Introduction

Abstract

In the quantum mechanical treatment of the stationary states of molecular systems (ground state and excited states) one uses frequently the following Hamiltonian:

$$H = \sum\limits_{i = 1}^N {\left( { - \frac{{{h^2}}}{{2m}})\nabla _i^2 - \sum\limits_{\alpha = 1}^{{N_\alpha }} {\frac{{{Z_\alpha }{e^2}}}{{{r_{\alpha i}}}}} } \right)} + \sum\limits_{i < j}^N {\frac{{{e^2}}}{{{r_{ij}}}} + \sum\limits_{\alpha < \beta } {\frac{{{Z_\alpha }{Z_\beta }{e^2}}}{{{R_{\alpha \beta }}}}} } $$

((1.1))

Here r _αi is the distance between electron i and nucleus α, Z _α e is the charge of nucleus α, r _ij is the distance between electrons i and j, and R _αβ is the distance between nuclei α and β. The use of (1.1) implies the following approximations: (a) We use the Born-Oppenheimer approximation⁽¹⁾; i.e., we consider the determination of the electronic wave function when the nuclear framework is momentarily fixed, (b) The Hamiltonian (1.1) is a nonrelativistic Hamiltonian; we have omitted the spin-orbit interaction term and various other small terms. As a consequence of approximation (b) this Hamiltonian is spin free, i.e., none of the terms appearing in it depends explicitly on the spin variables. The total spin operator and its z component commute with the Hamiltonian. They are constants of the motion and, therefore, each state of the system can be characterized by a simultaneous eigenfunction of these commuting operators. We can classify the eigenstates as singlets (S = 0), doublets (S = 1/2), triplets (S = 1), and so on; the use of the spin-free Hamiltonian (1.1) implies electronic states with definite multiplicities (2S + 1).