Perturbation theory solving many-electron Schrodinger

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. 

Because of interelectronic repulsions, the Schrodinger equation for many-electron atoms and molecules cannot be solved exactly. The two main approximation methods used are the variation method and perturbation theory. The variation method is based on the following theorem. Given a system with time-independent Hamiltonian //, then if

well-behaved function that satisfies the boundary conditions of the problem, one can show (by expanding

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Unfortunately, the stationary Schrodinger equation (1.13) can be solved exactly only for a small number of quantum mechanical systems (hydrogen atom or hydrogen-like ions, etc.). For many-electron systems (which we shall be dealing with, as a rule, in this book) one has to utilize approximate methods, allowing one to find more or less accurate wave functions. Usually these methods are based on various versions of perturbation theory, which reduces the many-body problem to a single-particle one, in fact, to some effective one-electron atom. 

The Schrodinger equation for the one-electron atom (Chapter 6) is exactly solvable. However, because of the interelectronic-repulsion terms in the Hamiltonian, the Schrbdinger equation for many-electron atoms and molecules cannot be solved exactly. Hence we must seek approximate methods of solution. The two main approximation methods, the variation method and perturbation theory, will be presented in Chapters 8 and 9. To derive these methods, we must develop further the theory of quantum mechanics, which is what is done in this chapter. 

A particularly valuable addition to the arsenal of methods has been density functional theory (DFT), which aims to predict molecular properties with greater accuracy than Hartree-Fock calculations. Rather than directly integrating the electronic Schrodinger equation to get the AT-electron wavefunction (where N is the number of electrons in the molecule), DFT methods solve instead for the overall electron density, po. The many-electron wavefunction is a function of 3N coordinates Xi,yi,Zj for each electron i), but po depends on just 3 coordinates x, y, and z. Only this density function is needed to calculate the energy, and many other molecular properties. By skirting the need for the explicit, many-electron wavefunction, DFT methods provide a fast alternative route to predicting properties of molecules. Other methods make use of perturbation theory, variational configuration interaction, and extensions of valence bond theory. 

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