  Considering the emblematic Hartree-Fock (HF) energy expression with single
Slater determinant and the ortho-normal molecular orbits (MO) in it, expressed
as a linear combination (LC) of atomic orbits (LCAO) basis set functions, the
HF energy expression is in fact a 4th order polynomial of the LCAO
coefficients, which is relatively easy to handle. The energy optimization via
the Variation Principle can be made with a Lagrange multiplier method to keep
the ortho-normal property and the Newton-Raphson (NR) method to find the
function minimum. It is an alternative to the widely applied HF self consistent
field (HF-SCF) method which is based on unitary transformations and eigensolver
during the SCF, and seems to have more convenient convergence property. This
method is demonstrated for closed shell (even number of electrons and all MO
are occupied with both, alpha and beta spin electrons) and restricted (all MOs
have single individual spatial orbital), but the extension of the method to
open shell and/or unrestricted cases is straightforward.
