  The Hartree-Fock equation is a fundamental equation in many-electron
problems. It is of practical importance in quantum chemistry to find solutions
to the Hartree-Fock equation. The self-consistent field (SCF) method is a
standard numerical calculation method to solve the Hartree-Fock equation. In
this paper we prove that the sequence of the functions obtained in the SCF
procedure is composed of a sequence of pairs of functions that converges after
multiplication by appropriate unitary matrices, which strongly ensures the
validity of the SCF method. A sufficient condition for the limit to be a
solution to the Hartree-Fock equation after multiplication by a unitary matrix
is given, and the convergence of the corresponding density operators is also
proved. The method is based mainly on the proof of approach of the sequence to
a critical set of a functional, compactness of the critical set, and the proof
of the \L ojasiewicz inequality for another functional near critical points.
