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The Variational Principle for Many-Electron Systems

The basic idea of Hartree-Fock now is to determine the lowest eigenenergy with corresponding eigenstate \bgroup\color{col1}$ \Psi$\egroup of an \bgroup\color{col1}$ N$\egroup-electron system not by solving the stationary Schrödinger equation \bgroup\color{col1}$ \hat{H}\Psi=\varepsilon\Psi$\egroup, but by minimizing the functional \bgroup\color{col1}$ F[\Psi]$\egroup. As these two are equivalent, nothing would have been gained. However, for \bgroup\color{col1}$ N$\egroup-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 \bgroup\color{col1}$ \Psi$\egroup, but just over a certain sub-class of states, i.e., those which can be written as a anti-symmetrized products of some single particle states \bgroup\color{col1}$ \vert\nu_i\rangle$\egroup, with the \bgroup\color{col1}$ \vert\nu_i\rangle$\egroup to be determined, i.e. Slater determinants \bgroup\color{col1}$ \vert\nu_1,...,\nu_N\rangle$\egroup. The determination of the \bgroup\color{col1}$ \vert\nu_i\rangle$\egroup leads to the Hartree-Fock equations. Note that here and in the following, \bgroup\color{col1}$ \vert\nu_i\rangle$\egroup 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 \bgroup\color{col1}$ \vert\nu_i\rangle$\egroup correspond to single particle wave functions \bgroup\color{col1}$ \psi_{\nu_i}({\bf r},\sigma)$\egroup. The label \bgroup\color{col1}$ \nu_i$\egroup includes the spin index. In quantum chemistry, these wave functions are sometimes called spin-orbitals, molecular orbitals, or shells.



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next up previous contents index
Next: Functional Derivative Up: Hartree-Fock Equations Previous: Lagrange Multiplier   Contents   Index
Tobias Brandes 2005-04-26