Matrix representation of vector potential: DVR and TDDVR formulations and dynamics

The inclusion of the geometric phase effects through the addition of vector potential is well known. We present the formulation of DVR and TDDVR matrix equations for any 2-D system with vector potential. The effective potential appears as the complex hermitian matrix in the DVR/TDDVR representation where in case of TDDVR, matrices associated with “classical” momentum also plays an important role in the dynamics. We derive the rigorous expressions of “classical” equations of motion from Dirac-Frenkel variational principle without introducing the “classical” path as such. Numerical calculations by using DVR/TDDVR equations have been carried out to obtain the signature of geometric phase on the reactive and non-reactive scattering processes. TDDVR appears to be better compromise between speed and accuracy than traditional quantum dynamics numerical methodologies (DVR/FFT).