, then there is no doubt we should assign the contribution by Kuchitsu, Fukuyama, and Morino in 1968 [169] as the beginning of the third period. In order to combine the outcomes of spectroscopy and GED in a consistent manner, we need to bring the results of both methods to the same geometrical basis.

In 1960, Oka [170] proposed a new type of structural

parameter, which he termed r z or the zero-point average structure. The same type of structure can be obtained from GED if we extrapolate r a to absolute zero: r0 = limT~ 0 (ra)

(6)

Thus, the r a structure introduced to account for the shrinkage effect, Eq. (5), being appropriately corrected, helps to relate spectroscopic and electrondiffraction distances. As already mentioned in Table 1, r O and r z represent the same physical quantity and it also puts an end to the variety of different structural types currently in use in the GED structure analysis although there are some more kinds of structure obtained from the ground state rotational constants [ 171 ]. The relation between different kinds of structures derived from spectroscopy and GED is shown in Fig. 5 using as an example a classical case of CH 4 and CD 4 molecules mentioned before [167].

(Fig. 5 is a

modified version of a figure taken from the review paper by Kuchitsu [172]). An examination of Fig. 5 shows that not only primary information taken from

120

MASTRYUKOV

CD4

CH4

rg

GED

ro re

rz

SP ro I

I

1.08

1.09

I 1.10

I

l.ll

r(C-X) X = H , D A FIG. 5

The C-H (open circles) and the C-D (black circles) distances of CH 4 and CD4 in electron diffraction (GED) and spectroscopic (SP) representations together with their equilibrium value, r e.

both methods (rg and r0) is sensitive to isotopic substitution but even r 0 and rz clearly show difference too. This happens because of the mass dependency of anharmonicity and it is only the equilibrium structure r e that is free of the isotope effects. The general strategy of the combination of spectroscopic and GED data is as follows. The rotational constants B 0 first are converted to B z using a harmonic force field B z = B 0 + 0.5 ~ - ( Z sharm

ELECTRON DIFFRACTION

121

while the electron diffraction r a or rg distances are converted to r o (see Table 1).

Then both sets of data are simultaneously treated by least-squares

procedure [ 169]:

II k=l

j=a,b,c

where the first term is identical with Eq. (3) and Ia, Ib and I c are principal moments of inertia calculated from the Az, B z, Cz rotational constants, respectively. In 1988 Kuchitsu, Nakata and Yamamoto [173] reviewed the state of the calculations at the time and collected data on 118 molecules studied by a combined (ED + MW) analysis.

Table 4 serves as a continuation of this

effort giving data on 35 molecules studied in the last decade [174-205]. However, this compilation is not exhaustive and serves only for illustrative reasons. It is also important to stress the point that these particular methods are found to be complementary, which means that their combination gives a better structure than any individual method alone.

C.

Theoretical Calculations As we saw in Section II C, in order to begin the GED structure analysis one needs a trial model. In early days this model was based on the chemical intuition of the researcher while nowadays it can be calculated using some of the methods of theoretical chemistry.

The role of theory is not limited,

however, to calculation of only the geometrical structure as is shown in Fig. 4: it gives also "Energy" and "Force Fields" (see more about theoretical

122

MASTRYUKOV

TABLE 4. Molecules studied by joint analysis of electron diffraction and microwave spectroscopy.

No.

Molecular formula

Name

Other methods useda Th

Reference 174

1.

B4H 10

Tetraborane(10)

2.

GeH6Si

Silylgermane

3.

CH3AsF2

Methyldifluoroarsine

Th

176

4.

CH3F2N

Methyldifluoroamine

Th

177

5.

CH40

Methanol

178

6.

CH6N2

Methyl hydrazine

179

7.

C2H3C1OS Methylchlorothioformate

8.

C2H3C13

1,1,1-Trichloroethane

C2H3F30

1,1,1-Trifluorodimethylether

10.

C2HsN

N-Methylmethyleneimine

81

11.

C2H6C12Si

Dimethyldichlorosilane

43

12.

C2H602

Ethane-1,2-diol

Th

183

13.

C2HsN2

Ethylenediamine

Th

184

14.

C3H403

Glycolicacid

185

15.

C3H6C1N

N-Chloroazetidine

186

16.

C3H7Br

1-Bromopropane

Th

187

17.

C3H7I

1-Iodopropane

Th

188

18.

C3H7N

Cyclopropylamine

19.

C3H7N

Propyleneimine

175

Th

180 181

Th

182

189 Th

190

ELECTRON DIFFRACTION

123

TABLE 4. (continued).

No.

Molecular formula

Name

Other methods useda

Reference

20.

C3H7NO

E-Propionaldehyde oxime

191

21.

C3H7NO

Z-Propionaldehyde oxime

Th

192

22.

C3HTNO

Propionamide

Th

193

23.

C3H7NO2

Methoxyacetamide

Th

194

24.

C3H9C1Si

Chlorotrimethylsilane

195

25.

C3H9N

Isopropylamin

196

26.

C4F6

Hexafluorocyclobutene

190

27.

C4H602

Methyl acrylate

Th

197

28.

C4H60 2

Cyclopropanecarboxylic acid

Th

198

29.

C4H80

2-Methylpropanal

Th

199

30.

CsHsN

Pyridine

Th

200

31.

C5H6S

2-Methylthiophene

Th

201

32.

C5H6S

3-Methylthiophene

Th

202

33.

C5H7S

N-Methylpyrrole

Th

203

34.

C6H10

Bicyclo[3.1.0]hexane

Th

204

35.

C7H5C10

4-Chloro-benzaldehyde

Th

205

aThe abbreviation Th means theoretical calculations (ab initio).

methods in Refs. [206-208]). In the case of a conformational mixture the calculation of energies provides a guess for a conformational ratio.

A

theoretical force field allows one to calculate the amplitudes of vibration and the shrinkage corrections discussed in Section IV A.

A quotation from a

124

MASTRYUKOV

paper by Oberhammer published in 1998 [209] gives some historical background: About 25 years ago experimental and theoretical studies of molecular structures and conformational properties were done quite separately.

Because theoretical methods have

also become applicable for reasonably sized molecules, experimental investigators started to take advantage of these methods and included molecular mechanics (MM), semiempirical, and later ab initio and/or density functional (DFT) calculations in their experimental analyses. Today, because computer programs are very easy to use and sufficient computer capacity is generally available, most experimental studies of gas phase structures by gas electron diffraction (GED), microwave (MW), or high-resolution infrared spectroscopy are combined with theoretical calculations. This combination of experimental and theoretical methods is to the advantage of both experiment and theory. The same author later, in Table 1, mentions four problems existing in the GED method and states that theory may be very useful in their solution: 1. Closely spaced distances 2. .

Location of hydrogen atoms High correlations between geometric parameters and vibrational amplitudes

4.

Several conformations

We can illustrate the first two problems using vinyl halide molecules. The location of hydrogen atoms in the parent molecule, ethene (I),

ELECTRON DIFFRACTION

125

H

H

\ /

/ C-'-C

H

H

H

X

\

/ H

I

/ C---, C

H

\ X

II

is described by just two parameters, i.e. the C-H bond lengths and the C=C-H bond angle. In vinyl halides (II) due to the reduced symmetry the three C-H bond lengths and the three C=C-H bond angles are no longer identical although the actual non-equivalence may be rather small and very difficult to measure. On the other hand, it is very easy to calculate this structure. A similar problem exists in vibrational spectroscopy. Fogarasi and Pulay [210] have provided a review of the ab initio calculations of force constants and they have shown that the less dominant force constants are usually more accurately determined from theory than from experiment. L. Sch~ifer was the first to begin to use in the late 1970s the results from molecular mechanics and ab initio calculations to supplement GED structure analysis. He also coined the acronym MOCED (Molecular Orbital Constrained Electron Diffraction) for such a combination [211,212].

The

major problem facing the direct incorporation of ab initio results (re structure) into electron-diffraction structure analysis is the same as it was for the (GED + MW) combination (see Sec. IV B): the different physical significance of the parameters involved. There were many different attempts to relate the electron-diffraction structures to equilibrium parameters extracted from ab initio and other

126

MASTRYUKOV

calculations [213-217]. However, in the MOCED analysis another approach is used: instead of operating with the absolute values of the parameters, their differences are constrained. Generally, the practice of constraining parameters or their differences is not ideal.

First, it implies that the fixed value is absolutely correct.

Second, fixing parameters can result in unrealistically low standard deviations for correlated parameters.

In order to solve these problems,

Rankin and his co-workers in 1996 [218] introduced still another approach, called SARACEN (Structure Analysis Restrained by Ab Initio Calculations for Electron DiffractioN). This "method hinges on two points: the use of calculated parameters as flexible restraints instead of rigid restraints, and choosing to refine all geometrical parameters as a matter of principle." In essence, the SARACEN method is the combination of the MOCED [211, 212] and predicate observation [55] (see Sec. II C) methodologies, with ab initio data being used to construct the predicate observations necessary to

complete the refinement. This method seems to be rather general allowing one to use any kind of additional independent information such as X-ray crystallography, which has never been used with GED. So far, SARACEN has been applied to the study of only a few molecules [ 174, 218-221 ] but no doubt it will find many applications in the future. Table 5 lists 24 molecules studied by GED assisted by Molecular Mechanics calculations [222-239]. This table is intended to be a continuation of Table 10-4 in the review by Geise and Pyckhout [ 117].

ELECTRON DIFFRACTION

127

TABLE 5. Examples of electron diffraction analyses assisted by molecular mechanics calculations.

Reference

No.

Formula

Name

1.

C3H602

1,3-dioxolane

222

2.

C4H8Br 2

1,4-Dibromobutane

223

3.

C4H8C12

1,4- Dichlorobutane

223

4.

C4H8F2

1,4- Difluorobutane

224

5.

C4H802

2-Methyl- 1,3-dioxolane

222

6.

C4HloO2

1,4-Butanediole

225

7.

C5HloO2

2,2-Dimethyl- 1,3-dioxolane

222

8.

C6H8

3-Methylene- 1,4-pentadiene

226

9.

C6H10

1,5-Hexadiene

227

10.

C6H140

Dilsopropyl ether

228

11.

C7H12

Norbomane

229

12.

C7H12

Cycloheptene

230

13.

CsH12

Bicyclo [3.3.0]oct- 1,5-ene

231

14.

C8H16

Cyclooctane

232

15.

CsH19N

Di-tert-butylamine

233

16.

C9H 18

Cyclononane

234

17.

C9H 18~

Di-tert-butylketone

233

18.

C9H20

Di-tert-butylmethane

233

19.

C10H18

Bicyclopentyl

235

20.

CloH30Si5

Decamethyl cyclopentasilane

236

128

MASTRYUKOV

TABLE 5. (continued). No.

Formula

Name

Reference

21.

C12H22

Bicyclohexyl

237

22.

C 12H24

Cyclododecane

238

23.

C18H21Sc

Tris(methlcyclopentadienyl)scandium

239

24.

C18H21Yb

Tris(methylcyclopentadienyl)ytterbium

239

Similarly, Table 6 is intended to be a continuation of Table 10-5 by the same authors [ 117]. However, the activity in the field of combination GED + ab initio (DFT) is more impressive than the present author imagined at the

start of work on this review.

The list of references grew during its

preparation beyond expectation. Therefore, it was decided to cover only the literature which appeared in 1995-1997 with the occasional papers published in 1998. Table 6 lists data for 102 molecules [240-313 ], about 60% of which were investigated in just two groups headed by H. Oberhammer and D. W. H. Rankin. The actual number of molecules studied with the incorporation of theoretical methods is larger: theory was used in 46 cases mentioned in Table 3 and 4.

ELECTRON DIFFRACTION

TABLE 6.

129

Examples of electron diffraction analyses assisted by

ab initio or DFT calculations.

No.

Reference

Formula

Name

1.

B4F3HsP

Tetraborane(8)-trifluoropho sphine ( 1/ 1)

240

2.

B6C12H8

1-(Dichloroboryl)pentaborane(9)

241

3.

C1FO3S

Chlorine fluorosulfate

242

4.

C1F5OS

Pentafluorsulfanyl hypochlorite

243

5.

C14H3NSi2

Bis(dichlorosilyl)amine

244

6.

F203S

Fluorine fluorosulfate

242

7.

F3Mn

Manganese trifluoride

245

8.

CFNOS

Sulfinyl cyanide fluoride

246

9.

CFNO3S

Fluorosulfonyl isocyanate

247

10.

CFNO 5

Fluorocarbonyl peroxynitrate

248

11.

CF2N2OS

Difluorosulfenylimine cyanide

249

12.

CF3NOS

Fluoroformylinimosulfur difluoride

250

13.

CF3NOS2

(Trifluoromethyl)sulfanyl sulfinylimine

251

14.

CF3NS

Sulfur cyanide trifluoride

246

15.

CF40 2

Bis(fluorooxy)difluoromethane

252

16.

CH4C1P

Chloromethylphosphine

253

17.

CHsC14NSi2

Bis(dichlorosilyl)methylamine

245

130

MASTRYUKOV

TABLE 6. No.

(continued). Reference

Formula

Name

18.

CH9NOSi2

O-Methyl-N,N-disilylhydroxylamine

254

19.

C2CIF30

Trifluoroacetyl chloride

255

20.

C2C1F3OS

(Chlorocarbonyl)trifluoromethylsulfane

256

21.

C2C1F3OS

Trifluorothioacetate chloride

257

22.

C2C12F20

Chlorodifluoroacetyl chloride

255

23.

C2C1202

Oxalyl chloride

258

24.

C2FNO2S

(Fluorocarbonyl)sulfenyl isocyanate

259

25.

C2F4OS

(Fluorocarbonyl)trifluoromethylsulfane

256

26.

C2F603

Bis(trifluoromethyl) trioxide

260

27.

C2F605S2

Trifluoromethanesulfonic anhydride

261

28.

C2HC12FO

Dichloroacetyl fuoride

262

29.

C2HF3OS

Trifluorothioacetic acid

257

30.

C2H5C13Si

(Chloromethyl)dichloromethylsilane

263

31.

C2H6AsF

Dimethylfluoroarsine

176

32.

C2H6CI2Ti

Dimethyldichlorotitanium(IV)

153

33.

C2H6FO2P

O-Methyl methylphosphonofluoridate

264

34.

C2H7NO

N,N-Dimethylhydroxlamine

265

35.

C2HloSi2

1,4-Disilabutane

266

ELECTRON DIFFRACTION

TABLE 6. No.

131

(continued). Reference

Formula

Name

36.

C3F60

Perfluoromethyl vinyl ether

267

37.

C3F6S

1-Trifluoromethylthiop- 1,2,2-trifluoroethane

268

38.

C3F804S2

Bis(trifluoromethylsulfonyl) difluoromethane

261

39.

C3H2C1202

Malonyl dichloride

269

40.

C3H2F6S2

Bis(trifluoromethylthio)methane

270

41.

C3H3F30

Methyl trifluorovinyl ether

267

42.

C3H3F3OS

Methyl trifluorothioacetate

257

43.

C3H3F3S

Trifluoromethyl vinyl sulfide

271

44.

C3H3F6NO

O-Methyl-N,N-bis(trifluoromethyl)hydroxy-

272

lamine 45.

C3H5NO

3-Aminoacrolein

273

46.

C3H6C12Si

Methyl(vinyl)dichlorosilane

274

47.

C3H6F2Si

Methyl(vinyl)difluorosilane

274

48.

C3H7C10

3-Chloropropane-l-ol

275

49.

C3HTN

N-Methylethylideneimine

276

50.

C3H8C12Si

(Chloromethyl)chlorodimethylsilane

263

51.

C3HsFN

(Fluoromethyl)dimethylamine

277

52.

C3H12Si2

1,4-disilapentane

266

53.

C3H14B4

1,2-Propano-2,4-tetraborane(10)

278

132

MASTRYUKOV

TABLE 6.

(continued). Reference

No.

Formula

Name

54.

C4F6OS2

Bis(trifluoromethylthio)ketene

279

55.

C4H2CI2N2

2,5-Dichloropyrimidine

218

56.

C4H2CI2N2

2,6-Dichloropyridazine

220

57.

C4H2C12N2

3,6-Dichloropyridazine

220

58.

C4H2CI2N2

4,6-Dichloropyridazine

220

59.

C4H6

1-Butyne

280

60.

C4H6Br202S

Trans-3,4-Dibromotetrahydrothiophene-1,1 -

281

dioxide 61.

C4H604Sn

Tin(II) acetate

219

62.

C4HTF3Si

(Trifluorosilylmethyl)cyclopropane

282

63.

C4HsBr2

1,3-Dibromobutane

283

64.

C4H80

(Z)-Methyl- 1-propenyl ether

284

65.

C4H9Br

1-Bromobutane

285

66.

C4H9C1

1-Chlorobutane

285

67.

C4H9I

1-Iodobutane

285

68.

C4H9NO

N,N-Dimethylacetamide

286

69.

C4H10Si

(Silylmethyl)cyclopropane

287

70.

C4H12OSb2

Bis(dimethylstibyl)oxane

288

71.

C4H12Sb2Se

Bis(dimethylstibyl)selane

288

ELECTRON DIFFRACTION

TABLE 6.

133

(continued).

No.

Formula

Name

72.

C4H16B4

2,3-Butano-2,4-tetraborane

278

73.

C5H60

Divinylketone

289

74.

CsH9F3Si

(Trimethylsilyl)trifluoroethene

290

75.

C5HloO

Diethylketone

291

76.

C5H100

Tetrahydrofurfuryl alcohol

292

77.

CsHloSi

Trimethylsilylacetylene

293

78.

C5H12OSi

3,3-Dimethyl-3-silatetrahydrofurane

294

79.

C5H12SSi

3,3-Dimethyl-3-silatetrahydrothiophene

295

80.

C5H12Si

Trimethylvinylsilane

296

81.

C6H602

5-Methylfuan-2-aldehyde

297

82.

C6H8

3-Ethenyl-3-methylcyclopropane

298

83.

C6H120

tert-Butyl vinyl ether

284

84.

C6H18N3P

Tris(dimethylamino)phosphine

299

85.

C6H18N3Sb

Tris(dimethylamino)antimony

299

86.

C7H7C10

2-Chloroanisole

300

87.

C7H7NO2

Methyl isonicotinate

301

88.

C7H7NO2

Methyl Nicotinate

302

89.

C7H7NO2

Methyl picolinate

302

Reference

134

MASTRYUKOV

TABLE 6.

(continued).

No.

Formula

Name

Reference

90.

C7H16Ge

1,1,3,3-Tetramethylgermacyclobutane

303

91.

C7H20N3P

Tris(dimethylamino)methylenphosphorane

304

92.

C8H604

4,6-Dihydroxyisophthanaldehyde

305

93.

CsH10

3,4-Dimethylenehexa1,5-diene

221

94.

C8H12S6

1,3,5,7-Tetramethyl-2,4,6,8,9,10-

306

hexathiaadamantane 95.

CsH16B10

1-Phenyl-1,2-dicarba-closo-dodecaborane(12)

307

96.

CsH18F4Si2

1,2-Di-tert-butyltetrafluorodisilane

308

97.

C8H18Si2

1,2-Bis(trimethylsilyl)acetylene

293

98.

CsH22B4Si2

1,2-Bis(trimethylsilyl)-1,2-dicarba-closo-

309

hexaborane 99.

C9H14Si

Trimethylsilylbenzene

310

100.

C11H23M~

Tris(dimethylamido)cyclopentadienyl-

311

molybdenum

D.

101.

C14H14N203

p-Azoxyanisole

312

102.

C70

Fullerene

313

Liquid Crystal NMR Spectroscopy All the previous combinations of GED with other techniques dealt with free molecules, i.e. molecules in the gas phase. However, as early as 1973 it was shown by Diehl and Niederberger [314] that liquid crystal NMR (LC

ELECTRON DIFFRACTION

13 5

NMR) spectra give structural information compatible with GED. Later, in 1979, a scheme was devised to obtain r~ structures from LC NMR data [315]. In principle, with a good harmonic force field, one should be able to combine ra structural information from both LC NMR and GED and determine a more precise molecular structure.

This idea was brought to

fruition in 1981 by Rankin and his co-workers [316]. Similar combinations of LC NMR with microwave spectroscopy [317] and ab initio calculations [318] are also known. In 1988, Rankin [319] wrote a review paper where he described in detail a combination, which can be symbolized as GED + LC NMR. At the time of writing, only two molecules were studied, difluorophosphine selenide [316] and difluorophosphine sulphide [320], so the major step forward was done only later. Quotations from this review by Ran~k~-[3 i9] can serve as a short introduction to the field. The form of the NMR spectrum of a compound dissolved in a normal isotropic solvent depends primarily on the chemical shifts of the spinning nuclei involved and on the coupling constants Jij between them.

These coupling

constants are indirect in that they depend on interactions between magnetic dipoles transmitted through the bonds and any intervening atoms. There is also a direct coupling Dij between two spinning nuclei, but this is not normally observed in NMR spectra. This coupling is through space, and its magnitude depends inversely on the cube of the distance between the two nuclei. However, it also depends on the angle of the vector joining the nuclei to the applied magnetic field, and under normal conditions the molecules tumble rapidly on the NMR timescale and thereby average

136

MASTRYUKOV

the direct coupling to zero. But in a liquid crystal solvent the solute molecules do not tumble freely, and consequently dipole-dipole couplings may be observed. Besides Rankin's group, there is only one more group headed by S. Konaka at Hokkaido University which is interested in structural applications of LC NMR, both alone [318,321,322] and in its combination with GED [323]. From the NMR spectrum measured in liquid crystal solvent, direct dipolar constants Dij can be obtained as was mentioned above in Rankin's quotation. Dipolar couplings Dij are related to structural parameters as

Dij =

8x 2

(3cos200,) ,

r3

(8)

where 0ij is the angle between the magnetic field and a vector rij connecting two nuclei i and j, ? is the gyromagnetic ratio and the bracket denote the average over intramolecular motion and reorientational molecular motion [317,319]. As it happens, the structural information derived from dipolar couplings is often complementary to that given by electron diffraction data. In the first place, the most accurate and easily measure NMR data relate to hydrogen nuclei, and so it is the hydrogen atom positions that are given most reliably by NMR methods. Thus electron diffraction and NMR spectroscopy together can give a complete structure for a molecule containing both hydrogen and heavier atoms. In a least-squares refinement based on the electron diffraction data alone, the quantity to be minimized is defined by Eq. (3), Sec II. C. When we

ELECTRON DIFFRACTION

137

use rotational constants as additional data, the original expression is extended to include the values of the extra observations as can be seen from Eq. (7), Sec. IV. B. Now we need to extend this expression to include the differences between the observed values of dipolar coupling constants and those calculated using the trial structure by Eq. (8). Table 7 lists 15 molecules [316,320,323-334] studied by a combination GED + LC NMR, which really means GED + IR + LC NMR because the conversion to an ra structure always requires a knowledge of the force field. In the majority of cases, rotational constants were also used and, finally, for perfiuorocyclopropene (No. 5) and v-picoline (No. 15) the highest integration of all methods was achieved: GED + IR +MW + ab initio + LC NMR. Therefore, these two molecules serve as an example of the self-consistent molecular model shown in Fig. 4.

VD

Concluding Remarks and Acknowledgements The development of the GED method in combination with other techniques has been presented.

Let us formally assume that the effectiveness of a

combination of GED with one particular technique is related to a number of molecules in Tables 2-7, which is 13, 37, 35, 24, 102 and 15, respectively (doing this we certainly ignore the fact that many molecules were studied by a combination of several techniques). Now, if we divide all these numbers by 13, which is the lowest number, and put the resulting ratios in the order we arrive at the following sequence: 1 (MS); 1.2 (LC NMR); 1.8 (MM) ; 2.7 (MW) ; 2.8 (IR) and 7.8 (ab initio ). This sequence helps us to realize once again the explosive

138

MASTRYUKOV

TABLE 7. Molecules studied by joint analysis of electron diffraction and liquid crystal NMR spectroscopy.

No.

Formula

Name

Other Methods useda

F2HPS

Difluorophosphine sulphide

F2HPSe

Difluorophosphine selenide

3.

CH3NSi

Silyl cyanide

MW

324

4.

C2H4Si

Silyl acetylene

MW

325

C3F4

Perfluorocyclopropene

IR, MW, ab initio

326

6.

C4H4N2

Pyrazine

MW

327

7.

C4H4N2

Pyrimidine

MW

327

8.

C4H4N2

Pyridazine

MW

328

9.

C4H40

Furan

MW

329

10.

C4H4S

Thiophene

MW

330

11.

C6H4C12

ortho-Dichlorobenzene

MW

331

12.

C6H4C12

meta-Dicheorobenzene

MW

332

13.

C6H4C12 para-Dichlorobenzene

RS

333

14.

C6H5C1

Chlorobenzene

MW

334

15.

C6HTN

y-Picoline

IR, MW, ab initio

323

MW

Ref.

320 316

aThe abbreviations have the following meaning: IR - infrared spectroscopy MW - microwave spectroscopy, and RS - rovibrational spectroscopy

growth of studies using ab initio calculations. The reason for this is simple: theory proved to be helpful in the GED structure analysis and it is easy to use. It leaves

ELECTRON DIFFRACTION

139

no doubt that ab initio calculations will remain the major additional tool in the hands of the electron diffractionists and those who use microwave spectroscopy [ 171 ] and LC NMR [318]. We also think that particularly the SARACEN method will find many applications in the future. In conclusion, the gas electron diffraction method was chosen as a specfic example in this chapter. However, the problem of combination of several different techniques seems to be a quite general concern. From what we have seen above, one can infer that there are just two basic conditions that determine how successful a particular combination of techniques may be.

These conditions are:

compatibility and complementarity. I would like to thank Professor James R. Durig for his invitation to write this Review.

I thank Professor Lev V. Vilkov for initiating my interest in a

combination of GED with various techniques. Professor James E. Boggs read and corrected the manuscript, which is much better thanks to his comments and strong opinions. Finally, I would like to acknowledge the people who have helped me write this Review.

First my wife, Patricia Hakes, for her constant encouragement.

Collecting the literature data was very much facilitated due to the GEDIS Letters, 1993-1997 prepared by Drs. Jtirgen and Natalja Vogt. I enjoyed discussions with Professors Manfred Fink and Denis A. Kohl, which clarified my ignorance in both scientific and historical matters. I thank N. B6wering, I. Hargittai, K. Hedberg, T. Iijima, S. Konaka, D. W. H. Rankin, L. Sch~ifer, P. Weber and K. R. Wilson for helpful correspondence.

I thank the Robert A. Welch Foundation for support.

Last, but no means least, my gratitude is extended to Linda Smitka and Conrie

140

MASTRYUKOV

Powell, in the UMKC Department of Chemistry Office, for typing and formatting this manuscript. NOTE A D D E D IN P R O O F

In section II.A, an apparatus built by Zewail and coworkers and using ultrafast electron diffraction (UED) was described and later in Section III.C a successful application of this technique to the photodissociation of CF3I was mentioned [34]. More recently, the same technique was applied to a study of the photodissociation of CH2I2 [335]. Related to ultrafast diffraction measurements in general, the many papers on ultrafast X-ray diffraction published by Wilson and coworkers [336-340] are of imerest, the X-ray and electron diffraction cases being very similar.

ELECTRON DIFFRACTION

141

REFERENCES

Stereochemical Applications of Gas-Phase Electron Diffraction. Part A: "The Electron Diffraction Technique," Part B: "Structural Information for Selected Classes of Compounds," (I. Hargittai, M. Hargittai, Eds.) New York, VCH, (1988). H. Mark in Fifty Years of Electron Diffraction (P. Goodman, Ed.) Dordrecht: Reidel, (1981), p 85.

,

0

K. Hedberg in Fifty Years of Electron Diffraction (P. Goodman, Ed.) Dordrecht: Reidel, (1981 ), p xiii.

4.

J. Vogt, Struct. Chem., 3, 147 (1992).

5.

J. Vogt, J. Mol. Spectrosc., 155, 413 (1992).

,

~

"Structure Data of Free Polyatomic Molecules," Landolt-BOrnstein, New Series. Vol. II/7 (Polyatomic Molecules), (K.-H. Hellwege, A. M. Hellwege, Eds.), (1976). "Structure Data of Free Polyatomic Molecules", Landolt-BOrnstein, New Series. Vol. II/15 (Polyatomie Molecules), (K.-H. Hellwege, A. M. Hellwege, Eds.) (1987). "Structure Data of Free Polyatomic Molecules", Landolt-BOrnstein, New Series. Vol. II/21 (Polyatomie Molecules), (K. Kuehitsu, Ed.) (1992).

go

0

"Structure Data of Free Polyatomic Molecules", Landolt-BOrnstein, New Series. Vol. II/23 (Polyatomie Molecules), (K. Kuehitsu, Ed.) (1995).

10. "Structure Data of Free Polyatomie Molecules", Landolt-BOrnstein, New Series. Vol. II/25A (Polyatomic Molecules), (K. Kuchitsu, Ed.) (1998). 11. B. Beagley, in Molecular Structure by Diffraction Methods, (G. A. Sim, L. E. Sutton, Eds.), (Specialist Periodical Reports), The Chemical Society, London, (1973) Vol. 1, p 59. 12. L. Sch~ifer, in Molecular Structure by Diffraction Methods, (L. E. Sutton, M. R. Truter, Eds.), (Specialist Periodical Reports), London, The Chemical Society, (1978) Vol. 6, p 1.

