Second-order many-body perturbation theory for the any particle molecular orbital method in the framework of Hartree products of localized wave functions

Abstract

This paper presents an alternative second-order many-body perturbation theory (MBPT(2)) for the Any Particle Molecular Orbital (APMO) method. Here, the reference wave function was constructed as a product of Slater determinants for light species (e.g., electrons) and Hartree products of localized orbitals for heavy species (e.g., muons, nuclei). The impact of this Hartree product approximation (HPA) on APMO/MBPT(2) energies was analyzed for the hydrogen molecule and hydrogen-bonded dimers. It was observed that the impact of the HPA on the first-order energies is negligible. In contrast, the HPA-induced changes in the nuclear-electron correlation terms led to reductions of the order of 10−2 a.u. in the second-order energies. APMO/MBPT(2)/HPA energies are closer to the full-CI energies than APMO/MBPT(2) energies. The Hartree product approximation (HPA) for heavy particles modifies virtual orbital energies, leading to increased correlation recovery similar to improved virtual orbital techniques in post-Hartree-Fock methods. Specifically, the HPA reduces virtual nuclear orbital energies, resulting in smaller energy denominators in the perturbation expansion and, thus, larger interspecies correlation corrections. However, the amount of interspecies correlation recovered is limited and system-dependent. The application to systems containing multiple hydrogen nuclei revealed that the HPA leads to reductions in the formal scaling of nuclear-electron and nuclear-nuclear integral transformations from quintic to quartic and cubic, respectively. These reductions allow APMO/MBPT(2) calculations for much larger systems. The application of the APMO/MBPT(2) method to muonic helium and lithium atoms and the muonic helium dimer revealed that the muon-electron correlation is of the order of only 1×10−6 a.u. This work provides a methodological and computational contribution, offering a pragmatic, low-cost approximation for exploratory studies of nuclear quantum effects in medium-sized systems. The primary advantage is the reduction in computational scaling, enabling exploration of larger systems and potential energy surfaces.