Equivalence restrictions and the brillouin condition for hartree-fock wave functions

The Brillouin condition, that matrix elements of single-replacement configuration interaction vanish, is a valuable consequence of the unrestricted Hartree–Fock method. This condition in general is not realized for restricted Hartree–Fock methods although these methods possess certain other practical advantages. The general applicability of the Brillouin condition is analyzed here in a formal manner that provides a useful framework for studying examples in which the condition is only approximately satisfied. A specific application to a symmetry- and equivalence-restricted Hartree–Fock calculation of the 1s-hole state of Ar+ is presented.