The exact electronic Schrodinger equation

The time-independent Schrodinger equation for a molecular system of arbitrary complexity can be easily written in the form 

We assume the Bom-Oppenheimer approximation so as to decouple the motion of the electrons from that of the nuclei. For a molecular system containing N electrons moving in the field of M fixed nuclei, the non-relativistic electronic Hamiltonian has 

The prime in (3.7) indicates that terms with i = y are omitted. To simplify our notation, we shall drop the subscript e from the electronic Hamiltonian operator since we are only eoneerned with the electronic structure problem in this volume. 

In order to gain some understanding of the nature of the many-body problem in atoms and molecules, let us consider an array of well-separated systems, a Unear array of helium atoms, for example. By weU-separated we mean that the systems are not interacting. For simplicity, let us begin by considering just two weU-separated systems. The total Hamiltonian operator for the supersystem may be written 

Here Mp is the number of nuclei associated with system P, whilst Np is the number of electrons associated with system P. Because the subsystems are well separated, there are no terms in the Hamiltonian operator (3.10) associated with interactions between them. We conjecture that the exact wave function for the supersystem may 

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We have not solved the exact electronic Schrodinger equation... 

We now follow the procedure that we adopted above for the exact electronic Schrodinger equation and conjecture that the model electronic wave function for the supersystem can be written... 

In this section, we have demonstrated that this requirement is satisfied by the solutions of the exact electronic Schrodinger equation and by the solutions of the eigen-problem for some approximate electronic Hamiltonian operator associated with an arbitrary independent particle picture. 

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