142

MASTRYUKOV

13. D. W. H. Rankin, H. E. Robertson, Chemical Society Specialist Periodical Reports, "Spectroscopic Properties of Inorganic and Organometallic Compounds," 25, 433 (1992); 26, 463 (1993); 27, 438 (1994); 28, 428 (1995); 29, 418 (1996); 30, 454 (1997). 14. L. V. Vilkov, V. S. Mastryukov, N. I. Sadova, Determination of the Geometrical Structure of Free Molecules, Moscow: MIR Publishers, (1983). 15. R. L. Hilderbrandt, in Molecular Structure by Diffraction Methods, (G. A. Sim, L. E. Sutton, Eds.), (Specialist Periodical Reports), London, The Chemical Society, (1975) Vol. 3, Chap. 2, Part 1 p 25. 16. H. Oberhammer, in Molecular Structure by Diffraction Methods, (G. A. Sim, L. E. Sutton, Eds.), (Specialist Periodical Reports), London, The Chemical Society, (1973) Vol. 4, Chap. 2, Part 1, p 24. 17. J. Tremmel, I. Hargittai, in Stereochemical Application of Gas-Phase Electron Diffraction, (I. Hargittai, M. Hargittai, Eds.), Part A, New York: VCH, (1988) p. 191. 18.

S. A. Shlykov, J. Tremmel, J. van Loock, H. J. Geise, J. Mol. Struct., 413-414, 579 (1997).

19. T. Iijima, W. Suzuki, Y. F. Yano, Jpn. J. Appl. Phys., 37, 5064 (1998). 20.

M. Fink, R. A. Bonham, Rev. Sci. Instrum., 41,389 (1970).

21.

R. A. Bonham, M. Fink, High Energy Electron Scattering, New York, Van Nostrand Reinhold, (1974).

22.

M. Fink, C. W. Schmiedekamp, D. Gregory, J. Chem. Phys., 71, 5238 (1979).

23.

J. J. McClelland, M. Fink, Phys. Rev., A31, 1328 (1985).

24.

J. D. Ewbank, L. Schafer, D. W. Paul, O. J. Benston, J. C. Lennox, Rev. Sci. Instrum., 55, 1598 (1984).

25.

J. D. Ewbank, L. Schafer, D. W. Paul, D. L. Monts, W. L. Faust, Rev. Sei. Instrum., 57, 967 (1986).

ELECTRON DIFFRACTION

143

26.

J. D. Ewbank, D. W. Paul, L. Sch~ifer, L. Bakhtiar, Appl. Spectrosc., 43, 415 (1989).

27.

J. D. Ewbank, J. Y. Luo, J. T. English, D. L. Monts, D. W. Paul, Q. Dou, L. Sch~ifer, Rev. Sci. Instrum., 63, 3352 (1992).

28.

J. D. Ewbank, D. W. Paul, L. Sch~ife.r, K. Siam, D. L. Monts, W. L. Faust, Rev. Sei. Instrum., 59, 1144 (1988).

29.

A.A. Ischenko, J. D. Ewbank, L. Sch~ifer, J. Phys. Chem., 98, 4287 (1994).

30.

P. Maggard, V. A. Lobastov, L. Sch~ifer, J. D. Ewbank, A. A. Ischenko, J. Phys. Chem., 99, 13115 (1995).

31.

M. Volkmer, C. Meier, A. Mihill, M. Fink, N. BOwering, Phys. Rev. Lett., 68, 2289 (1992).

32.

M. Volkmer, C. Meier, J. Lieschke, A. Mihill, M. Fink, N. BOwering, Phys. Rev. A., 53, 1457 (1996).

33.

J. C. Williamson, M. Dantus, S. B. Kim, A. H. Zewail, Chem. Phys. Lett., 196, 529 (1992).

34.

M. Dantus, S. B. Kim, J. C. Williamson, A. H. Zewail, J. Phys. Chem. 98, 2782 (1994).

35.

J. D. Geiser, P. M. Weber, in Time-Resolved and X-ray Diffraction, (P. M. Rentzepis, Ed.) SPIE, San Diego, (1995) p. 136.

36.

M. Davis, Electron Diffraction in Gases, New York, Marcel Dekker, (1971).

37.

H. M. Seip, in Molecular Structures by Diffraction Methods (Specialist Periodical Report), (G. A. Sim, L. E. Sutton, Eds.), Vol. 1, Part 1, Chap. 1, London, The Chemical Society, (1973).

38.

I. Hargittai, in Stereochemical Applications of Gas-Phase Electron Diffraction, (I. Hargittai, M. Hargittai, Eds.), Part A, The Electron Diffraction Technique, New York, VCH, (1988).

144

MASTRYUKOV

39.

I. Hargittai, in Accurate Molecular Structures, Their Determination and Importance, (A. Domenicano, I. Hargittai, Eds.), Oxford, Oxford University Press, (1992), p. 95.

40.

M. Fink, in L. Bergmann, C. Schaefer, Constituents of Matter: Atoms, Molecules, Nuclei and Particles, Walter de Gruyter, Berlin, New York, (1997) p. 385.

41.

A. W. Ross, M. Fink, R. L. Hilderbrandt, in International Tables for X-ray Crystallography, (A. J. C. Wilson, Ed.), Vol. C, Boston, Kluwer Academic Publishers, (1992), p. 245.

42.

L. S. Bartell, K. Kuchitsu, H. M. Seip, Acta Cryst., A32, 1013 (1976).

43.

A. V. Belyakov, V. S. Zavgorodnii, V. S. Mastryukov., Zh. Strukt. Khim., 30, 34 (1989); Russ. J. Struct. Chem., 30, 27 (1989).

44.

L. Hedberg, 5~ Austin Symp. Gas Phase Mol. Struct., Austin, TX, USA, 1974, p. 37.

45.

K. Hedberg, M. Iwasaki, Acta Cryst., 17, 529 (1964).

46.

B. Andersen, H. M. Seip, T. G. Strand, R. Stolevik, Acta Chem. Stand., 23, 3224 (1969).

47.

K. Hedberg, in Critical Evaluation of Chemical and Physical Structural Information, (D. R. Lide, M. A. Paul, Eds.) Washington, DC, National Academy of Sciences, (1974) p. 77.

48.

S. J. Cyvin, V. S. Mastryukov, J. Mol. Struct., 30, 333 (1976).

49.

V. S. Mastryukov, S. J. Cyvin, J. Mol. Struct., 29, 15 (1975).

50.

V. S. Mastryukov, E. L. Osina, L. V. Vilkov, S. J. Cyvin, J. Struct. Chem. (USSR), 17, 64 (1976).

51.

V. S. Mastryukov, J. Struct. Chem. (USSR), 17, 69 (1976).

52. V. S. Mastryukov, E. L. Osina, J. Struet. Chem. (USSR), 17, 147 (1976). 53.

G. Gundersen, K. Hedberg, J. Chem. Phys., 51, 2500 (1969).

ELECTRON DIFFRACTION

145

54.

L. Hedberg, K. Hedberg, J. Phys. Chem., 86, 598 (1982).

55.

L. S. Bartell, D. J. Romenesko, T. C. Wong, in Molecular Structure by Diffraction Methods', (L. E. Sutton, G. A. Sim, Eds.), Specialist Periodical Reports, Vol. 3, London, The Chemical Society, (1975)p. 72.

56.

P. Audit, M. Rouault, C. R. Aead. Sei. Paris, 265B, 1100 (1967).

57.

P. Audit, M. Rouault, Entropie, 18, 22 (1967).

58.

B. Raoult, J. Farges, M. Rouault, C. R. Aead. Sei. Paris, 267B, 942 (1968).

59.

P. Audit, Bull. Am. Phys. Soe. Ser. 2, 13, 834 (1968).

60.

P. Audit, J. Phys., 30, 192 (1969).

61.

B. Raoult, J. Farges, Rev. Sei. Instrum., 44, 430 (1973).

62.

L.S. Bartell, R. K. Heenan, M. Nagashima, J. Chem. Phys., 78, 236 (1983).

63.

R.K. Heenan, E. J. Valente, L. S. Bartell, J. Chem. Phys., 78, 243 (1983).

64.

R.K. Heenan, L. S. Bartell, J. Chem. Phys., 78, 1265 (1983).

65.

R.K. Heenan, L. S. Bartell, J. Chem. Phys., 78, 1270 (1983).

66.

L.S. Bartell, Chem. Rev., 86, 492 (1986).

67.

L.S. Bartell, J. Huang, J. Phys. Chem., 98, 7455 (1994).

68.

L.S. Bartell, J. Phys. Chem., 99, 1080 (1995).

69.

J. Huang, L. S. Bartell, J. Phys. Chem., 99, 3924 (1995).

70.

L.S. Bartell, J. Phys. Chem., B101, 7573 (1997).

71.

J.W. Hovic, L. S. Bartell, J. Phys. Chem., B102, 534 (1998).

72. N. BSwering, M. Volkmer, C. Meier, J. Lieschke, M. Fink, Z. Phys. D, 30, 177 (1994).

146

73.

MASTRYUKOV

C. Meier, M. Volkmer, J. Lieschke, M. Fink, N. BSwering, Z. Phys. D, 30, 183 (1994).

74. N. BSwering, M. Volkmer, C. Meier, J. Lieschke, R. Dreier, J. Mol. Struct., 348, 49 (1995). 75.

M. Volkmer, C. Meier, J. Lieschke, R. Dreier, M. Fink, N. BSwering, Phys. Rev. A., 56, R1690 (1997).

76. N. BOwering, in Selected Topics in Electron Physics, (D. Campbell, H. Kleinpoppen, Ed.), New York, Plenum Press, (1996) p. 193. 77. N. B6wering, in Photonic, Electronic and Atomic Collisions, (F. Aumayr, H. P. Winter, Ed.), Singapore, World Scientific, (1998) p. 259. 78. M. Fink, A. W. Ross, A. J. Fink, Z. Phys. D., 11, 231 (1998). 79. A. Mihill, M. Fink, Z. Phys. D., 14, 77 (1989). 80. J. Haeger, S. Zech, D. Glatzer, M. Fink, H. Waltner, Rev. Sci. Instrum, 65(2), 407 (1994). 81. H. Fujiwara, T. Egawa, S. Konaka, J. Mol. Struet., 344, 217 (1995). 82. R.E. Kennerly, Ph.D. Dissertation, Austin, Texas, 1975. 83.

S. Boh~itka, I. Berecz, J. Tremmel, I. Hargittai, Proc. 7th Intern. Vac. Cong., 3rd Intern. Conf. Solid Surfaces, Vienna, 1977, p. 193-196; cited by Ref. 84.

84. M.M. Arvedson, D. A. Kohl, Chem. Phys. Lett., 64, 119 (1979). 85. M. Hargittai, M. Kolonits, L. G/SdSrhfizy, Chem. Phys. Lett., 257, 321 (1996). 86. A. G. Girichev, N. I. Giricheva, G. V. Girichev, N. Vogt, J. Vogt, J. Mol. Struct., 384, 175 (1996). 87. N. Vogt, G. V. Girichev, J. Vogt, A. G. Girichev, J. Mol. Struet., 352-353, 175 (1995). 88.

G. Schultz, J. Tremmel, I. Hargittai, N. D. Kagramanov, A. K. Maltsev, O. M. Nefedov, J. Mol. Struct., 82, 107 (1982).

ELECTRON DIFFRACTION

89.

147

I. Hargittai, G. Schultz, J. Tremmel, N. D. Kagramanov, A. K. Maltsev, O. M. Nefedov, J. Am. Chem. Soe., 105, 2895 (1983).

90.

G. Schultz, J. Tremmel, I. Hargittai, I. Berecz, S. Boh~tka, N. D. Kagramanov, A. K. Maltsev, O. M. Nefedov, J. Mol. Struet., 55, 207 (1979).

91.

I. N. Belova, N. I. Giricheva, G. V. Girichev, N. V. Petrova, S. A. Shlykov, Zh. Strukt. Khim, 37, 257 (1996)/J. Struet. Chem. (Engl. Transl.), 37, 224 (1996).

92.

I.N. Belova, N. I. Giricheva, S. A. Shlykov, Zh. Strukt. Khim., 37(4), 708 (1996).

93. N. I. Giricheva, G. V. Girichev, S. A. Shlykov, V. A. Titov, T. P. Chusova, J. Mol Struet., 344, 127 (1995). 94. N. I. Giricheva, N. V. Belova, G. V. Girichev, S. A. Shlykov, J. Mol. Struet., 352353, 167 (1995). 95.

L.S. Bartell, M. A. Kacner, S. R. Goates, J. Chem. Phys., 75, 2730 (1981).

96.

L.S. Bartell, M. A. Kacner, S. R. Goates, J. Chem. Phys., 75, 2736 (1981).

97.

L.S. Bartell, M. A. Kacner, J. Phys. Chem., 88, 3485 (1984).

98.

L.S. Bartell, M. A. Kacner, J. Chem. Phys., 81, 280 (1984).

99.

S.R. Goates, L. S. Bartell, J. Chem. Phys., 77, 1866 (1982).

100. S. R. Goates, L. S. Bartell, J. Chem. Phys., 77, 1874 (1982). 101. L. S. Bartell, W. N. Vance, S. R. Goates, J. Chem. Phys., 80, 3923 (1984). 102. L. S. Bartell, J. F. Stanton, J. Chem. Phys., 81, 3792 (1984). 103. A. A. Ischenko' V. V. Golubkov, V. P. Spiridonov, A. V. Zgurskii, A. S. Akhmanov, M. G. Vabischevich, Appl. Phys. B, 32, 161 (1983). 104. A. A. Ischenko, V. V. Golubkov, V. P. Spiridonov, A. V. Zgurskii, A. S. Akhmanov, Vestn. Mosk. Univ. Khim., 25, 385 (1985). 105. V. S. Mastryukov, Vestn. Mosk. Univ. Khim., 32, 106 (1991); Moscow Univ. Chem. Bull., 46, 30 (1991).

148

MASTRYUKOV

106. A. A. Ischenko, V. P. Spiridonov, L. Schiifer, J. D. Ewbank, J. Mol. Struct., 300, 115(1993). 107. J. D. Ewbank, J. Y. Luo, J. T. English, R. Liu, W. L. Faust, L. Sch~ifer, J. Phys. Chem., 97(34), 8745 (1993). 108. A. A. Ischenko, L. Schiifer, J. Y. Luo, J. D. Ewbank, J. Phys. Chem., 98, 8673 (1994). 109. A. P. Rood, J. Milledge, J. Chem. Soc. Faraday Trans. 2, 80, 1145 (1984). 110. A. A. Ischenko, J. D. Ewbank, L. Schiifer, J. Mol. Struct., 320, 147 (1994). 111. J. D. Ewbank, L. Schlifer, A. A. Ischenko, J. Mol. Struct., 321,265 (1994). 112. J. C. Williamson, A. H. Zewail, J. Phys. Chem., 98, 2766 (1994). 113. A. A. Ischenko, L. Schiifer, J. D. Ewbank, J. Mol. Struct., 376, 157 (1996). 114. A. A. Ischenko, V. A. Lobastov, L. Sch~ifer, J. D. Ewbank, J. Mol. Struct., 377, 261 (1996). 115. M. Ben-Nun, T. J. Martinez, P. M. Weber, K. R. Wilson, Chem Phys. Lett., 262,

405 (1996). 116. J. D. Geiser, P. M. Weber, J. Chem. Phys., 108, 8004 (1998). 117. H. J. Geise, W. Pyckhout, in Stereochemical Application of Gas-Phase Electron Diffraction, (I. Hargittai, M. Hargittai, Eds.), Part A; New York: VCH, (1988) Chap. 10, p. 321. 118. I. L. Karle, J. Karle, J. Chem. Phys., 17, 1052 (1949). 119. P. Debye, J. Chem. Phys., 9, 55 (1941). 120. R. W. James, Z. Physik, 33, 737 (1932). 121. J. Karle, I. L. Karle, in Fifty Years of Electron Diffraction, (P. Goodman, Ed.), Dordrecht: Reidel, (1981) p. 243. 122. Y. Morino, J. Chem. Phys., 18, 395 (1950).

ELECTRON DIFFRACTION

149

123. E. B. Wilson, J. C. Decius, P. C. Cross, Molecular Vibrations, McGraw-Hill, New York, (1955). 124. Y. Morino, in Fifty Years of Electron Diffraction, (P. Goodman, Ed.), Dordrecht: Reidel, (1981) p. 136. 125. S. J. Cyvin, in Molecular Vibrations and Mean Universitetsforlaget, Amsterdam, Oslo, Elsevier, (1967).

Square

Amplitudes,

126. K. Kuchitsu, S. J. Cyvin, in Molecular Structure and Vibrations, (S. J. Cyvin, EO.), Amsterdam, Elsevier, (1972) Chapter 12. 127. R. Stolevik, H. M. Seip, S. J. Cyvin, Chem. Phys. Lett., 15, 263 (1972). 128. D. Christen, J. Mol. Struct., 48, 101 (1978). 129. L. Hedberg, I. M. Mills, J. Mol. Spectrosc., 160, 117 (1993). 130. K. Iijima, T. Miwa, T. Matsuyoshi, T. Sakaizumi, O. Ohashi, J. Mol. Struct., 352/353, 161 (1995). 131. T. Matsuyoshi, K. Iijima, J. Mol. Struct., 378, 1996 (1996). 132. R. L. Hilderbrandt, Comp. Chem., 1, 179 (1977). 133. V. S. Mastryukov, N. L. Allinger, 14th Austin Symp. Mol. Struct., Austin, Texas, USA, 1992, p. 49. 134. K.-G. Martinsen, N. Vogt, H. V. Volden, V. S. Lyutsarev, J. Vogt, J. Mol. Struct., 385, 159 (1996). 135. A. J. Downs, T. M. Greene, L. A. Harman, P. F. Souter, P. T. Brain, C. R. Pulham, D. W. H. Rankin, H. E Robertson, M. Hofmann, P. v. R. Schleyer, Inorg. Chem., 34, 1799 (1995). 136. J. Molnfir, M. Kolonits, M. Hargittai, J. Mol. Struct., 413-414, 441 (1997). 137. J. Molnfir, M. Kolonits, M. Hargittai, R. J. M. Konings, A. S. Booij, Inorg. Chem., 35, 7639 (1996).

150

MASTRYUKOV

138. T. H. Johansen, K. Hagen, R. Stolevik, M. Ernst, K. Hassler, J. Mol. Struet., 372, 161 (1995). 139. N. Vogt, A. Haaland, K.-G. Martinsen, J. Vogt, Acta Chem. Stand., 47, 937 (1993). 140. N. Vogt, M. Hargittai, M. Kolonits, I. Hargittai, Chem. Phys. Lett., 199, 441 (1992). 141. J. Moln~, R. J. M. Konings, M. Kolonits, M. Hargittai, J. Mol. Struet., 375, 223 (1996). 142. J. Moln~, C. J. Marsden, M. Hargittai, J. Phys. Chem., 99, 9062 (1995). 143. N. Vogt, A. Haaland, K.-G. Martinsen, J. Vogt, J. Mol. Speetrosc., 163, 515 (1994). 144. A. Haaland, K.-G. Martinsen, O. Swang, H. V. Volden, A. S. Booij, R. J. M. Konings, J. Chem. Soe., Dalton Trans,, 185 (1995). 145. R. J. M. Konings, A. S. Booij, A. Kov~ics, G. V. Girichev, N. I. Giricheva, O. G. Krasnova, J. Mol. Struet., 378, 121 (1996). 146. D. Scheffler, I. Schaper, H. Willner, H.-G. Mack, H. Oberhammer, Inorg. Chem, 36, 339 (1997). 147. H.-G. Mack, C. O. Della V6dova, H. Willner, J. Mol. Struet., 291, 197 (1993). 148. G. A. Arguello, B. Jtilicher, S. E. Ulic, H. Willner, B. Casper, H.-G. Mack, H. Oberhammer, Inorg. Chem., 34, 2089 (1995). 149. Y. Wang, J. Tremmel, J. de Smedt, C. van Alsenoy, H. J. Geise, B. J. van der Veken, J. Phys. Chem. A, 101, 5919 (1997). 150. H.-G. Mack, C. O. Della V6dova, H. Oberhammer, J. Phys. Chem., 96, 9215 (1992). 151. D. Christen, H.-G. Mack, C. J. Marsden, H. Oberhammer, G. Schatte, K. Seppelt, H. Willner, d. Am. Chem. Soe., 109, 4009 (1987).

ELECTRON DIFFRACTION

151

152. G. Wu, S. A. Shlykov, C. van Alsenoy, H. J. Geise, J. Phys. Chem., 99, 8589 (1995). 153. G. S. McGrady, A. J. Downs, D. C. McKean, A. Haaland, W. Scherer, H. P. Veme, H. V. Volden, lnorg. Chem., 35, 4713 (1996). 154. I. F. Shishkov, L. V. Vilkov, L. V. Khristenko, P. N. Skancke, J. Mol. Struet., 376, 103 (1996). 155. G. Wu, S. A. Shlykov, C. van Alsenoy, H. J. Geise, E. Sluyts, B. J. van der Veken, J. Phys. Chem., 100, 11620 (1996). 156. V. A. Naumov, M. B. Zuev, D. W. H. Rankin, H. E. Robertson, J. Mol. Struet.,

318, 151 (1994). 157. Z. H. Peng, J. Tremmel, G. Wu, C. van Alsenoy, H. J. Geise, J. Phys. Chem., Y9, 10201 (1995). 158. Y. Wang, J. de Smedt, I. Coucke, C. van Alsenoy, H. J. Geise, J. Phys. Chem., 99, 43(1993). 159. N. Kuze, H. Takeuchi, T. Egawa, S. Konaka, S. Q. Newton, L. Sch~ifer, J. Mol. Struet., 291, 11 (1993). 160. N. Kuze, N. Kuroki, H. Takeuchi, T. Egawa, S. Konaka J. Mol. Struet., 301, 81 (1993). 161. H. Takeuchi, M. Sugino, T. Egawa, S. Konaka, J. Phys. Chem., 97, 7511 (1993). 162. T. Egawa, H. Moriyama, H. Takeuchi, S. Konaka, K. Siam, L. Sch~ifer, J. Mol. Struet., 298, 37 (1993). 163. V. A. Naumov, L. L. Tuzova, J. de Smedt, C. van Alsenoy, H. J. Geise, J. Mol. Struet., 344, 117 (1995). 164. M. V. Popik, V. P. Novikov, S. Samdal, L. V. Vilkov, M. A. Tafipolsky, J. Mol. Struet., 376, 173 (1996). 165. D. Hnyk, R. S. Fender, H. E. Robertson, D. W. H. Rankin, M. Btihl, K. Hassler, K. Schenzel, J. Mol. Struet., 346, 215 (1995).

152

MASTRYUKOV

166. A. G. Robiette, in Molecular Structure by Diffraction Methods, (G. A. Sim, L. E. Sutton, Eds.) (Specialist Periodical Reports), London, The Chemical Society, (1973) Vol. 1, Part 1, p. 160. 167. L. S. Bartell, K. Kuchitsu, R. J. de Neui, J. Chem. Phys., 35, 1211 (1961). 168. J. E. Boggs, J. Mol. Struct., 97, 1 (1983). 169. K. Kuchitsu, T. Fukuyama, Y. Morino, J. Mol. Struct., 1, 463 (1967-1968). 170. T. Oka, J. Phys. Soc. Japan, 15, 2274 (1960). 171. J. Demaison, G. Wlodarczak, H. D. Rudolph, Adv. Mol. Struct. Res., 3, 1 (1997). 172. K. Kuchitsu, in: MTP International Review of Science, (G. Allen, Ed.) Phys. Chem. Series 1, Vol. 2, Chap. 6, Oxford, Medical, Technical Publ. Co., (1972). 173. K. Kuchitsu, M. Nakata, S. Yamamoto, in Stereochemical Application of GasPhase Electron Diffraction Part A: The Electron Diffraction Technique (I. Hargittai, M. Hargittai, Eds.) New York: VCH, (1988) p. 227. 174. P. T. Brain, C. A. Morrison, S. Parsons, D. W. H. Rankin, J. Chem. Soe., Dalton Trans,, 4589 (1996). 175. H. Oberhammer, T. Lobreyer, W. Sundermeyer, J. Mol. Struct., 323, 125 (1994). 176. A. J. Downs, T. M. Greene, G. S. McGrady, N. Townson, P. T. Brain, C. R. Pulman, D. W. H. Rankin, H. E. Robertson, B. A. Smart, I. L. Alberts, Inorg. Chem., 35, 6952 (1996). 177. K. Hagen, K. Hedberg, E. O. John, R. L. Kirchmeier, J. M. Shreeve, J. Phys. Chem., 102,4, 5106 (1998). 178. T. Iijima, J. Mol. Struct., 212, 137 (1989). 179. N. Murase, K. Yamanouchi, T. Egawa, K. Kuchitsu, J. Mol. Struct., 242, 409 (1991). 180. Q. Shen, R. Krisak, K. Hagen, J. Mol. Struct., 346, 13 (1995). 181. T. Iijima, R.-I. Wada, J. Mol. Struct., 221, 7 (1990).

ELECTRON DIFFRACTION

153

182. R. Ktihn, D. Christen, H.-G. Mack, R. Minkwitz, D. Konikowski, H. Oberhammer, J. Mol. Struct., 3 76, 217 (1996). 183. M.R. Kazerouni, L. Hedberg, K. Hedberg, J. Amer. Chem. Sot., 119, 8324 (1997). 184. M. R. Kazerouni, L. Hedberg, K. Hedberg, J. Amer. Chem. Sot., 116, 5279 (1994). 185. K. Iijima, M. Kato, B. Beagley, J. Mol. Struct., 295, 289 (1993). 186. H. Fujiwara, T. Egawa, H. Takeuchi, S. Konaka, J. Mol. Struct., 301, 113 (1993). 187. K. Hagen, R. Stolevik, Struct. Chem., 6, 75 (1995). 188. K. Hagen, R. Stolevik, P. C. Saeboe, J. Mol. Struct., 346, 75 (1995). 189. T. Iijima, T. Kondou, T. Takenaka, J. Mol. Struct., 445, 23 (1998). 190. H. Fujiwara, T. Egawa, H. Takeuchi, S. Konaka, J. Mol. Struet., 380, 63 (1996). 191. K. Iijima, M. Matsuoka, T. Sakaizumi, O. Ohashi, Bull. Chem. Soc. Japan, 69, 2481 (1996). 192. K. Iijima, O. Ohashi, J. Mol. Struct., 291, 159 (1993). 193. K.-M. Marstokk, H. MSllendal, S. Samdal, J. Mol. Struct., 3 76, 11 (1996). 194. K.-M. Marstokk, H. Mtillendal, S. Samdal, Acta Chem. Stand., 50, 845 (1996). 195. T. Iijima, T. Shimoda, H. Hattori, J. Mol. Struct., 350, 57 (1995). 196. L. Hedberg, K. Hedberg, J. Phys. Chem, 97, 10349 (1993). 197. T. Egawa, S. Maekawa, H. Fujiwara, H. Takeuchi, S. Konaka, J. Mol. Struct., 352353, 193 (1995). 198. K.-M. Marstokk, H. MSllendal, S. Samdal, Acta Chem. Stand., 45, 37 (1991). 199. K. Aarset, L. G. Faksnes, I. Nyghrd, K. Hagen, J. Phys. Chem., 98, 2848 (1994). 200. W. Pyckhout, N. Horemans, C. van Alsenoy, H. J. Geise, D. W. H. Rankin, J. Mol. Struet., 156, 315 (1987).

154

MASTRYUKOV

201. M. Tanabe, N. Kuze, H. Fujiwara, H. Takeuchi, S. Konaka, J. Mol. Struct., 372,

173 (1995). 202. M. Tanabe, N. Kuze, H. Takeuchi, S. Konaka J. Mol. Struct., 380, 205 (1996). 203. N. Kurai, H. Takeuchi, S. Konaka, J. Mol. Struct., 318, 143 (1994). 204. Q. Shen, V. S. Mastryukov, J. E. Boggs, J. Mol. Struet., 352, 181 (1995). 205. H. M611endal, S. Gundersen, M. A. Tafipolsky, H. V. Volden, J. Mol. Struet., 444, 47 (1998). 206. J. E. Boggs, in Stereochemical Applications of Gas-Phase Electron Diffraction. Part B." Structural Information for Selected Classes of Compounds, (I. Hargittai, M. Hargittai, Eds.) New York, VCH, (1988) p. 455. 207. J. E. Boggs, in Accurate Molecular Structures, Their Determination and Importance, (A. Domenicano, I. Hargittai, Eds.) Oxford, Oxford University Press, (1992) p. 322. 208. N. L. Allinger, in Accurate Molecular Structures, Their Determination and Importance, (A. Domenicano, I. Hargittai, Eds.) Oxford, Oxford University Press, (1992) p. 336. 209. H. Oberhammer, J. Comp. Chem., 19, 123 (1998). 210. G. Fogarasi, P. Pulay, in Vibrational Spectra and Structure, (J. R. Durig, Ed.), 14, 125 (1985). 211. L. Schiller, J. Mol. Struct., 100, 51 (1983).

212. L. Sch~ifer, J. D. Ewbank, K. Siam, N. S. Chiu, H. L. Sellers, in Stereochemical

Applications of Gas-Phase Electron Diffraction, Part A, The Electron Diffraction Technique, (I. Hargittai, M. Hargittai, Eds.) VCH, New York, (1988) p. 301. 213. R. L. Hilderbrandt, D. A. Kohl, J. Mol. Struet. (Theoehem), 85, 25 (1981). 214. D. A. Kohl, R. L. Hilderbrandt, J. Mol. Struct. (Theochem), 85, 325 (1981). 215. J. de Smedt, F. Vanhouteghem, C. van Alsenoy, H. J. Geise, L. Sch~ifer, J. Mol. Struet. (Theoehem), 259, 289 (1992).

