Symmetry-Projected Nuclear-Electronic Hartree-Fock: Eliminating Rotational Energy Contamination

TL;DR

This paper introduces a symmetry projection method for nuclear-electronic Hartree-Fock wave functions that enforces rotational and parity symmetries, reducing energy contamination and enabling targeted rotational state calculations.

Contribution

The paper presents a novel symmetry projection technique that restores rotational and parity symmetries in nuclear-electronic Hartree-Fock methods, improving accuracy for molecular rotational states.

Findings

Eliminates rotational energy contamination in wave functions.

Enables targeting specific rotational states.

Demonstrates effectiveness on H₂ and H₃⁺ molecules.

Abstract

We present a symmetry projection technique for enforcing rotational and parity symmetries in nuclear-electronic Hartree-Fock wave functions, which treat electrons and nuclei on equal footing. The molecular Hamiltonian obeys rotational and parity-inversion symmetries, which are, however, broken by expanding in Gaussian basis sets that are fixed in space. We generate a trial wave function with the correct symmetry properties by projecting the wave function onto representations of the three-dimensional rotation group, i.e., the special orthogonal group in three dimensions SO(3). As a consequence, the wave function becomes an eigenfunction of the angular momentum operator which (i) eliminates the contamination of the ground state wave function by highly excited rotational states arising from the broken rotational symmetry, and (ii) enables the targeting of specific rotational states of the molecule. We demonstrate the efficiency of the symmetry projection technique by calculating energies of the low-lying rotational states of the H$_2$ and H$_3^+$ molecules.