Abstract
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.
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This work is presented as an appreciation of Tim Clark’s achievements in quantum chemistry, developing new genuine methods and concepts.
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This paper belongs to the Topical Collection Tim Clark 70th Birthday Festschrift
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Glushkov, V.N., Assfeld, X. On the orthogonality of states with approximate wavefunctions. J Mol Model 25, 148 (2019). https://doi.org/10.1007/s00894-019-4019-2
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DOI: https://doi.org/10.1007/s00894-019-4019-2