ELECTRON DIFFRACTION

155

216. B. Ma, J.-H. Lii, H. F. Schaefer, N. L. Allinger, J. Phys. Chem., 100, 8763 (1996). 217. B. Ma, N. L. Allinger, J. Mol. Struct., 413-414, 395 (1997). 218. A. J. Blake, P. T. Brain, H. McNab, J. Miller, C. A. Morrison, S. Parsons, D. W. H. Rankin, H. E. Robertson, B. A. Smart, J. Phys. Chem., 100, 12280 (1996). Q

219. C. A. Morrison, B. A. Smart, S. Parsons, D. W. H. Rankin, H. E. Robertson, H. McNab, J. Chem. Sot., Perkin Trans. 2, 857 (1997). 220. B. A. Smart, L. E. Griffiths, C. R. Pulham, H. E. Robertson, N. W. Mitzel, D. W. H. Rankin, J. Chem. Soe., Dalton Trans., 1656 (1997). 221. P. T. Brain, B. A. Smart, H. E. Robertson, M. J. Davis, D. W. H. Rankin, W. J. Henry, I. Gosney, J. Org. Chem., 62, 2767 (1997). 222. Q. Shen, T. L. Mathers, T. Raeker, R. L. Hilderbrandt, J. Am. Chem. Sot., 108, 6888 (1986). 223. K. Aarset, K. Hagen, R. Stolevik, J. Phys. Chem., 98, 5249 (1994). 224. K. Krosley, K. Hagen, K. Hedberg, J. Mol. Struct., 352, 87 (1995). 225. M. Traetteberg, K. Hedberg, J. Amer. Chem. Soc, 116, 1382 (1994). 226. A. Almenningen, A. Gatial, D. S. B. Grace, H. Hopf, P. Klaeboe, F. Lehrich, C. J. Nielsen, D. L. Powell, M. Traetteberg, Aeta Chem. Seand., A42, 634 (1988). 227. G. Schultz, I. Hargittai, J. Mol. Struet., 346, 63 (1995). 228. H. Takeuchi, M. Fujii, S. Konaka, M. Kimura, J. Phys. Chem., 91, 1015 (1987). 229. N. L. Allinger, H. J. Geise, W. Pyckhout, L. A. Paquette, J. C. Gallucci, J. Am. Chem. Sot., 111, 1106 (1989). 230. L. I. Ermolaeva, V. S. Mastryukov, N. L. Allinger, A. Almenningen, J. Mol. Struct., 196, 151 (1989). 231. V. S. Mastryukov, E. Y. Archipova, M. Traetteberg, A. Almenningen, Acta Chem. Stand., 43, 238 (1989).

156

MASTRYUKOV

232. O. V. Dorofeeva, V. S. Mastryukov, N. L. Allinger, A. Almenningen, J. Phys. Chem., 89, 252 (1985). 233. S. Liedle, H. Oberhammer, N. L. Allinger, J. Mol. Struet., 317, 69 (1994). 234. O. V. Dorofeeva, V. S. Mastryukov, N. L. Allinger, A. Almenningen, J. Phys. Chem., 94, 8044 (1990). 235. I. F. Shishkov, V. S. Mastryukov, K. Chen., N. L. Allinger, S. Gundersen, S. Samdal, H. V. Volden, J. Mol. Struet., 3 76, 133 (1996). 236. Q. Shen, C. Bickford, J. Mol. Struet., 446, 223 (1998). 237. O. V. Dorofeeva, V. S. Mastryukov, A. Almenningen, A. Horn, P. Klaeboe, L. Yang, N. L. Allinger, J. Mol. Struet., 263, 281 (1991). 238. E. G. Atavin, V. S. Mastryukov, N. L. Allinger, A. Almenningen, R. Seip, J. Mol. Struet., 212, 87 (1989). 239. R. Blom, A. Hammel, A. Haaland, J. Weidlein, T. V. Timofeeva, Yu. T. Struchkov, J. Organometal. Chem., 462, 131 (1993). 240. P. T. Brain, D. W. H. Rankin, H. E. Robertson, R. Greatrex, M. A. Fox, A. Nikrahi, M. Biahl, lnorg. Chem., 36, 1048 (1997). 241. A. J. Downs, T. M. Greene, P. T. Brain, C. R. Pulham, D. W. H. Rankin, H. E. Robertson, M. Hofinann, P. v. R. Schleyer, J. Chem. Sor Dalton Trans., 2193 (1995). 242. F. Aubke, B. Casper, H. S. P. Mtiller, H. Oberhammer, H. Willner, J. Mol. Struet., 346, 111 (1995). 243. A. Komath, N. Hartfeld, H. Oberhammer, Inorg. Chem., 36, 5156 (1997). 244. H. Fleisher, D. Hnyk, D. W. H. Rankin, H. E. Robertson, M. Btihl, W. Thiel, Chem. Ber., 128, 807 (1995). 245. M. Hargittai, B. R6ffy, M. Kolonits, C. J. Marsden, J. L. Heully, J. Am Chem. Soe., 119, 9042 (1997).

ELECTRON DIFFRACTION

157

246. H.-G. Mack, H. Oberhammer, Jo Jacobs, M. Kronberg, H. Willner, Inorg. Chem., 35, 806 (1996). 247. C. O. Della V6dova, E. H. Cutin, H.-G. Mack, H. Oberhammer, J. Mol. Struct., 380, 167 (1996). 248. D. Scheffler, I. Schaper, H. Willner, H.-G. Mack, H. Oberhammer, lnorg. Chem., 36, 339 (1997). 249. E. H. Cutin, C. O. Della V6dova, H.-G. Mack, H. Oberhammer, J. Mol. Struct., 354, 165 (1995). 250. C. Leibold, E. H. Cutin, C. O. Della V6dova, H.-G. Mack, R. Mews, H. Oberhammer, J. Mol. Struet., 375, 207 (1996). 251. R. M. Romano, C. O. Della V6dova, H.-G. Mack, H. Oberhammer, J. Mol. Struct., 440, 43 (1998). 252. K. I. Gobbato, H.-G. Mack, H. Oberhammer, C. O. Della V6dova, J Amer. Chem. Soe., 119, 803 (1997). 253. A. J. Downs, T. M. Greene, P. T. Brain, D. W. H. Rankin, H. E. Robertson, M. Hofmann, P. v. R. Schleyer, J. Mol. Struet., 352-353, 135 (1995). 254. N. W. Mitzel, E. Breuning, A. J. Blake, H. E. Robertson, B. A. Smart, D. W. H. Rankin, J. Am. Chem. Soe., 118, 2664 (1996). 255. K. I. Gobbato, C. Leibold, S. Centeno, C. O. Della V6dova, H.-G. Mack, H. Oberhammer, J. Mol. Struet., 380, 55 (1996). 256. K. I. Gobbato, H.-G. Mack, H. Oberhammer, S. E. Ulic, C. O. Della V6dova, H. Willner, J. Phys. Chem., 101A, 2173 (1997). 257. K. I. Gobbato, C. O. Della V6dova, H.-G. Mack, H. Oberhammer, Inorg, Chem., 35, 6152 (1996). 258. D. D. Danielson, L. Hedberg, K. Hedberg, K. Hagen, M. Traetteberg, J. Phys. Chem., 99, 9374 (1995). 259. K. I. Gobbato, S. E. Ulic, C. O. Della V6dova, H.-G. Mack, H. Oberhammer, Chem. Phys. Lett., 266, 527 (1997).

158

MASTRYUKOV

260. K. I. Gobbato, M. F. Klapdor, D. Mootz, W. Poll, S. E. Ulic, H. Willner, H. Oberhammer, Angew. Chem., 107, 2433 (1996). 261. R. Haist, H.-G. Mack, A. Waterfeld, G. L. Gard, H. Oberhammer, J. Mol. Struet.,

380, 213 (1996). 262. P. T. Brain, D. W. H. Rankin, H. E. Robertson, M. Biahl, J. Mol. Struet., 384, 123 (1996). 263. T. H. Johansen, K. H. Wold, K. Hagen, R. Stolevik, J. Mol. Struet., 384, 215 (1996). 264. A. Rauk, I. F. Shishkov, L. V. Vilkov, K. F. Koehler, R. G. Kostyanovsky, J. Am Chem. Soe., 117, 7180 (1995). 265. N. W. Mitzel, B. A. Smart, S. Parsons, H. E. Robertson, D. W. H. Rankin, J. Chem. Soe., Perkin Trans., 2, 2727 (1996). 266. N. W. Mitzel, B. A. Smart, A. J. Blake, H. E. Robertson, D. W. H. Rankin, J. Phys. Chem., 100, 9339 (1996). 267. C. Leibold, S. Reinenmann, R. Minkwitz, P. R. Resnik, H. Oberhammer, J. Org. Chem. 62, 6160 (1997). 268. A. Oleta, A. Haas, H. Oberhammer, Chem. Bet., 128, 803 (1995). 269. H.-G. Mack, C. O. Della V6dova, J. Mol. Struet., 346, 51 (1995). 270. M. Kom, A. Haas, H. Oberhammer, Chem. Ber., 128, 461 (1995). 271. M. Lieb, H. Oberhammer, Chem. Bet. 130, 131 (1997). 272. B. Casper, J. Jakob, R. Minkwitz, H. Oberhammer, Chem. Ber., 129, 653 (1996). 273. A. D. Richardson, K. Hedberg, K. B. Wiberg, P. R. Rablen, J. Mol. Struet., 445, 1 (1998). 274. T. H. Johansen, K. Hagen, R. Stolevik, K. Hassler, J. Phys. Chem., I O1A, 3580 (1997). 275. A. D. Richardson, K. Hedberg, J. Mol. Struct., 413-414, 61 (1997)i

ELECTRON DIFFRACTION

159

276. H. Fujiwara, T. Egawa, S. Konaka, J. Amer. Chem. Sot., 119, 1346 (1997). 277~ D. Christen, H.-G. Mack, S. Rtidiger, H. Oberhammer, J. Amer. Chem. Sot., 118, 3720 (1996). 278. P. T. Brain, M. J. Picton, D. W. H. Rankin, H. E. Robertson, R. Greatrex, M. A. Fox, M. Buhl, Inorg. Chem., 34, 2841 (1995). 279. M. Kom, H.-G. Mack, H. W. Praas, C. O. Della V6dova, H. Oberhammer, J. Mol. Struct., 352-353, 145 (1995). 280. O. Bastiansen, P. Bakken, E. Kloster-Jensen, S. Samdal, M. Traetteberg, J. Mol. Struct., 352-353, 77 (1995). 281. A. J. Blake, P. T. Brain, I. Gosney, R. O. Gould, D. W. H. Rankin, H. E. Robertson, P. Trickey, M. Biahl, J. Chem. Sot., Perkin Trans., 2, 2293 (1995). 282. M. Dakkouri, J. Mol. Struct., 413-414, 133 (1997). 283. K. Aarset, K. Hagen, R. Stolevik, J. Phys. Chem., 99, 11089 (1995). 284. C. Leibold, H. Oberhammer, J. Am. Chem. Sot., 120, 1533 (1998). 285. K. Aarset, K. Hagen, R. Stolevik, P. C. Saeboe, Struet Chem., 6, 197 (1995). 286. H.-G. Mack, H. Oberhammer, J. Amer. Chem. Soe., 119, 3567 (1997). 287. M. Dakkouri, T. Hermann, J. Mol. Struct., 346, 239 (1995). 288. A. Haaland, V. I. Sokolov, H. V. Volden, H. J. Breuning, M. Denker, R. R6sler, Z. Naturforsch, 5213, 296 (1997). 289. I. F. Shishkov, L. V. Vilkov, P. N. Skancke, J. Mol. Struct., 346, 95 (1995). 290. S. K. Ritter, R. L. Kirchmeier, J. M. Shreeve, H. Oberhammer, Inorg. Chem., 35, 4067 (1996). 291. N. Kuze, E. Kojima, H. Fujiwara, H. Takeuchi, T. Egawa, S. Konaka, J. Mol. Struet., 3 75, 231 (1996). 292. K. B. Borisenko, S. Samdal, I. F. Shishkov, L. V. Vilkov, J. Mol. Struet., 448, 29 (1998).

160

MASTRYUKOV

293. M. Traetteberg, P. Bakken, J. F. Liebman, M. Hulce, J. Mol. Struet., 346, 101 (1995). 294. A. Yu. Gromov, I. F. Shishkov, A. Skancke, L. V. Vilkov, A. V. Yesipenko, S. V. Kirpichenko, J. Mol. Struet., 352-353, 115 (1995). 295. K. B. Borisenko, S. Samdal, E. N. Suslova, V. A. Sipachev, I. F. Shishkov, L. V. Vilkov, Acta Chem. Stand., 52, 975 (1998). 296. E. M. Page, D. A. Rice, R. Walsh, K. Hagen, J. Mol. Struet., 403, 199 (1997). 297. I. F. Shishkov, L. V. Vilkov, I. Hargittai, J. Mol. Struet., 352-353, 157 (1995). 298. P. Bakken, I. Bolesov, I. Kutchuk, A. de Meijere, V. S. Mastryukov, M. Traetteberg, J. Mol. Struct., 445, 107, (1998). 299. A. V. Belyakov, P. E. Baskakova, A. Haaland, L. K. Krannich, O. Swang, Zh. Obsheh. Khim., 67, 263 (1997). 300. M. V. Popik, N. P. Novikov, S. Samdal, L. V. Vilkov, M. A. Tafipolsky, Zh. Strukt. Khim., 36, 863 (1995)/J. Struet. Chem., (Engl. Transl.) 36, 784 (1995). 301. H. Kiyono, N. Kuze, H. Fujiwara, H. Takeuchi, T. Egawa, S. Konaka, J. Mol. Struet., 3 76, 145 (1996). 302. H. Kiyono, R. Tatsunami, T. Kurai, H. Takeuchi, T. Egawa, S. Konaka, J. Phys. Chem., 102,4, 1405 (1998). 303. A. Haaland, S. Samdal, T. G. Strand, M. A. Tafipolsky, H. V. Volden, B. J. J. van de Heisteeg, O. S. Ackermann, F. Bickelhaupt, J. Organometal. Chem., 536-537, 217 (1997). 304. N. W. Mitzel, B. A. Smart, K.-H. Dreihupl, D. W. H. Rankin, H. Schmidbaur, J. Amer. Chem. Soe., 118, 12673 (1996). 305. K. B. Borisenko, K. Zauer, I. Hargittai, J. Phys. Chem., 100, 19303 (1996). 306. G. Sz~isz, A. Kov~cs, I. Hargittai, I. Jeon, G. P. Miller, J. Phys. Chem. A, 102, 484 (1998).

ELECTRON DIFFRACTION

161

307. P. T. Brain, D. Donohoe, D. Hnyk, D. W. H. Rankin, D. Reed, H. E. Robertson, A. J. Welch, M. Hofmann, P. v. R. Schleyer, Inorg. Chem., 35, 1701 (1996). 308. K. Hassler, R. Zink, N. W. Mitzel, D. W. H. Rankin, H. E. Robertson, B. A. Smart, J. Chem. Soe., Dalton Trans., 2475 (1997). 309. J. A. Maguire, K. J. Lu, C. J. Thomas, T. G. Gray, Y. Wang, N. S. Hosmane, H. Binder, M. Wanitschek, H. Borrmann, A. Simon, H. Oberhammer, Chem. Eur. J., 3, 1059 (1997). 310. A. R. Campanelli, F. Ramondo, A. Domenicano, I. Hargittai, J. Organomet. Chem., 536-537, 309 (1997). 311. J. C. Green, R. P. G Parkin, X. Yan, A. Haaland, W. Scherer, M. A. Tafipolsky, J. Chem. Soe., Dalton Trans., 3219 (1997). 312. N. Kuze, M. Ebizuka, H. Fujiwara, H. Takeuchi, T. Egawa, S. Konaka, G. Fogarasi J. Phys. Chem., 102A, 2080 (1998). 313. K. Hedberg, L. Hedberg, M. Biihl, D. S. Bethune, C. A. Brown, d. Amer. Chem. Soe., 119, 5314 (1997).

314. P. Diehl, W. Niederberger, J. Magn. Reson., 9, 495 (1973). 315. S. Sykora, J. Vogt, H. B6siger, P. Diehl, J. Magn. Resort., 36, 53 (1979). 316. A. S. F. Boyd, G. S. Laurenson, D. W. H. Rankin, J. Mol. Struct., 71, 217 (1981). 317. P. Diehl, in Accurate Molecular Structures, Their Determination and Importance, (A. Domenicano, I. Hargittai, Eds.) Oxford, Oxford University Press, (1992) p. 299. 318. M. Kon, H. Kurokawa, H. Takeuchi, S Konaka, J. Mol. Struct., 268, 155 (1992). 319. D. W. H. Rankin, in Stereochemical Applications of Gas-Phase Electron Diffraction. Part A, The Diffraction Technique, (I. Hargittai, M. Hargittai, Eds.), New York, VCH, (1988) p. 451. 320. P. D. Blair, S. Cradock, D. W. H. Rankin, J. Chem. Soc., Dalton Trans., 755 (1985).

162

MASTRYUKOV

321. H. Takeuchi, R. Watanabe, J. Enmi, T. Egawa, S. Konaka, Struct. Chem., 9, 33 (1998). 322. H. Takeuchi, S. Konaka, J. Mol. Struet., 471, 73 (1998). 323. K. Inoue, N. Kuze, M. Tanabe, H. Takeuchi, T. Egawa, S. Konaka, J. Mol. Struct.,

413-414, 81 (1997). 324. P. D. Blair, A. J. Blake, R. W. Cockman, S. Cradock, E. A. V. Ebsworth, D. W. H. Rankin, J. Mol. Struet., 193, 279 (1989). 325. C. A. Brookman, S. Cradock, D. W. H. Rankin, H. E. Robertson, P. Vefghi, J. Mol. Struct. 216, 191 (1990). 326. B. T. Abdo, I. L. Alberts, C. J. Attfield, R. E. Banks, A. J. Blake, P. T. Brain, A. P. Cox, A. Heppeler, C. A. Morrison, V. Murtagh, C. R. Pulham, D. W. H. Rankin, H. E. Robertson, J. Am. Chem. Soe., 118, 209 (1996). 327. S. Cradock, P. B. Liescheski, D. W. H. Rankin, H. E. Robertson, J. Am. Chem. Soe., 110, 2758 (1988). 328. S. Cradock, C. Purves, D. W. H. Rankin, J. Mol. Struet., 220, 193 (1990). 329. P. B. Liescheski, D. W. H. Rankin, J. Mol. Struct. 196, 1 (1989). 330. P. B. Liescheski, D. W. H. Rankin, J. Mol. Struct. 178, 227 (1988). 331. P. B. Liescheski, D. W. H. Rankin, H. E. Robertson, Aeta Chem. Seand., 42A, 338 (1988). 332. D. G. Anderson, S. Cradock, P. B. Liescheski, D. W. H. Rankin, J. Mol. Struet., 216, 181 (1990). 333. S. Cradock, P. B. Liescheski, D. W. H. Rankin, J. Magn. Resort., 91, 316 (1991). 334. S. Cradock, J. M. Muir, D. W. H. Rankin, J. Mol. Struet. 220, 205 (1990). 335. J. C. Williamson, J. Cao, H. Ihee, H. Frey, A. H. Zewail, Nature, 386, 159 (1997). 336. F. Rfiksi, K. R. Wilson, Z. Jiang, A. Ikhlef, C. Y. C6t6, J. C. Kieffer, J. Chem. Phys., 104, 6066 (1996).

ELECTRON DIFFRACTION

163

337. M. Ben-Nun, J. Cao, K. R. Wilson, J. Phys. Chem., 101A, 8743 (1997). 338. J. Cao, K. R. Wilson, d. Phys. Chem., 102A, 9523 (1998). 339. C. P. J. Barty, M. Ben-Nun, T. Guo, F. R~tksi, C. Rose-Petruck, J. A. Squier, K. R. Wilson, V. V. Yakovlev, P. M. Weber, Z. Jiang, A. Ikhlef, J. C. Kieffer, in TimeResolved Diffraction, (J. Helliwell, P. M. Rentzepis, Eds.), New York, Oxford University Press, 1998, p. 44. 340. C. Rose-Petruck, R. Jimenez, T. Guo, A. Cavalleri, C. W. Widers, F. Rfiksi, J. A. Squier, B. C. Walker, K. R. Wilson, C. P. J. Barty, Nature, 398, 310 (1999).

166

I.

GRONER

INTRODUCTION The determination of molecular structures has been one of the objectives of high resolution spectroscopy for a long time. This technique, together with gas phase electron diffraction, is the only practical method available to study structures of molecules in the gaseous state. In that state at sufficiently low pressures (when collisional lifetimes are sufficiently long), molecules are free from interactions with other species and can therefore be studied isolated from the environment. Modem high resolution molecular spectroscopy has a tremendous resolving power

and precision.

Conventional

Stark effect modulated

microwave

spectroscopy has a typical resolution of 1 in 105 and precision of 1 in 106. A similar precision is achieved by high resolution Fourier transform infrared spectroscopy. Fourier transform microwave spectroscopy and laser based infrared spectroscopy have a precision close to 1 in 108. If these spectroscopic methods are coupled with the supersonic molecular beam technique, the resolution approaches 1 in 107.

Moreover, the theoretical models for fairly rigid molecules are

reasonably well understood.

With these models, the spectra observed for such

molecules can be reproduced within the experimental precision.

Many fitting

parameters of these models - the spectroscopic constants - are obtained with high precision. Among them are the rotational constants, which sometimes have 8 or more significant digits. Most models used to fit the observed spectra are based (in principle) on a geometrical model for the molecule in question.

Thus, one might come to the

conclusion that structural parameters of molecules should be obtainable with a

QUEST FOR EQUILIBRIUM STRUCTURE

167

precision that matches the precision of the spectroscopic constants. However, it has been known for a long time that the relation between the model parameters and the geometrical structural parameters is a complicated one because the rotating molecules are not rigid. Even in the vibrational ground state, molecules vibrate. They execute many vibrational periods during one rotation. This is the cause of vibration-rotation interactions, which make the extraction of precise structu~:al parameters from the spectroscopic constants quite difficult. The goal of this article is to present a review of the methods and techniques used to determine gas phase molecular structures from high resolution spectroscopic data. The review covers the period from about 1980 to 1998 with the emphasis on the structures of fairly rigid molecules. The particular problems of determining structures of floppy molecules with low-frequency large-amplitude intemal motions or of Van der Waals complexes are not addressed. Likewise, the determination of structures by a combined analysis of spectroscopic and diffraction data is not a topic of this review. The other principal technique to obtain structures of molecules in the gas phase, electron diffraction, is reviewed in another article in this volume [ 1]. Significant progress has been made during the last two decades in the determination of molecular structures in the gas phase.

The theory and the

development of new methods will be reviewed in Section II. However, the big advances have really been made in the experimental field. In the "old days", the determination of molecular structures in the gas phase from spectroscopic data used to be a problem for a small band of microwave spectroscopists.

The

development of new experimental methods has moved the problem of determining

168

GRONER

accurate and precise molecular structures into the mainstream. Today, rotationally resolved spectral data are collected not only in the microwave region, but also in the infrared and UV/VIS regions. In the microwave region, significant progress has been made with the introduction of pulsed excitation of molecules and the subsequent analysis of the free induction decay signal in the time domain by the Fourier transform. Rotational spectra of molecules with extremely small dipole moments such as those induced by isotopic substitution [2] or centrifugal distortion in molecules without permanent dipole moments can now be measured by exciting gases in waveguides with short powerful microwave pulses [3]. The combination of MW Fourier transform spectroscopy with pulsed supersonic molecular beams has reduced the spectral line width to about 1 kHz. Because of the cooling of the species in the molecular beam, the sensitivity has increased and the spectroscopy of molecules at very low number densities is now routine. With the increased sensitivity and the simplified spectrum achieved by the molecular beam technique, it is now possible to study many isotopic modifications of molecules in natural abundance such as those substituted by 13C (1.10%), 33S (0.75%), 15N (0.37%), 180 (0.20%).

Under favorable circumstances, even D

substituted isotopomers can be detected in natural abundance (0.015%) [4]. Also the spectroscopy of many unstable species such as weakly bound Van der Waals complexes of molecules and atoms or unstable reaction intermediates and radicals as well as of species with low volatility (e. g. by laser induced evaporation) has become feasible.

A review of new experimental methods based on Fourier

transform microwave spectroscopy by Dreizler [5] has 100 references to methods and applications. Fourier transform IR, near IR or visible region spectroscopy is

QUEST FOR EQUILIBRIUM STRUCTURE

169

now a viable source of rotationally resolved spectroscopy, at least for small molecular species. The various laser sources make it possible to study rotationally resolved spectra in the IR and optical regions of the spectrum. These methods, particularly when combined with the molecular beam technique, have been very valuable for nonpolar molecules. Thanks to new techniques, rotationally resolved spectroscopy of ions is no longer the big problem it used to be particularly in combination with the generation of ions in electrical discharges [6]. Reviews of the methods to determine molecular structures have been published in the past [7-9]. In section II of this article, the basic theory of the vibration-rotation interactions are summarized and the various methods to determine the structure of molecules are reviewed, beginning with the

rs

method.

The short introduction to the least squares fitting technique and the r 0 and pseudoKraitchman methods is followed by an overview of newer developments based on the mass-dependence ab initio

(rm)

method. Section II concludes with methods employing

derived quantities, methods based on empirical correlations and a short

summary on the

rz

method. In section III, complete structure determinations of

molecules are listed that have been published from 1980 to 1998. I hope that the information presented in this section is fairly complete for the period, although I know that it is impossible not to miss something. Also in this section, a number of interesting examples of problems and their resolutions are illustrated.

170

II.

GRONER

THEORY AND METHODS

A.

Vibration-Rotation Interactions The theory of vibration-rotation interactions has been developed over the last 50 years by many prominent researchers. It has been presented in many texts on the subject e.g. [ 10], among them a rather complete summary by Aliev and Watson [11 ]. It is based on classical perturbation theory in the form of a sequence of contact transformations.

The results relevant to the

rotational constants are summarized here. The effective rotational constant about the fl axis in the vibrational state characterized by the vibrational quantum numbers v=(v 1 ... v k ...) with degeneracies (d 1 ... d k ...), B~, is given by [121 dk

dk

d/

e~ = ~,, - Zk ~l(v~ + T ) + Zk,/ r~(v, + T)(~, + T)...

(1)

Bfl is the rotational constant at the equilibrium geometry. The parameters or,p and 7"~ are called vibration-rotation constants. The indices k and l sum over all distinct vibrational (normal) frequencies. So far, no general equations are available for the constants 7"~. The constants a~contain harmonic and anharmonic components ak~ = Crka (harm) + akp (anh)

(2)

that are defined by [ 13 ] ~z~(harm) = - 2(Bff)2

pr 2

3o~ + (o~

(3)

QUEST FOR EQUILIBRIUM STRUCTURE

cok

Zku,a~a,

171

C03,n/2 "

In these equations, cok is the harmonic frequency of the normal coordinate Qk and ( ~ = - ( ~ is a Coriolis coupling constant. I,r is the equilibrium moment

of inertia about the axis 7 and a~r = \( ~cOl, / or1 is the inertial derivative with

respect to the normal coordinate Qk at equilibrium.

The cubic potential

derivative in dimensionless normal coordinates, kkl m, is related to the cubic force constant Okl m by

r

\ OQkOQtOQme= kklm(YkYlYm)l'2 hc

(5)

where

Yk = hCCOk/ h2"

(6)

The case of strong Coriolis resonance (e. g. if com= cok) must be treated appropriately and the term containing (~ in Eq. (4) be must be modified (see [ 1 1]). It is less obvious from Eq. (5) that a similar procedure must be applied to the anharmonic term in the case of Fermi resonance (if 2cok ~ COrn)[13]. The effective rotational constant for the vibrational ground state is therefore given by (neglecting the 7 terms)

dk - Z

=

B~ -

ct p

--i

(7)

The effective moment of inertia in the vibrational state v defined as h2

I~ - 2hcB~

(8)

172

GRONER

is related to the moment of inertia at equilibrium by dk

I~ ~ Ifl + ~ c~ (v, + -~-)

(9)

k

where c ~ = I a a~

(10)

For the vibrational ground state we have therefore

IoP~, lfl + ~ c~ Tdk = IP +cp Another

complication

(11)

arises

from the

fact that the rotational

constants B~ are usually obtained as effective fitting parameters of a reduced rotational Hamiltonian [14].

Not only do the numerical values of the

rotational constants depend on the exact form of the reduced Hamiltonian, they also contain small contributions from quartic and higher order centrifugal distortion terms. Watson [14] has proposed to always determine the so-called determinable combinations of these constants. The values of these combinations are independent of the form of the reduction, although they still contain small contributions from the distortion terms.

Up to the

quartic centrifugal distortion terms, the determinable combinations of the rotational constants are B p = B p - 2T ~r where

(12)

T ~`r are the quartic terms in the expansion of the rotational

Hamiltonian Hrot = EBPd~ +ETBr(J~ +d2r)+. . . . P

P,r

(13)

QUEST FOR EQUILIBRIUM STRUCTURE

1 73

Molecular structures determined from equilibrium moments of inertia, Ifl, are called

re

structures. Within the Born-Oppenheimer approximation,

they are well defined as the structures associated with local minima of the potential energy surface. As such they have no isotope effect, and the

re

structures are invariant to the particular choice of the isotopic data set. Molecular structures determined directly from the observed ground state moments of inertia, I0a, are called r 0 structures. They depend strongly on the specific set of isotopomer data used because the isotopic dependence of the vibration-rotation interaction terms ~e is different from the dependence of the moments of inertia. This quickly leads to contradictions in the results from different isotopic species.

