Force constants virial theorem

For a harmonic oscillator with frequency v, the potential energy equals M7kJu2 where k. is the force constant. Using the virial theorem, which states... 

From the chemical point of view, we must say these equations are not tractable and provide no useful information. In common, the study carried out by many authors (Salem, 1963b Byers-Brown, 1958 Byers-Brown and Steiner, 1962 Bader, 1960b Murrell, 1960 Berlin, 1951 Ben-ston and Kirtman, 1966 Davidson, 1962 Benston, 1966 Bader and Bandrauk, 1968b Kern and Karplus, 1964 Cade et al., 1966 Clinton, 1960 Phillipson, 1963 Empedocles, 1967 Schwendeman, 1966) on the force constants is based on the application of the virial and the Hellmann-Feynman or the electrostatic theorems. In particular, the Hellmann-Feynman theorem provides the expression for ki which relates the harmonic force constant to the properties of molecular charge distribution p(r), i.e., it follows (Salem, 1963b) that... 

The calculated potential-energy curve E (R ) for diatomic molecules in the Bom-Oppenheimer approximation does not, in general satisfy the virial theorem, if approximate wave functions are used. There are two methods which can be used to compute force constants and frequencies, viz. the methods of McLean (81) and PhiUipson (82). 

It is perhaps worthwhile to return again to the poor results which Ross and PhilUpson (97) obtained when certain "popular wave functions which appear in the literature are used to calculate the force constants through the virial theorem Eq. (5.18). In these cases a fixed function of a certain form is used instead of the correct wave function to determine the energy, e. g. the familiar HeiUer-London function (92) is used ... 

The reason why the Heitler-London method gives such a bad Hellmann-Feynman force-constant is thus that R is "attached to the nuclei so that 9 / R is not zero the same applies to the Weinbaum function 94), to the Wang function (95), and to the Coulson function 96) for Hg. To yield better force results, the variable parameters ) must be "detached from the nuclei and their values determined at each internuclear configuration. A wave function in which the parameters are determined by the variational procedure is called a floating function by Hurley 93, 97, 30, 31, 32) (this is eqtuvalent to Hall s stable wave functions 88)). This procedure can be extended to the scale factors, as discussed by McLean 81) and Lowdin 83). The vibrational frequency of H2 determined by Ross and Phillipson using the differentiation of the virial theorem (which assumed that all the variable parameters are variationally... 

Empedocles [701) proposed a general semi-empirical approach using data from the united-atom and separated-atom models and calculate the quadratic (harmonic), cubic and quartic force constants kz, kz and ki from the differentiation of the virial theorem electronic kinetic energy expression Eq. (4.27), where E is the total energy... 

PhiUipson 146) has given an extension of his diatomic molecule force-constant theory to polyatomic molecules, using the scaling procedure and the virial theorem no numerical calculations have appeared. 

A different type of structural parameter was considered recently by the present authors, namely X representing a homogeneous macroscopic strain defined as the linear scaling of all particle positions as x- 1+e)x. The e is a constant 3x3 strain tensor, and e=0 corresponds to some reference configuration. The conjugate force is in this case defined as the macroscopic stress a, and an explicit general expression denoted the "stress theorem" is derived by Nielsen and Martin (1983). The result is a generalization of the quantum virial theorem (Born et al., 1926),... 

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