Fewest switches surface hopping with decoherence in the Marcus inverted regime: correct rates but wrong thermal populations

Abstract

Fewest switches surface hopping (FSSH) is a well benchmarked dynamical method for simulating nonadiabatic systems. In particular, the literature shows that for the spin-Boson model Hamiltonian, FSSH with appropriate corrections usually captures the detailed balance well and obtains rate constants within a factor of 2 compared to numerically exact results. In this study, we show that in the deep inverted Marcus regime, the augmented-FSSH (AFSSH, one version that includes decoherence) yields reasonably accurate rate constants but incorrect thermal populations over a broad range of parameters. We present an analytical derivation to understand the AFSSH behavior, and therefore, show that AFSSH obtains correct rate constants owing to the resonance of the time derivative coupling with the exothermicity, but obtains an incorrect thermal population owing to the self-consistency issue. The presented derivation provides an analytical expression for the quantum correction factor for AFSSH simulations in the Marcus inverted regime.

Figures (4)

• Figure 1: (a) Electronic adiabatic PES and nonadiabatic coupling vector as a function of classical coordinate. The red dots correspond to the position of hops between two states along the reaction coordinate. (b) Long time diabatic population dynamics for parameters of column 2 of Table \ref{['table_param']} obtained from HEOM, AFSSH and Marcus rate theory. Boltzmann population is shown as a dashed black line. (c) Comparison of rate constants obtained from high temperature FGR (Eq. \ref{['eq_k_FGR_highT']}), FGR formula (Eq. \ref{['eq_k_FGR']}), HEOM, AFSSH and Marcus formula (Eq. \ref{['eq_k_Marc']}) for the variation of reorganization energy. (d) Comparison of rate constants obtained from high temperature FGR theory, FGR theory, HEOM and AFSSH for variation in diabatic coupling constant.
• Figure 2: Comparison of the rate constants and long time populations obtained from AFSSH, HEOM, analytical rate theory (Eqs. \ref{['k_rate_final']} and \ref{['k_rate_final1']}) and FGR (Eqs. \ref{['eq_LT_limit1']} and \ref{['eq_LT_limit2']}) as a function of (a) exothermicity, (b) reorganization energy, and (c) temperature with parameters taken from Table \ref{['table_param']}. Lower panels shows comparison of the thermal population while upper panel show the rate constants.
• Figure 3: Short time dynamics of FSSH adiabatic populations and $|c_i|^2$ (in the adiabatic basis) compared with those calculated using FGR rate constants (Eqs. \ref{['k_fgr_no_deco']} and \ref{['k_fgr_reverse']}) and AFSSH rate constants (Eqs. \ref{['k_rate_final']} and \ref{['k_rate_final1']}). (a) Dynamics initialized on surface 2 with $c_ {\color{black}2}(t=0)=1$, (b) dynamics initialized on surface 1 with $c_ {\color{black}1}(t=0)=1$. Parameters are from column two of Table \ref{['table_param']}.
• Figure 4: (a) Comparison of diabatic population of state 1 obtained from surface hopping with and without decoherence and HEOM for the parameters in the second column of Table \ref{['table_param']}. (b) Long-time population comparison between FSSH, AFSSH, and HEOM for a range of temperatures. Values of other variables are given in table \ref{['table_param']}, column 2.