Alternately, data from many isotopic

molecules can be averaged by the least-squares method.

The results still

depend somewhat on the isotopic data set and they have low relative precision typically in the range of 1%. Most methods to determine molecular structures try to compensate for or correct the effect of the vibration-rotation terms. As a result, there are now multitudes of methods available with as many special notations as methods. Their isotope dependence varies in practice from zero to full (for r 0 structures).

Their error limits are sometimes difficult to determine.

The

goals of some of the newer methods are internal consistency and maximum relative errors (difference to

re

structure) of O. 1% or less.

174

GRONER

B.

re

Structures The

structure of a molecule is obtained from the equilibrium

re

moments of inertia

Ie

or the rotational constants

B e.

Unless the molecule has

the formula XY n and sufficiently high symmetry, the equilibrium moments of inertia of more than one isotopomer are required because the number of independent structure parameters is larger than the number of independent moments of inertia of a single molecule. For each isotopomer, the constants aft have to be determined by analyzing the vibration-rotation spectrum for each vibrational fundamental or the pure rotational spectrum in the corresponding excited state. For asymmetric rotor molecules with more than three atoms, this task requires therefore a tremendous amount of data. Moreover, attention must be paid to the non-trivial problems of Coriolis and Fermi resonances because inadequate treatment of these effects may lead to incorrect vibration-rotation constants and therefore incorrect equilibrium constants [9].

For these reasons, the number of accurate equilibrium

structures for molecules with more than two atoms is still rather small although some progress has been made during the last two decades.

C.

Traditional

rs

method

In the traditional substitution method, the structural parameters are calculated from the Cartesian position coordinates of the nuclei. The squares of these coordinates are obtained from isotopic differences of principal moments of inertia by using equations first formulated by Kraitchman [ 15]. To apply this method, one needs the principal moments of inertia of a

QUEST FOR EQUILIBRIUM STRUCTURE

175

reference molecule, the parent, and those of all species that have just one of the nuclei substituted by an isotope. Chutjian [16] and Nygaard [17] have derived equations that are applicable in the case of substitutions of complete sets of symmetrically equivalent nuclei. 1.

Single substitution The equations formulated by Kraitchman [15] are based on the relations Px =Em~ x2

(14)

i

I,, = Py + P~

(15)

P. =(Iy + I , - / ~ ) / 2

(16)

Em, x, =0

(17)

i

~m,x,y, =0

(18)

i

and those obtained by cyclic permutation of x, y, and z in equations (14)-(18). The first equation defines the planar moments in terms of the atomic masses m i and the principal coordinates of the parent isotopomer, equation (15) defines the principal moment of inertia I x in terms of the planar moments P (also called second moment), whereas equations (17) and (18) are the first moment condition (also called the center-of-mass condition) and the product of inertia condition, respectively. In the general case, the Cartesian coordinates of atom k in the principal axes system of the parent isotopomer are obtained from the planar moments of the parent and the daughter isotopomer,

176

GRONER

where atom k with mass mass

x~ =

mk+Am , as

].,/-l(p;_ Px)

mk

has been substituted by an isotope with

follows:

=]./ AP~ 1+

(P>,-P:)(P:-Px)

1+

P>,-Px

P:-P,,

(19)

where APx = P ' - Px

.

(20)

The primed quantities refer to the daughter isotopomer, and the reduced mass of isotopic substitution,/z, is defined as

/~-

MAm M'

"

(21)

M and M' are the molecular masses of the parent and daughter species, respectively. Kraitchman [15] also derived several expressions for cases where atom k lies on a symmetry axis or in a symmetry plane. In these cases, one or two of the AP vanish exactly for rigid molecules

(r e

structure).

The resulting relations between the A/can be used to formulate several expressions for a single case. Because of the vibrational contributions to ground state moments of inertia, these relations do not hold exactly for the

(vibrating)

molecules

and,

therefore,

the

resulting

rs

coordinates are not identical. Usually, the averages of all possibilities have been reported as

rs

coordinates.

Rudolph [18] has shown that

these averages are almost identical to the results obtained from

QUEST FOR EQUILIBRIUM STRUCTURE

177

Chutjian-type expressions. For the atom k in the symmetry plane axy, z k vanishes by symmetry and x k is obtained from

=/tI'AP~ 1+ pyAPy _ p,, )

~

(22)

with/ax =/~. The coordinate Yk is obtained by interchanging x and y in these equations.

For the atom k on the symmetry axis z, the

components xk and Yk vanish while z k is given by z,2 = / u ; ' ( e " - e , ) =

' aP,

.

(23)

According to Rudolph [19], the coordinates of atom k in the principal axes system of the substituted molecule can be obtained by changing the sign of Am and interchanging the primed and unprimed quantities in equations (14-23). The complete transformation between rk = (XkYk Zk) and

r k = Rr~ + t

rk' = (xk' yk' zk') can be described by and

r~ = R,(rk - t)

(24)

where t and the elements of R are given by [ 19] t

Am

,

k . R~y, =-Am pxxxY------~' _ py,

(25)

(26)

Equations (24)-(26) can be used to transform any vector between the two principal axes systems. Except for the signs of the components of r k and rk', R and t are completely defined by the planar moments P and P' of the parent and the daughter isotopomers, respectively.

178

GRONER

2.

Symmetrical multiple substitution Chutjian [ 16] derived expressions for molecules in which a whole set of symmetrically equivalent atoms are substituted.

These

expressions were reformulated in terms of the planar moments by Nygaard [ 17]. For double substitution of a pair of atoms related by the symmetry plane Sxy, Xk and Yk are obtained from equation (22) with/Ac = ~t2. ktn is defined by

~=n

Mare M'

(27)

where Am is, as before, the isotopic mass difference of a single atom. The coordinate z k is obtained from equation (23) by setting ktz = 2Am. The equations for substitution of a pair of equivalent atoms related to each other by a C2 operation are also obtained from the equations (22-23) above by setting/.tx = 2Am and flz = Ft2. For a substitution of n atoms (n > 2) related by a Cn axis (= z axis), the coordinate Zk is obtained from equation (23) with ~tz = Pn. The radial coordinate x k (= distance from the z axis) is obtained from

x, =~t,-'(e; 2

where

F

- e,) = ~t,-'ae,

(28)

/.tx = nAm.

Because the principal moment Iz (about the unique principal axis in symmetric rotors) cannot usually be obtained from rotation or

QUEST FOR EQUILIBRIUM STRUCTURE

179

vibration-rotation spectra, the expression for the coordinate Zk is usually written as, assuming A/z = 0, z ,2= ~ t ;1 ( I ' - L ) = , , - ' A / ,

.

(29)

For the same reason, the radial coordinate x k cannot be obtained from symmetrical substitution of symmetric rotors. One way to overcome this problem is to use single isotopic substitution of an off-axis atom making the substituted molecule a slightly asymmetric near-prolate or -oblate rotor. For many of these near symmetric rotors it is difficult to determine lz' precisely. An alternate method described by Li et al. [20] circumvents the need for I z and Iz'.

For a molecule of C3v

symmetry, they combined data from the symmetric rotor parent, the di-substituted asymmetric rotor and the tri-substituted symmetric rotor. In fact, data from any three of the four species (symmetric rotor parent, mono-, di- and tri-substituted isotopomers) can be used to determine z k and the radial coordinate x k. The expressions are listed in Table 1. The differences A/are defined as zxi'=l,-L

,

A/,,- I,,-L

,

(30)

A/'"= 1""- I x . The single, double or triple primes attached to the data refer to the mono-, di- or tri-substituted species, respectively. The expressions in Table 1 are valid for the situation where the parent and the heavier tri-

180

GRONER

T A B L E 1. r s Coordinates of C 3 symmetric rotors from multiple substitutions a.

zk2

Combination P, S, D

x,2

- M ' M " + 2 M"AI[,'

4 M ' A I " - 2 M"AI[,'

3 MAm

3 MAm

P, S, T

- 3 M'A/" + 2 MA/t~" M'" 3 MAm 2 M - M'"

3 M'A/" - .A,r . . . I,,, b 2M 3 MAm 2 M - M'"

P, D, T

3M A/~- ~r^1,,,

M'"

- 3 M A/~ +2 .hX'''Af''' ... ~

3 MAm

2 M'" - M

3 MAm

Ft

Fr

,t vat L a u t b

?F

?~

2M 2 M'" - M

ap, S, D and T refer to the parent and the mono-, di- and tri-substituted species, respectively.

substituted isotopomer are prolate rotors. If the tri-substituted species is oblate, A/b'' must be substituted by A/a" in all expressions given in the table; if the parent molecule is an oblate rotor, A/c' must be replaced by A/b' likewise. Similar expressions may be developed for other symmetric rotor molecules. Wilson and Smith [21] derived equations for general multiple substitutions that can be used if the coordinates of only one of several substituted atoms are unknown.

Pierce [22] and others [23,24] have

introduced the double substitution method to locate nuclei near a principal axis by taking second differences of ground state moments of inertia.

This method has not found widespread application maybe

because of the necessity for very precise data for a larger set of isotopomers.

A formula to calculate an internuclear distance in a

linear molecule directly from the moments I 0 of a parent molecule, two monosubstited species and the disubstituted isotopomer ([25],

QUEST FOR EQUILIBRIUM STRUCTURE

181

citing a private communication by J. K. G. Watson) is related to the double substitution method.

3.

Error estimates for r s coordinates The errors of substitution coordinates can be derived from the errors of the principal moments of inertia or of the planar moments [26]. &k -

K /.oq

with K = 0(APx) / 2 .

The error 5(AP) has two different contributions.

(31)

The first kind is

associated with the experimental error of the planar moments; it is usually negligible.

The second kind, more severe and systematic,

originates from model errors.

The equations for the substitution

coordinates are based on the assumption that the molecules are rigid. Since they are not, the experimental ground state constants contain vibrational contributions that have a different mass-dependence. As a consequence, the substitution coordinates do not satisfy the basic relations (17-18). Another aspect of the same problem is the question of how close the

rs

coordinates are to the equilibrium coordinates

r e.

Costain [26] has shown that = (re +

0)/2

(32)

is a good first order approximation for diatomic molecules. However, no simple formula is available for polyatomic molecules because the relations are so complex and the

rs

coordinates must be compared on a

182

GRONER

case by case basis with experimentally determined

re

coordinates.

Costain [27] has advocated that equation (31) be used to estimate the uncertainty of the

(re)

structure of molecules, and he proposed to use

the average value K = 0.0012 u.A 2 (from the data for N20 ) to estimate the

rs/r e

discrepancies.

Schwendeman [28] proposed to use K =

0.0015 u-A 2 instead. It seems doubtful whether equation (31) really tells us anything about the errors of the used to check the consistency of

rs

re

coordinates. It can only be coordinates.

Even if the

substitution coordinates were completely consistent and fulfilled the basic assumptions (17-18) it would not automatically mean that they were

re

coordinates.

van Eijck [29] reported the results of such a check by calculating AP values for substituted molecules where AP should vanish for symmetry reasons, e. g. substitution of atoms in a symmetry plane or on a symmetry axis. By analyzing over 430 observations reported in the literature up to about 1979, he found that 2K= ~/((AP) 2)

(33)

decreased with increasing mass of the substituted atom. Costain's [27] estimate for K proved to be adequate for 10B/liB, 14N/15N and 160/180 substitution but somewhat too large for the substitution of 12C/13C or any of the heavier atoms (S, C1, Br, Se). However, K for H/D substitutions should be increased considerably to about 0.0032

QUEST FOR EQUILIBRIUM STRUCTURE

183

uA 2. Moreover, van Eijck [29] had to exclude the data of 13 H/D substitutions in methyl groups because they did not fit the pattern of the other H/D substituted species. For the methyl group substitutions, K as defined in equation (33) and (AP) were 0.016 u'A 2 and +0.025 u.A 2, respectively, whereas they were 0.0032 u'A 2 and -0.0004 u.A 2, respectively, for the other 131 H/D substituted species.

Van Eijck

found a correlation between AP and the barrier to intemal rotation of the methyl groups.

D.

Least-squares methods Structural parameters are determined usually by fitting them to experimental data by the least-squares method. traditional

rs

The exceptions are the

method and cases in which the number of rotational constants or

moments of inertia equals the number of structural parameters. An excellent introduction to the least-squares method in spectroscopic applications has been written by Albritton, Schmeltekopf and Zare [30]. Least-squares methods have been used to determine molecular structures successfully first by N6sberger et al. [31 ], Schwendeman [28], and Typke [32].

N6sberger et al. fitted structural parameters (internal

coordinates) to isotopic differences of moments of inertia.

To solve the

normal equations, they used the singular value decomposition of real matrices to calculate the pseudo-inverse of such matrices with the option to omit nearzero singular values in illdetermined systems.

Schwendeman [28] fitted

internal coordinates to moments of inertia or isotopic differences of these

184

GRONER

moments.

The latter was called a pseudo-Kraitchman (or pseudo-rs) fit

because the results were almost identical to the traditional

rs

method.

Discrepancies arose only if some parameters had to be kept constant because the available data set was not complete.

Schwendeman also showed that

essentially the same structures were obtained when Cartesian coordinates were used as adjustable parameters.

Typke [32] described a least-squares

program to fit Cartesian coordinates to isotopic differences using the Kraitchman formulas. He specifically designed the program so that data from multiply substituted species could be used together with data from monosubstituted species. The essential definitions and results of the least-squares procedure are summarized in the following equations. The least squares problem is defined as, adopting the notation of Rudolph [33], y = X 13+ r

(34)

| = a2M

(35)

were y is the vector of the observables, 13is the vector of the parameters to be determined, the Jacobian X contains the partial derivatives of the observables with respect to the parameters, and r is the vector of the residuals. | is the covariance matrix of the observations, a the standard deviation and M the inverse of the generalized weight matrix.

The unknowns are 13, r and a,

whereas y, M and X are given. The solution of this problem is described by [331 --" (X~/1-1 X ) -1

~'vI-ly

(36)

QUEST FOR EQUILIBRIUM STRUCTURE

185

O~ = 62 (~,~-'X)-~

(37)

c 2 = r ~ l - ' r ( n - m)-'

(38)

r=y-X[3

(39)

In these last equations, [3, cr and r are now the expectation values (solutions) of the quantities defined above, and 013 is the covariance matrix of the parameters, n is the number of observables and m the number of parameters. Because the functional dependence of the observables (rotational constants, principal moments of inertia or planar moments) on the structural parameters is strongly non-linear in most cases, an iterative process is essential. Typically, one begins with an assumed structure and expands the moment of inertia functions in terms of the parameters of this structure in a Taylor series up to the linear term.

a/, ) ~ z(Pko Xpk

I~ (obs) = I~ (Pko) + k

OPko

(40)

with Apk = P k - PkO. The differences Ii(obs)- li([30) become the observables y and the changes in the parameters Z$pkcorrespond to the parameters [3. The next approximation for the parameters is then obtained as PkO'= PkO + APk.

(41)

The procedure is repeated with the new parameter set until the least-squares fit converges. A systematic presentation of least-squares methods applied to the determination of molecular structures has been published by Rudolph [33].

186

GRONER

Investigating a number of related cases, he distinguished between structural parameters fitted to rotational constants, moments of inertia or planar moments or differences thereof. A survey of common problems occurring during least-squares fits of molecular structures and possible remedies has been given by Demaison et al. [9].

E.

r 0 and r s

Structures

According to Rudolph [33], structures determined by fitting structural parameters directly to the experimentally determined rotational constants B 0, the moments of inertia I 0 or the planar moments P0 of a series of isotopomers are called r 0 structures. If the covariances of the observables are transformed correctly, the structures resulting from any of these fits are identical. However, if covariances are not transformed, these structures are not identical nor equivalent (for examples, see [34]).

Therefore, one should always

specify what kind of covariance matrix is used with a particular fit. PseudoKraitchman (p-Kr) or pseudo-r s fits are those in which the structures are fit to differences of rotational constants, moments of inertia or planar moments. The structures called

rM

and rap again are equivalent to each other if the

covariances are transformed correctly. On the other hand, the structure rAB is theoretically different from the rA/and rap structures because of a lack of a linear relation between the AB and A/. In practice, rAB structures are very close to other p-Kr structures. The rle structure is obtained when the isotopic moments I 0 are fitted to structural parameters and three e parameters. It is

QUEST FOR EQUILIBR/UM STRUCTURE

187

based on the assumption that the vibrational contributions to the moments of inertia (one for each axis), e, are the same for all isotopic species. The rlr structure is identical to the ra/structure. Its big advantage is the fact that it gives much better predictions of moments of inertia (or rotational constants) than the rA/structure because of the explicit inclusion of the vibrational contributions. A much more detailed account of the particular problems of leastsquares methods in the determination of molecular structures has been given by Rudolph [35]. In this review, Rudolph calls all methods mentioned in the preceding paragraph r 0 or r0-derived methods.

He reserves the term

rs

method strictly to least-squares methods based on the equations of Kraitchman [15], Chutjian [16] and Nygaard [17] and their extensions to multiply substituted species [32,35,21 ] despite the fact that the p-Kr methods rA/, rap, rAB and ric result in structures much closer to the traditional

rs

method than to the r 0 methods. It is a little disconcerting that the notation tbr some of these structures is not uniform. named

re, I

Originally, the tic method was

by Rudolph [33]; it has been sometimes called

rl, ~,

even by the

original author [35]. In this review, the latest notation [9] tie is used.

188

GRONER

F.

r m Structures and beyond

1.

r m Structures

Watson [36] introduced the mass-dependent coordinates r m. If the vibration-rotation constant for a substituted species, 6', is expanded as a function of Amk

~'=~+

Amk+

Omk2 (Amk +...

,

the square of the substitution coordinate becomes for a linear molecule Io - Io

2

- zZ~ +

+ --~

+-~ Omk2

Am k +....

(43)

Retaining only the first two terms in this expansion, the substitution moment of inertia defined as Is

=

E m i z i s2

(44)

i

is given by

I s = I, + ~ m,

.

(45)

Using Euler's theorem on homogeneous functions, Watson [36] showed that the sum in this equation is equal to d2 and, therefore, Is = (le + I0)/2

.

(46)

He defined the mass-dependence estimate of the equilibrium moment of inertia, Ira, as I m = 2Is - 10

(47)

QUEST FOR EQUILIBRIUM STRUCTURE

189

and called the structural parameters fitting these estimates

rm

parameters. Analogous arguments lead to identical definitions of the mass-dependence estimates

Im

for all principal moments of inertia for

nonlinear molecules. To apply the

rm

method, one needs to determine the moments

for a sufficient number of isotopomers. To calculate

I s,

Is

data from the

first moment equations or product of inertia conditions cannot be used, and negative squares of the position coordinates, should they arise, have to be retained. The big advantage of the traditional

rs

rm

method over the

method is the complete cancellation of the first order

term of the mass dependence of e. Therefore, the moments

Im

are

expected to be much better approximations of I e than I s. However, the rm

method suffers from several disadvantages: (i) It can only be

applied if all atoms of the molecule can be substituted by isotopes. (ii) It is not applicable to hydrogen containing molecules because the large relative mass changes involved in H atom substitution render the approximations invalid. (iii) The ground state moments of inertia of many isotopic species of a molecule must be available.

(iv) The

neglected terms in the expansion of c are not as small as expected. Watson [36] obtained good agreement between the

rm

and

re

structures for the test molecules CO, SO 2 and N20. As expected, the results were not so good for HCN because of the large relative mass changes associated with H/D substitutions. Rather unexpectedly, the

190

GRONER

results for OCS were somewhat disappointing because rather large discrepancies between the

r e and

rm

parameters were obtained. In a

follow-up paper, Smith and Watson [37] tested the procedure on synthetic data for OCS. They concluded that neglected higher order terms in the expansion of c as function of the changes in mass (third term in Eq. (42)) were responsible for the unsatisfactory result for OCS and that the good results for N20 were fortuitous.

The test

calculations for OCS suggested one way to salvage the original method: The extrapolation of the provides

I s and

rs

coordinates to Amk = 0 (Eq. (43))

moments that are virtually free of the dependence

Im

on the finite mass changes. The resulting 4.10 -5 A with the

rm

re

rm

parameters agree within

parameters. However, this route is definitely not

practical in most cases because every atom would have to be substituted by two different isotopes.

2.

rc

Structures Kuchitsu and coworkers [38,39] introduced the

an extension of the

rm

rc

method that is

procedure. They recognized that the influence

of the third term in Eq. (43) could be eliminated almost completely by averaging "complementary sets" of isotopic data. dichlorine monoxide, they obtained the rm

parameters obtained for

rc

In the case of

structure by averaging the

35C12160 and 37C12180.

The isotopic

mass changes are positive for all atoms in the first of these species but

QUEST FOR EQUILIBRIUM STRUCTURE

191

negative in the second. Almost identical parameters were obtained by averaging the

rm

parameters from

35C12180and 37C12160. These

parameters agreed very well with the best

re

rc

structure which was

obtained by a careful analysis of microwave data [40].

The same

method has also been applied to phosgene, COCI 2 [39]. Even though at least some of the parameters are indeed very close to the

re

parameters, this method demands even more isotopic data than the

rm

method.

It can therefore be applied only in very special cases for

small and, preferably, symmetrical molecules. The potential range of applications of the

rc

method has been extended a little further by

Nakata and Kuchitsu who introduced an additivity rule for isotopic data [41].

For different types of small molecules, they defined F

functions from the moments of inertia and, using additivity rules discovered for isotopic changes for

rz

distances, they derived

additivity rules for these functions such as 2

3

F12=F+~(F~-F) i=1

and F123=F+ff'~(F~-F)

.

(48)

i=1

F12 and F123 are the F functions for the isotopomers where atoms 1 and 2 or atoms 1 through 3 have been substituted, respectively, whereas F and

F i

are the F functions for the parent molecule and the

molecule where atom i has been substituted by an isotope, respectively.

That rule allows it to estimate the F functions and,

192

GRONER

therefore, the moments of inertia of multiply substituted species from the moments of monosubstituted molecules.

3.

rmP

Structm'es Another, more versatile approximation to the

rm

method has been

introduced by Harmony and coworkers [42-46]. To reduce the number of isotopomers that need to be investigated to derive the

rm

structure,

they used the fact that the ratio of the substitution moment of inertia, I s,

to the respective ground state moment, I0, p = Is/I0

,

(49)

is essentially isotope invariant. They used ,o (one for each principal axis) as determined from moments

Is

I s and I 0

and, therefore,

this procedure the

Im

for other isotopic species. They called

method.

rmP

for one isotopomer to estimate the

The structure is determined by the

least-squares method from a sufficient number of moments

ImO

defined as /m0 = ( 2 ; -

As in the

rm

1)/0

9

(50)

method, first or second moment conditions cannot be used

to evaluate the moments

I s.

Like the

rm

method, it is therefore

restricted to molecules in which all atoms can be substituted. advantage compared to the

rm

method is that it can be used whenever a

complete substitution structure is available. significant

reduction

of

Its

the

necessary

This results in a number

of

isotopic

QUEST FOR EQUILIBRIUM STRUCTURE

193

modifications. Molecules containing hydrogen atoms have to be dealt with by a special procedure since even the r m method gives unsatisfactory results in these cases. A detailed review of the theory of the rm9 method and its applications has been prepared by Harmony for this volume [47].

He recommended that rmO structures should be

determined whenever all the necessary isotopic data are available to determine a complete r s structure. Particularly the error limits of the rmO method have been criticized by Demaison et al. [48]. They claimed that the error estimates were far too optimistic because the errors of the parameters p were not taken into consideration. Furthermore, the fact that the moments 1rap are all multiplied by the factor (2p - 1) generates a significant amount of correlation between these moments that affects not only the error limits but also the numerical values of the rmP parameters. The rmP method has also been criticized by Cooksy et al. [49] because the correction by the factor (2p - 1) does not allow for the proper mass dependence of the quantities e.

4.

rm(1) and

rm(2) Structures

In the paper proposing the r m structure, Watson [36] showed that the vibrational contribution to the ground state moment of inertia, e, is a homogeneous function of degree 89 in the masses and he used this argument to derive the formula for the r m structure (section II.F.1).

194

GRONER

Demaison and Nemes [50] proposed that the relationship between log c and log Ie should be nearly linear with a slope of 89 for all diatomic molecules.

The proposition is based on the equation for diatomic

molecules [ 10] 3B, 3h ro = r~(1--~-(1 +a~)) = re --~-(1 +al)(r2f/O -v2

(51)

where co is the harmonic vibrational frequency, a 1 the first Dunham coefficient, f the harmonic force constant and/.t the reduced mass. Since a 1 and the product re2f are nearly invariant among diatomic molecules [50], c should depend linearly on /fl/2.

To test this

proposition, they investigated the correlation between 6 and I 0 for diatomic and polyatomic molecules. For 76 diatomic molecules, they obtained by weighted least-squares fitting the equation

(52)

log 6 = 0.611 (1 O) log I 0 - 1.776(25)

with a correlation coefficient of 0.93. This correlation held for a range of almost three orders of magnitude in I 0.

Considering the

approximations and the small difference between 10 and I e, the deviation of the slope from 89 is perfectly acceptable.

For 61

polyatomic molecules, they report a similar correlation as log 6 = 1.247(5) log/0 - 2.651(13)

(53)

with a correlation coefficient of 0.98. The much stronger dependence of c on I 0 for the polyatomic molecules is rather surprising. The straight line shown in Fig. 2 of ref.

QUEST FOR EQUILIBRIUM STRUCTURE

195

[50] in relation to the log r versus log I 0 data points seems to be the line described by the equation but it does not look like it were the line fitting the data.

Of course, looks are sometimes deceptive and

unreliable, particularly for a weighted fit if the weights are unknown. If the data for the H2X molecules were excluded, the slope of the line would likely be much less than 1 instead of more than 1. Le Guennec et al. determined equilibrium rotational constants and moments of inertia for FC10 3 [51] and 74GeH3F [52] and calculated the vibration-rotation contributions r for each principal axis. They used them to estimate e' for other isotopomers of FCIO 3 and 74GeH3F using the relation d = e ( I ' / 1 ) 1/2

(54)

.

They corrected the isotopic ground state moments of inertia with these estimates to obtain equilibrium moments from which they determined the approximate

re

structure for both molecules.

A near equilibrium structure for CH3CN was derived by Le Guennec et al. [53] using the same approximation in a different way. They approximated I e in effect with

(55)

le ~ 10--C(10) ''~-

and fitted

re

structural parameters and the proportionality constants c

(one for each axis) to these estimates. They compared the results with those obtained by other methods (r 0, r s,

r i c , rmO ).

196

GRONER

More recently, a method almost identical to this last one by Le Guennec [53] has been used by Watson and coworkers [49,54]. They approximated the ground state principal moment I 0 as I 0 = I m + C(Im)l/2

where

Im

(56)

,

is calculated in the manner of a rigid rotor from the structural

parameters. The parameters and the proportionality constants c (one for each axis) were determined by least-squares fitting. parameters were called

rm(1) because

The

the method is mass-dependent

and because one extra parameter per axis was used. In the case of the NO dimer [54], the rm(1) NO distance was reported to be within 0.0007 A of the

re

distance of the NO monomer.

The details of the rm(1) method have been published only very recently [55] where Watson et al. have given a detailed account of numerous calculations with a variety of models to approximate I 0. All methods tested are based on the correct mass dependence of G [36]. The general method is based on the approximation I 0 = I m + C(Im) 1/2 + d ( m 1 m 2 ...

mN/M)I/(2N-2) + 8L

where I 0 is the effective ground state moment of inertia and moment of inertia calculated from the

rm

parameters,

(57)

. Im

the

c and d are

fitting parameters (one each for each axis), M is the total molecular mass, and representing

mk

are the atomic masses,

Laurie

corrections

for H

sL is a symbolic term atom

distances

(Laurie

QUEST FOR EQUILIBRIUM STRUCTURE

197

corrections are applied in a different form, however, see below). N is the number of atoms. The first correction term (with the coefficient c) essentially scales the moment of inertia with the correct mass dependence. It corrects for the fact that e tends to be positive, making r 0 bond lengths generally longer than the r e distances.

The second

term with the scaling coefficient d corrects for the "small coordinate anomaly" which is so troublesome in the r s method. It is based on the fact that, in model calculations, the contribution to 6 from most atoms with small principal coordinates tends to be negative.

Models that

include only the c-terms are called rm(1), those with both the c- and dterms rm(2). If the Laurie corrections are applied, the models become rm(1L) and rm(2L) , respectively. According to Watson et al. [55], the

structural parameters as well as c and d tend to be strongly correlated if the molecule does not have atoms with small principal coordinates. If Laurie corrections have to be applied, parameters are also likely correlated, particularly in the rm(2L) model. The Laurie corrections are applied by defining the effective XH distance rmeff(XH ) as M rmeff(XH) = rm(XH ) + 6 . m. ( M - m.

)/1/2

(58)

The parameter 5 H is a fitting parameter, the expression in parentheses is the inverse of the reduced mass of the H atom vibrating against the rest of the molecule.

198

GRONER

Watson et al. [55] also described a procedure to take into account the axis rotation induced by isotopic substitution. Instead of three c parameters (one for each axis), six c parameters are necessary to correct the moments of inertia of a general asymmetric rotor molecule. Watson et al. [55] applied these methods to a number of molecules. The rm(2) internuclear distances determined for the linear molecules N20 , CO2, OCS, OCSe, CICN, and BrCN agree within 0.001 A or better with the reported r e distances. The rm(2) structures for C1BO and CIBS can be compared only with r m a n d r s structures, respectively. Excellent agreement between the r e and rm(2) structures was obtained for the nonlinear molecules SO 2 and 0 3. The agreement is not quite as good for nonlinear C1NO and COC12. In the case of CINO, experimental ot constants were available for only two isotopomers to determine the equilibrium rotational constants.

