Overcoming positivity violations for density matrices in surface hopping

Abstract

Fewest-switches surface hopping (FSSH) has emerged as one of the leading methods for modeling the quantum dynamics of molecular systems. While its original formulation was limited to adiabatic populations, the growing interest in the application of FSSH to coherent phenomena prompts the question how one should construct a complete density matrix based on FSSH trajectories. A straightforward solution is to define adiabatic coherences based on wavefunction coefficients. In this Paper, we demonstrate that inconsistencies introduced in the density matrix through such treatment may lead to a violation of positivity. We furthermore show that a recently proposed coherent generalization of FSSH results in density matrices that satisfy positivity, while yielding an improved accuracy throughout much (but not all) of parameter space.

Dynamics of mixed quantum–classical spin systems *

“…´ D d 3 q, respectively. Notice that, unlike other quantum-classical models widely popular in chemical physics [14,52], here both the classical and the quantum density are positive. In more generality, the model in (31) satisfies the consistency criteria outlined at the beginning of §2.…”

“…While most of these models continue to be used in several test cases with a certain success, their underlying equations suffer from well-known consistency issues. The possible violation of Heisenberg's uncertainty principle in the most popular method [14] is only one example [2].…”

“…Among the most popular alternatives to the Ehrenfest model, the surface hopping algorithm was recently shown to violate Heisenberg's uncertainty principle since the quantum density matrix ρ may become sign-indefinite [14]. The same violation of the Heisenberg principle is also allowed by the quantum-classical Liouville equation on phase-space [5,15,41,52].…”

“…However, the interaction dynamics of quantum and classical degrees of freedom remains a challenging question since the currently available models suffer from several consistency issues [2]. In some cases [4,8,22,49], the Heisenberg principle is lost due to the fact that the quantum density matrix is allowed to change its sign. In some other cases [12,25], the model does not reduce to uncoupled quantum and classical dynamics in the absence of a quantumclassical interaction potential.…”

Consistent Construction of the Density Matrix from Surface Hopping Trajectories

“…We express the amplitude of coherence as false| ρ i j false[ cc false] false( boldR , t false) false| = false[ ρ i i false[ as false] false( boldR , t false) ρ j j false[ as false] false( boldR , t false) ] 1 / 2 By design, ρ ii [cc] ( R , t ) = ρ ii [as] ( R , t ) and thus the constraint in eq is automatically maintained. In particular, the positivity violation problem of trajectory-analyzed populations is also circumvented . Moreover, we expect that this approach will improve the accuracy of coherence, as the populations based on active states in surface hopping dynamics are much more reliable compared with those by electronic wave functions.…”