Non-adiabatic coupling matrix elements in a magnetic field: Geometric gauge dependence and Berry phase

Non-adiabatic coupling matrix elements (NACMEs) are important in quantum chemistry, particularly for molecular dynamics methods such as surface hopping. However, NACMEs are gauge dependent. This presents a difficulty for their calculation in general, where there are no restrictions on the gauge function except that it be differentiable. These cases are relevant for complex-valued electronic wave functions, such as those that arise in the presence of a magnetic field or spin-orbit coupling. In addition, the Berry curvature and Berry force play an important role in molecular dynamics in a magnetic field and are also relevant in the context of spin-orbit coupling. For methods such as surface hopping, excited-state Berry curvatures will also be of interest. With this in mind, we have developed a scheme for the calculation of continuous, differentiable NACMEs as a function of the molecular geometry for complex-valued wave functions. We demonstrate the efficacy of the method using the H2 molecule at the full configuration-interaction (FCI) level of theory. In addition, ground- and excited-state Berry curvatures are computed for the first time using FCI theory. Finally, Berry phases are computed directly in terms of diagonal NACMEs.