Elimination of the Translational Kinetic Energy Contamination in pre-Born-Oppenheimer Calculations

TL;DR

This paper introduces a straightforward method to remove translational kinetic energy contamination from pre-Born-Oppenheimer calculations in laboratory-fixed Cartesian coordinates, ensuring accurate total energy computations for molecular systems.

Contribution

It develops a practical strategy using coordinate transformations and basis function properties to eliminate translational contamination in ECG-based pre-Born-Oppenheimer calculations.

Findings

Successfully eliminated translational contamination in H2 molecule calculations.

Validated the method on the two lowest rotational energy levels of H2.

Provided a general approach applicable to other molecular systems.

Abstract

In this paper we present a simple strategy for the elimination of the translational kinetic energy contamination of the total energy in pre-Born--Oppenheimer calculations carried out in laboratory-fixed Cartesian coordinates (LFCCs). The simple expressions for the coordinates and the operators are thus preserved throughout the calculations, while the mathematical form and the parametrisation of the basis functions are chosen so that the translational and rotational invariances are respected. The basis functions are constructed using explicitly correlated Gaussian functions (ECGs) and the global vector representation. First, we observe that it is not possible to parametrise the ECGs so that the system is at rest in LFCCs and at the same time the basis functions are square-integrable with a non-vanishing norm. Then, we work out a practical strategy to circumvent this problem by making use of the properties of the linear transformation between the LFCCs and translationally invariant and center-of-mass Cartesian coordinates as well as the transformation properties of the corresponding basis function parameter matrices. By exploiting these formal mathematical relationships we can identify and separate the translational contamination terms in the matrix representation of the kinetic energy operator in the LFCC formalism. We present numerical examples for the translational contamination and its elimination for the two lowest rotational energy levels of the singlet hydrogen molecule, corresponding to para- and ortho-H2, respectively, treated as four-particle quantum systems.