Molecular intensities

The nature of the intemuclear distance, r, is the object of interest in this chapter. In Eq. (5.1) it has the meaning of an instantaneous distance i.e., at the instant when a single electron is scattered by a particular molecule, r is the value that is evoked by the measurement in accordance with the probability density of the molecular state. Thus, when electrons are scattered by an ensemble of molecules in a given vibrational state v, characterized by the wave function r /v(r), the molecular intensities, Iv(s), are obtained by averaging the electron diffraction operator over the vibrational probability density. 

In Eq. (5.2), the function i iv(r) 2/r = P(r)/r is an example of a so-called radial distribution (RD) function, in the form in which it is obtained from gas-electron diffraction, in this case for a particular vibrational state of a diatomic molecule. It is seen that the molecular intensity curve is the Fourier transform of Pf. The reverse, by inversion, the RD function is the Fourier transformation of the molecular intensities ... 

Molecular ensembles usually exist in some distribution over several vibrational states. Denoting the population of the vth state by pv, the total molecular intensity function can then be calculated by summing over the states ... 

Fig. 5.3 Average molecular intensities, /molO), (left) and their Fourier transform (right) calculated for C = S in a Boltzmann distribution at 2000K, using the data shown in Figs. 5.1 and 5.2.

For polyatomic molecules the situation is somewhat more complex but essentially the same. The effect of intramolecular motion upon the scattering of fast electrons by molecular gases was first described by Debye3 for the particular case of a molecular ensemble at thermal equilibrium. The corresponding average molecular intensity function can be expressed in the following way ... 

Lj-atom pairs. Again, a Fourier transform of the molecular intensities yields the RD-cxnrve, Pj/r, which is a composite curve for polyatomics, containing a series of probability maxima at different distances, some of them possibly closely spaced and overlapping. 

This expression is given here, without derivation, for the molecular contribution to the total electron scattering intensities (molecular intensities) ... 

Figures 4 and 5 show not only the experimental distributions but also the distributions calculated for the best model of tetramethylsilane, which is characterized by the following bond lengths and bond angle and Td symmetry and staggered methyl conformation, Si-C 1.877(4)A, Si-H 1.110(3)A, and Si-C-H 111.0(2)°. These are so-called average parameters lyide infra). The radial distribution is convenient to visually inspect the validity of a model and to read off some principal intemuclear distances, but the quantitative determination of all the parameters is done on the basis of the molecular intensities. The refinement of parameters usually starts from an initial set of parameters. The expression of the molecular intensities is a non-linear relationship, a good choice of the initial parameters will ensure that the calculation reaches the global rather than a local minimum.

Figure 4. Molecular intensities (E - experimental and T - theoretical) for tetramethylsilane. ...

The fa intemuclear distance in the expression of molecular intensities (vide supra) is an effective parameter without rigorous physical meaning. It is, however, related in a good approximation to the thermal average... 

The radiation nozzle system has been used for studying a series of transition metal dihalide molecules. Typical molecular intensity distributions are shown in Fig. 4 for manganese(II) chloride. The quickly damping character of the intensity distribution relates to the large-amplitude motion in the molecule due to the high temperature ( 750 °C) conditions of the experiment. Fig. 5 shows the radial distribution from the same experiment which also well demonstrates the straightforward manner of structure determination of such simple molecules. 

The purpose of classical methods is to obtain transferable molecular intensity parameters, to calculate reliable intensities within reasonable computation times, and to investigate large interesting molecules. 

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 ... 

FIG. 2 Electron-dif action intensity curves. The two iqipermost curves are the total intensities, I(s), obtained at two different nozzle-to-plate distances. Empirical background is shown. Below is a composite molecular intensity curve sM(s). The bottom curve is the difference between e q)eriment and theory multiplied by a factor of two. 

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... 

FIGURE 1 Reduced molecular intensity for BF,(Kutchitsu and Konaka, 1966). Values observed from three photographic plates are shown in dots. The solid curve is calculated from the rotational constants measured by high-resolution spectroscopy with corrections for vibrational effects. The lower curve is the difference between the observed and theoretical curves for one of the photographic plates showing the magnitude of random experimental error, which is within the estimated limits of experimental error shown by the broken lines. 

The contribution of the long H H distances to the molecular intensity was too small to allow the relative orientation of the methyl groups to be determined. 

The stractural analysis was carried out using cumulant expansion for molecular intensities. The effects of higher-order cumulants, of multiple scattering, and of different spectroscopic anharmonic force fields on the results of the analysis were investigated. 

The stmctural analysis was carried out using cumulant expansion for molecular intensities. 

Structure Analysis Theoretical Expressions.—Modified Molecular Intensity Functions and Radial Distribution Functions. Various research groups use slightly diHerent methods in structure analysis. The author usually applies the /(s) I values for two atoms in the molecule k and /, see later how to choose these atoms) to compute a modified molecular intensity function ... 

