CHAPTER 1
Part I
ELECTRON DIFFRACTION
Introduction to Part I: Electron Diffraction Studies on Vapours
BY L. E. SUTTON
The part which electron diffraction studies on vapours can play in the elucidation of molecular structure is commonly regarded as much more restricted than that of X-ray crystallographic studies. This view needs a great deal of qualification. It is true that much more complex structures can be tackled by the latter method because of the three-dimensional information which can be obtained from a crystal. Thermally unstable species must be examined at low temperatures and therefore often in crystalline form. Ionic species can usually be examined only in crystals. However, for compounds which are not too complex and which can be vapourized satisfactorily, electron diffraction can and, broadly speaking, usually does give more accurate determinations of interatomic distances than X-ray crystallography. Hydrogen atoms can be more readily placed by electron diffraction. For the examination of species which exist only in the vapour phase, for finding the shape and size of a free molecule not constrained by packing, or for the study of gas-phase equilibrium, e.g. of conformational isomers, and the evaluation of ΔH and ΔS values, it is necessarily the method to use.
I have been vividly reminded of the progress in techniques by sorting through some old electron diffraction photographs taken in the 1930s. The great potential increase in accuracy and reliability which stemmed from the development of rotating sector cameras just after the War has been brilliantly realized. This has come about by combining instrumental observations of scattering intensity, which the new cameras made possible, with more rigorous theoretical treatments and the resources of high-speed computing. Error analysis has become vastly more sophisticated. The accuracy in favourable cases is as high as that of microwave spectroscopy, indeed the limiting factors are now seen to be the same in both methods, viz. the correct treatment of the effects of vibration.
The further hopes which once may have been entertained, that electron diffraction could be used independently of spectroscopy to derive the complete vibrational characteristics of molecules, have not been realized. Instead, it has become apparent that the best way to determine a structure precisely is to combine electron diffraction and spectroscopy.
There has been no shortage of excellent reviews of methodology; and there have been extensive reviews of results, but these have usually been restricted to some particular fields or interests. In the present Report the aim has been to have a fresh look at the fundamentals of the scattering process and then a very critical look at accuracy. This has been done by Dr. H. M. Seip of the Oslo school. It has proved impossible this year to review the development of apparatus adequately so this is being left for a future volume. A search of the literature of the past six to seven years has shown that the amount of interesting structural information which has come from electron diffraction studies is far greater than had been realized; so a serious attempt has been made by Dr. B. Beagley to present a comprehensive survey of it all, emphasizing the general structural principles which emerge. This has been a major task, but it has brought together a wealth of chemically significant facts and ideas. Dr. A. G. Robiette has reviewed the present state of the interplay between electron diffraction and spectroscopy which has been one of the growing points in structural studies and which promises to be of even greater importance. Here also it has proved impossible to cover the whole field adequately this year; but a general survey has been given. Most of the major topics have thus been covered, but there is one task which ought to be attempted when the time is ripe. This is to make a systematic comparison of structures of molecules as free molecules and in the crystal so that more understanding of the effects of interaction, in the most general sense, of a molecule with its environment may be developed.
I should like to thank the contributors, Dr. Seip, Dr. Beagley and Dr. Robiette for their spirited and willing help. The number of people working in this field is still relatively limited and there is a good deal of personal co-operation which has, as always, made it a pleasure to be a participant on, or a bit beyond the fringe.
Regrets have often been expressed that there have been no later volumes of "Tables of Interatomic Distances and Configuration in Molecules and Ions" after the original volume, published in 1958, and the supplementary one published in 1965. I decided long ago that this task must be given over to a professional, full-time team. Readers will know that, under the general direction of Mrs. Olga Kennard who was one of the Sectional Scientific Editors of these volumes, three bibliographical volumes for crystal structures have already been published in the series "Molecular Structures and Dimensions". It is good news that, in this same series, a numerical volume on "Interatomic Distances, 1960–65, Organic and Organometallic Crystal Structures" will appear early in 1973 and that a bibliographical volume on vapour-phase structure determinations by diffraction and spectroscopic methods, covering a long period of time, will appear in the summer of 1973. There is a longer term plan for publishing numerical results of vapour-phase investigations. Dr. Barbara Starck of the Sektion Strukturdokumentation, Zentrum Chemie-Physik-Mathematik, at the University of Ulm, is planning a volume in the Landolt–Börnstein series which will include microwave and electron diffraction data. Professor K. Kuchitsu and Professor E. Hirota are co-operating in this venture.
In this Part of the Report, the general aim has been to describe what has happened since about 1965. The four chapters have been written rather differently, as befits the different topics. Those on theory and accuracy (Chapter 1) and on the Interaction of Spectroscopy and Electron Diffraction (Chapter 4) are not intended to give a comprehensive review of all that has been written in the time span covered, but to present the main ideas that have developed and the conclusions that have emerged. On the other hand, the chapters on results (2 and 3) are intended to be Comprehensive. More exactly, all relevant papers abstracted in Bulletin Signaletique from January 1965 until August 1972 are mentioned, and additional material has been added when it was available. Dr. Beagley has endeavoured, nevertheless, to avoid producing a mere list of bond lengths, by developing discussions based on the simpler approaches which commonly are still used by authors to rationalize their results.
