The Valence Force Model

We now turn to a consideration of the valence force model. This model has been widely used in analyzing molecular vibrations [4.19,20] but has been extended to study phonons in covalent or partly covalent crystals [4.21,22]. To illustrate the valence force field, we first consider the H2O molecule (Fig.4.18). The valence coordinates are bond distance changes and interbond angle changes. For the H O molecule, we define three internal coordinates r, where r = Ar j and r = Ar 2 the changes of the bond distances and 3 change of the bond angle a and the 

It is often possible to obtain a reasonable first approximation by neglecting the cross terms associated with Ar2jAr22 Ar2 (rAa), etc., because the force constants F, F are usually considerably smaller than the diagonal elements F and F of the F-matrix. Furthermore, F is roughly ten times smaller than F. For the H 0 molecule, F = 7.8 mdyn/ and F = 0.69 mdyn/  

Since the internal coordinates r represent changes of distances and angles, they are unaltered by pure translations or rotations of the molecule. Therefore, the potential energy is automatically invariant to those motions. 

Another advantage of the valence force field is that it is often possible to transfer force constants from one molecule to another molecule or crystal which contain similar bonds. 

For molecular vibrations there is a well-established technique called the 6F technique, which was developed by WILSON et al. [4.19]. An internal coordinate r can always be expressed in terms of the components of the Cartesian displacement coordinates u of the molecule (k = 1,2.N a = x,y,z)  

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Kuchitsu and co-workers5 7 were the first to introduce what is perhaps the simplest and most generally useful model, in which they assume all anharmonic force constants in curvilinear co-ordinates to be zero with the exception of cubic and quartic bond-stretching constants. These may be estimated from the corresponding diatomics, or from a Morse function, or they may be adjusted to give the best fit to selected spectroscopic constants to which they make a major contribution. This is often called the valence-force model. It is clear from the results on general anharmonic force fields quoted above that this model is close to the truth, and in fact summarizes 80 % of all that we have learnt so far about anharmonic force fields. 

Efforts have also been made to devise more sophisticated models, with a few more parameters, because some of the spectroscopic constants generally show a sensitivity to features of the force field which are not present in the valence force model. They range from extended Urey-Bradley models,7 through semi-empirical valency models,1 to an ad hoc introduction of extra anharmonic... 

Sulphur Trioxide. The He I photoelectron spectrum of SO3 has been measured and compared with the spectra of SO2 and BF3. The results indicate a substantial stabilization of the sulphur lone-pairs in SO2 and the possible involvement of central atom anharmonic force-field for SO3, based on the valence force model, has been investigated. Gas-phase Raman and i.r. spectra together with a band-contour calculation have been used to establish beyond doubt the assignment of at 497.5 and at 530.2 cm for SO3. 

A model which is often used for covalent or partly covalent crystals is the valence force model. This model has originally been developed for vibrations in molecules [4.19,20] but the formalism can be extended to crystals as well [4.21,22]. The potential energy is expressed in terms of changes in bond distances, bond angles and other so-called internal coordinates, we shall illustrate this model for the linear (SN) chain and discuss the results for diamond and 3-AgI. 

The valence force model has been applied to diamond. Bond stretching and bond bending coordinates are introduced and the model involves 6 independent force constants (one stretching, one bending and 4 interaction cbnstants). These 6 parameters were fitted to the observed phonon spectrum of diamond with the results shown in Fig.4.20, where it may be noted that a good fit has been achieved. 

Dynamical Matrix for the Valence Force Model The potential energy is... 

More recently, Costain and Sutherland 139> using a valence force model proposed the following potential for inversion vibrations in AX3 systems ... 

Valence-Force Model. The simplest harmonic oscillator model for the potential energy (/of a tetrahedral molecule such as CCL can be written as... 

Although the cohesive forces in such an idealized metal as described would be nondirectional (as in ionic solids), the orientation effects of d orbitals contribute a directional-covalent component to the bonding in transition metals that requires a more sophisticated definition for metallic bonding. The intemuclear distances in the close packed, or nearly close packed, stmcmres of most metalhc elements ate small enough that the valence orbitals on the metal atoms can overlap (in the valence-bond model) or combine to form COs (in the MO or Bloch model). 

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. 

N02,239,247,248,256 and C102256 should be regarded as tentative since they are based on various model potentials (as indicated in the footnotes) and on much less extensive sets of data. Table 3 lists the values of the cubic constants in the valence-force coordinate space (fyt) and in the dimensionless normal coordinate space (kg/s") as obtained for... 

A valence force model has been used to analyze our data and the Raman and IR data of Stolz et al. It turns out... 

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