Vibrational model force fields

This situation changed drastically when it was discovered in the 1990s that density functional (DF) methods do a much better job of modeling force fields than (affordable) wave function based methods. Already within the local density approximation (LDA) of DF theory, vibrational frequencies were predicted with... 

This Report is arranged as follows. Section 2 is concerned with the representation of force fields the definition of force constants, choice of units, etc. Section 3 is a brief discussion of the theory and interpretation of diatomic vibration-rotation spectra, and is intended to act as an introduction to the greater complications of polyatomic molecules. Section 4 is concerned with the theoretical and mathematical problems involved in relating the spectra of a polyatomic molecule to its force field, and in trying to calculate the force field from observed data. Finally, in Section 5 we discuss some of the calculations carried out at this time, with examples, and we consider some of the problems involved in finding useful model force fields. 

Before it is possible to interpret on a rigorous basis the behavior of the carbonyl stretching frequencies of a series of isostructural and isoelectronic complexes complete vibrational analyses are necessary. However, it is only within the last few years that far-infrared 137) and laser Raman 84) spectrometers have become available generally. Hence, in the general absence of the data they have provided, earlier complete analyses were limited to the spectra of simple metal carbonyls (for which such information was available). Even for these complexes, the number of force constants exceeds the number of observable frequencies, and model force fields had to be used. Since Urey-Bradley type force fields proved to be unsuitable for carbonyl complexes 86,105, 106), Jones 80-82) developed a resonance interaction valence force field which reduced the number of force constants by interrelating several on the basis of orbital overlap. This approach is not readily adaptable to less symmetrical substituted carbonyl complexes. Alternative models had, therefore, to be investigated. 

In Eq. (29), A represents the matrix containing A(=47r c Vj ) values along its diagonal, and, in Eq. (30), E represents the unit matrix. From Eq. (29) it is evident that the elements Ljj are dependent upon the force and interaction constants and thus upon the details of the assumed force field. In general, then, to interpret the relative or absolute intensities of carbonyl stretching vibrations, it is necessary first to assume some model force field and then evaluate the elements of the L matrix. 

The difficulty with constrained force field calculations is that from their very nature it is not possible to be certain of the extent to which the constraints imposed are justified. If one s motive is to obtain the true force field-for comparison with a priori calculations, for example -they should be treated with suspicion if, on the other hand, one wishes to calculate the expected vibration frequencies of a related molecule as an aid to assignment, then there is no doubt that model force field calculations are often successful. Perhaps the most important point is that constraints-when imposed—should be clearly specified in reporting the force field. 

At this point, spectroscopists and molecular modellers part company because they have very different aims. Spectroscopists want to describe the vibradons of a molecule to the last possible decimal point, and their problem is how a force field should be determined as accurately as possible from a set of experimental vibrational frequencies and absorption intensities. This problem is well understood, and is discussed in definitive textbooks such as that by Wilson, Decius and Cross (1955). 

Raman intensities of the molecular vibrations as well as of their crystal components have been calculated by means of a bond polarizibility model based on two different intramolecular force fields ([87], the UBFF after Scott et al. [78] and the GVFF after Eysel [83]). Vibrational spectra have also been calculated using velocity autocorrelation functions in MD simulations with respect to the symmetry of intramolecular vibrations [82]. 

Vibrational spectroscopy has played a very important role in the development of potential functions for molecular mechanics studies of proteins. Force constants which appear in the energy expressions are heavily parameterized from infrared and Raman studies of small model compounds. One approach to the interpretation of vibrational spectra for biopolymers has been a harmonic analysis whereby spectra are fit by geometry and/or force constant changes. There are a number of reasons for developing other approaches. The consistent force field (CFF) type potentials used in computer simulations are meant to model the motions of the atoms over a large ranee of conformations and, implicitly temperatures, without reparameterization. It is also desirable to develop a formalism for interpreting vibrational spectra which takes into account the variation in the conformations of the chromophore and surroundings which occur due to thermal motions. 

Entropies and heat capacities can thus now be calculated using more elaborate models for the vibrational densities of states than the Einstein and Debye models discussed in Chapter 8. We emphasize that the results are only valid in the quasiharmonic approximation and can only be as good as the accuracy of the underlying force-field calculation of such properties can thus be a very sensitive test of interatomic potentials. 

A potentially much more adaptable technique is force-field vibrational modeling. In this method, the effective force constants related to distortions of a molecule (such as bond stretching) are used to estimate unknown vibrahonal frequencies. The great advantage of this approach is that it can be applied to any material, provided a suitable set of force constants is known. For small molecules and complexes, approximate force constants can often be determined using known (if incomplete) vibrational specha. These empirical force-field models, in effect, represent a more sophisticated way of exhapolating known frequencies than the rule-based method. A simple type of empirical molecular force field, the modified Urey-Bradley force field (MUBFF), is introduced below. 

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