An Introduction to Molecular Integral Evaluation

Abstract

The expectation value of the electronic energy of a n-electron system may be written as the ratio of two integrals,

$$E = < \psi \left| {{\text{ }}{H_e}} \right|\psi > / < \psi |\psi >$$

integration being taken over the 3n spatial and n spin coordinates of the electrons. Procedures for integration over the spin variables are not the concern of the present work, and we proceed under the assumption that the spin integrations of the energy expectation value have been completed. We are thus left with the evaluation of two integrals over the 3n spatial coordinates. A rather widely used procedure is to construct the trial form of the many electron wavefunction, Ψ, from a set of functions of the coordinates of one electron. We will denote the basis set of one electron functions by {Φ}. Notice that each basis function defines a set of n building blocks to be used in the construction of the wavefunction, since each function may be written in the coordinates of any of the n electrons.