  It was realized from the early days of Chemical Physics (Rev. Mod. Phys. 35,
496 (1963)) that the energy $E_{HF}$ of the Slater determinant (SlDet) $|\Phi
_{HF}\rangle$, obtained by the single particle Hartree-Fock (HF) equation, does
not coincide with the minimum energy of the functional $\langle\Phi|H|\Phi
\rangle$ where $|\Phi \rangle$ is a SlDet and $H$ is the many particle
Hamiltonian. However, in most textbooks, there is no mention of this fact. In
this paper, starting from a Slater determinant $|\Phi \rangle$ with its spin
orbitals calculated by the standard HF equation or other approximation, we
search for the maximum of the functional $|\langle\Phi ^{\prime }|H|\Phi
\rangle|$, where $|\Phi ^{\prime }\rangle$ is a SlDet and $H$ is the exact
Hamiltonian of an atom or a molecule. The element $|\langle\Phi _{1}|H|\Phi
\rangle|$ with $|\Phi _{1}\rangle$ the maximizing $|\Phi ^{\prime }\rangle$
gives a value larger than $\langle\Phi|H|\Phi\rangle$. The next step is to
calculate the corresponding maximum overlap $\langle\Phi_{2}|H|\Phi
_{1}\rangle|$ and subsequently $|\langle\Phi _{n+1}|H|\Phi _{n}\rangle|$ until
$|\langle\Phi _{m+1}|H|\Phi _{m}\rangle -\langle\Phi _{m-1}|H|\Phi
_{m}\rangle|\leq\varepsilon$, where $\varepsilon $ determines the required
numerical accuracy. We show that the sequence $a_{n}=|\langle\Phi
_{n+1}|H|\Phi_{n}\rangle|$ is ascending and converges. We applied this
procedure for determining the eigenstate energies of several configurations of
H$_{3}$, the Lithium atom, LiH and Be. After comparing our values with those of
the configuration interaction we found that our deviations are in the range
10$^{-5}~$to $10^{-8}$ and the ground state energy is significantly below that
of the standard HF calculations.