The

discrepancy in the case of COCI 2 is suspected by the authors to arise from inconsistencies in the rotational constants because of the quadrupole coupling.

For these twelve molecules not containing

hydrogen, the standard deviations of the rm(2) fits are significantly smaller than o(ro)/O(rm(2) )

the

standard

deviations

of r 0 fits,

with

ranging from as low as 20 to several hundreds.

ratios with the standard deviations from r s fits,

ratios The

O(rs)/O(rm(2)), range

from 2 to 30. r m ( 1 L ) structures were determined for the linear hydrides

QUEST FOR EQUILIBRIUM STRUCTURE

199

HCN, HCO +, HBO, HBS, HNC, HCNO, HCCCI, HCCCN because none of them (except HCNO) has an atom near the center of mass. The differences of the internuclear distances between the rm(1L) and r e structures are smaller than 0.002 A for the first five molecules. No r e structure is yet available for HCNO. For the last two linear hydrides, the rm(1L) distances between non-hydrogen atoms agree with the r e values within 0.003 A, the HC distances within 0.007 A. The ratios of the standard deviations for different kinds of fits are much less spectacular.

The ratios o(ro)/O(rm(1L)) for these molecules were

between 14 and 50 for the first four linear hydrides, but only between 1.5 and 8 for the second group of four. The ratios even smaller.

a(rs)/a(rm(1L)) were

Because data sets were rather limited, only rm(2)

structures were determined for the nonlinear hydrides HOCI, HNCO, HCOOH, H2CCC12, H2CS and H2CCC. The rotation coefficient Cab was included for H2CCO for a rm(2r) structure while Cab and the Laurie correction were used for H2CO resulting in a rm(2Lr) structure. The ratios a(ro)/a(rm(2)) ranged from 2.5 (for H2CCC12) and 12 (for HCOOH) to 300 whereas a(rs)/a(rm(2)) H2CCC12 and HCOOH) and 46.

was between 1.3 (for

For this group of molecules, the

results are of mixed quality. For HNCO and H2CO, the agreement with the r e structural parameters is quite good although the fit for formaldehyde is only moderately good. H2CS gives a better fit but a

200

GRONER

rather large difference to the

re

CS bond length. For H2CCC12 and

HCOOH, there is agreement with the error limits.

Differences to the

re

re

structure within rather large

parameters are rather large

compared to the uncertainties for HOCI but the error limits of the

re

parameters are also large. The fits for H2CCO and H2CCC are less good but the data sets for these molecules are rather limited. The big advantage of the rm(1) and rm(2) methods is the fact that they are applicable also to molecules containing elements with only a single stable isotope.

Watson et al. [55] observed that the rm(1)

method requires the same amount of isotopic data as the a n d rA1

Corrections based on Ab

methods tic

and should therefore be adopted systematically instead. The

same is true, of course, for the

G.

rs

initio

ab initio

rmP

method.

calculations

calculations have reached a stage where they become

extremely useful to spectroscopists [56-63]. They can be used not only to predict

re

structures but also to provide reasonably accurate force fields and

dipole moments. Results of

ab initio

calculations are used now in a number

of ways to determine molecular structures from spectroscopic data. review by Demaison et al. [9] contains an assessment of the modem methods that are relevant to molecular structure research.

The

ab &itio

QUEST FOR EQUILIBRIUM STRUCTURE

1.

201

Empirical corrections One type of application involves the estimate of empirical offsets by comparing

ab

initio

re

distances with experimental r 0 or

re

distances and to use these offsets as constraints in fits of structural parameters for similar molecules. Le Guennec et al. [64] used two different basis sets, a double-zeta basis with a single set of polarization functions (DZP) and a triple-zeta basis with two sets of polarization functions on C and N atoms (TZ2P), at three different levels of theory (SCF, MP2 and QCISD). For each type of calculation, they derived offsets for C-H, C-C, C-C and C-=N bonds with which they predicted the

re

structure of propyne, HC-CCH 3. Similar offsets determined for

smaller basis sets (6-31G(d), 6-311G(d,p) and 6-31 l+G(d,p)) and electron correlation at the MP2 level have been used for acrylonitrile [65] and a series of dicyanides [48]. van der Veken et al. [66] used a scaling method to adjust structural parameters from

ab

initio

calculations (e. g. MP2/6-

31 l++G(d,p)) to fit experimental rotational constants. By optimizing scaling factors (one for a group of similar parameters, e. g. all C-H distances), they were able to maintain the ratios of parameters at the values predicted by the

ab

initio

calculations.

This new method

promises to be very useful for the determination of the structures of conformer pairs because (1) complete structures for conformer pairs

202

GRONER

are rare (see Section III.B.6), and (2) conformer ratios of parameters can be maintained.

2.

Vibration-rotation constants from a b

initio

calculations

In another application, the quadratic and cubic force constants are calculated by

ab initio

methods. From the force field, the

re

geometry

and the atomic masses, it is possible to calculate the vibrational frequencies and the vibration rotation constants aft (Eq. (2-4)). If they are sufficiently accurate, they can be used in place of the experimental aft constants to estimate the equilibrium constants Be"o from the observed constants in the vibrational ground state, Bg.

This

procedure has been followed by Bailleux et al. [67] for silaethene, CH2SiH 2.

They obtained the vibration-rotation constants by a

CCSD(T) calculation with the triple-zeta basis TZ2Pf. They corrected the experimental ground state constants for seven isotopomers (expressed in terms of Watson's determinable parameters) by the initio a f t

constants and obtained an

well with their rather sophisticated pV(Q,T)Z).

re

ab

structure that agreed remarkably

a b initio r e

structure (CCSD(T)/cc-

The differences were at most 0.0005 A for the bond

lengths and 0.10 ~ for the HCSi and HSiC angles.

This excellent

agreement is evidence that this level of calculation produces good

re

structures and that quadratic and cubic force fields at somewhat lower levels yield good vibration rotation constants.

QUEST FOR EQUILIBRIUM STRUCTURE

H.

203

Other methods

1.

rz

Structures Although this article deals primarily with experimental structures

approximating

re

structures, the

rz

or average structure in the

vibrational ground state is mentioned here for completeness. It is the structure calculated from the averaged positions of the nuclei. It is equivalent to the r O structure that can be obtained from electron diffraction data. The

rz

structural parameters are obtained from the

rotational constants Bzfl defined as dk

(59)

By = BoP+ Z a / ( h a r m ) T = B ] - Z a[(ar~) k

In practice,

rz

k

structures are obtained from spectroscopic data by

correcting the rotational constants Bg with aft(harm)calculated from a suitable harmonic force field.

The analysis of electron

diffraction data also requires a complete normal coordinate treatment to calculate the mean vibrational amplitudes from the harmonic force field. Therefore, the results of joint analyses of electron diffraction data and microwave data are often reported as Isotopic differences of

rz

rz

structures.

bond lengths, 5r, are usually estimated

by a formula derived by Kuchitsu [68,69] 5r = (3/2) a 5(u 2) - 5/s

(60)

204

GRONER

where (u 2) and K are the zero-point mean-square amplitudes of a given bond and its perpendicular amplitude (both obtained from the force

field),

and

a

is the

Morse

anharmonicity parameter,

approximated from the corresponding bond in the respective diatomic molecule. The r z bond length and the mean-square amplitudes are otten used to estimate the r e distance by Kuchitsu's equation [68,69] re = r z - ( 3 / 2 ) a ( u 2 ) + K

2.

.

(61)

Isolated C-H stretching frequencies McKean has demonstrated that there is an excellent correlation between isolated C-H stretching frequencies and r 0 C-H bond lengths [70] resulting in the relationship [71] ro(CH)/A = 1.3982 - 0.0001023 * vis(cH)/cm-1

.

(62)

The correlation has been corroborated by low level ab initio calculations (HF/4-21G) [71].

Although the theoretical (re) bond

lengths are systematically off, the variations of unequal C-H distances in similar environments agree with those determined by the bond length- frequency relationship (Eq. 62). Demaison and Wlodarczak have evaluated the correlation between experimental r e distances and vis(cH) and obtained [72] re(CH)/A = 1.3009(53)- 7.175(173)'10 -5 * vis(cH)/cm-1

form 26 data points.

(63)

They also pointed out correlations between

re(CH) distances and deuterium quadrupole coupling constants,

QUEST FOR EQUILIBRIUM STRUCTURE

205

rg(CH) distances from electron diffraction experiments, and NMR coupling constants j(13C-H) measured in liquid crystal solvents. However, the data sets for these other correlations are not very large nor of sufficient quality to allow them to be used for predictions. Nevertheless, they are useful indicators.

III.

M O L E C U L A R S T R U C T U R E S F R O M S P E C T R O S C O P Y SINCE 1980

A.

Introduction to Tables 2-6 Molecular structures determined since 1980 are listed in the following five tables. As a rule, only molecules are listed for which complete structures have been determined by rotationally resolved spectroscopy.

With a few

exceptions, structures from joint analyses of rotational and diffraction data are omitted. Likewise, no data for diatomic molecules and ions or Van der Waals complexes are included.

For most molecules, complete structures

have been determined for the first time. However, the structures of a number of molecules have been redetermined with more precise rotational constants, by different methods, from different isotopic species or from larger isotopic data sets. In some cases, particularly since about 1990, papers cited provide only the most recent isotopic data, which made it possible to determine a complete structure. In all tables, the species are arranged by the sum formula according to the Hill system (alphabetical order of element symbols except for species containing carbon in which case C followed by H precede the other elements).

The tables contain the sum formula, the structure formula, a

206

GRONER

name, the types of structures determined, the principal experimental method and the literature citation. Table 2 lists the information for molecular ions (without names). Table 3 contains the data for nonpolar molecules. Table 4 lists the information for molecules for which partial or complete structures have been determined for more than one conformation.

In addition, the

number of isotopic modifications studied for each conformer is given (molecules where only one isotopomer was used were excluded). Table 5 contains the data for radicals, carbenes and other exotic species (again without names). Table 6 is devoted to closed shell polar molecules. In the columns identifying the experimental method, "MW" stands for any method studying the pure rotational spectrum of a molecule except for rotational Raman spectroscopy marked by the "rot. Raman" entry. "FTIR" stands for Fourier transform infrared spectroscopy, "IR laser" for any infrared laser system (diode laser, difference frequency laser or other).

"LIF"

indicates laser induced fluorescence usually in the visible or ultraviolet region of the spectrum. "joint" marks a few selected cases where spectroscopic and diffraction data were used to determine the molecular structure. A method enclosed in parentheses means that the structure has been derived from data that were collected by this method in earlier publications.

The type of

structure determined is shown by the symbols identifying the various methods discussed in section II.

"rs" refers

to determinations using the

Kraitchman/Chutjian expressions or least squares methods fitting only isotopic differences of principal or planar moments (with or without first

QUEST FOR EQUILIBRIUM STRUCTURE

207

TABLE 2. Ions.

Sum formula

Structure formula

BFH +

HBF +

CBrN +

BrCN +

CC1N+

C1CN+

CHO + CHO +

Structure

Experimental method

Reference

MW

[73,74]

rs

LIF

[75]

rs

LIF

[76]

HCO +

MW

[77]

HOC +

MW

[78]

rs

MW

[79]

rs/r 0

MW

[8o]

MW

[81]

rs

IR laser

[82]

re*

IR laser

[83]

CHO2 +

HOCO +

CHS +

HCS +

CH2N+

HCNH +

CH 3+

CH 3+

r 0 rz

IR laser

[84]

C2HN2 +

HNCCN +

ro rs

MW

[85]

C2H2+

C2H2+

r o rz

IR laser

[86]

C4H2 +

HCCCCH +

rs

LIF

[87]

C12H+

H2C1+

re

IR laser

[88]

HN2 +

HN2 +

rs

MW

[89]

re

IR laser

[90]

HOSi +

HOSi +

r0

IR laser

[91,92]

H2F+

H2F+

r0

IR laser

[93]

H2 O+

H2 O+

r0

MW

[94]

H3 O+

H3 O+

r0

IR laser

[95]

H3 S+

SH3 +

re

IR laser

[96]

r o rz

MW

[97]

208

GRONER

TABLE 2. (continued).

Sum formula

Structure formula

Structure

Experimental method

Reference

H4N+

NH4 +

ro

IR laser

[98]

CI2H-

C1HCI-

r 0 re*

IR laser

[99l

F2H-

FHF-

re

IR laser

[100,101]

HO-

OH- (OD-)

re

IR laser

[102]

H2N-

N H 2-

r0

IR laser

[103]

TABLE 3. Nonpolar molecules.

Sum formula

Structure formula

Name

Structure

Exp. method Ref.

BH 3

BH 3

borane

r0

FT-IR

[104]

B2H6

B2H6

diborane

ro rs rz re*

FTIR

[105]

CF4

CF4

tetrafluoromethane

r0

IR laser

[106]

CO2

CO2

carbon dioxide

ro rm re

FTIR

[107]

CO2

CO2

carbon dioxide

rc

CSe2

CSe2

carbon diselenide

ro re

FTIR

[108]

C2C12

C2C12

dichloroacetylene

ro rs

FTIR

[109]

C2F2

C2F2

difluoroacetylene

ro re

FTIR

[110]

C2H2F2

CHFCHF

trans- 1,2-difluoroethylene

rslro

FTIR

[111 ]

C2H4

CH2CH2

ethylene

rz

MW

[112]

C2H4F2

CH2FCH2F trans- 1,2-difluoroethane

rslro

FTIR

[ 113]

C2H6

CH3CH 3

rz

MW

[114]

ethane

[41]

[115]

rmP

C2N2

NCCN

cyanogen

rs/ro

rot. Raman

[ 116]

C3H6

C3H6

cyclopropane

ro rz re*

MW

[117]

C4H6

C4H6

s-trans- 1,3-butadiene

r0(P)

MW

[118]

QUEST FOR EQUILIBRIUM STRUCTURE

209

TABLE 3. (continued).

Sum formula

Structure formula

Name

Structure

Exp. method Ref.

C4H8

C4H8

cyclobutane

r0

MW

[1 19,120,121]

C6H6

C6H6

benzene

r0

rot. Raman

[1 22]

r~(p)

MW

[123]

ro

FTIR

[124]

r m re*

FTIR

[125]

C6H12

C6H12

cyclohexane

rsr~

MW

[126]

C8H8

C8H8

cubane

ro

MW

[121]

F2Kr

KrF2

krypton difluoride

re

FTIR

[127]

F2Xe

XeF2

xenon difluoride

re

FTIR

[127,128]

F4Xe

XeF 4

xenon tetrafluoride

r0

FTIR

[129]

F5P

PF 5

phosphorus pentafluoride

r0

MW

[130]

F6S

SF6

sulfur hexafluoride

ro

IR laser

[131]

GeH4

GeH4

gerlTlane

re*

MW

[132]

H2N2

HNNH

trans-diazene

ro rz re*

FTIR

[133]

H4Si

Sill 4

silane

re*

MW

[134]

H4Si

Sill 4

silane

re*

MW

[132]

H4Sn

Sna 4

stannane

re*

MW

[132]

02

02

dioxygen

Fs Fm

MW

[135]

O3S

so3

sulfm" trioxide

ro re

MW

[136]

P4

P4

phosphorus

r0

rot. Raman

[137]

210

o o

o o

n~ 0

U~

r~

0

0

Z

L--~

~ ~ &

~

~

~ ~

U--~

~

~

2 =

0

0

0

~

=

~

0

~

~

0

r~ (N ..

L--~

L--~

~

L--~

GRONER

~

.,-

o

~

~

~~

o

:r162

~

.~0

0

rO

0 ~

0

~

Im ~

0

e',l

...

0

~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

e e ) ) e e e e

0

o ~

0 ..~

0

~ ~.-

0

o ~

0

rn

~~

(N

rO

r~

~

"-'

rO

r~

0

0

~ r ~

0

~

0 0

rO

n~ o

o

0

Z

~9

~

~

~

&

o -

-u

L_~

9

oo

~

~

o

0

o

~

~

~

9~

2

&&

~

~

o

N

o~

oo

r

~

~

&&

~

.,.~

..

~

.~

~

~&

o

-u

,_~

Z

r

~n

o~

0

o

oo

0

N

m

o.~

r

-~

&&&&&

L__a

QUEST FOR EQUILIBRIUM STRUCTURE

n~

o o

0

P....

N

0

o~

Z

o o o o ,.o

211

212

GRONER

TABLE 5. Radicals and carbenes.

Sum formula

Structure formula

Structure

Experimental method

Ref.

AsH2

AsH2

rz

MW

[159]

BaHO

BaOH

r0

LIF

[160]

CCaN

CaNC

rs

LIF

[161]

CC12

CCI2

r0

MW

[162]

CF3

CF3

r0

IR laser

[163]

CFeO

FeCO

rs

MW

[164]

CHTi

TiCH

r0

LIF

[165]

CH30

CH30

rs

MW

[166]

CMgN

MgNC

r0

MW

[167]

C2H

C2H

rs

MW

[168]

C2HN

HCCN

rs

MW

[169]

CEHESi

SiC2H2 (cyclic)

rs

MW

[170]

C3H

HCCC

rs

MW

[171]

C3H

C3H (cyclic)

rs

MW

[172]

C3HO

HCCCO

rors rmP rm(1) MW

C3H2

H2CCC

ro rs re*

MW

[173]

C3H2

C3H2 (cyclic)

rs

MW

[174]

C3N

CCCN

ro rs

MW

[175]

C30

C30

rs

MW

[176]

C3S

C3S

rs

MW

[177]

C4H

HCCCC

ro rs

MW

[175]

C4H2

H2CCCC

rors

MW

[178]

C5H5

C5H5

ro

LIF

[179]

C102

C102

ro rs re

MW/IR laser

[180]

C102

C102

re

MW

[181]

C1S2

CIS2

rs/ro

MW

[182]

C12Ge

GeC12

re

MW

[183]

C12Si

SiCI2

re

MW

[184]

C12Si

SiC12

re

[49]

[41]

QUEST FOR EQUILIBRIUM STRUCTURE

213

TABLE 5. (continued).

Experimental

Sum formula

Structure formula

Structure

F2P

PF2

r0

MW

[185]

H2Si2

SiH2Si

r0

MW

[186]

rs

MW

[1871

method

Ref.

H2Si2

SiHSiH

r0(P)

MW

[188]

NO2

NO 2

re

MW

[189]

O2P

PO 2

r0

MW

[190]

moment or product of inertia conditions).

Procedures where rotational

constants, moments of inertia or second moments were fitted are referred to as "r0".

"rs/rO"

refers to cases where the majority of the structural parameters

have been determined by the

rs

method but that a few parameters have been

obtained by the r 0 method or from second moment equations. In very few case, mostly in Table 3 (conformers), partially determined structures are indicated by "(p)". The structures marked by "re*" are from

rz

structures derived

structures by Kuchitsu's estimate (Section II.H) or by using at least

some vibration rotation constants (z obtained from

B.

re

ab initio

calculations.

Selected examples In this section, a few selected examples are used to demonstrate one or another of the newer methods. The first one (dimethyl sulfoxide) illustrates the advantage of using data from (extra) multisubstituted isotopomers. The

214

0

Ob

o

~

o

t___.a

= = =

t___a

t___J

0

~

~

..2o _,o

,.=

t__J

o

r

~

r

= = = = = =

~

"

~

0

r

=

=

t___a

=

t___J

=

t___a

m

=

~

I=I

~ I,,,~

m

=

~

o

~

o

o

=

L._J

I,.

t-----a

o

GRONER

o.

o

0

m

~

m

z

u

0

o

~

"

~~&z o

=

~

r..q

Z

1-1

o ~

O

~ o

~ O

1~

~ O

I~

~~~

O O O

11~

~~

.

~

~:~, ~

QUEST FOR EQUILIBRIUM STRUCTURE

O

O

~ .~

r.D r,D r,.D r,D r,D r,D r,D r.,D r,.D r,D r,D r,D ~

~

ll,ml

ll,-i ~

~

l.-i

~,~

~

w

O~~~~

O

"~

r,D r.,D r,.D r.,D r,D r,D ~

215

216

o o

r,r.l

[-.

0

Z

o

o

~

r

~ ~ ~ "

o~ ~

o

~ ~ "~

~

~

N-~ ~ ~ ~.~

N

~

~~

N

~

O

N

O

"

~

~

-

Z

o

~ Z

o

o

~ . - r.~ r.D ~

r~

GRONER

MO

r,.D

_

o~~ ~= o~~N~

~ff ~~

~

~ ~ ~ ~ ~

,.~ b~ ~

r..D r,D r,D r,D r,D r,D r..D r,D r,D r,D r,D U

~

O

r,D

~~~ Z

r..D r..D U

~~,., U

r

0

Z

~

"~

-~

0

~

e'q

~

o

"~"

~

~ ~

e'q

~ ~

O

~

0 0 ~ U ~

0

r

~

~ ~

~

r ~

~

e'q

r

~

.~

~

~

C',l ~

r

r i~1

e'q L~

,..,

e'q

~

e'q L~J

e'q

.? .~

~

~

r~ r~

~'~'~

0

N

N

~

N

~

r~ N

~

r~ ~ ~

0 ~

~0

~

~ ~

~

~

~

QUEST FOR EQUILIBRIUM STRUCTURE

,.-t,

o o

0

e'q ~

~

C',,I

~

L___J

e'q

f~ "~ 0 o - -

r~

~

L-.----I

rj

r~

e'q

f~ o

r

~

~

r~

~

0,1

r

~

0

r

~

rj ~

~

~

e',,I L.__J

" 0

-

r

217

218

o~ 0 0

e'q

9~ ~

~

~-B~_

ro

Oq

C~

_~

F~ F~

ro

ro

~

~ =

eXl

~

.9~

... - ~

~, ~

ro

ro

0,1

ro

C'q

C'q

Oq

t"q

o

~

~ o o -,~

o _ _

~

~, o

ro

C~

o

e'q

~

0,1

~

ff

OR

rO ro

0,1

~, . . . .

0,1

ro

eq

~,

-~

r

ro

ffffffffffffffffffffffffffff

ro

Oq

e'q

~

C'q

e'q

~

~~ _~.

0,1

~

C'q

.-~

Ol

rO ro

GRONER

Oq

ro

0,1

-~

0,1

~

~

O,t

ro

_'~

ffff

0,1

ro

~

O,,I

ro

r

Z

0

(",1 ~0 ~0 i'M t",l I____.a ~

~ ~0 ~P ' I

~ ~0

I"~

~

~0 ~0 ~,1 P L-----I ~,I

~

~

~

~0 ~0 ~ ~ ~ I--------I i-------I ~P q

~

(:~

~

O.

I~ P',I L---J

1~

O.

o~,,,1

t"'. P',I ~

~,~

O.

0

0 0

~0

~

r,.)

~

~.

~z

r,.)

,~

~

r,.)

r,.)

r,.)

r,.)

~

,~

z~ o ~ ~ ~ z

z

r,.)

r

0 r,D r,D

zz

~ -':,,~" .~.~- ~. ~ ~' ~ .~ ~

0

0

tCh

~

O.

r,D r,.)

o~

~

t "~ I" ~ I"-. t". ~0 ~1 ~ ~ ~ L----I L----J L----J L--...I ~' ~ "

~

~

r,D ~

r,.)

~ ~.~.

t'. t". I~ r t1______1 'N t N_ J I___...a L

~ ~

~

o

~

~

r,.)

o ~ ~ o ~

.~ ~

t"t",l L__.I

QUEST FOR EQUILIBRIUM STRUCTURE

,,--k ,'1::1

o e~

0

r,D r,.)

219

220

,,.4,

o~.~ 0 0

.1

o

0

Z

~o

ff

N

exl

~

e'q

e'q

e~l

e'q

e'q

e'q

0,1

~

"~ ~

e~

~

e~

e~

e~

e~

"

r~

~ ~

o~

~,

~

o ~ ~ SS SS ~

~o

~

~

rj

=

r...)

~ ~

= =

~

rj

GRONER

e~

"~

0

=

r~ 0 SS

r,.) r~

0 ~

=

~

e'q

o

r...)

0 N

r~

~

~

~~

r~

0 ~

ffffffffff=

rj

ff

0 ~

r...p r j

rj

~

rj

rj

o0 m ~ ~ 0 ~ ~ SS N ~ r..p ~

~

0 N ~

0 ~

r..P r@ rJ

O

Z

~

-

t" ~

~

~'~

~

~

~

~ a

-~

~ U ~ ~ -~~. ~~

_~.~.~

~

0

Z~ ~_

o~

~

eq

"'~

~1

C

~

~

~

0~0~0 ~~

e~

QUEST FOR EQUILIBRIUM STRUCTURE

,..a,

~162

[,..

~

~ e~

~

~

~

"~"

~,~

t~

~

~ ~162

.~-~ ~,~~ , . y,~ m

~ ~ ~~ ~~ Z

~~~Z

~',

~

~

-

221

222

0

.1 [..

o

o

Z

~

~

~

_ _ - ~

~

41,

~

~

"~ "~

~, o

...~ ..~ ..~

~=

_o

~=

~'~

~

e~

.........

~

~

~

0 0 0 0 0 0 0 0 0 ~

ooo~~~~o~ooo8~~

0 0 0 ~

,,,~

.~

~,

GRONER

.~'~

~0

~~~ 0

o

Z

o

~

~

~::~

~=~

r~

~

~

~

0e,I 0eq ~~ ~e,I ~~'N ~~ ~m ~m 0e,I 0eq 0~ I::I::I I::I::I I:::I::I I::I::I I::I:::I ~ l:X::l I::I::I Z Z Z

~

QUEST FOR EQUILIBRIUM STRUCTURE

n~

o o

~

~

,N

0

o 0

r~

0

~

~____o n=~

0

m

0

m

~ ~ ~r ~ 0 0 0

223

224

GRONER

second example (hydrogen pseudohalides) shows how the use of the substitution moments of inertia (instead of the ground state moments) and the comparison of molecules with similar structures help to overcome the "small coordinate problem".

The next example demonstrates the difficulty of

applying some of the newer methods (rmP and

rm(1)) in the problematic case

of the HC30 radical and the big effect of the H/D correction in the rmP method. In the case of vinyl alcohol and vinyl fluoride, it is shown how lower level ab initio calculations can help to locate a problem in the determination of molecular structure.

The fluoroacetylene example

demonstrates that the determination of accurate vibration-rotation parameters is a formidable challenge and that high level ab initio calculations can help to detect unknown resonances among the vibrational levels. The last section addresses the question of structural differences between conformers of the same molecule.

1.

Dimethyl sulfoxide Dimethyl sulfoxide, (CH3)2SO , represents a case where the single substitution method resulted in a very asymmetric methyl group [348] although the H/D substitution does not take place in a symmetry plane [29]. Typke recently reanalyzed the same data by also considering the data of isotopomers in which one or both methyl groups had been deuterated [272]. He found that, in addition to 4 overdetermined sets, 10 different "minimal" sets of isotopic data could be constructed from which a complete substitution structure could be determined. For each

QUEST FOR EQUILIBRIUM STRUCTURE

set, he determined the

rs

225

coordinates by a least-squares fit and then

calculated the rotational constants of all isotopomers for which data were available. All data sets that contained the rotational constants of one particular monodeuterated species gave significantly worse predictions for the remaining isotopic species. He could not determine the cause of the abnormal results particularly since the assignment and the fitting of the spectrum of that one isotopomer had been carefully double-checked [349].

2.

The hydrogen pseudohalides HNCO, HNCS and HN 3 The b coordinates of all three non-hydrogen atoms were imaginary when the Kraitchman equations were used to locate the position of the nuclei in isothiocyanic acid, HNCS [218].

To

determine a reasonable structure for this planar molecule, the authors used the first momem condition for the b coordinates and the product of inertia condition. The second moment condition was used also but with the substitution moment of inertia moment I0. reliable

rs

Is

instead of the ground state

The authors argued that this procedure gives a more

structure instead of a

rs/r 0

hybrid, particularly since Watson

[36] demonstrated the physical significance of the substitution momem Is.

However, it can only be applied if two or more atoms have to be

located by means of auxiliary conditions. The structure of isocyanic acid, HNCO, was determined by the same method, except that the first moment condition had to be invoked also for the a coordinates [217].

226

GRONER

The same procedure was not applicable for hydrazoic acid, HN3, because of missing data for HN15NN [334]. Because the Kraitchman values for the b coordinate of the N atom in both HNCS [218] and HNCO [217] agree very well with the results obtained by the procedure described above, it was assumed that the b coordinate for N 1 (bound to H) in HN 3 can be obtained with sufficient accuracy from the Kraitchman equations, particularly since the magnitude for the coordinate is even larger in HN 3 [334]. The resulting slightly transbent r s structures for the three molecules agree nicely with each other, with N-H distances between 0.993 and 1.015 A and angles at the central atom of the near linear chain between 171.3 and 173.8 ~

3.