There is now a general and lively appreciation of the limitations imposed by intramolecular vibrations on the observation and even on the definition of interatomic distances, and much thought has been given to trying to solve the difficulties. This topic is discussed extensively in Chapter 4. It is necessary here to remind readers in a preliminary fashion of the need to be careful and critical in comparing and in using reported distances. To facilitate comparisons the bond lengths quoted in Chapters 2 and 3 are rg [i.e. rg(0)] values unless otherwise stated. Dr. Beagley has converted literature values to rg values where these are not given specifically. A standard convention has also been adopted for estimates of error: they are all standard deviations. Errors are given in a variety of ways in the literature, so Dr. Beagley has attempted to convert them all to a common basis. When there have been difficulties in deciding what type of distance has been used or the meaning of an error estimate, he has given his subjective estimates. Lastly, it is an interesting fact that every paper reviewed uses the Ångstrom unit, no doubt because it is so convenient, and this unit is therefore used throughout this part of the Report (1 Å = 10-10 m; so 1 nm = 10 Å).
1 Theory and Accuracy
BY H. M. SEIP
1 Theory
Introduction. — The theory necessary to carry out a fairly accurate electron-diffraction investigation was developed many years ago. The bases of the theory are to be found in papers by Rutherford, Debye, and Ehrenfest. Thus most of the theory was available when Davisson and Germer and Thomson made their famous experiments on the diffraction of electrons by crystals and so verified de Broglie's relation, and when Mark and Wierl carried out the first experiments on scattering of electrons by gases. An important contribution to the theory was also made by Faxén and Holtsmark in 1927, but their results were not applied to structure determinations by gas electron diffraction until about two decades ago.
Several review papers dealing with structure determination by gas electron diffraction have been published. Many of these include a discussion of the scattering theory commonly applied. Parts of the theory are also found in many textbooks on quantum mechanics and on scattering theory. Recently, the first textbook devoted exclusively to structure determinations by gas electron diffraction has also appeared. It seems therefore unnecessary to derive here the usual expressions applied in structure analysis. However, the most important expressions obtained by considering the atoms as spherically symmetrical force fields and using the independent atom approximation, are given below, and the approximations involved are discussed in some detail. A discussion of electron diffraction as a tool for studying conformational equilibria and determining thermodynamic properties forms the last part of this Section. The study of molecules with large amplitudes of vibration is mentioned, though this topic is also discussed in Chapter 4.
Basic Theory. — Scattering by a Spherically Symmetrical Force Field. To obtain the expressions for the intensity of fast electrons scattered elastically by a molecule, it is useful to begin by considering the scattering of electrons by spherically symmetrical force fields. The Schrodinger equation for one electron may be written
[MATHEMATICAL EXPRESSION OMITTED] (1)
where k2 = 2mE/h2 (k = p/h = 2π/λ where λ is the electron wavelength), and U(r)= (2m/h2)V (r)(V is the potential energy for the electron in the field). The asymptotic form of the acceptable solution of (1) is
[MATHEMATICAL EXPRESSION OMITTED] (2)
where the wavefunction for the incident beam is eikz. By the use of basic quantum mechanics it is found that the intensity is
[MATHEMATICAL EXPRESSION OMITTED] (3)
where θ is the scattering angle, and Io is the intensity of the incident beam. f(θ), the scattering amplitude, may be determined by the partial wave method. The result is a sum over Legendre polynomials (Pi)
[MATHEMATICAL EXPRESSION OMITTED] (4)
f (θ) is a complex quantity, and η(θ) is the argument of f(θ).
To be able to carry out this summation, the phase shifts in the partial waves, δl, must be known. An approximate expression, known as Born's phase shift formula, is
[MATHEMATICAL EXPRESSION OMITTED] (5)
jl is the spherical Bessel function of order l. This expression is valid for small δ. If we expand
[MATHEMATICAL EXPRESSION OMITTED] (6)
and neglect δl2 and higher terms, the combination of equations (4) and (5) gives
[MATHEMATICAL EXPRESSION OMITTED] (7)
[MATHEMATICAL EXPRESSION OMITTED] (8)
The expression (7) is known as the scattering amplitude in the first Born approximation. Notice that the scattering amplitude is real in this approximation.
The potential V (r) may be expressed in terms of the electron density, p (r'), if the scatterer is an atom, i.e.
[MATHEMATICAL EXPRESSION OMITTED] (9)
Combination of (9) and (7) gives
[MATHEMATICAL EXPRESSION OMITTED] (10)
where F(s) is the atom form factor
[MATHEMATICAL EXPRESSION OMITTED] (11)
and ao the Bohr radius.
