Brian C. Hoffman: SCF Theory

Basic Principles and
Hartree-Fock Theory

Spin orbitals

In section 1.5.1 it was demonstrated that a product of spin orbitals, a Hartree product, adequately describes a molecular system in which the mutual repulsions of the electrons are neglected. Other than indicating that a spin orbital is a one-electron wave function dependent upon both the space and spin coordinates of the electron, section 1.5.1 states nothing about the mathematical form of a spin orbital. A spin orbital, $\chi$ , consists of a spatial molecular orbital, $\psi(r)$ , multiplied by either the $\alpha(\omega)$ or the $\beta(\omega)$ spin function to indicate $+ \frac{1}{2}$ and $- \frac{1}{2}$ spin states respectively.

$\begin{displaymath}\chi(x) = \left\{ \begin{array}{c}\psi(r) \alpha(\omega)\\\psi(r) \beta(\omega)\\\end{array} \right.\end{displaymath}$

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The variable $\omega$ is an arbitrary spin variable commonly used for keeping track of integrations over spin. The spin functions $\alpha(\omega)$ and $\beta(\omega)$ are orthogonal; thus, $\langle \alpha\vert \alpha \rangle = \langle \beta \vert \beta \rangle = 1$ and $\langle \alpha\vert \beta \rangle = \langle \beta \vert \alpha \rangle = 0$ . If the set of spatial orbitals are also orthogonal, then any two spin orbitals generated from this set must also be orthogonal.

The spatial molecular orbital $\psi$ contains all the information about the distribution of the electron in space and is the only place where a set of variational parameters can be included in a Hartree product wave function. Technically, the form of the spatial orbital can be any mathematical function with a sufficient degree of flexibility; however, theorists typically choose a wave function based on a physical picture of the system. Relating the trial function to the model of the physical system allows theorists to make useful chemical interpretations of the trial wave function and many of the terms which occur in the ab initio calculations. The most common method of constructing spatial orbitals is to take a linear combination of atomic orbitals to form the molecular orbitals with the variational parameters being the linear combination coefficients, the so called LCAO-MO approach.

In ab initio methods two formalisms exist for combining the spatial and spin functions: restricted and unrestricted. The restricted formalism is closest to the standard picture taught in general chemistry where every orbital contains both a spin-up and a spin-down electron, i. e. electrons with opposite spin are selectively paired into a single spatial orbital. In the unrestricted formalism,[8] electrons with $\alpha$ and $\beta$ spin are allowed to occupy completely separate spatial orbitals. The advantage of the restricted formalism is that the wave functions which are constructed in this fashion can be made to be eigenfunctions of the S² operator. The unrestricted formalism has the advantage that there are more variational parameters which allows for a lower expectation value of the energy than the restricted formalism. However, the unrestricted formalism is subject to ``spin contamination'' because the wave function is not an eigenfunction of the S² operator.

This page maintained by Brian C. Hoffman