HC30 radical The structure of the HC30 radical has been determined by Cooksy et al. [49] by the r O, r s (more precisely: rIe), rmP and rm(1) methods from the data of the parent molecule and of all singly substituted species. The authors concluded that the rmP method was not suitable for this molecule but they did not include the recommended correction for the H/D substitution [45]. Harmony included the correction in a new rmP analysis for the preferred "a" form but excluded the illdetermined rotational constants A of the isotopic modifications [350]. The resulting structure is shown in Table 7 together with the

QUEST FOR EQUILIBRIUM STRUCTURE

227

TABLE 7. Structure of the HCCCO radical ("a" shape) a. r(HC l)

r(C 1C2)

r(C2C3)

r(C3O)

0(HCC)

0(CCC)

0(CCO)

rmp b

1.052(18) 1.230(36) 1.363(36) 1.220(21) 172(1)

165(1)

136.5(6)

rm(1) b

1.060(18) 1.219(3)

1.387(5)

1.192(2)

168(7)

163(2)

136.5(6)

rmp c

1.055

1.222

1.376

1.186

179.0

175.0

140.6

ab initio d

1.061

1.190

1.447

1.165

179.1

175.0

129.8

aDistances r in A, angles 0 in degrees. bRef. [49]. cWith the H/D correction [350]. dMP2/6-311G(2p,2d) [49].

results by Cooksy et al. [49]. The bond distances are much closer to the rm(1) structure than the original rmO analysis but the bond angles are not. Two of the rmP bond angles are very close to values predicted by the ab initio calculation reported in ref. [49].

According to

Harmony [350], the nearly linear structure of the HCCC part of the molecule almost guarantees low frequency anharmonic bending motions, which are troublesome in the rm9 method, particularly if H atoms are involved. Therefore, the error limits on the heavy atom rmP distances could be as large as 0.005 A instead of the more common 0.001 to 0.002 A.

4.

Vinyl alcohol and vinyl fluoride A complete r s structure of syn-vinyl alcohol has been published by Rodler and Bauder [263].

The authors noted the difficulties of

determining an accurate structure because of the close proximity of the

228

GRONER

central C atom and the attached H atom to the b principal axis. They also noted a similar difficulty in the case of vinyl fluoride.

Their

remark about the surprisingly large discrepancy between the experimental CCH angle and earlier ab initio predictions prompted Radom and coworkers [34,351,352] to undertake a systematic ab initio study of substituted ethenes. For a number of related molecules, they compared experimental r 0 structures with ab initio predictions for different basis sets and levels of electron correlation and established empirical correction terms. These predicted r 0 structures agreed well with the experimental structures for vinyl chloride and bromide [352] but not for vinyl fluoride and vinyl alcohol.

Because the many

experimental structures proposed for vinyl fluoride differed sometimes substantially from each other, they also redetermined the experimental structures exploring different methods (r 0, r 1, r B, rAl, rAB ) and weighting schemes.

They found sometimes very large differences

between the methods and weighting schemes for vinyl fluoride and alcohol [351 ] but not for vinyl chloride or bromide. They concluded that the predicted r 0 structures agree with the experiment for vinyl chloride and bromide and that they are the best estimates of the r 0 structure for vinyl fluoride and alcohol.

The problem with the

structure of the latter two molecules is caused entirely by the two atoms close to the b axis.

QUEST FOR EQUILIBRIUM STRUCTURE

5.

229

Fluoroacetylene Borro and Mills [244] have determined the equilibrium structure of fluoroacetylene using experimental B e constants for HCCF and DCCF. B e constants for the 13C substituted species were estimated by scaling the differences Be-B 0 with the factor obtained from ab initio calculations [353].

While the CC and CF distances agreed within

0.0006 and 0.0016 A, respectively, with the ab initio based predictions [353], the CH distance was 0.0036 A shorter. Botschwina and Seeger [245] pointed out that this discrepancy arose from an incorrect value for the vibration-rotation constant ct 1 for the HCCF isotopomer (the difference tXl(theor)-c~l(exp) was only 515 % for DCCF but 31.2 % for HCCF). They also referred to the relation between the C-H stretching frequency and the equilibrium bond length [72] to support their value of 1.0591 A. Borro and Mills discovered ([244] note added in proof) that a quartic term in the potential not considered before brings the vibration-rotation constant a l and therefore the equilibrium structure more in line with the ab initio predictions.

This incidence

demonstrates that determining sufficiently accurate vibration rotation constants is not an easy job and that good ab initio predictions are able to provide valuable support in this endeavor.

230

GRONER

6.

Conformers It has been recognized some time ago [354,355] that the rotational constants of different conformers of one molecule sometimes can only be explained if structural parameters other than torsional dihedral angles have significant conformational differences. In these and other early studies [356], only a limited number of isotopomers have been studied and definitive confirmation of such differences has been published only in the last two decades.

It seems that Hirota's

investigation of 3-fluoropropene [357] was the first almost complete study of the structures Unfortunately, complete

of both conformers rs

of a molecule.

or r 0 structural parameters for both

conformers are available only for a relatively small number of molecules ethanethiol,

such

as

3-fluoropropene,

CH3CH2SH

[358-360],

CH2CHCH2F ethaneselenol,

[148,357],

CH3CH2SeH

[142], ethylmethyl sulfide, CH3CH2SCH 3 [152,153], 1-fluoropropane, CH3CH2CH2F [150], ethylphosphine, CH3CH2PH 2 [146], and 1,2difluoroethane, FCH2CH2F [113,139]. In other cases, the structure is complete

for

only

one

conformer

as

for

ethylfluorosilane,

CH3CH2SiH2F [143,150], thiohydroxylamine, NH2SH [158], sulfur diimide, HNSNH [ 156], propanal, CH3CH2CHO [ 149], and hydrogen trisulfide, HSSSH [157]. Most often however, only partial structures have been determined for both conformations as for difluoroacetic acid, CH2FCOOH [138], 2-propaneselenol, (CH3)2CHSeH [154],

231

QUEST FOR EQUILIBRIUM STRUCTURE

ethanamine,

CH3CH2NH 2

2-methylpropanal,

[144,145],

(CH3)2CHCHO [155], ethylphosphonic difluoride, CH3CH2PSF 2 [141 ], isopropyl difluorophosphine, (CH3)2CHPF2 [151 ], and ethyl difluorophosphine, CH3CH2PF 2 [140]. Two papers have compared structural parameters between different conformers of a number of molecules [150,361].

The most significant differences so far have

been associated with skeletal bond angles. It seems that 1,4- or 1,5interactions are responsible for sizable structural changes.

IV.

CONCLUSION The quest for the equilibrium structures of molecules has been advanced significantly by new methods. The rlc and related methods offer a more versatile and better balanced way to determine structures equivalent to the traditional structure.

The

rmP

rs

method provides a somewhat better modeling of vibration-

rotation interaction; it can however only be used if all nuclei can be substituted by isotopes. The

rc

method seems to give the best results but its applications are

restricted to relatively small molecules.

The

rm(1)/rm(2) method looks very

promising; because it is a new method, only few applications have been published. Ab initio

methods have become extremely useful to structural research. The

CCSD(T)/cc-pV(Q,T)Z calculations seem to give very good molecules.

re

structures for small

If the budget allows it, it is easier to obtain sufficiently accurate

vibration-rotation constants from force fields calculated by the CCSD(T)/TZ2Pf method than by the traditional method of analyzing all vibrational fundamentals of

232

GRONER

numerous isotopomers.

Even lower level calculations (MP2/6-31G(d,p)) are

sufficiently reliable to establish trends and correlations and to estimate corrections for the structural parameters. New experimental techniques have increased the capacity of rotationally resolved spectroscopy tremendously. They are generating experimental data at an unprecedented rate and with excellent precision.

As a result of all these

developments, we know a lot more with much greater precision and accuracy about many more molecules. This includes equilibrium structures of polyatomic molecules.

ACKNOWLEDGEMENT The author wishes to thank Dr. M. D. Harmony for valuable comments and the permission to use unpublished data and to Dr. J. K. G. Watson for sending me the manuscript on the

rm(1) and rm(2) methods prior to publication.

QUEST FOR EQUILIBRIUM STRUCTURE

233

REFERENCES

V. S. Mastryukov, in Vibrational Spectra and Structure, (J. R. Durig, Ed.) Vol. 24, Amsterdam: Elsevier, 1999. ~

~

~

5. ~

0

~

9.

E. Hirota, in Molecular Spectroscopy: Modern Research, (K. N. Rao, Ed.), Vol. III., Orlando: Academic Press, 1985, A. Bauder, in Vibrational Spectra and Structure, (J. R. Durig, Ed.), Vol. 20, Amsterdam: Elsevier, 1993. H. Dreizler, U. Andresen, J.-U. Grabow, D. H. SuRer, Z. Naturforsch., 53A, 887 (1998). H. Dreizler, Ber. Bunsenges. Phys. Chem., 99, 1451 (1995). T. A. Miller, V. E. Bondybey, (Eds.), Molecular Ions: Spectroscopy, Structure and Chemistry, Amsterdam: North-Holland, 1983. M. D. Harmony, V. W. Laurie, R. L. Kuczkowski, R. H. Schwendeman, D. A. Ramsay, F. J. Lovas, W. J. Lafferty, A. G. Maki, J. Phys. Chem. Ref. Data, 8, 619 (1979). L. Nemes, in Vibrational Spectra and Structure, (J. R. Durig, Ed.), Vol. 13, Amsterdam: Elsevier, 1984. J. Demaison, G. Wlodarczak, H. D. Rudolph, Adv. Mol. Struct. Res., 3, 1 (1997).

10. W. Gordy, R. L. Cook, Microwave Molecular Spectra, New York: Wiley, 1984. 11. M. R. Aliev, J. K. G. Watson, in Molecular Spectroscopy: Modern Research, (K. N. Rao, Ed.), Vol. III, Orlando: Academic Press, 1985. 12. I. M. Mills, in Molecular Spectroscopy: Modern Research, (K. N. Rao, C. W. Mathews, Eds.), Vol. I, New York: Academic Press, 1972. 13. D. Papou~ek, M. R. Aliev, Molecular Vibrational-Rotational Spectra, Amsterdam: Elsevier, 1982. 14. J. K. G. Watson, in Vibrational Spectra and Structure, (J. R. Durig, Ed.), Vol. 6, Amsterdam: Elsevier, 1977.

234

GRONER

15. J. Kraitchman, Am. J. Phys., 21, 17 (1953). 16. A. Chutjian, J. Mol. Spectrosc., 14, 361 (1964). 17. L. Nygaard, J. Mol. Spectrosc., 62, 292 (1976). 18. H.D. Rudolph, J. Mol. Spectrosc., 89, 460 (1981). 19. H.D. Rudolph, J. Mol. Spectrosc., 89, 430 (1981). 20.

Y.S. Li, K. L. Kizer, J. R. Durig, J. Mol. Spectrosc., 42, 430 (1972).

21.

E.B. Wilson, Z. Smith, J. Mol. Spectrosc., 87, 569 (1981).

22.

L. Pierce, J. Mol. Spectrosc., 3, 575 (1959).

23.

L.C. Krisher, L. Pierce, J. Chem. Phys., 32, 1619 (1960).

24.

R.E. Penn, L. W. Buxton, J. Chem. Phys., 67, 831 (1977).

25.

C. Kirby, H. W. Kroto, J. Mol. Spectrosc., 83, 130 (1980).

26.

C.C. Costain, J. Chem. Phys., 29, 864 (1958).

27.

C.C. Costain, Trans. Am. Cryst. Assoc., 2, 157 (1966).

28.

R. H. Schwendeman, in Critical Evaluation of Chemical and Physical Structural Information, (D. R. Lide, M. A. Paul, Eds.), Washington, D.C.: National Academy of Sciences, 1974.

29.

B.P. van Eijck, J. Mol. Spectrosc., 91, 348 (1982).

30.

D. L. Albritton, A. L. Schmeltekopf, R. N. Zare, in Molecular Spectroscopy: Modern Research, (K. N. Rao, Ed.), Vol. II, New York: Academic Press, 1976.

31.

P. N/Ssberger, A. Bauder, H. H. Gtinthard, Chem. Phys., 1, 418 (1973).

32.

V. Typke, J. Mol. Spectrosc., 69, 173 (1978).

33.

H.D. Rudolph, Struct. Chem., 2, 581 (1991).

34.

B.J. Smith, D. Coffey, L. Radom, J. Chem. Phys., 97, 6113 (1992).

QUEST FOR EQUILIBRIUM STRUCTURE

35.

H.D. Rudolph, Adv. Mol. Struct. Res., 1, 63 (1995).

36.

J.K.G. Watson, J. Mol. Spectrose., 48, 479 (1973).

235

37. J.G. Smith, J. K. G. Watson, J. Mol. Speetrose., 69, 47 (1978). 38.

M. Nakata, M. Sugie, H. Takeo, C. Matsumura, T. Fukuyama, K. Kuchitsu, J. Mol. Spectrose., 86, 214 (1981).

39. M. Nakata, T. Fukuyama, K. Kuchitsu, H. Takeo, C. Matsumura, J. Mol. Speetrose., 83, 118 (1980). 40.

M. Sugie, M. Ayabe, H. Takeo, C. Matsumura, J. Mol. Struct., 352-353, 259 (1995).

41.

M. Nakata, K. Kuchitsu, J. Mol. Struct., 320, 179 (1994).

42.

M.D. Harmony, W. H. Taylor, J. Mol. Spectrosc., 118, 163 (1986).

43.

M.D. Harmony, R. J. Berry, W. H. Taylor, J. Mol. Spectrosc., 127, 324 (1988).

44.

R.J. Berry, M. D. Harmony, J. Mol. Spectrosc., 128, 176 (1988).

45.

R.J. Berry, M. D. Harmony, Struct. Chem., 1, 49 (1990).

46.

H.S. Tam. J.-I. Choe, M. D. Harmony, J. Phys. Chem., 95, 9267 (1991).

47.

M. D. Harmony, in Vibrational Spectra and Structure, (J. R. Durig, Ed.) Vol. xx, Amsterdam: Elsevier, 1999.

48.

J. Demaison, G. Wlodarczak, H. Rtick, K. H. Wiedenmann, H. D. Rudolph, J. Mol. Struct., 3 76, 399 (1996).

49.

A. L. Cooksy, J. K. G. Watson, C. A. Gottlieb, P. Thaddeus, J. Chem. Phys., 101, 178 (1994).

50.

J. Demaison, L. Nemes, J. Mol. Struet., 55, 295 (1979).

51.

K. Burczyk, H. BUrger, M. Le Guennec, G. Wlodarczak, J. Demaison, J. Mol. Speetrose., 148, 65 ( 1991).

236

GRONER

52. M. Le Guennec, W. D. Chen, G. Wlodarczak, J. Demaison, R. Eujen, H. Btirger, J. Mol. Spectrose., 150, 493 (1991). 53.

M. Le Guennec, G. Wlodarczak, J. Burie, J. Demaison, J. Mol. Speetrosc., 154, 305 (1992).

54. A.R.W. McKellar, J. K. G. Watson, B. J. Howard, Mol. Phys., 86, 273 (1995). 55. J.K.G. Watson, A. Roytburg, W. Ulrich, J. Mol. Spectrosc., 196, 102 (1999). 56. J.M.L. Martin, P. R. Taylor, Spectrochim. Acta, 53,4, 1039 (1997). 57.

K.A. Petersen, Speetrochim. Aeta, 53.4, 1051 (1997).

58.

C.E. Dateo, T. J. Lee, Speetroehim. Aeta, 53,4, 1065 (1997).

59.

P. Botschwina, A. Heyl, M. Oswald, T. Hirano, Speetroehim. Aeta, 53,4, 1079 (1997).

60.

J. Koput, S. Carter, Speetroehim. Aeta, 53.4, 1091 (1997).

61.

A.G. Csfiszhr, I. M. Mills, Speetroehim. Aeta, 53A, 1101 (1997).

62.

C. Puzzarini, R. Tarroni, P. Palmieri, Spectrochim. Acta, 53,4, 1123 (1997).

63.

P. Palmieri, C. Puzzarini, R. Tarroni, A. O. Mitrushenkov, Speetrochim. Acta, 53.4, 1139 (1997).

64.

M. Le Guennec, J. Demaison, G. Wlodarczak, C. J. Marsden, J. Mol. Speetrose., 160, 471 (1993).

65.

J. Demaison, J. Cosldou, R. Bocquet, A. G. Lesarri, J. Mol. Speetrose., 167, 400 (1994).

66.

B. J. van der Veken, W. A. Herrebout, D. T. Durig, W. Zhao, J. R. Durig, J. Phys. Chem. A, 103, 1976 (1999).

67.

S. Bailleux, M. Bogey, J. Demaison, H. Btirger, M. Senzlober, J. Breidung, W. Thiel, R. Fajgar, J. Pola, J. Chem. Phys., 106, 10016 (1997).

68.

K. Kuchitsu, J. Chem. Phys., 44, 906 (1966).

QUEST FOR EQUILIBRIUM STRUCTURE

237

69.

K. Kuchitsu, J. Chem. Phys., 49, 4456 (1968).

70.

D.C. McKean, Chem. Soc. Rev., 7, 399 (1978).

71.

D.C. McKean, J. E. Boggs, L. Sch~ifer, J. Mol. Struet., 116, 313 (1984).

72.

J. Demaison, G. Wlodarczak, Struet. Chem., 5, 57 (1994).

73.

S. Saito, S. Yamamoto, K. Kawaguchi, J. Chem. Phys., 86, 2597 (1987).

74.

G. Cazzoli, C. D. Esposti, L. Dore, P. G. Favero, J. Mol. Speetrose., 121, 278 (1987).

75.

M. R6sslein, M. A. Hanratty, J. P. Maier, Mol. Phys., 68, 823 (1989).

76. M. RSsslein, J. P. Maier, J. Phys. Chem., 93, 7342 (1989). 77.

R. C. Woods, R. J. Saykally, T. G. Anderson, T. A. Dixon, P. G. Szanto, J. Chem. Phys., 75, 4256 (1981).

78.

C.S. Gudeman, R. C. Woods, Phys. Rev. Lett., 48, 1344 (1982).

79.

M. Bogey, C. Demuynck, J. L. Destombes, J. Mol. Spectrose., 115, 229 (1986).

80. M. Bogey, C. Demuynck, J. L. Destombes, A. Krupnov, J. Mol. Struet., 190, 465 (1988). 81.

J. Tang, S. Saito, Astrophys. J., 451, L93 (1995).

82.

T. Amano, K. Tanaka, J. Mol. Speetrose., 116, 112 (1986).

83. D.-J. Liu, S.-T. Lee, T. Oka, J. Mol. Speetrose., 128, 236 (1988). 84. M. W. Crofton, M.-F. Jagod, B. D. Rehfuss, W. A. Kreiner, T. Oka, J. Chem. Phys., 88, 666 (1988). 85.

G. Cazzoli, C. D. Esposti, F. Scappini, J. Chem. Phys., 97, 6187 (1992).

86. M.-F. Jagod, M. R6sslein, C. M. Gabrys, B. D. Rehfuss, F. Scappini, M. W. Croton, T. Oka, J. Chem. Phys., 97, 7111 (1992). 87.

J. Lecoultre, J. P. Maier, M. R6sslein, J. Chem. Phys., 89, 6081 (1988).

238

88.

GRONER

S.K. Lee, T. Amano, K. Kawaguchi, M. Oldani, J. Mol. Speetrose., 130, 1 (1988).

89. P. G. Szanto, T. G. Anderson, R. J. Saykally, N. D. Piltch, T. A. Dixon, R. C. Woods, J. Chem. Phys., 75, 4261 (1981). 90.

J. C. Owrutsky, C. S. Gudeman, C. C. Martner, L. M. Tack, N. H. Rosenbaum, R. J. Saykally, J. Chem. Phys., 84, 605 (1986).

91.

H.E. Warner, A. Fox, T. Amano, D. K. Bohme, J. Chem. Phys., 91, 5310(1989).

92. N. Moazzen-Ahmadi, A. R. W. McKellar, H. E. Warner, T. Amano, J. Chem. Phys., 91, 5313 (1989). 93. E. Sch~ifer, R. J. Saykally, J. Chem. Phys., 81, 4189 (1984). 94.

S.E. Strahan, R. P. Mueller, R. J. Saykally, J. Chem. Phys., 85, 1252 (1986).

95.

M.H. Begemann, R. J. Saykally, J. Chem. Phys., 82, 3570 (1985).

96.

T. Nakanaga, T. Amano, J. Mol. Spectrose., 133, 201 (1989).

97. M. Araki, H. Ozeki, S. Saito, J. Mol. Spectrose., 192, 228 (1998). 98.

M.W. Croflon, T. Oka, J. Chem. Phys., 86, 5983 (1987).

99.

K. Kawaguchi, d. Chem. Phys., 88, 4186 (1988).

100. K. Kawaguchi, E. Hirota, J. Chem. Phys., 87, 6838 (1987). 101. K. Kawaguchi, E. Hirota, J. Mol. Struet., 352-353, 389 (1995). 102. B. D. Rehfuss, M. W. Crotton, T. Oka, J. Chem. Phys., 85, 1785 (1986). 103. L. M. Tack, N. H. Rosenbaum, J. C. Owrutsky, R. J. Saykally, J. Chem. Phys., 85, 4222 (1986). 104. K. Kawaguchi, J. Chem. Phys., 96, 3411 (1992). 105. J. L. Duncan, J. Harper, Mol. Phys., 51,371 (1984). 106. C. W. Patterson, R. S. McDowell, N. G. Nereson, R. F. Begley, H. W. Galbraith, B. J. Krohn, J. Mol. Speetrose., 80, 71 (1980).

QUEST FOR EQUILIBRIUM STRUCTURE

239

107. G. Graner, C. Rossetti, D. Bailly, Mol. Phys., 58, 627 (1986). 108. H. BUrger, H. Willner, J. Mol. Speetrose., 128, 221 (1988). 109. D. McNaughton, Struct. Chem., 3, 245 (1992). 110. H. Btirger, M. Senzlober, S. Sommer, J. Mol. Spectrosc., 164, 570 (1994). 111. N. C. Craig, O. P. Abiog, B. Hu, S. C. Stone, W. J. Lafferty, L.-H. Xu, J. Phys. Chem., 100, 5310 (1996). 112. E. Hirota, Y. Endo, S. Saito, K. Yoshida, I. Yamaguchi, K. Machida, J. Mol. Spectrosc., 89, 223 (1981). 113. N. C. Craig, A. Chen, K. H. Suh, S. Klee, G. C. Mellau, B. P. Winnewisser, M. Winnewisser, J. Phys. Chem. A, 101, 9302 (1997). 114. E. Hirota, Y. Endo, S. Saito, J. L. Duncan, J. Mol. Spectrose., 89, 285 (1981). 115. M. D. Harmony, J. Chem. Phys., 93, 7522 (1990). 116. H. G. M. Edwards, H. R. Mansour, J. Mol. Struct., 160, 209 (1987). 117. Y. Endo, M. C. Chang, E. Hirota, J. Mol. Spectrosc., 126, 63 (1987). 118. W. Caminati, G. Grassi, A. Bauder, Chem. Phys. Lett., 148, 13 (1988). 119. B. Vogelsanger, W. Caminati, A. Bauder, Chem. Phys. Lett., 141,245 (1987). 120. W. Caminati, B. Vogelsanger, R. Meyer, G. Grassi, A. Bauder, J. Mol. Spectrosc., 131, 172 (1988). 121. E. Hirota, Y. Endo, M. Fujitake, E. W. Della, P. E. Pigou, J. S. Chickos, J. Mol. Struct., 190, 235 (1988). 122. H. G. M. Edwards, J. Mol. Struct., 161, 23 (1987). 123. M. Oldani, R. Widmer, G. Grassi, A. Bauder, J. Mol. Struct., 190, 31 (1988). 124. J. Pliva, J. W. C. Johns, L. Goodman, J. Mol. Spectrosc., 140, 214 (1990). 125. J. Pliva, J. W. C. Johns, L. Goodman, J. Mol. Spectrosc., 148, 427 (1991).

240

GRONER

126. J. Dommen, T. Brupbacher, G. Grassi, A. Bauder, J. Am. Chem. Soe., 112, 953 (1990). 127. H. Btirger, R. Kuna, S. Ma, J. Breidung, W. Thiel, J. Chem. Phys., 101, 1 (1994). 128. H. Biirger, S. Ma, J. Mol. Speetrosc., 157, 536 (1993). 129. H. Btirger, S. Ma, J. Breidung, W. Thiel, J. Chem. Phys., 104, 4945 (1996). 130. C. Styger, A. Bauder, J. Mol. Speetrose., 148, 479 (1991). 131. B. Bobin, J. C. Bord6, J. Bord6, C. Br6ant, J. Mol. Speetrose., 121, 91 (1987). 132. K. Ohno, H. Matsuura, Y. Endo, E. Hirota, J. Mol. Speetrose., 118, 1 (1986). 133. J. Demaison, F. Hegelund, H. BUrger, J. Mol. Struet., 413-414, 447 (1997). 134. K. Ohno, H. Matsuura, Y. Endo, E. Hirota, J. Mol. Speetrose., 111, 73 (1985). 135. G. Cazzoli, C. D. Esposti, P. G. Favero, G. Severi, Nuovo Cim. Soe. Ital. Phys. B, 62B, 243 (1981). 136. V. Meyer, D. H. SuRer, H. Dreizler, Z. Naturforsch., 46A, 710 (1991). 137. N. J. Brassington, H. G. M. Edwards, D. A. Long, J. Raman Spectrose., 11, 346 (1981). 138. B. P. van Eijck, A. A. J. Maagdenberg, G. Janssen, T. J. van Goethem-Wiersma, J. Mol. Speetrose., 98, 282 (1983). 139. H. Takeo, C. Matsumura, Y. Morino, J. Chem. Phys., 84, 4205 (1986). 140. J. R. Durig, B. Galabov, R. D. Johnson, P. Groner, J. Mol. Struet., 477, 241 (1999). 141. J. R. Durig, R. D. Johnson, H. Nanaie, T. J. Hizer, J. Chem. Phys., 88, 7317 (1988). 142. J. Nakagawa, H. Okutani, M. Hayashi, J. Mol. Spectrosc., 94, 410 (1982). 143. M. Hayashi, M. Imachi, M. Oyamada, J. Mol. Struct., 74, 97 (1981). 144. E. Fischer, I. Botskor, J. Mol. Speetrosc., 91, 116 (1982).

QUEST FOR EQUILIBRIUM STRUCTURE

241

145. E. Fischer, I. Botskor, J. Mol. Speetrose., 104, 226 (1984). 146. P. Groner, R. D. Johnson, J. R. Durig, J. Chem. Phys., 88, 3456 (1988). 147. K. Yamanouchi, M. Sugie, H. Takeo, C. Matsumura, M. Nakata, T. Nakata, K. Kuchitsu, J. Phys. Chem., 91, 823 (1987). 148. J. R. Durig, M. Zhen, H. L. Heusel, P. J. Joseph, P. Groner, T. S. Little, J. Phys. Chem., 89, 2877 (1985). 149. J. Randell, A. P. Cox, K. W. Hillig, M. Imachi, M. S. LaBarge, R. L. Kuczkowski, Z. Naturforsch., 43,4, 271 (1988). 150. M. Hayashi, M. Fujitake, J. Mol. Struet., 146, 9 (1986). 151. J. R. Durig, M.-S. Cheng, Y. S. Li, P. Groner, A. E. Stanley, J. Phys. Chem., 93, 3492 (1989). 152. M. Hayashi, M. Adachi, J. Nakagawa, J. Mol. Spectrosc., 86, 129 (1981). 153. M. Adachi, J. Nakagawa, M. Hayashi, J. Mol. Spectrosc., 91,381 (1982). 154. J. Nakagawa, A. Nagayama, M. Hayashi, J. Mol. Spectrosc., 99, 415 (1983). 155. O. L. Stiefvater, Z. Naturforsch., 41A, 641 (1986). 156. R. D. Suenram, F. J. Lovas, W. J. Stevens, J. Mol. Spectrosc., 112, 482 (1985). 157. M. Liedtke, K. M. T. Yamada, G. Winnewisser, J. Hahn, J. Mol. Struct., 413-414, 265 (1997). 158. F. J. Lovas, R. D. Suenram, W. J. Stevens., J. Mol. Spectrosc., 100, 316 (1983). 159. H. Fujiwara, S. Saito, J. Mol. Spectrosc., 192, 399 (1998). 160. S. Kinsey-Nielsen, C. R. Brazier, P. F. Bemath, J. Chem. Phys., 84, 698 (1986). 161. C. T. Scurlock, D. A. Fletcher, T. C. Steimle, J. Chem. Phys., 101, 7255 (1994). 162. M. Fujitake, E. Hirota, J. Chem. Phys., 91, 3426 (1989). 163. C. Yamada, E. Hirota, J. Chem. Phys., 78, 1703 (1983).

242

GRONER

164. Y. Kasai, K. Obi, Y. Ohshima, Y. Endo, K. Kawaguchi, J. Chem. Phys., 103, 90 (1995).

165. M. Barnes, A. J. Merer, G. F. Metha, J. Mol. Spectrosc., 181, 168 (1997). 166. T. Momose, Y. Endo, E. Hirota, T. Shida, J. Chem. Phys., 88, 5338 (1988). 167. M. A. Anderson, L. M. Ziurys, Chem. Phys. Lett., 231, 164 (1994). 168. M. Bogey, C. Demuynck, J. L. Destombes, Mol. Phys., 66, 955 (1989). 169. F. X. Brown, S. Saito, S. Yamamoto, J. Mol. Spectrosc., 143,203 (1990). 170. M. Izuha, S. Yamamoto, S. Saito, Can. J. Phys., 72, 1206 (1994). 171. M. Kanada, S. Yamamoto, S. Saito, Y. Osamura, J. Chem. Phys., 104, 2192 (1996). 172. S. Yamamoto, S. Saito, J. Chem. Phys., 101, 5484 (1994). 173. C. A. Gottlieb, T. C. Killian, P. Thaddeus, P. Botschwina, J. Fliigge, M. Oswald, J. Chem. Phys., 98, 4478 (1993). 174. M. Bogey, C. Demuynck, J. L. Destombes, H. Dubus, J. Mol. Speetrose., 122, 313 (1987). 175. M. C. McCarthy, C. A. Gottlieb, P. Thaddeus, M. Horn, P. Botschwina, J. Chem. Phys., 103, 7820 (1995). 176. R. D. Brown, P. D. Godfrey, P. S. Elmes, M. Rodler, L. M. Tack, J. Am. Chem. Soe., 107, 4112 (1985). 177. Y. Ohshima, Y. Endo, J. Mol. Speetrose., 153, 627 (1992). 178. M. J. Travers, W. Chen, S. E. Novick, J. M. Vrtilek, C. A. Gottlieb, P. Thaddeus, J. Mol. Speetrose., 180, 75 (1996). 179. L. Yu, J. M. Williamson, T. A. Miller, Chem. Phys. Lett., 162, 431 (1989). 180. H. Jones, J. M. Brown, J. Mol. Speetrosc., 90, 222 (1981).