The expression (10) is also easily obtained by the use of Green's function. Let G(r,r') be a solution of
[MATHEMATICAL EXPRESSION OMITTED] (12)
Using the properties of Dirac's δ-function it is seen that
[MATHEMATICAL EXPRESSION OMITTED] (13)
satisfies equation (1). The solution of (12) may be obtained by complex integration
[MATHEMATICAL EXPRESSION OMITTED] (14)
The first Born approximation to ψ(r)is obtained by substituting
[MATHEMATICAL EXPRESSION OMITTED] (15)
in the integrand in (13), i.e. by assuming that the scattered wave is negligible compared to the incident wave. Better approximations may (presumably) be obtained by iteration. The first Born approximation to ψ(r), combined with the asymptotic form of (14) andequation (2), give f(Bθ) [equation (10)].
The intensity of electrons scattered by a spherically symmetrical atom is thus approximately [see equations (3) and (10)]
[MATHEMATICAL EXPRESSION OMITTED] (16)
This relation may also be obtained from Rutherford's expression for the scattering by a point charge.
Until Glauber and Schomaker showed that the first Born approximation is far from valid for molecules with atoms differing considerably in atomic numbers, the expression (10) was used in all electron-diffraction investigations. The calculation of accurate scattering amplitudes by means of equation (4) has greatly contributed to the increased reliability of the electron-diffraction method. This is further discussed below (pp. 21 and 40), but it may be mentioned here that |f| and η values for a series of atoms were first published by Ibers and Hoerni, who used the WKB method. More accurate values, based partly on numerical solution of (1), have recently been published (cf. Figure 1).
Scattering by Molecules; Independent Atom Approximation. The intensity of electrons scattered elastically by a molecule is in the Debye–Ehrenfest theory obtained using the first Born approximation for the scattering amplitudes and assuming that the atoms scatter independently. In the independent atom approximation applied today complex scattering amplitudes [see equation (4)] are used.
For a rigid system of M atoms considered to be independent scattering centres, the intensity of electrons scattered elastically becomes
[MATHEMATICAL EXPRESSION OMITTED] (17)
[R is used here for the distance between the points of diffraction and observation, although r was used in the previous section, e.g. equations (3) and (16). This is done to avoid confusion with the interatomic distances rij].
If it is assumed that the inelastically scattered intensity is completely incoherent (see p.26), the total intensity is given by
[MATHEMATICAL EXPRESSION OMITTED] (18)
[MATHEMATICAL EXPRESSION OMITTED] (19)
and Si is the inelastic scattering factor for atom i.
Ib is often referred to as the background.
[MATHEMATICAL EXPRESSION OMITTED] (20)
The structure-dependent part [Im(S)] may be written
[MATHEMATICAL EXPRESSION OMITTED] (21)
[cf. equation (4)].
Equation (21) is only valid for a rigid system. To include the interatomic vibrations the intensity expression must be replaced by
[MATHEMATICAL EXPRESSION OMITTED] (22)
where Pij(r)dr gives the probability that the distance between the atoms i and j is between r and r + dr.
In most cases Gaussian distance distributions are assumed, i.e.
[MATHEMATICAL EXPRESSION OMITTED] (23)
where uij is the root-mean-square amplitude of vibration, and rij now denotes the mean distance between the atoms i and j. To perform the integration in (22) we expand
[MATHEMATICAL EXPRESSION OMITTED] (24)
Using only the first term in the expression, Im(s) becomes
[MATHEMATICAL EXPRESSION OMITTED] (25)
where
[MATHEMATICAL EXPRESSION OMITTED] (26)
Using the two first terms in (24) and the approximation
[MATHEMATICAL EXPRESSION OMITTED]
the result is
[MATHEMATICAL EXPRESSION OMITTED] (27)
It has been shown that if a distance distribution corresponding to the ground state of the Morse oscillator is used for Pij(r), the expression (27) is to a first approximation replaced by
[MATHEMATICAL EXPRESSION OMITTED] (28)
κij which may be called the asymmetry constant, is related to the constant a in the Morse potential and to the mean amplitude. In many cases one may use
[MATHEMATICAL EXPRESSION OMITTED] (29)
a is often about 2 Å-1 for bond distances. A more accurate expression derived for diatomic molecules, which gives the correct temperature dependence, is
[MATHEMATICAL EXPRESSION OMITTED]
where x = hv/kT.
Structure Analysis; Theoretical Expressions. — Modified Molecular Intensity Functions and Radial Distribution Functions. Various research groups use slightly different methods in structure analysis. The author usually applies the |f(s)| values for two atoms in the molecule (k and l, see later how to choose these atoms) to compute a modified molecular intensity function
[MATHEMATICAL EXPRESSION OMITTED] (30)
[MATHEMATICAL EXPRESSION OMITTED] (31)
and M is the number of atoms in the molecule.
In the first Born approximation, equation (10), we get
[MATHEMATICAL EXPRESSION OMITTED] (32)
Figure 2 shows some examples of g(s) calculated from (31). The variation in g(s) is large only when the atoms i and j differ considerably in atomic numbers.
The choice of |fk| and |fl| in equation (30) is somewhat arbitrary and was more important before modern computers became available. The g functions were then usually assumed to be constant with s. Equations (31) and (32) show that this approximation may be fairly good in most cases by proper choice of the atoms k and l. Today one may very well do least-squares refinements (see p. 19 and p. 44) without modifying the molecular intensity curve.
