Approximation of the ground and excited states of atoms and molecules with a single determinant by means of the exact many-particle Hamiltonian

Abstract

In the present work, we search for the maximum of the functional $|⟨{\mathrm{\Phi }}^{\mathrm{\prime }}|\hat{H}|\mathrm{\Phi }⟩|$, where $|{\mathrm{\Phi }}^{\mathrm{\prime }}⟩$ is a Slater Determinant (SlDet) and $\hat{H}$ is the exact Hamiltonian of an atom or a molecule, starting from a SlDet $|\mathrm{\Phi }⟩$, with its spin orbitals calculated by the standard Hartree-Fock (HF) equation or other approximation or any determinant. The element $|⟨{\mathrm{\Phi }}_1|\hat{H}|\mathrm{\Phi }⟩|$ with $|{\mathrm{\Phi }}_1⟩$ the maximising $|{\mathrm{\Phi }}^{\mathrm{\prime }}⟩$ gives a value greater or equal to $|⟨\mathrm{\Phi }|\hat{H}|\mathrm{\Phi }⟩|$. Next we calculate the corresponding maximum overlap $|⟨{\mathrm{\Phi }}_2|\hat{H}|{\mathrm{\Phi }}_1⟩|$ and finally $|⟨{\mathrm{\Phi }}_{n+1}|\hat{H}|{\mathrm{\Phi }}_n⟩|$ until $|⟨{\mathrm{\Phi }}_{n+1}|\hat{H}|{\mathrm{\Phi }}_n⟩-⟨{\mathrm{\Phi }}_n|\hat{H}|{\mathrm{\Phi }}_{n-1}⟩|\le \epsilon$, where ε determines the desired numerical accuracy. We show that the sequence $⟨{\mathrm{\Phi }}_{n+1}|\hat{H}|{\mathrm{\Phi }}_n⟩$ converges to an extremum $⟨\mathrm{\Phi }|\hat{H}|\mathrm{\Phi }⟩$. Having found the SlDet with the lowest energy we repeat the procedure in the orthogonal subspace for finding the first and the higher excited states. We applied this method in order to determine the energies of several configurations of H${}_3$, the Lithium atom, LiH and Be.

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Acknowledgments

The authors would like to thank Dr N. Lathiotakis for his help with the numerical implementations and useful discussions. The authors would like also to give a special thank you to the anonymous reviewers for their constructive comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).