Model Hamiltonian in the Kirkwood Approximation

The most rigorous method for constructing the operator U in the continuum model is based on the Kirkwood theory. In this case, the distribution of point charges is replaced in the classic expression of Eq. (3.4) with the corresponding quantum mechanical distribution. In other words, in Eq. (3.4) and r and the angle jk between these vectors will have to be regarded not as constants but rather as variable quantities. Then, using the atomic units, U can be written as a sum of the nuclear-nuclear, nuclear-electron, and electron-electron terms  

It is assumed that in this system there are P nuclei and s electrons, and the positions of the nuclei R, are fixed. After solving the Schrodinger equation with the Hamiltonian in Eq. (3.7), we obtain the wave function of the solute molecule depending on the dielectric permittivity of the solvent. 

The complete continuum approach was employed in the Kirkwood model on an ab initio level with the basis set of the floating Gauss functions in 1976 [17]. Around that time, a similar formalism for taking the solvent into account was included in the CNDO/2 method [18]. However, such calculational schemes did not gain wide acceptance by reason of excessive expenditure of computer time, difficulties in evaluating some integrals and overall drawbacks inherent in the macroscopic approximation. Eventually some simplified techniques were developed, each of which takes usually one of the components in Eq. (3.8) into account. Next the simplest of these will be considered. 

Strictly speaking, the Born equation of Eq. (3.5) can be used only for calculating the solvaton energy of an ion. However with some additional assumptions, it is possible to construct from Eq. (3.5) a model Hamiltonian for neutral molecules. Such a scheme was first suggested by Klopman [19], while its nonselfconsistent version had already been employed in Ref. [20]. 

Let each charge of the solute molecule induce in the medium with the polarizability a a counter-charge  

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