Bonding 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. 

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... 

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. 

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. 

Bohr s hydrogen atom model of 1913 had provided inspiration to a few physicists, like Kossel, who were interested in chemical problems but to very few chemists concerned with the explanation of valence. First of all, the Bohr atom had a dynamic character that was not consistent with the static and stable characteristics of ordinary molecules. Second, Bohr s approach, as amended by Kossel, could not even account for the fundamental tetrahedral structure of organic molecules because it was based on a planar atomic model. Nor could it account for "homopolar" or covalent bonds, because the radii of the Bohr orbits were calculated on the basis of a Coulombic force model. Although Bohr discussed H2, HC1, H20, and CH4, physicists and physical chemists mainly took up the problem of H2, which seemed most amenable to further treatment. 11... 

Unfortunately, most of the structural information of IR spectra is contained in the often very crowded region of 500-1600 cm which was hardly exploited for diagnostic purposes except in the case of very small molecules with few vibrations, or for pattern matching of spectra of reactive intermediates obtained independently from different precursors. The reason for this was that the prediction of IR spectra was only possible on the basis of empirical valence force fields, and the unusual bonding situations that prevail in many reactive intermediates made it difficult to model the force fields of such species on the basis of force constants obtained from stable molecules. 

The ionic model divides the forces acting on atoms into an electrostatic component that can be calculated using classical electrostatic theory and a short-range component that is determined empirically. The previous chapter explored the properties of the classical electrostatic field. This chapter explores the properties of the empirically determined short-range force which is represented in the electrostatic model by the bond capacitance, C,y, defined in eqn (2.8). Rather than try to determine the values of Cy directly, it is better to step back and look at the way in which the bond valence model developed historically. Its connection with the electrostatic model of Chapter 2 will then become apparent. 

A second empirical observation related to the short-range forces of the ionic model is the observation that in many crystals the experimental bond valences also obey eqn (3.4) (Brown 9%lb) ... 

A closer comparison of bond valence and electron density models is not possible because of the different underlying assumptions of the models. The forces in the bond valence model act between structureless point atoms, but the forces in the electron density model are exerted by electrons on nuclei and vice versa. This basic difference makes it difficult to compare the two models in greater detail. They are best seen as complementary, the electron density model providing important information about the nature of the bonding between the atoms, the bond valence model providing a simple tool for predicting structure and properties, particularly in cases where the structure is complex. 

While more physically based models provide a picture of the underlying forces that lead to chemical bonding, the bond valence model reduces the rules of chemistry to their simplest mathematical form. In this form it is able to provide insights into the behaviour of the many complex systems found in acid-base chemistry. 

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