Overlap integral Perturbation theory

Bardeen considers two separate subsystems first. The electronic states of the separated subsystems are obtained by solving the stationary Schrodinger equations. For many practical systems, those solutions are known. The rate of transferring an electron from one electrode to another is calculated using time-dependent perturbation theory. As a result, Bardeen showed that the amplitude of electron transfer, or the tunneling matrix element M, is determined by the overlap of the surface wavefunctions of the two subsystems at a separation surface (the choice of the separation surface does not affect the results appreciably). In other words, Bardeen showed that the tunneling matrix element M is determined by a surface integral on a separation surface between the two electrodes, z = zo. 

By ab initio we refer to quantum chemical methods in which all the integrals of the theory, be it variational or perturbative, are exactly evaluated. The level of theory then refers to the type of theory employed. Common levels of theory would include Hartree-Fock, or molecular orbital theory, configuration interaction (Cl) theory, perturbation theory (PT), coupled-cluster theory (CC, or coupled-perturbed many-electron theory, CPMET), etc. - We will use the word model to designate approximations to the Hamiltonian. For example, the zero differential overlap models can be applied at any level of theory. The distinction between semiempirical and ab initio quantum chemistry is often not clean. Basis sets, for example, are empirical in nature, as are effective core potentials. The search for basis set parameters is not usually considered to render a model empirical, whereas the search for parameters in effective core potentials is so considered. 

In an HMO calculation the reductions due to deviation from maximum (pd)v overlap may be introduced by the use of perturbation theory ll9) instead of solving the secular determinant directly. As usual, the Hiickel bond-intergral parameter (3 v is assumed to be proportional to the overlap integral of the corresponding bond 6>7). The reduction is therefore proportional in our case to the cosine of the angle of deviation y (10). 

The symbol 0(5 ) stands for all the terms that vanish with the fourth power of the overlap integrals or faster. The valence repulsion already appears in the first order of the perturbation theory (besides the electrostatic energy as a result of the Pauli exclusion principle. ... 

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