The Variational Principle for Many-Electron Systems

The basic idea of Hartree-Fock now is to determine the lowest eigenenergy with corresponding eigenstate $\Psi$ of an $N$-electron system not by solving the stationary Schrödinger equation $\hat{H}\Psi=\varepsilon\Psi$, but by minimizing the functional $F[\Psi]$. As these two are equivalent, nothing would have been gained. However, for $N$-electron systems either of these methods has to be done approximately anyway and the argument is now that the minimization procedure is the better starting point.

Idea: do not carry out the minimization of the functional over all possible states $\Psi$, but just over a certain sub-class of states, i.e., those which can be written as a anti-symmetrized products of some single particle states $\vert\nu_i\rangle$, with the $\vert\nu_i\rangle$ to be determined, i.e. Slater determinants $\vert\nu_1,...,\nu_N\rangle$. The determination of the $\vert\nu_i\rangle$ leads to the Hartree-Fock equations. Note that here and in the following, $\vert\nu_i\rangle$ does not refer to any fixed set of basis states but to the states to be determined from the Hartree-Fock equations.

Definition: The single particle states $\vert\nu_i\rangle$ correspond to single particle wave functions $\psi_{\nu_i}({\bf r},\sigma)$. The label $\nu_i$ includes the spin index. In quantum chemistry, these wave functions are sometimes called spin-orbitals, molecular orbitals, or shells.