On the inclusion of cusp effects in expectation values with explicitly correlated Gaussians

Abstract

This paper elaborates the integral transformation technique of [K. Pachucki, W. Cencek, and J. Komasa, J. Chem. Phys. 122, 184101 (2005)] and uses it for the case of the non-relativistic kinetic and Coulomb potential energy operators, as well as for the relativistic mass-velocity and Darwin terms. The techniques are tested for the ground electronic state of the helium atom and new results are reported for the ground electronic state of the H + 3 molecular ion near its equilibrium structure.

Lower Bounds for Nonrelativistic Atomic Energies

“…These operators appear also in the Foldy−Wouthuysen relativistic perturbation theory 58 and are known to have nonfavorable convergence properties when using a Gaussian basis due to the lack of a proper representation of the wave function cusps by Gaussian functions. 52,53 Figure 3 shows the convergence behavior for the terms in H 2 and indeed shows that the "singular" operators are responsible for the slowdown in the convergence of the ⟨H 2 ⟩ expectation value calculated with the ground-state Ritz eigenvector. In principle, it should be possible to use (and extend) the integral transformation (IT) technique 52,53 to include the analytic short-range behavior near the cusps of the exact wave function during the evaluation of ⟨H 2 ⟩.…”

“…They explicitly account for pair correlation and allow for a rapid convergence of the energy with the basis set size, while the integrals needed to compute the Hamiltonian and overlap matrices can be evaluated in a closed, analytic form. It is necessary to mention that ECGs fail to describe the exact behavior of the nonrelativistic wave function at the particle−particle coalescence points 52,53 and for large particle separations. For almost all lower bound implementations, matrix elements are required not only for the nonrelativistic Hamiltonian, H, but also for the Hamiltonian squared, H 2 .…”

“…52,53 Figure 3 shows the convergence behavior for the terms in H 2 and indeed shows that the "singular" operators are responsible for the slowdown in the convergence of the ⟨H 2 ⟩ expectation value calculated with the ground-state Ritz eigenvector. In principle, it should be possible to use (and extend) the integral transformation (IT) technique 52,53 to include the analytic short-range behavior near the cusps of the exact wave function during the evaluation of ⟨H 2 ⟩. Generalization of the IT technique should be straightforward to 1/r 2 -type terms along the lines of ref 53, but some further considerations may be necessary for the IT evaluation of Ψ Ψ p rj k 1 2 i -type "mixed" kinetic-Coulomb terms.…”