Valence force model

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... 

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... 

Tentative values baaed on a modified valence-force model with 11 parameters. 

General quartic force field with three quartic constraints. c Tentative values based on a modified valence-force model with nine parameters. 

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

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. 

Mecke(4,122) has therefore suggested what may be conveniently called a valence force model. In it the potential energy is assumed to be of the following kind When the distance between two atoms joined by a valence bond is changed, there shall appear restoring forces along this valence bond, the valence forces when, on the other hand, the angle between two... 

Constants kjjk were also predicted from a van der Waals model and from a valence-force model [6,41 ]. A complete set of constants kjjkSimilar to those above and quartic constants were calculated from a model potential, containing three independent parameters for the enharmonic part, by a Monte-Carlo procedure [2]. 

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. 

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... 

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. 

Fig.4.20. Dispersion relations for diamond along (100) and (111) directions. The calculations (solid lines) are based on a valence force model described in the text [4.21]...

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

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