The Valence Force Field

The three independent elastic constants describe the rigidity of the system under uniform distortions but not under arbitrary nonuniform distortions. The elastic energy of the sy.stem,, is a function of the positions of all of the atoms in the crystal, or in particular, of the components , of the displacements of each of the atoms from equilibrium there are 6N such components for the atom pairs. For sufficiently small distortions we can expand that energy for small 

All dE, JdUj vanish identically, since they arc evaluated at the equilibrium positions. The d E, Jdu ()uj arc constants of the crystal and function as spring constants or force constants. Many arc related by symmetry, but there arc nevertheless a number (of the order of the number of atoms in the crystal) that are independent and they must be known if one wishes to calculate all the normal modes of vibration of the crystal. 

It is possible in principle to calculate all of these modes from the theory of the electronic structure, which is equivalent to the calculation of all the force constants. Indeed we will see that this is possible in practice for the simple metals by using pseudopotential theory. In covalent solids, even within the Bond Orbital Approximation, this proves extremely difficult because of the need to rotate and to optimize the hybrids, and it has not been attempted. The other alternative is to make a model of the interactions, which reduces the number of parameters. The most direct approach of this kind is to reduce the force constants to as few as possible by symmetry, and then to include only interactions with as many sets of neighbors as one has data to fit- for example, interactions with nearest and next-nearest neighbors. This is the Born-von Karman expansion, and it has somewhat surprisingly proved to be very poorly convergent. This simply means that in all systems there arc rather long-ranged forces. 

A second possibility is to model the interactions on the basis of theoretical or intuitive ideas in order to reduce the number of independent parameters. Perhaps the simplest such model for the diamond or zincblende slruclure is to assume the bond-stretching force constant Cq introduced in Eq. (8-1) and bond-bending interactions, which are written 

In all such models the more constants one introduces, the more experimental constants one c 4it and, it is always hoped, the more accurately also can one predict other quantities, though the process has not proved rapidly convergent. The particular model that will be used here fits nicely with the spirit of the 

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IR absorptions of these species were assigned to fundamental modes by comparison with the spectra of stable perfluoroorganic compounds. Normal coordinate analysis of the perfluoroethyl radical was performed and the valence force field of C2F5 was calculated (Snelson et al., 1981). 

In the construction of the matrix F of Eq. (63), the symmetrical equivalence of the two O-H bonds was taken into account. Nevertheless, it contains four independent force constants. As the water molecule has but three fundamental vibrational frequencies, at least one interaction constant must be neglected or some other constraint introduced. If all of the off-diagonal elements of F are neglected, the two principal constants, f, and / constitute the valence force field for this molecule. However, to reproduce the three observed vibrational frequencies this force field must be modified to include the interaction constant... 

We saw in Section 8-C that the strain 64 introduced four complications in the problem that were not present in the strain ei- The valence force field bypasses three of these, but leaves us with internal displacements. These are of interest in their own right and must be included if we wish to predict C44 in terms of the valence force field and parameters obtained from c, and c,2. That will be an interesting prediction since it gives some measure of the validity of the valence force field model, so let us proceed with it. 

The calculation of vibration spectra in terms of force constants is similar to the calculation of energy bands in terms of interatomic matrix elements. Force constants based upon elasticity lead to optical modes, as well as acoustical modes, in reasonable accord with experiment, the principal error being in transverse acoustical modes. The depression of these frequencies can be understood in terms of long-range electronic forces, which were omitted in calculations tising the valence force field. The calculation of specific heat in terms of the vibration spectrum can be greatly simplified by making a natural Einstein approximation. 

The large difference between the angular force constants Ci determined in the two ways appears to be the principal defect in application of the valence force field theory to the treatment of covalent solids. The defect is not readily repaired by the... 

This formula, however, has the same form—(l/2)C[Z > (5a - as the valence force field introduced in Eq. (8-18), except that there we associated contributions with angles both at the metallic and nonmetallic atoms. We must therefore divide this expression by two before equating it to (l/2)C,E5,>/j(5ap if we wish to identify it with the C given in Eq. (8-22) doing this, we rewrite Eq. (8-22) as... 

The Valence Force Field and Organic Molecular Mechanics Computations... 

The theoretical basis for a full valence force field treatment of extended lattices lies with the work of Kleinman and Spitzer published in 1962. ° These authors, who developed their force field to calculate the vibrational frequencies of quartz, felt that the most accurate way to represent the vibrational motion in a quartz crystal was to include the relative motion of oxygen and silicon atoms. The valence force field was the most effective method for treating this localized picture. 

The valence force field for XeF, contains 20 force constants, but only nine fundamentals are observed the system is clearly underdetermined, and some assumptions must be made about the force field. Our approach initially involved the transfer of interaction constants from related molecules. A detailed treatment by Curtis has given uncertainty limits for the force constants of IF, and XeOF, moreover, frU is usually assumed to be zero in systems with massive central atoms. IF, was also considered since it has unusual vibrational properties (v,> vi> v,y, the most noticeable feature of its force field is f <0. Good agreement with the experimental frequencies of XeF was obtained in calculations using a simple seven parameter force field / = 4.3, /, = 4.0,... 

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]. 

Linearity of BeFg was confirmed by electric-deflection studies (1 ) of the vapor and by infrared studies (j ) of matrix-isolated BeF. Infrared absorptions were found near 330 and 1540 cm" in neon matrices, leading to gas-phase estimates of 345 and 1555 cm" (13). The latter absorption was observed (14) at 1520 cm" in the infrared spectra of the vapor at 1000°C. We adopt a compromise of 1530 cm" and use this value in the valence-force-field approximation to estimate 670 cm" for the symmetric stretching mode. 

The vibrational frequencies for the three lower lying states were calculated via the valence force field method using the force constants calculated by O Neil, Schaeffer, and Bender ( ). The geometry and vibrational levels for the c A state were estimated. The uncertainty in the entropy mainly reflects the uncertain position of the low lying electronic levels. 

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