Single determinant approximation for ground and excited states with accuracy comparable to that of the configuration interaction

Abstract: 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.