Valence force field potentials

Fig, 5,8,2) reveal the presence of non-central interactions, which are typical of covalent materials this is explained in terms of angular Valence Force Field potentials (see text). (From Ref. 39.)... 

The quadratic-valence force field potential function for bent triatomic molecules XY2 is defined by... 

Three-body and higher terms are sometimes incorporated into solid-state potentials. The Axilrod-Teller term is the most obvious way to achieve this. For systems such as the alkali halides this makes a small contribution to the total energy. Other approaches involve the use of terms equivalent to the harmonic angle-bending terms in valence force fields these have the advantage of simplicity but, as we have already discussed, are only really appropriate for small deviations from the equilibrium bond angle. Nevertheless, it can make a significant difference to the quality of the results in some cases. 

We refer to models where we write the total potential energy in terms of chemical endties such as bond lengths, bond angles, dihedral angles and so on as valence force field models. 

Molecular Energetics. Molecular energies can be computed in a variety of ways including empirical fixed valence potentials, full force field potentials, and semi-empirical molecular orbital techniques (CNDO-2, INDO, MINDO-3, MNDO, PCILO). 

The potential energy expressions used for force field calculations are all descendants of three basic types originating from vibrational spectroscopy (5) the generalized valence force field (GVFF), the central force field, and the Urey-Bradley force field. General formulations for the relative potential energy V in these three force fields are the following ... 

We have seen that for our calculations essentially two types of force fields have to be considered VFF- and UBFF-expressions. The main difference with repect to spectroscopic force fields consists in the superposition of nonbonded interactions. The force fields used so far for our purposes are almost exclusively simple valence force fields without cross terms, and a veriety of UB-force fields. Only recently could experiences be gathered with a valence force field that includes a number of important cross terms (79). Vibrational spectroscopic force fields of both types have been derived and tested with an overwhelming amount of experimental data. The comprehensive investigations of alkanes by Schachtschneider and Snyder (26) may be mentioned out of numerous examples. The insights gained from this voluminous spectroscopic work are important also when searching for suitable potentials for our force-field calculations. 

The above discussion has outlined the theoretical approach to the determination of the fundamental vibration frequencies of a molecule. The practical solution of the problem as formulated above presents, however, certain more or less serious difficulties. For example, the completely general potential function of equation (4) is generally not usable even for small molecules, because it contains more independent constants than can be determined from the experimental data. However, by making certain assumptions about the nature of the force field in the molecule, the number of constants can be reduced. One assumption which often works quite well in practice is that of a valence force field [Herzberg 76)]. This assumes that contributions to the potential energy... 

The focus in the reaction dynamics studies was on the N02 elimination channel, but they also studied the HONO elimination reactions [70]. They based the potential energy surface on experimental data but performed some minimal basis set ab initio calculations to determine geometries, force fields, torsional potentials, and some information about the reaction paths. The representations of the global potential energy surfaces were based on valence force fields for equilibrium structures with arbitrary switching functions operating on the potential parameters to effect smooth and (assumed) proper behavior along the reaction paths. Based on the available experiments [71-73], they assumed that the primary decomposition reaction is simple N-N bond rupture to eliminate N02. 

This type of representation of the potential energy in terms of the internal (valence) degrees of freedom is called a Valence Force Field. Valence force fields have long been used in vibrational spectroscopy in order to carry out normal mode analysis[j ]. Basically what the terms in equation (2) express are the energies required to deform each internal coordinate from some unperturbed... 

Force constant calculations are facilitated by applying symmetry concepts. Group theory is used to find the appropriate linear combination of internal coordinates to symmetry-adapted coordinates (symmetry coordinates). Based on these coordinates, the G matrix and the F matrix are factorized, which makes it possible to carry out separate calculations for each irreducible representation (c.f. Secs. 2.133 and 5.2). The main problem in calculating force constants is the choice of the potential function. Up until now, it has not been possible to apply a potential function in which the number of force constants corresponds to the number of frequencies. The number of remaining constants is only identical with the number of internal coordinates (simple valence force field SVFF) if the interaction force constants are neglected. If this force field is applied to symmetric molecules, there are often more frequencies than force constants. However, the values are not the same in different irreducible representations, a fact which demonstrates the deficiencies of this force field (Becher, 1968). 

In the general valence force field (GVFF), all // constants are taken into account. To n normal vibrations of the same symmetry correspond n stretching or deformation constants and ( -l)/2 interaction constants. The GVFF is the most appropriate potential function from a chemical point of view. However, for a few small molecules all force constants of the GVFF have been calculated reliably, since with decreasing symmetry and with increasing molecular size the number of force constants by far exceeds the number of frequencies. Restrictions to the potential function have therefore been recommended for many problems. 

Infrared spectra are potentially a very useful source of information on cis and trans effects. The frequency of a vibration is determined by the kinetic and potential energies of the nuclei involved (theoretically the whole complex, since all vibrations will couple to a certain extent) and the observed frequencies can be analyzed in terms of a generalized valence force field or some more approximate model to eliminate the kinetic energy and yield force constants (for stretching, bending, twisting, etc.) and interaction constants l iO)... 

For a more detailed understanding of complex fluid molecules, a valence force field [54] representing many-body interactions is commonly used. In this approach, the potential energy of a system is expressed as a sum of valence (or bonded) and nonbonded interactions... 

Nonbonded interactions consist of van der Waals (VDW) and electrostatic potentials. Examples of the valence force field approach include UFF or DREIDING [54], MM2/MMP2 [55], AMBER [56], and CHARMM [57]. The parameters of the potentials can be determined from either experiments or ab initio quantum chemical methods [58]. 

At normal temperatures the lattice dynamics involves predominantly low amplitude atomic motions that are well described in a harmonic approximation. Therefore, potential models widely used in the theory of molecular vibration, such as a generalized valence force field (GVFF) model, may be of use for such studies. In a GVFF the potential energy of a system is described with a set... 

Many such model force fields have been discussed in the literature. [For general discussions, see Herzberg (1945), Wilson et al. (1955), Woodward (1972), and Califano (1976). For discussions of the Urey-Bradley force field, see the review by Duncan (1975). For discussions of the entirely different consistent force field approach, see Lifson and Warshel (1968), Warshel et al. (1970), and Burkert and Allinger (1982).] We have chosen to use a simplified general valence force field (SGVFF), which has been defined as one which contains the minimum possible number of interaction constants compatible with a good fit of the spectra (Califano, 1976). Such a force field has been demonstrated to be very effective for hydrocarbons (Schachtschneider and Snyder, 1963). For this form the potential energy of Eq. (63) is written explicitly as... 

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