QUEST FOR EQUILIBRIUM STRUCTURE

243

181. K. Miyazaki, M. Tanoura, K. Tanaka, T. Tanaka, J. Mol. Spectrosc., 116, 435 (1986). 182. M. Fujitake, E. Hirota, Can. J. Phys., 72, 1043 (1994). 183. M. J. Tsuchiya, H. Honjou, K. Tanaka, T. Tanaka, J. Mol. Struet., 352-353, 407 (1995). 184. M. Fujitake, E. Hirota, Speetrochim. Acta, 50A, 1345 (1994). 185. S. Saito, Y. Endo, E. Hirota, J. Chem. Phys., 85, 1778 (1986). 186. M. Bogey, H. Bolvin, C. Demuynck, J. L. Destombes, Phys. Rev. Lett., 66, 413 (1991).

187. M. Bogey, H. Bolvin, M. Cordonnier, C. Demuynck, J. L. Destombes, A. G. Cs~sz~ir, J. Chem. Phys., 100, 8614 (1994). 188. M. Cordonnier, M. Bogey, C. Demuynck, J. L. Destombes, J. Chem. Phys., 97, 7984 (1992). 189. Y. Morino, M. Tanimoto, S. Saito, E. Hirota, R. Awata, T. Tanaka, J. Mol. Speetrose., 98, 331 (1983). 190. K. Kawaguchi, S. Saito, E. Hirota, N. Ohashi, J. Chem. Phys., 82, 4893 (1985). 191. H. Btirger, J. Demaison, P. Dr6an, C. Gerke, L. Halonen, H. Harder, H. Mader, J. Radtke, H. Ruland, J. Mol. Spectrosc., 180, 85 (1996). 192. T. A. Cooper, S. Firth, H. W. Kroto, J. Chem. Soe. Farad. Trans., 87, 1 (1991). 193. Y. Kawashima, H. Takeo, M. Sugie, C. Matsumura, E. Hirota, J. Chem. Phys., 99, 820 (1993). 194. K. Kawaguchi, Y. Endo, E. Hirota, J. Mol. Spectrosc., 93, 381 (1982). 195. M. Sugie, H. Takeo, C. Matsumura, Spectrochim. Acta, 50,4, 1379 (1994). 196. Y. Kawashima, K. Kawaguchi, Y. Endo, E. Hirota, J. Chem. Phys., 87, 2006 (1987). 197. A. G. Robiette, M. C. L. Gerry, J. Mol. Spectrosc., 80, 403 (1980).

244

GRONER

198. Y. Kawashima, Y. Endo, K. Kawaguchi, E. Hirota, Chem. Phys. Lea., 135, 441 (1987). 199. M. Sugie, H. Takeo, C. Matsumura, J. Mol. Spectrosc., 123, 286 (1987). 200. L. R. Thome, R. D. Suenram, F. J. Lovas, J. Chem. Phys., 78, 167 (1983). 201. J. R. Durig, A. B. Mohamad, P. L. Trowell, Y. S. Li, J. Chem. Phys., 75, 2147 (1981). 202. Y. Koga, H. Takeo, S. Kondo, M. Sugie, C. Matsumura, G. A. McRae, E. A. Cohen, J. Mol. Spectrosc., 138, 467 (1989). 203. J. Demaison, J. Cosl6ou, H. Biirger, E. B. Mkadmi, J. Mol. Spectrosc., 185, 384 (1997). 204. M. Le Guennec, G. Wlodarczak, W. D. Chen, R. Bocquet, J. Demaison, J. Mol. Speetrose., 153, 117 (1992). 205. F. Tamassia, C. D. Esposti, L. Dore, G. Cazzoli, J. Mol. Spectrosc., 174, 59 (1995). 206. K. D. Hensel, M. E. Lam, M. C. L. Gerry, H. Willner, J. Mol. Spectrosc., 151, 184 (1992). 207. N. Heineking, W. J~iger, M. C. L. Gerry, J. Mol. Speetrose., 158, 69(1993). 208. P. Groner, J. R. Durig, D. D. DesMarteau, S.-H. Hwang, d. Chem. Phys., 110, 9411 (1999). 209. S. Firth, S. Khalaf, H. W. Kroto, J. Chem. Soc. Farad. Trans., 88, 3393 (1992). 210. S. Yamamoto, M. Nakata, K. Kuchitsu, J. Mol. Speetrose., 112, 173 (1985). 211. C. D. Esposti, P. G. Favero, S. Serenellini, G. Cazzoli, J. Mol. Struct., 82, 221 (1982). 212. H. W. Kroto, J. F. Nixon, N. P. C. Simmons, J. Mol. Speetrose., 82, 185 (1980). 213. D. T. Cramb, Y. Bos, H. M. Jemson, M. C. L. Gerry, C. J. Marsden, J. Mol. Struet., 190, 387 (1988). 214. R. W. Davis, M. C. L. Gerry, J. Mol. Spectrose., 97, 117 (1983).

QUEST FOR EQUILIBRIUM STRUCTURE

245

215. P. Groner, J. R. Durig, D. D. DesMarteau, J. Chem. Phys., 105, 7263 (1996). 216. S. Carter, I. M. Mills, N. C. Handy, J. Chem. Phys., 97, 1606 (1992). 217. K. Yamada, J. Mol. Speetrose., 79, 323 (1980). 218. K. Yamada, M. Winnewisser, G. Winnewisser, L. B. Szalanski, M. C. L. Gerry, J. Mol. Speetrose., 79, 295 (1980). 219. J. Vogt, M. Winnewisser, Ber. Bunsenges. Phys. Chem., 88, 444 (1984). 220. H. W. Kroto, J. F. Nixon, O. Ohashi, K. Ohno, N. P. C. Simmons, J. Mol. Speetrose., 103, 113 (1984). 221. Z. Kisiel, L. Pszcz61kowski, L. B. Favero, W. Caminati, J. Mol. Speetrose., 189, 283 (1998). 222. U. P. Verma, K. M611er, J. Vogt, M. Winnewisser, J. J. Christiansen, Can. J. Phys., 63, 1173 (1985). 223. R. W. Davis, A. G. Robiette, M. C. L. Gerry, E. Bjamov, G. Winnewisser, J. Mol. Speetrose., 81, 93 (1980). 224. M. Oldani, T.-K. Ha, A. Bauder, J. Am. Chem. Soe., 105, 360 (1983). 225. A. P. Cox, S. D. Hubbard, H. Kato, J. Mol. Speetrose.' 93, 196 (1982). 226. R. D. Brown, P. D. Godfrey, D. McNaughton, P. R. Taylor, J. Mol. Speetrose., 12o, 292 (1986). 227. C. Kirby, H. W. Kroto, J. Mol. Speetrose., 83, 1 (1980). 228. J. R. Durig, G. M. ARia, P. Groner, H. Beckers, H. Btirger, J. Chem. Phys., 88, 545 (1988). 229. M. D. Allen, T. C. Pesch, J. S. Robinson, A. J. Apponi, D. B. Grotjahn, L. M. Ziurys, Chem. Phys. Lett., 293, 397 (1998). 230. J. A. DuckeR, A. G. Robiette, M. C. L. Gerry, J. Mol. Speetrosc., 90, 374 (1981). 231. D. Priem, J. Cosl6ou, J. Demaison, I. Merke, W. Stahl, W. Jerzembeck, H. BUrger, J. Mol. Speetrose., 191, 183 (1998).

246

GRONER

232. H. W. Kroto, J. F. Nixon, K. Ohno, J. Moi. Speetrosc., 90, 367 (1981). 233. P. D. Godfrey, R. D. Brown, A. N. Hunter, J. Mol. Struct., 413-414, 405 (1997). 234. J. R. Durig, J. F. Sullivan, A. B. Mohamad, S. Cradock, Y. S. Li, J. Chem. Phys., 84, 5796 (1986). 235. M. Hayashi, S. Kaminaka, M. Fujitake, S. Miyazaki, J. Mol. Speetrose., 135, 289 (1989). 236. Y. Shiki, M. Oyamada, M. Hayashi, J. Mol. Speetrose., 92, 375 (1982). 237. M. Hayashi, M. Fujitake, K. Kishioka, J. Moi. Speetrose., 135, 305 (1989). 238. L. Bizzocchi, C. D. Esposti, L. Dore, J. Mol. Struct., 443, 211 (1998). 239. J. J. van Vaals, W. L. Meerts, A. Dymanus, J. Mol. Speetrose., 106, 280 (1984). 240. J.-G. Lahaye, R. Vandenhaute, A. Fayt, J. Mol. Spectrosc., 123, 48 (1987). 241. M. Le Guennec, G. Wlodarczak, J. Demaison, H. BUrger, M. Litz, H. Willner, J. Mol. Speetrose., 15 7, 419 (1993). 242. M. Nakata, K. Kuchitsu, J. Mol. Struet., 352-353, 219 (1995). 243. M. Le Guennec, G. Wlodarczak, J. Demaison, H. BUrger, O. Polanz, J. Mol. Speetrose., 158, 357 (1993). 244. A. F. Borro, I. M. Mills, J. Mol. Struet., 320, 237 (1994). 245. P. Botschwina, S. Seeger, J. Mol. Struet., 320, 243 (1994). 246. R. D. Brown, P. D. Godfrey, K. H. Wiedenmann, J. Mol. Speetrose., 136, 241 (1989). 247. U. Andresen, N. Heineking, H. Dreizler, J. Mol. Speetrose., 137, 296 (1989). 248. H. Takeo, M. Sugie, C. Matsumura, J. Mol. Struet., 190, 205 (1988). 249. G. LaBrecque, C. W. Gillies, T. T. Raw, J. W. Agopovich, J. Am. Chem. Soe., 106, 6171 (1984).

QUEST FOR EQUILIBRIUM STRUCTURE

247

250. O. L. Stiefvater, Z. Naturforsch., 43.4, 597 (1988). 251. O. L. Stiefvater, H. D. Rudolph, Z. Naturforsch., 44A, 95 (1989). 252. O.L. Stiefvater, Z. Naturforsch., 44A, 29 (1989). 253. R. D. Brown, P. D. Godfrey, D. McNaughton, A. P. Pierlot, W. H. Taylor, J. Mol. Spectrosc., 140, 340 (1990). 254. M. Hayashi, C. Ikeda, T. Inagusa, J. Mol. Spectrosc., 139, 299 (1990). 255. M. Hayashi, T. Inagusa, J. Mol. Struct., 220, 103 (1990). 256. I. Merke, L. Poteau, G. Wlodarczak, A. Bouddou, J. Demaison, J. Mol. Spectrosc., 177, 232 (1996). 257. M. Hayashi, N. Inada, Y. Niide, J. Mol. Struct., 352-353, 325 (1995). 258. P. Groner, C. L. Tolley, J. R. Durig, Chem. Phys., 142, 381 (1990). 259. M. Hayashi, T. Inagusa, J. Mol. Spectrosc., 138, 135 (1989). 260. C. A. Rego, A. P. Cox, J. Chem. Phys., 89, 124 (1988). 261. T. Sakaizumi, K. Sekishita, K. Furuya, Y. Tetsuda, K. Kaneko, O. Ohashi, I. Yamaguchi, J. Mol. Speetrose., 161, 114 (1993). 262. M. Sugie, M. Kato, C. Matsumura, H. Takeo, J. Mol. Struct., 413-414, 487 (1997). 263. M. Rodler, A. Bauder, J. Am. Chem. Sot., 106, 4025 (1984). 264. B. P. van Eijck, J. van Opheusden, M. M. M. van Schaik, E. van Zoeren, J. Mol. Spectrosc., 86, 465 (1981). 265. C. E. Blom, A. Bauder, J. Am. Chem. Sot., 104, 2993 (1982). 266. T. Inagusa, M. Hayashi, J. Mol. Spectrosc., 129, 160 (1988). 267. M. Hayashi, M. Fujitake, T. Inagusa, S. Miyazaki, J. Mol. Struct., 216, 9 (1990). 268. J. Nakagawa, H. Kato, M. Hayashi, J. Mol. Spectrosc., 90, 467 (1981).

248

GRONER

269. J. R. Durig, A. B. Mohamad, G. M. Attia, Y. S. Li, S. Cradock, J. Chem. Phys., 83, 10(1985). 270. T. Inagusa, M. Fujitake, M. Hayashi, J. Moi. Speetrose., 128, 456 (1988). 271. M. Hayashi, Y. Morimoto, N. Inada, J. Mol. Speetrose., 171, 328 (1995). 272. V. Typke, J. Mol. Struet., 384, 35 (1996). 273. M. Hayashi, N. Nakata, S. Miyazaki, J. Mol. Speetrose., 135, 270 (1989). 274. Y. Shiki, A. Hasegawa, M. Hayashi, J. Mol. Struet., 78, 185 (1982). 275. Y. Shiki, N. Ibushi, M. Oyamada, J. Nakagawa, M. Hayashi, J. Mol. Speetrose., 87, 357 (1981). 276. M. C. L. Gerry, F. Stroh, M. Winnewisser, J. Mol. Speetrosr

140, 147 (1990).

277. T. Brupbacher, R. K. Bohn, W. J~iger, M. C. L. Gerry, T. Pasinszki, N. P. C. Westwood, J. Mol. Speetrose., 181, 316 (1997). 278. T. Okabayashi, M. Tanimoto, K. Tanaka, J. Moi. Speetrose., 174, 595 (1995). 279. R. N. Beauchamp, C. W. Gillies, N. C. Craig, ,I. Am. Chem. Soe., 109, 1696 (1987). 280. J. Randell, A. P. Cox, I. Merke, H. Dreizler, J. Chem. Soc. Farad. Trans., 86, 1981 (1990). 281. R. D. Brown, R. Champion, P. S. Elmes, P. D. Godfrey, J. Am. Chem. Soe., 107, 4109(1985). 282. T. Ogata, K. Fujii, M. Yoshikawa, F. Hirota, J. Am. Chem. Soe., 109, 7639 (1987). 283. K. H. Wiedenmann, I. Botskor, H. D. Rudolph, J. Mol. Speetrose., 113, 186 (1985). 284. R. N. Beauchamp, J. W. Agopovich, C. W. Gillies, J. Am. Chem. Sot., 108, 2552 (1986). 285. J. M. Colmont, G. Wlodarczak, D. Priem, H. S. P. Miiller, E. H. Tien, R. J. Richards, M. C. L. Gerry, J. Moi. Speetrose., 181,330 (1997).

QUEST FOR EQUILIBRIUM STRUCTURE

249

286. D. McNaughton, O. I. Osman, H. W. Kroto, J. Mol. Struet., 190, 195 (1988). 287. J. R. Durig, P. Groner, J. Lin, B. J. van der Veken, J. Chem. Phys., 96, 8062 (1992). 288. D. Christen, J. H. Griffiths, J. Sheridan, Z. Naturforsch., 37.4, 1378 (1982). 289. C. E. Blom, G. Grassi, A. Bauder, J. Am. Chem. Soe., 106, 7427 (1984). 290. S. L. Baughcum, R. W. Duerst, W. F. Rowe, Z. Smith, E. B. Wilson, J. Am. Chem. Soe., 103, 6296 ( 1981). 291. M. Rodler, A. Bauder, J. Mol. Struet., 117, 141 (1984). 292. Y. Niide, J. Sci. Hiroshima Univ., Ser. A, 48, 23 (1984). 293. H. Nanaie, G. A. Guirgis, J. R. Durig, Speetroehim. Aeta, 49A, 2039 (1993). 294. H. Takeo, M. Sugie, C. Matsumura, J. Mol. Struet., 352-353, 267 (1995). 295. J. M. Vacherand, B. P. van Eijck, J. Burie, J. Demaison, J. Mol. Spectrosc., 120, 118 (1986). 296. H. M. Badawi, P. Lorencak, K. W. Hillig, M. Imachi, R. L. Kuczkowski, d. Mol. Struet., 162, 247 (1987). 297. M. Imachi, R. L. Kuczkowski, J. Mol. Struet., 96, 55 (1982). 298. M. Fujitake, M. Hayashi, d. Mol. Struet., 127, 21 (1985). 299. M. S. LaBarge, H. Keul, R. L. Kuczkowski, M. Wallasch, D. Cremer, J. Am. Chem. Soe., 110, 2081 (1988). 300. C. Ikeda, T. Inagusa, M. Hayashi, J. Mol. Speetrose., 135, 334 (1989). 301. M. Meyer, J.-U. Grabow, H. Dreizler, H. D. Rudolph, J. Mol. Speetrose., 151, 217 (1992). 302. M. Hayashi, C. Ikeda, d. Mol. Struet., 223, 207 (1990). 303. M. Hayashi, H. Kato, M. Oyamada, J. Mol. Speetrose., 83, 408 (1980).

250

GRON-ER

304. K. J. Epple, H. D. Rudolph, J. Mot Spectrosc., 152, 355 (1992). 305. M. Hayashi, M. Adachi, J. Mol. Struct., 78, 53 (1982). 306. M. Imachi, J. Nakagawa, M. Hayashi, J. Mol. Struct., 102, 403 (1983). 307. K. K. Chatterjee, J. R. Durig, S. Bell, J. Mot Struct., 265, 25 (1992). 308. M. Winnewisser, E. W. Peau, Acta Phys. Hung., 55, 33 (1984). 309. S. G. Kukolich, S. M. Sickafoose, J. Chem. Phys., 105, 3466 (1996). 310. F. Karlsson, Z. Smith, J. Mot Spectrosc., 81, 327 (1980). 311. T. Ogata, J. Phys. Chem., 96, 2089 (1992). 312. D. G. Borseth, P. Lorencak, H. M. Badawi, K. W. Hillig, R. L. Kuczkowski, J. Mol. Struct., 190, 125 (1988). 313. H. Kato, J. Nakagawa, M. Hayashi, J. Mol. Speetrose., 80, 272 (1980). 314. B. J. Drouin, P. A. Cassak, P. M. Briggs, S. G. Kukolich, J. Chem. Phys., 107, 3766 (1997).

315. G. O. Sorensen, A. Tang-Pedersen, E. J. Pedersen, J. Mol. Struct., 101, 263 (1983). 316. O. L. Stiefvater, Z. Naturforsch., 43,4, 147 (1988). 317. O. L. Stiefvater, Z. Naturforsch., 43,4, 155 (1988). 318. T. Sakaizumi, H. Ishii, J. Mol. Speetrose., 162, 458 (1993). 319. M. D. Harmony, S. N. Mathur, J.-I. Choe, M. Kattija-Ari, A. E. Howard, S. W. Staley, J. Am. Chem. Soe., 103, 2961 (1981). 320. G. Knuchel, G. Grassi, B. Vogelsanger, A. Bauder, J. Am. Chem. Soe., 115, 10845 (1993). 321. H. S. P. Mtiller, M. C. L. Gerry, J. Chem. Soe. Farad. Trans., 90, 2601 (1994). 322. F. Meguellati, G. Graner, H. Btirger, K. Burczyk, H. S. P. Mtiller, H. Willner, J. Mol. Speetrose., 164, 368 (1994).

QUEST FOR EQUILIBRIUM STRUCTURE

251

323. H. S. P. MUller, M. C. L. Gerry, J. Mol. Speetrosc., 175, 120 (1996). 324. J. Preusser, M. C. L. Gerry, J. Chem. Phys., 106, 10037 (1997). 325. J. Demaison, G. Wlodarczak, J. Burie, H. Btirger, J. Mol. Speetrose., 140, 322 (1990). 326. W. D. Anderson, M. C. L. Gerry, R. W. Davis, J. Mol. Spectrosc., 115, 117 (1986). 327. C. M. Deeley, J. Mol. Speetrose., 122, 481 (1987). 328. G. Cazzoli, C. D. Esposti, P. Palmieri, G. Simeone, J. Mol. Spectrosc., 97, 165 (1983). 329. M. Birk, R. R. Friedl, E. A. Cohen, H. M. Pickett, S. P. Sander, J. Chem. Phys., 91, 6588 (1989). 330. F. Mata, N. Carballo, J. Mol. Struet., 101,233 (1983). 331. S. Firth, R. W. Davis, J. Mol. Speetrose., 179, 32 (1996). 332. H. Najib, N. Ben Sari-Zizi, H. Biirger, A. Rahner, L. Halonen, J. Mol. Spectrosc., 159, 249 (1993). 333. N. M. Lakin, T. D. Varberg, J. M. Brown, J. Mol. Spectrosc., 183, 34 (1997). 334. B. P. Winnewisser, J. Mol. Spectrosc., 82, 220 (1980). 335. A. P. Cox, M. C. Ellis, C. J. Attfield, A. C. Ferris, J. Mol. Struct., 320, 91 (1994). 336. E. Kagi, K. Kawaguchi, Astrophys. J., 491, L129 (1997). 337. M. Bogey, B. Delcroix, A. Waiters, J.-C. Guillemin, J. Mol. Speetrose., 175, 421 (1996). 338. J.-M. Flaud, C. Camy-Peyret, P. Arcas, H. Btirger, H. Willner, J. Mol. Spectrosc.,

167,383(1994). 339. J.-M. Flaud, P. Arcas, H. Biirger, O. Polanz, L. Halonen, J. Mol. Spectrosc., 183, 310 (1997). 340. K. Kijima, T. Tanaka, J. Mol. Spectrosc., 89, 62 (1981).

252

341. G. A. McRae, M. C. L. Gerry, E. A. Cohen, J. Mol. Speetrosr

GRONER

116, 58 (1986).

342. L. Fusina, G. Di Lonardo, P. De Natale, J. Chem. Phys., 109, 997 (1998). 343. S. G. Kukolich, J. Mol. Speetrose., 98, 80 (1983). 344. J. Lindenmayer, H. D. Rudolph, H. Jones, J. Mol. Speetrosc., 119, 56 (1986). 345. J. Lindenmayer, J. Mol. Speetrosc., 116, 315 (1986). 346. J.-M. Flaud, W. J. Lafferty, J. Mol. Spectrosc., 161,396 (1993). 347. J. M. Colmont, J. Demaison, J. Cosl6ou, J. Mol. Speetrose., 171,453 (1995). 348. W. Feder, H. Dreizler, H. D. Rudolph, V. Typke, Z. Naturforseh., 24,4, 266 (1969). 349. V. Typke, Z. Naturforsch., 33,4, 842 (1978). 350. M. D. Harmony, private communication (1999). 351. B. J. Smith, L. Radom, J. Am. Chem. Sot., 112, 7525 (1990). 352. D. Coffey, B. J. Smith, L. Radom, J. Chem. Phys., 98, 3952 (1993). 353. P. Botschwina, M. Oswald, J. Fltigge, ,~. Heyl, R. Oswald, Chem. Phys. Lett., 209, 117 (1993). 354. S. S. Butcher, E. B. Wilson, J. Chem. Phys., 40, 1671 (1964). 355. J. M. Riveros, E. B. Wilson, J. Chem. Phys., 46, 4605 (1967). 356. R. K. Kakar, C. R. Quade, J. Chem. Phys., 72, 4300 (1980). 357. E. Hirota, J. Chem. Phys., 42, 2071 (1965). 358. M. Hayashi, H. Imaishi, K. Kuwada, Bull. Chem. Soe. Jpn., 47, 2382 (1974). 359. R. E. Schmidt, C. R. Quade, J. Chem. Phys., 62, 3864 (1975). 360. J. Nakagawa, K. Kuwada, M. Hayashi, Bull. Chem. Soc. Jpn., 49, 3420 (1976). 361. J. R. Durig, P. Groner, J. F. Sullivan, J. Mol. Struct., 173, 1 (1988).

CHAPTER 4

THE ACCURACY

OF MOLECULAR

GEOMETRY

BY QUANTUM CHEMICAL

PREDICTIONS

METHODS

Stephen Bell Department of Chemistry, University of Dundee, Dundee DD 1 4HN, Scotland, U.K.

I.

I N T R O D U C T I O N ................................................................................................. 254

II.

C O M P U T A T I O N A L M E T H O D S ....................................................................... 256

III. P R E S E N T A T I O N O F R E S U L T S .......................................................................

IV.

258

S M A L L M O L E C U L E S ........................................................................................ 260

A~ Bond Lengths .................................................................................................. 260 B.

V.

Bond Angles .................................................................................................... 270

H Y D R O C A R B O N S .............................................................................................. 273 A.

C - C Single Bonds Lengths in Alkanes and Alkyl Moieties ........................... 273

B.

C - C Bonds Adjacent to C=C, C - C , Oxygen and Fluorine Atoms ................. 277

C.

C=C Double Bonds Lengths ........................................................................... 283

D.

C - C Triple Bonds Lengths ............................................................................. 288

E.

C - H Bond Lengths ......................................................................................... 291

F.

Bond Angles in Hydrocarbons ........................................................................ 302

VI. M O L E C U L E S C O N T A I N I N G N I T R O G E N , O X Y G E N O R F L U O R I N E .... 313 A.

CN, C - O and C - F Bond Lengths ................................................................... 313

B.

C=O Bond Lengths and Associated Bond Angles .......................................... 322

C.

Molecules with Central Oxygen Atoms .......................................................... 333

VII. T O R S I O N A L A N G L E S ....................................................................................... 336 VIII. C O N C L U S I O N S ................................................................................................... 344 R E F E R E N C E S ..............................................................................................................

253

348

254

I.

BELL

INTRODUCTION There are relatively few experimental techniques for the determination of accurate molecular structures of free molecules in their ground states in the gas phase, even considering the subdivisions of spectroscopy: microwave, infrared and Raman. The common techniques come under either spectroscopic or electron diffraction methods. In contrast there are several quantum chemical (QC) methods and very many basis sets and even more different combinations of these. Relatively few combinations are used in this study and an attempt is made to treat several molecules with the same standard method/basis set combinations. Because of this, little attempt is made to make a comprehensive review of the extensive literature. There have been numerous studies of the effects of different methods and different basis sets. General studies coveting a wide range of molecules of special note relevant to this study are those by B. Johnson et al. [ 1] and B. Ma et al. [2], although the former does not include one of the methods used extensively in this study. Many papers study these effects for single molecules individually or for single types of molecule.

The effects of basis sets/methods on small hydride

molecules were considered many years ago and displayed in a graphical manner [3 ] in plots which are two-dimensional and relate two internal coordinates. In QC methods, energy minimization is used usually without constraints and hence the structures obtained are predictions of r e structures in the spectroscopic sense.

The distinction between different types of experimental

structures are given in the Landold-B6rnstein review series [4] and also in other chapters of this publication. However, experimental r e structures are not available

ACCURACY OF MOLECULAR GEOMETRY

255

for many molecules (many more than stated by Ma et al. [2]). In principle, r s, r 0, rg, etc. structures could be found by QC methods. In other words, the r e structures obtained theoretically could be corrected for zero-point and vibrational averaging. In spectroscopy studies, rg and r z are not of much value in analysis of spectra. The differences between r e, r s, and r 0 structures may be fairly large for X-H bonds but are sometimes within experimental error for bonds between first-row atoms and certainly within the differences obtained by different methods/basis sets. The geometrical structures of hydrocarbons are easy to predict if no multiple bonds are present apart for some effects on the C-H bond lengths. By "easy to predict" we mean small calculations using small basis sets and simple methods so that no great expenditure of computer time is required. The greatest difficulties arise in predicting the structures of molecules with electronegative atoms and/or multiple bonds, i.e. molecules with lone pairs and 7t bonds. In this study, the results are categorized by BOND TYPE for several molecules possessing the same type of bond rather than giving the full structure, i.e. all the geometric parameters, of each molecule in a class.

Most of the

molecules considered are organic with only a few small molecules containing no carbon atom. Hence, almost all the bond types included in the study have a carbon at one end of the bond or both ends. Additionally, the breadth of the study is restricted by considering only a few QC methods and a few standard sets of basis functions. Since Gaussian 94 [5] and Gaussian 98 [6] have been used, the Gaussian basis sets normally used by

256

BELL

exponents of these programs will be considered. There may be better methods and basis sets for the purpose of predicting structures but these are commonly used. It would be informative for this study to include the effects of method/basis set on molecular vibrational frequencies and the accuracy of prediction of frequencies. It is clear that structural parameters, at least bond lengths, are related to stretching vibrational frequencies through a Badger type of relationship. Therefore, overshort or overlong bond lengths as obtained by some QC methods lead to stretching vibrational frequencies that are either ridiculously high or low respectively. Also cubic anharmonicity should be calculated if a structure type other than r e were to be considered. Consideration of these matters for a large number of molecules, as in this study, has not been done as it would lead to a much larger project. The torsional angles of included in this study.

gauche conformers of a number of molecules are

However, the prediction of torsional potential energy

functions for these molecules are not included although geometrica.l structures at conformation minima are indeed related to torsional potentials. Further systematic study of these problems is required.

II.

