On the orthogonality of states with approximate wavefunctions

A variational solution to the eigenvalue problem for the Hamiltonian H, with orthogonality restrictions on eigenvectors of H to the vector H ∣ Φ₀〉, where ∣Φ₀〉 is an approximate ground-state vector, is proposed as a means to calculate excited states. The asymptotic projection (AP) method proposed previously is further developed and applied to solve this problem in a simple way. We demonstrate that the AP methodology does not require an evaluation of the matrix elements of operator H², whereas conventional approaches-such as the elimination of off-diagonal Lagrange multipliers method, projection operator techniques, and other methods-do. It is shown, based on the results obtained for the single-electron molecular ions H₂⁺, HeH²⁺, and H₃²⁺, that applying the new method to determine excited-state wavefunctions yields the upper bounds for excited-state energies. We demonstrate that regardless of whether the orthogonality constraint for states (〈Φ| Φ₀〉 = 0) is applied, the zero-coupling constraint (〈Φ| H| Φ₀〉 = 0) is imposed, or both of these restrictions are enforced simultaneously, practically the same excited-state energy is obtained if the basis set is almost complete. For the systems considered here, all schemes are capable of giving a sub-μhartree level of accuracy for the ground and excited states computed with different basis sets.