Concepts of functional analysis, namely, regular points, tangent subsp
aces, constraint surfaces, Lagrangian matrix restricted to the tangent
subspace of a constraint surface, are presented in connection with th
e Hartree-Fock (HF) problem. The energy functional in LCAO approximati
on is considered to be a polynomial function of several variables subj
ect to subsidiary conditions. General HF equations and instability con
ditions for the unrestricted Hartree-Fock (UHF) solutions are derived
from this standpoint.