Consistent Construction of the Density Matrix from Surface Hopping Trajectories

Abstract

Proper construction of the density matrix based on surface hopping trajectories remains a difficult problem. Due to the well-known overcoherence in traditional surface hopping simulations, the electronic wave function cannot be used directly. In this work, we propose a consistent density matrix construction method, which takes the advantage of occupation of active states to rescale the coherence calculated by wave functions and ensures the intrinsic consistency of the density matrix. This new trajectory analysis method can be used for both Tully’s fewest switches surface hopping (FSSH) and our recently proposed branching corrected surface hopping (BCSH). As benchmarked in both one- and two-dimensional standard scattering models, the new approach combined with BCSH trajectories achieves highly accurate time-dependent spatial distributions of adiabatic populations and coherence compared to exact quantum results.

Surface Hopping with Reliable Wave Function by Introducing Auxiliary Wave Packets to Trajectory Branching

“…Two independent definitions of the off-diagonal density matrix elements were established for the fact that different surfaces carry different trajectories, and their averages can also be considered. Recently, we proposed a new trajectory analysis approach that uses both occupation of active states and wave functions . The amplitude of coherence is determined by the occupation of active states and the phase of coherence is defined according to the wave functions, so that the intrinsic consistency of the density matrix for a pure state is fulfilled.…”

“…Similar to eq , the spatial distribution of coherence can be obtained according to the statistical average of coherences from surface hopping trajectories, ρ i j ( x , t ) = 1 N t r a j ∑ n = 1 N t r a j w i , n false( t false) w j , n * false( t false) δ false( bold-italicx − bold-italicx n false) In the fully quantum dynamics, the quantum coherence and populations for a pure state should always satisfy the intrinsic relationship, false| ρ i j ( x , t ) false| 2 = ρ i i ( x , t ) ρ j j ( x , t ) This equation, however, is not automatically fulfilled for the density matrix distribution constructed by surface hopping trajectories. Here, we rescale the amplitude of the off-diagonal density matrix element (i.e., coherence) using the diagonal elements (i.e., populations) according to eq .…”