On the inclusion of the diagonal Born-Oppenheimer correction in surface hopping methods

TL;DR

This paper evaluates the impact of including the diagonal Born-Oppenheimer correction (DBOC) in surface hopping methods for nonadiabatic dynamics, finding it beneficial under certain conditions but detrimental when DBOC diverges.

Contribution

It provides a comprehensive assessment of DBOC's role in nonadiabatic surface hopping methods across various models, clarifying when its inclusion improves or worsens simulation accuracy.

Findings

DBOC inclusion improves accuracy when nuclear kinetic energy is lower than DBOC.

Including DBOC is harmful when DBOC energy diverges or is very high.

Surface hopping methods' effectiveness depends on the interplay between diagonal and off-diagonal couplings.

Abstract

The diagonal Born-Oppenheimer correction (DBOC) stems from the diagonal second derivative coupling term in the adiabatic representation, and it can have an arbitrary large magnitude when a gap between neighbouring Born-Oppenheimer (BO) potential energy surfaces (PESs) is closing. Nevertheless, DBOC is typically neglected in mixed quantum-classical methods of simulating nonadiabatic dynamics (e.g., fewest-switch surface hopping (FSSH) method). A straightforward addition of DBOC to BO PESs in the FSSH method, FSSH+D, has been shown to lead to numerically much inferior results for models containing conical intersections. More sophisticated variation of the DBOC inclusion, phase-space surface-hopping (PSSH) was more successful than FSSH+D but on model problems without conical intersections. This work comprehensively assesses the role of DBOC in nonadiabatic dynamics of two electronic state problems and the performance of FSSH, FSSH+D, and PSSH methods in variety of one- and two-dimensional models. Our results show that the inclusion of DBOC can enhance the accuracy of surface hopping simulations when two conditions are simultaneously satisfied: 1) nuclei have kinetic energy lower than DBOC and 2) PESs are not strongly nonadiabatically coupled. The inclusion of DBOC is detrimental in situations where its energy scale becomes very high or even diverges, because in these regions PESs are also very strongly coupled. In this case, the true quantum formalism heavily relies on an interplay between diagonal and off-diagonal nonadiabatic couplings while surface hopping approaches treat diagonal terms as PESs and off-diagonal ones stochastically.