Brian C. Hoffman: SCF Theory

Basic Principles and
Hartree-Fock Theory

Hartree Product Wave Functions

The Average Value Theorem together with the Variational Principle provide an approximate method of solving the electronic Schrödinger equation for the wave function and energy of a molecular system by guessing a trial function and minimizing the expectation value of the energy with respect to parameters in that trial function until both the trial function and energy are as close as possible to the exact answer. The question then becomes, what choice of trial function is best from the viewpoints of both accuracy and feasibility? In order to answer this question, first consider the much simpler problem of solving the electronic Schrödinger equation with a modified Hamiltonian which does not include contributions from the electrostatic repulsions between electrons. In the case of a one-electron system, the Hamiltonian would take the form

$\begin{displaymath}\hat{h}(i) = - \frac{1}{2} \nabla_{i}^{2} + \sum_{A}{M} \frac{1}{r_{iA}}.\end{displaymath}$

(12)

Within the BO approximation, this one-electron problem is solvable and one can thereby accurately treat such simple systems as the hydrogen molecule cation. The solutions to the one-electron Schrödinger equation are the molecular spin orbitals, denoted by $\chi$ , and they satisfy the equation

$\begin{displaymath}\hat{h}(i) \chi_{j}(x_i) = \epsilon_{j} \chi_{j} (x_i).\end{displaymath}$

(13)

The interpretation of (1.13) is that electron i with combined spatial and spin coordinates x_i occupies spin orbital $\chi_j$ with a total energy of $\epsilon_{j}$ .

Maintaining the assumption that the electrons in a molecule do not interact, the Hamiltonian for an N-electron system can be constructed as a simple sum of one-electron Hamiltonians:

$\begin{displaymath}\hat{H} = \sum_{i}^{N} \hat{h(i)}.\end{displaymath}$

(14)

Mathematically, the eigenfunction associated with a sum of independent operators is the product of the eigenfunctions of those operators. Therefore, the eigenfunction of the N-electron Hamiltonian given in equation (1.14) is a product of spin orbitals called a Hartree product, [5,6,7] $\Psi^{HP}$ , which yields an eigenvalue equal to the sum of the eigenvalues corresponding to each molecular spin orbital, E_total:

$\begin{displaymath}\Psi^{HP} = \chi_i(x_1) \chi_j(x_2) \cdots \chi_n(x_N),\end{displaymath}$

(15)

$\begin{displaymath}E_{total} = \langle \Psi^{HP} \vert \hat{h} \vert \Psi^{HP} \rangle = \epsilon_{i} +\epsilon_{j} \cdots + \epsilon_{n}.\end{displaymath}$

(16)

Hartree product wave functions suffer from several major flaws that serve to make them physically unrealistic. First, Hartree products do not satisfy the Pauli Antisymmetry Principle which states that the sign of any many-electron wave function must be antisymmetric (i. e. change sign) with respect to the interchange of the coordinates, both space and spin, of any two electrons. The Antisymmetry Principle is a postulate of quantum mechanics derived from relativistic arguments and is, in the end, equivalent to the more standard statement of the Pauli Principle which prevents two electrons with the same spin from occupying the same spatial orbital. Second, Hartree products force a particular electron to occupy a given spin orbital despite the fact that electrons are indistinguishable from one another. Lastly, because the Hartree product wave function is constructed on the assumption that the electrons are non-interacting, there exists a non-zero probability of finding two electrons occupying the exact same point in space.

This page maintained by Brian C. Hoffman