C O M P U T A T I O N A L METHODS All the calculations have been made with the Gaussian 94 [5] or the Gaussian 98 [6] program. Molecular structures for all molecules studied here have been calculated by geometry optimization within the procedures of the restricted Hartree-Fock method (HF) but good agreement with experimental structures is obtained only accidentally with HF when using small basis sets, so that "Post-SCF

ACCURACY OF MOLECULAR GEOMETRY

257

methods" are required that include at least some of the correlation energy. Of these, the Moller-Plesset perturbation method to second order (MP2) is used for all molecules and all basis sets employed. In recent year, methods based on Density Functional Theory (DFT) have become very popular and in particular the hybrid DFT method employing Becke's three-parameter exchange functional B3LYP is the most successful. B3LYP is also applied for all molecules and all basis sets employed. Since some of these are genuine ab initio electronic structure methods and some are parameterized density functional theory methods, they will be referred to collectively as quantum chemical or quantum computational (QC) methods. The basis sets used are the simple basis sets without polarization functions 3-21G and 6-31G, but these give molecular structures in good agreement with experiment only accidentally and in particular give poor agreement for Post-SCF methods. The 3-21G basis set is only included because it still may be necessary to calculate very large molecules with this basis. The extended sets with polarization functions 6-31G(d) and 6-31G(d,p) are used for most molecules.

For smaller

molecules, the much more extended triple-split and polarized basis set 6311+(3df,2p) is used in combination with the MP2 and B3LYP methods only. Other basis set types, especially the Huzinaga-Dunning DZ [7] and TZ [8] Gaussian bases, may be better for some of the molecular types considered but the emphasis of this work is more on method than on basis. All of the MP2 calculations on smaller molecules are done with all electron excitations considered, i.e. MP2(Full). For molecules with more than four firstrow atoms, electron excitations are done with "frozen core", that is electron

258

BELL

excitation from the inner-shell l s-like MOs is not included; some small molecules are also done with frozen core for the very extended basis set used. In any case, the shortening of bond distances on changing from "frozen core" (FC) to "Full" is very precisely predictable for each basis set. The amount of shortening will be given in some cases. Some other ab initio methods of recovering correlation energy are also considered; these are the methods MP3, MP4(SDTQ), CISD, QCISD, and CCSD. These are all done with "Full" inclusion of electron correlation unless otherwise indicated.

Bond lengths are obtained with these methods using the 6-31G(d,p)

basis set only or, if no hydrogens are present, the equivalent 6-31G(d) basis. Only one other hybrid DFT method (B1LYP) using Becke's one-parameter exchange functional is considered but other DFT methods are not included in view of the poor geometrical predictions with some of them as shown by Johnson et al. [ 1]. The geometrical structures obtained by other researchers have not been included because of the need for a consistency of treatment for all the molecules; only a few specific cases will be cited in the self-imposed constraint of considering the effects of method/basis set only. Certain aspects of the geometrical structures of about 40 molecules or conformers are included with at least the same three methods and the same three basis sets.

III.

PRESENTATION OF RESULTS Concerning the presentation of results, even with the idea of restricting the study to tabulations by BOND TYPES, there is still the choice as to whether to list in tables either by METHOD first, as the outer do-loop, then by BASIS SET

ACCURACY OF MOLECULAR GEOMETRY

259

secondly, as the inner do-loop, or by BASIS SET first, in the outer do-loop, then by METHOD secondly, in the inner do-loop.

There is both a need to make

comparisons among basis sets with ONE method, e.g. to study shortening of bonds with increased size of basis set, and also to make comparison among methods for ONE basis set, e.g. to draw attention to the lengthening of bonds with MP2 as against HF or to bonds of certain types being shorter with B3LYP than with MP2. The latter arrangement has been chosen in order to emphasize the superiority of one method against another, especially of B3LYP against MP2 for some bond types. The results of energy minimization by optimizing geometrical parameters, mainly bond lengths and bond angles, are presented in tables in the order described. In many cases, it is difficult to make comparison between digits in tables, especially if they are not adjacent, and therefore for most of the molecules the optimal parameter values are also plotted diagrammatically when comparisons can be made at a glance. These diagrams are effectively only one-dimensional plots; each component diagram is for one parameter of one molecule only as ordinate while the abscissa intervals are used to separate different basis sets and computational methods spaced for labeling purposes.

To avoid an excessive

number of diagram labels, all of the points are strict!y in the same order as in the tables with only "3" for HF/3-21G, "H" for HF, "M" for MP2, and "B" for B3LYP actually labeled. The largest basis set used, 6-31 l+G(3df,2p), is identified as EBS (extended basis set) in the diagrams. Each diagram has a horizontal reference line which is drawn at the value of the parameter for the best available experimental structure. The diagams also

260

BELL

make evident the convergence of the computed value of a molecular parameter on to the experimental equilibrium parameter value, when that is well determined, as the basis set is increased and better theoretical treatments are used. Since there are large numbers of tables and figures, a method of numbering is required. Although for some tables it has been necessary to have more than one figure and in one case there is more than one table for a particular bond type, for ease of reference the table numbers and figure numbers have been "synchronized". Where more than one table or figure is required the extra one has an added letter A~

IV.

SMALL MOLECULES

A.

Bond Lengths The re, r s and/or r 0 structures known for the four small hydride molecules HF, H 2 0 , NH3, CH 4 [9-17], and values for all these are given in Table 1 and Fig. 1 which illustrate that for bond lengths often the relationship r e < r s < r 0 is valid.

The differences in bond lengths are significant for

hydrides (approx. 0.01 A for HF where there is no uncertainty about derivation of structural parameter from observation, and 0.007 A for CH4). The bond length difference between re and r 0 for water is small but this may be due to the derivation treatment for the r 0 value using two rotational constants only.

A C C U R A C Y OF M O L E C U L A R G E O M E T R Y

261

T A B L E 1. Experimental and computed bond lengths of small hydrides molecules. Basis

Method

Expt

re

HF 0.91681 a

H20 0.95781 b

NH 3 1.0124 d

CH4 1.0870g

1.0138 e

rs r0

0.92563 a

0.9581 c

1.0162 f

1.0940 h

3-21G

HF

0.9374

0.9667

1.0026

1.0829

6-31G

HF

0.9209

0.9496

0.9913

1.0821

MP2

0.9470

0.9746

1.0100

1.0953

B3LYP

0.9493

0.9759

1.0059

1.0932

CISD 6-31G(d)

6-31G(d,p)

6-31 l+G(3df,2p)

0.9727

HF

0.9109

0.9476

1.0025

1.0837

MP2

0.9339

0.9686

1.0168

1.0897

B3LYP

0.9338

0.9687

1.0194

1.0933

CISD

0.9317

0.9662

HF

0.9005

0.9431

1.0009

1.0835

MP2

0.9210

0.9608

1.0115

1.0843

B3LYP

0.9254

0.9653

1.0180

1.0919

MP3

0.9179

0.9580

1.0107

1.0843

MP4

0.9206

0.9609

1.0138

1.0867

B 1LYP

0.9239

0.9639

1.0167

1.0910

CISD

0.9177

0.9574

1.0102

1.0846

QCISD

0.9202

0.9604

1.0131

1.0865

CCSD

0.9200

0.9603

1.0130

1.0863

HF

0.8970

0.9398

MP2

0.9171

0.9578

1.0092

1.0839

B3LYP

0.9220

0.9608

1.0134

1.0881

CISD

0.9105

0.9517

aRef. [9].

CRef. [10].

eRef. [13].

gR f.

bRef. [ 11 ].

dRef. [12].

fRef. [141.

hRef. [ 15].

262

BELL

Bond Length

Bond Length o o

o co

o --~

, !

.o co

p to 0

.o to "~

!

i

,

I

Expt

.o to tO

i

I

Expt ~

!

.o to CO

o co ~

!

1

@ to 01 "

'

~~ i=

o ,

co

6-31 G

.........~

co

6-31G

4r/

I

6-31G(d)

6-31G(d)

w

O r El (/1 ..o ul

!

~r o

w

i

CD 6-31G(d,p)

6-31G(d,p)

E~ EBS

EBS

Bond Length

Bond Length _~ o o0 . . .

o 4~

._= o to i

.

o o3 i

.o to ol I

o Co

,

Expt

,x0,

o -,,4 I,

co

6-31G

6-31G

6-31G(d)

~

6-31G(d)

=

i33

o E m~

I/l

=

6-31G(d,p)

~"

>.

6-31G(d,p)

, ~ : ~ -r.

(3

O

4"

EBS

~, ............

w

EBS

00

.......

FIG. 1. Experimental and computed bond lengths of small hydride molecules. In all the figures, "3" stands for HF/3-21G, "H" for HF, "M" for MP2, and "B" for B3LYP, and EBS stands for 6-311 +G(3df,2p).

ACCURACY OF MOLECULAR GEOMETRY

263

However, for bonds not involving hydrogen, the differences should be small because of mass effects. If some differences are large for non-hydrides it is probably because of improper treatment. All types of ab initio calculations of these small molecules have been made previously times without number [e.g. 1,2]. The bond lengths obtained by the treatments used in this chapter given in Table 1 and Fig. 1 should be compared with r e values rather than any of the other experimental structural types and especially in cases where the difference between structural determinations is significant. The following are some well-known general observations. The normal effect for most molecules of increasing basis set size, especially by the addition of polarization functions, is usually the shortening of bonds. This is seen clearly in the two-parameter diagrams by Bell [3 ]. In the current diagrams, this effect is seen clearly for Hartree-Fock calculations on the HF molecule and H20 as the points are lower than for other methods and so are isolated in the diagrams, but NH 3 and CH 4 aIe exceptions. Additionally, the spread of bond lengths decreases from HF to CH4; this would be even more obvious if all these diagrams were on the same scale.

The inclusion of correlation energy by whatever method normally

lengthens bonds.

"Full" correlation always gives bond lengths slightly

shorter than "frozen core", for example, for H20, MP2(FuI1)/6-31G(d) gives the O-H bond length as 0.9696 A in comparison with the MP2(FC) value of 0.9697 A, or for NH3, the MP2(Full)/6-31G(d) procedure yields 1.0168 A

264

BELL

and MP2(FC) yields 1.0171 A for the N-H distance.

Because these

differences are small and sometimes also the differences of particular bond types in different molecules are also small, bond lengths have been converged to 0.0001 A and are quoted thus in the tables. It is clear from Fig. 1 that the predicted bond lengths using MP2 and B3LYP converge asymptotically toward the experiment r e value as the basis set increases, with B3LYP usually a little longer than MP2 which is very near to experimental r e value for these molecules. For the more electronegative first-row central atoms, F and O, bond lengths calculated with MP4 are very like those with MP2, but those with MP3 are a little shorter. Also in these molecules, the bond lengths obtained with the CISD method are like those from MP3 calculations and those from QCISD calculations are similar to those from applying the MP4 (and so MP2) method. These conclusions are not really true for CH 4. In the component diagram for CH4 in Fig. 1, the bond length does not exhibit the large excursions of the other molecules with more electronegative atoms. The bond length from B 1LYP calculations are always very like those obtained from B3LYP and, at least for small molecules, there is not a large difference in computing time. Since, as far as geometrical structure is concerned, these two methods give such similar results, it might be just as good to use B 1LYP only. Only a few molecules containing second-row atoms are included in this study and results for HC1 and H2S are shown in Table 2 and Fig. 2. Excellent

ACCURACY OF MOLECULAR GEOMETRY

265

TABLE 2. Bond lengths of small second-row hydrides. Basis

Method

Expt

re

HC1

H2S

1.27455 a

1.3356 b

rs r0

1.3376 c 1.28392 a

rav

1.3372 d 1.3518 c

3-21G

HF

1.2935

1.3505

6-31G

HF

1.2953

1.3531

MP2

1.3175

1.3749

B3LYP

1.3206

1.3787

HF

1.2662

1.3264

MP2

1.2800

1.3395

B3LYP

1.2895

1.3496

HF

1.2656

1.3273

MP2

1.2682

1.3291

B3LYP

1.2861

1.3478

MP3

1.2691

1.3304

B1LYP

1.2846

1.3462

CISD

1.2694

1.3308

MP2

1.2713

1.3318

B3LYP

1.2800

1.3414

6-31G(d)

6-31G(d,p)

6-31 l+G(3df,2p) aRef. [9]. bRef. [18].

CRef. [19]. dRef. [20].

experimental structures are of course known and especially for a diatomic molecules [9,18-20]. Of course, calculations made without d functions in the basis set give ridiculous bond lengths but they are only included for consistency of treatment. For hydrides with second-row central atoms and for larger basis sets, MP2 calculations give bond lengths shorter than

266

BELL

B M h

HCI

O 3

~._ 03

'(~H

~

O T-r

B

o

?

re

H~

~.

,

~ M , ,

9M

I

Method/Basis

B M

H2S t'~

3 I

I re

~

t

LU r

rO

"t~ v

"I:3 v

O

(.9

r

CO

cb o

~

cb =

B

=

LU

--

Method/Basis

FIG. 2.

Bond lengths of small second-row hydride molecules.

ACCURACY OF MOLECULAR GEOMETRY

267

experimental and those by B3LYP calculation longer {by 0.01 A with 631G(d,p)}, but the MP2 estimate is more like the experimental r e value as noted by Ma et al. [2]. However, the B3LYP value appears to approach the MP2 value as the basis set is increased further to 6-31 l+G(3df,2p). Bond lengths obtained by MP3 and CISD calculations are again similar and also near to those by the MP2 method and the results from the B 1LYP method are again close to those from B3LYP calculation. A few small molecules, F2, H202, and N2H 4, involving single bonds between first-row atoms other than carbon are included here (Table 3 and Fig. 3). Only the X-X bond lengths are given here and not the X-H bond lengths or the HXX angles which are considered later with others of similar bond type (see Tables 24 and 25). These X-X bond lengths exhibit even large swings on changing computational method than the X-H bonds above (Notice that the scale interval of the diagrams here is 0.04 A in contrast to the previous 0.01 A).

For F 2 and H20 2, MP2 with unpolarized basis set is

particularly poor (error 0.085 A, 6%) relative to the r e or r 0 structural values [9,21,22].

However, there is little to choose between the bond lengths

yielded by the MP2 and B3LYP methods in terms of agreement with experimental r e or r 0 parameters for polarized basis sets. The N-N bond length of N2H 4 does not vary so much with basis set and method as for the molecules with more electronegative atoms, and for polarized basis sets the bond lengths by the two major methods are very similar to each other and near to the experimental value [23,24].

268

BELL

TABLE 3. Bond lengths of other small molecules.

Basis

Method

F2

Expt

re

1.41193 a

r0

1.41745 a

H20 2

N2H4

1.463 b

1.447 e 1.449 d

rg 3-21G

HF

1.4025

1.4731

1.4496

6-31G

HF

1.4125

1.4623

1.3964

MP2

1.5034

1.5685

1.4378

B3LYP

1.4678

1.5313

1.4035

HF

1.3449

1.3965

1.4134

MP2

1.4206

1.4681

1.4379

B3LYP

1.4035

1.4558

1.4367

HF

1.3957

1.4112

MP2

1.4665

1.4349

B3LYP

1.4558

1.4360

6-31G(d)

6-31G(d,p)

6-311 +G(3 df,2p)

aRef. [9]. bRef. [22].

MP3

1.4142 e

1.4502

B1LYP

1.4002 e

1.4525

CISD

1.4045 e

1.4406

MP2

1.3956

1.4409

1.4242

B3LYP

1.3947

1.4464

1.4321

MP4 r [23]. dRef. [24].

1.4572 edone as 6-31G(d).

For all three molecules in Table 3, the optimized bond lengths obtained using the extended basis set 6-31 l+G(3df,2p), labeled EBS for short, yields curiously short bond lengths in comparison with other basis sets but especially in comparison with experimental. It appears that in these and some other molecules below employing such larger basis sets may not provide one

A C C U R A C Y OF M O L E C U L A R G E O M E T R Y

1.50

J

M

269

,

F2

1.46

C --I 'ID

M 1.42

0

rn

M

re

3 B

X ILl

1.38

(.9 03

1.34 Method/Basis 1.58

H202 1.54 --

i

B

01 r -,I "0

1.50 --

m

1.46'

"-"

(3

03

C 0

M v

rO

9

M

v

H

0

M x LLI

1.42 --

H 1.38 = MethodiBasis 1.58

....

N2H4 1.54 --

'4...'

Q. 1.50 --

.J "O

C

O Q3 1.46 --

z,

X LU

(.9

(.9

03

03

m

3

,

M

Z

M

B

rO 1.42

M

H

H

H 1.38 Method/Basis

FIG. 3.

Bond lengths of other small molecules.

270

BELL

with improved molecular structures and may, therefore, be a waste of computer resources.

B.

Bond Angles For molecules with electronegative central atoms, bond angles are even more sensitive to basis set and method than bond length [3]. For H20 and NH3, the optimized angles given in Table 4 and Fig. 4 with HF/6-31G method/basis differ from the experimental values by 7 ~ (Compare this 7% error with the largest bond length error found in H20 of 1.8%). For small hydrides with only two internal parameters, the combination of bond length predictions and band angle predictions can be seen together in plots like those by Bell [3]. However, all the procedures currently used, except those with unpolarized basis sets, give good comparison with experimental angles. The differences between the angles calculated with the MP2(FulI) and MP2(FC) models is usually quite small, in the cases here only 0.03 ~ Another way of presenting the NH 3 angles is as out-of-plane angles since these vary over a wider range (33 ~ to 63 ~ for the calculations in Table 3) and hence are a more sensitive measure of deviation from planarity. One second row hydride, H2S, is included with first row hydrides in Table 4 and Fig. 4. The bond angle in H2S does not undergo such large excursions in the diagrammatic presentation as those of H20 or NH3. The B3LYP method appears to give slightly better angles than the MP2 method but both give excellent bond angles with the 6-31G(d,p) basis set.

Once

ACCURACY OF MOLECULAR GEOMETRY

271

TABLE 4. Experimental and computed bond angles of small hydrides. Basis

Method

Expt

re

H 20 104.478 a

rs

r0

105.28 b

NH3

H 2S

106.67 c

92.11 f

107.23 d

91.6g

107.8 e

92.31 h 92.13g

rav

3-21G

HF

107.68

112.40

95.81

6-31G

HF

111.55

116.13

96.16

MP2

109.29

114.26

94.89

B3LYP

108.30

116.18

94.05

CISD

109.37

HF

105.50

107.18

94.37

MP2

104.00

106.36

93.33

B3LYP

103.65

105.76

92.78

CISD

104.26

HF

105.97

107.58

94.38

MP2

103.87

106.12

92.84

B3LYP

103.74

105.75

92.66

MP3

104.26

106.12

92.76

MP4

103.88

105.76

B1LYP

103.86

105.84

92.79

CISD

104.34

106.21

92.92

QCISD

104.08

105.94

CCSD

104.09

105.96

HF

106.39

MP2

104.55

107.29

92.14

B3LYP

105.20

107.25

92.45

CISD

105.08

6-31G(d)

6-31G(d,p)

6-31 l+G(3df,2p)

92.92

aRef. [11].

CRef. [12].

eRef. [14].

gRef. [19].

bRef. [10].

dRef. [13].

fRef. [18].

hRef. [20].

272

BELL

112.0

H20 angles 110.0

o.

108.o C < "O C O m

e

v

(9 ~" 03 I r

(9 1-

3

03

m LU H

H

106.0 r0

rJ

M

1 04.0 o. x UJ

M

e

102.0

B

Method/Basis

~

116.0

114.0

B

NH3 angles

112.0 01 c ,< "o o El

CL 110.0 ..

108.0

v"o (9

,..., Q. X UJ

(9 ,_ 03 ! (D

r| (O

i

09 rn UJ

H

H

B

r0 106.0 .: re B 104.0

Method/Basis

97.0

3

95.0

H~ angles

H

M

=, O1

B H

(,.9 ~

H

(.9

< "O C O El

e~ X

93.0 W

B rO

re

~

B

B

,,,,* v

rs

91.0 ,

FIG. 4.

Method/Basis

Experimental and computed bond angles of small hydride molecules.

ACCURACY OF MOLECULAR GEOMETRY

273

again the 6-31 l+G(3df,2p) or EBS in the diagram yields angles a little out of line with the angles predicted by the other basis sets.

V.

HYDROCARBONS A.

C-C Single Bonds Lengths in Alkanes and Alkyl Moieties Good experimental structures are available only for a few simple alkanes.

Among the references for ethane [25-29], the r e structure

determination by Duncan et al. [27] is most relevant as the QC methods make predictions of equilibrium structures,

r z [28] and rmP [29] structures for

ethane are also given in Table 5 and Fig. 5. An early r s structure [30] and an rmP [29] structure of propane are available for propane, but the best structure available for butane appears to be an rg structure [31-33] which may not be very relevant to spectroscopic determinations considered here.

By alkyl

moiety is meant a saturated carbon chain of at least two C atoms, i.e. CH 3CH 2- or longer, so that it is sufficiently isolated from groups that influence adjacent bonds as in section 4B. The isolated C-C bond lengths of 1-butene by Kondo et al. [34] and Bouchy et al. [35] are not in disagreement for this bond. The pattern of bond lengths obtained by quantum chemical (QC) methods is almost identical for all of the alkanes and alkyl moieties given. The structures of alkanes are always easily predicted (see CH 4 above) i.e. the vertical spread of points in the diagrams is always less than for the X-X

274

BELL

T A B L E 5.

B o n d lengths o f C - C single bonds in alkanes or alkyl moieties.

Basis

Method

Expt

re

ethane

t-butane

1.526 d 1.522 b

1.521 b 1.536g rgl.533 e

r z 1.5351 e HF

1-butyne

1.530 f

r0

3-21G

c-l-butene

1.528 a

rs rm~

propane

1.5424

1.5408

1.5403

1.5382

1.5467

1.5290

1.5363

1.5445

1.5520

1.5346

1.5457

1.5264

1.5324

1.5227

1.5307

1.5299

1.5402

1.5408 6-31G

HF

1.5299

1.5313

MP2

1.5446

1.5454

1.5333 1.5310 1.5466 1.5452

B3LYP

1.5353

1.5375

1.5401 1.5370

6-31G(d)

HF

1.5274

1.5283

MP2

1.5244

1.5245

1.5297 1.5282 1.5250 1.5245

B3LYP

1.5305

1.5322

1.5341 1.5319

6-31G(d,p)

HF

1.5268

1.5278

MP2

1.5218

1.5225

1.5235

B3LYP

1.5300

1.5317

1.5337

1.5294

1.5258

1.5277 1.5208

1.5287

1.5292

1.5397

1.5226 1.5314

6-311 +G(3df,2p)

MP3

1.5243

MP4

1.5265

B 1LYP

1.5302

CISD

1.5225

MP2

1.5195

B3LYP

1.5277

aRef. [27].

eRef. [28].

eRef. [32].

bRef. [29].

dRef. [30].

fRef. [341.

gRef. [35].

ACCURACY OF MOLECULAR GEOMETRY

275

C-C Bond Length

C-C Bond Length ..=

..~

01 to !

"

..=

01 o3

01 4~

I

'

..=

-.=

01 to

01 (.~

I

'

,

Expt w Expt

--~ 01 01 ,,

--.

~' I

,s

,

o0

~

6-31G

6-31G

6-alG(d) -r-

6-31G(d,p)

"~

EBS

130

l~ C-C Bond Length

C-C Bond Length 01 to

01 ~

I

01 4~

"

O1 !,,3

01 (.~

I

I

t

O1 4~ '

I

01 01 '"

E..ty

Expt

3

6-31G

6-31G co~1~, ~

6.31G

~ 1:o .=,,

6-31G(d)~ ~co

6-31G(d,p) --...

FIG. 5.

=

Bond lengths of C-C single bonds in alkanes or alkyl moieties.

--

276

BELL

bonds in Table 3 and Fig. 3 so that all basis sets with polarization functions give similar bond lengths, although as above, using the MP2 method with unpolarized bases gives C--C lengths which are long.

The addition of p

functions to hydrogen atoms makes very little difference to HF, MP2, B3LYP bond lengths and this will be seen to be true also for other CC bond types. For ethane, other calculation models have been used including MP3, MP4, CISD, and B1LYP, and the MP2 method with the large basis 631 l+G(3df,2p). The calculated C-C bond lengths straddle the experimental r e value with the MP2 and B3LYP predictions of r e on each side and with the B3LYP optimal value nearer [2] for all basis sets up to the largest. The MP2 values show a wider variation with change of basis set. The rmP bond length is shorter than the r e structure value and is more like some MP2 values. Since there is an almost identical pattern in these diagrams, it may be possible for this particular bond type to use the computational result to assess the experimental. The r s bond length in propane seems a little too short and that in cis-l-butene is a little too long. The middle and end C-C bonds in butane are not identical by QC methods although assumed to be so experimentally probably because of the lack of electron diffraction data [33]. The full lines in Fig. 5 are for the end C-C bonds and the dotted lines for the middle bond; it is clear that even theoretically they are very similar and all follow the usual pattern. By comparison with ethane and propane, an r e or r s C-C bond length for butane should probably be around 1.525 A.

ACCURACY OF MOLECULAR GEOMETRY

277

In table column headings and in the diagrams, the identification of conformers is kept as short as possible to avoid clutter in small diagrams or crowding in tables: italic c stands for s-cis or synplanar, italic t for s-trans or antiplanar, and italic g for gauche which may be synclinal or anticlinal depending on the rotational isomerism exhibited. The C-C single bond two CC bonds away from the triple bond in 1butyne (or 1 pentyne) is not in the diagram because of the few points calculated and because the pattem is like other alkyl moieties except that the C-C bond is slightly longer by about 0.01 A than in ordinary alkanes.

C

B.

C

C

C

C

C

C-C Bonds Adjacent to C=C, C-=C, Oxygen and Fluorine Atoms The C-C bonds which are called single bonds according to Lewis-theory but which are adjacent to double bonds, triple bonds, fluorine atoms, or oxygen atoms including the carbonyl group, are all affected by the neighboring bonds or groups and are significantly shorter than the usual alkyl C-C bond. Adjacent C--O bonds and C-F bonds show the smallest effect but the shortening appears to be systematic in all the fluorides and the alcohol listed.

278

BELL

Adjacent C=C double bonds and carbonyls show a slightly larger and similar effect, but the largest effect of all seems to be shown with C-C triple bonds. Experimemal r s or r 0 structures are available for all the molecules with double bonds in Table 6, viz. propene [36-39]; cis-l-butene [34,35]; and cis2-butene [40-42].

The experimental and calculated C-C bond lengths are

displayed in Fig. 6 for propene and 1-butene and the patterns of bond length variation with method/basis is closely parallel. The plot for 2-butene is not given because of the complete similarity of the 2-butene diagram to the 1butene pattern except for the numbers being consistently smaller by about 0.004 .A for 2-butene. However, since the C-C bond length in the r 0 structure of 1-butene by Bouchy et al. [35] of 1.519 A is not realistic, it is not shown in the table or diagram either. The experimental C--C bond lengths in ethanal [43,44] and propanal [45] are quite similar to those of the alkenes and the pattern of variation of the calculated bond lengths is also quite similar. Some extra methods have been applied to ethanal. For two of these molecules, the MP2 method gives bond lengths closer to the r s or r 0 values and for the other two the B3LYP model appears to be better. In Table 7, the bond lengths of the C-C bonds adjacem to the C-F or C-O bonds of some fluorohydrocarbons and ethanol are given. Since r e, r s or r 0 structures are available for 1,2-difluoroethane [46-49], 1-fluoropropane

ACCURACY OF MOLECULAR GEOMETRY

279

T A B L E 6. Bond lengths of C-C single bonds adjacent to double bonds. Basis

Method

Expt

rs

propene 1.501 a

r0

c-l-butene

c-2-butene

ethanal

propanal

1.501 c 1.507 c

rg 1.506b

ravl.51 ld

1.497 e

1.5005 f

1.509h

r z 1.515g

3-21G

HF

1.5095

1.5137

1.5104

1.5069

1.5083

6-31G

HF

1.5015

1.5074

1.5028

1.4946

1.5001

MP2

1.5163

1.5219

1.5176

1.5135

1.5183

B3LYP

1.5049

1.5117

1.5057

1.5031

1.5087

HF

1.5026

1.5078

1.5043

1.5042

1.5077

MP2

1.4976

1.5022

1.4987

1.5016

1.5047

B3LYP

1.5021

1.5080

1.5034

1.5083

1.5121

HF

1.5019

1.5073

1.5030

1.5068

MP2

1.4960

1.5011

1.5005

1.5041

B3LYP

1.5013

1.5074

1.5072

1.5112

6-31G(d)

6-31G(d,p)

6-31 l+G(3df,2p)

1.4972

MP3

1.5028

MP4

1.5072

CISD

1.5001

MP2

1.4965

B3LYP

1.5010

aRefs. [37,38]. bRef. [39].

CRef. [34]. dRef. [4].

eRef. [40,42]. fRef. [43].

gRef. [44]. hRef. [45].

[50], and trans ethanol [51 ], the usual type of diagram is draw for only these in Fig. 7.

The patterns of variation of calculated C-C bond length with

method and basis are very like those for the aldehydes in Fig. 6.

For

difluoroethane, the MP2 value for the 6-31G(d) and 6-31G(d,p) bases are very close to the r s value. The C-C bond in gauche difluoroethane is shorter

280

BELL

C-C Bond Length ..=

..=

...=

01 o

oi ~

oi bo

-,

r

, ;

C-C Bond Length _=

r

Ol

.Ix

w i,,,

"

I

=

Expt

Expt r

fh

~

6-31G Z-~

6-31G

6-31G(d)

=, , . ~ -I-

E