Chapter 14 - Many-electron wavefunctions and model Hamiltonians

Abstract

In this chapter we examine how many-electron wavefunctions satisfying Pauli’s principle can be constructed starting from an orbital basis, and how appropriate model Hamiltonians can be introduced to find definite molecular orbital (MO) approximations to the molecular energy. An N-electron wavefunction is built as an N-term determinant (Slater det) of orthonormal spin-orbitals, and the reduction from the 4N-dimensional configuration space of the wavefunction to the ordinary three-dimensional space+spin is obtained in terms of one- and two-particle density matrices. Electron and spin densities are defined, and the average values of one- and two-electron operators are derived. The Hartree–Fock theory for closed shells is then introduced as the best independent particle model, and Hall–Roothaan self-consistent field equations are variationally derived from the appropriate functionals. Localization of the delocalized canonical MOs is discussed next, while Hückel and semiempirical extended Hückel’s theory, complete neglect of differential overlap, intermediate neglect of differential overlap and Zerner or spectroscopic intermediate neglect of differential overlap methods are introduced in the last part of this chapter.