Surface Hopping, Electron Translation Factors, Electron Rotation Factors, Momentum Conservation, and Size Consistency

TL;DR

This paper proposes modifications to the surface hopping algorithm to ensure conservation of linear and angular momentum by incorporating electron translation and rotation factors, improving the physical accuracy of nonadiabatic simulations.

Contribution

It introduces electron translation factors and a novel electron rotation factor to address momentum conservation issues in surface hopping algorithms.

Findings

Disallowing nuclear linear momentum changes with translation basis.

Disallowing nuclear angular momentum changes with a rotating basis.

New basis modifications improve momentum conservation in surface hopping.

Abstract

For a system without spin-orbit coupling, the (i) nuclear plus electronic linear momentum and (ii) nuclear plus orbital electronic angular momentum are good quantum numbers. Thus, when a molecular system undergoes a nonadiabatic transition, there should be no change in the total linear or angular momentum. Now, the standard surface hopping algorithm ignores the electronic momentum and indirectly equates the momentum of the nuclear degrees of freedom to the total momentum. However, even with this simplification, the algorithm still does not conserve either the nuclear linear or the nuclear angular momenta. Here, we show that one way to address these failures is to dress the derivative couplings (i.e. the hopping directions) in two ways: (i) we disallow changes in the nuclear linear momentum by working in a translating basis (which is well known and leads to electron translation factors [ETFs]); and (ii) we disallow changes in the nuclear angular momentum by working in a basis that rotates around the center of mass (which is not well-known and leads to a novel, rotationally removable component of the derivative coupling that we will call electron rotation factors [ERFs] below, cf. Eq. 96). The present findings should be helpful in the short term as far as interpreting surface hopping calculations for singlet systems (without spin) and then developing new surface hopping algorithm in the long term for systems where one cannot ignore the electronic orbital and/or spin angular momentum.