The choice of f and fi in equation (30) is somewhat arbitrary and was more important before modem 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 /. Today one may very well do least-squares refinements (see p. 19 and p. 44) without modifying the molecular intensity curve. ... 

The notation is slightly simplified compared to previous equations. The summation is over all non-equivalent distances in the molecule and nn the number of equivalent distances. If gify) may be regarded as a constant and Kii = 0, the modified molecular intensity is a superposition of damped sine curves. 

Figure 3 Experimental circles) and theoretical molecular intensity curves for C33tSt. The differences between experimental and theoretical values are also shown. The experimental curve is composed of data taken at two nozzle-to-plate distances...

Calcidation of Experimental Molecular Intensity Functions The total intensity, which is the sum of the molecular intensity and the background [c/. equation (18)], is obtained experimentally, usually firom photographic plates. In routine work the structure information is supposed to lie in Ita(s) only and to derive this function in a background, IiJls), must be found. The theoretical background, I s), may be calculated from (20). However, the theoretical background cannot in general be used for all s values because of systanatic errors (see p. 39). 

When K s) in equation (43) has been determined, the author usually calculates a modified molecular intensity function by subtracting K s) from nd multiplying the result with the expression necessary to obtain the experimental function corresponding to equation (30) or (36) ... 

Hoemi studied the double scattering in diatomic molecules. He used the second Bora approximation and foimd no contribution to the total intensity which interfered with the molecular intensity. This result has recently been confirmed by Yates and Tenney using a theory for multiple scattering developed by Glauber. 

Figure 9 Levelled intensity for Hj calculated with Weinbaum wavefunction (A) and in the independent atom approximation (B). C is the approximate background which has to be used to give the same molecular intensity from A as obtained from B...

Intensity Errors. A number of the error sources mentioned in Table 2 (especially 4—10) will leave the zero points in the molecular intensity function [equation (21)] almost unchanged, while the relative intensity values may be more seriously changed. Such errors will primarily affect the mean amplitudes of vibration. However, it should be emphasized that the correlation between some distances and amplitudes may be large in molecules with many nearly equal distances. Table 3 shows some of the correlation coefficients (c/. p. 45) in hexafluorobenzene. In such cases the distance parameters may also be affected to a large extent by these errors. 

An error in the determination of the centre of the diffraction pattern wfll primarily reduce the amplitudes of the molecular intensity curve. Since the effect is largest close to the plate centre, the u values obtained may be too small. It is usually possible to make the error insignificant. A similar effect may be induced by deviations from radial symmetry in the pattern caused by electrostatic charges. 

The amplitudes of the molecular intensity curve may be 10—25 % of the total Intensity curve for small s values (s =a5 A ) and decrease as s increases for s s%i25A the corresponding number may be 1—3%. A standard deviation of 0.1 % in the total intensity may thus give standard deviations in the molecular intensities of less than 1 % of the amplitudes for small s values and about 5 % for s f 25 A. ... 

Figure 14 The curves drawn in full line show the theoretical molecular intensities for perfluoroDewarbenzene. Experimental average values are given by circles. The data were recorded with nozzle-to-plate distances of about 50.0, 25.0, and 15.1 cm. As is 0.125 A for the upper curve and 0.25 for the others. The lower part of the Figure shows the difference between experimental and theoretical intensities full line) as well as standard deviations dots) obtained experimentally by using five plates for the shortest nozzle-to-plate distance and four plates for the other data sets. [cf. equation (89)]. Note the difference in scale for the upper and lower part of the Figure. (B. Ahlquist, B. Andersen, and H. M. Seip, unpublished results).

Finally, some papers which carry the theory of vibrational averaging to higher levels of approximation should be mentioned. Bartell, in his 1955 paper " on the Morse oscillator probability distribution, considers the effect of an increase of temperature on r. Bartell s theory is extended by Bartell and Kuchitsu and by Kuchitsu these papers show in particular that the effective mean amplitude obtained by refinement on the molecular intensity curve is not quite equal to the harmonic mean amplitude calculated from the harmonic force field. Bonham and co-workers have calculated the effect of temperature on both a Morse oscillator and an oscillator in an RKR potential energy curve. In their final paper an informative series of diagrams shows how the quantum-mechanical average passes into the classical average at high temperature. 

The main information about molecular vibrations is contained in the term exp[—(V2(,/i )]. Here ly is the root-mean-square amplitude of vibration for the pair of atoms i and j. This exponential term is a damping term and leads to a decrease of intensity of the oscillations along the radius of a diffraction pattern. This can be nicely seen in the molecular intensity curve of the molecule P4 shown in Figure 10.10(a) as all P-P distances in this molecule are equivalent there is only one sine contribution. 

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