Detailed balance, internal consistency, and energy conservation in fragment orbital-based surface hopping

Abstract

We have recently introduced an efficient semi-empirical non-adiabatic molecular dynamics method for the simulation of charge transfer/transport in molecules and molecular materials, denoted fragment orbital-based surface hopping (FOB-SH) [J. Spencer et al., J. Chem. Phys. 145, 064102 (2016)]. In this method, the charge carrier wavefunction is expanded in a set of charge localized, diabatic electronic states and propagated in the time-dependent potential due to classical nuclear motion. Here we derive and implement an exact expression for the non-adiabatic coupling vectors between the adiabatic electronic states in terms of nuclear gradients of the diabatic electronic states. With the non-adiabatic coupling vectors (NACVs) available, we investigate how different flavours of fewest switches surface hopping affect detailed balance, internal consistency, and total energy conservation for electron hole transfer in a molecular dimer with two electronic states. We find that FOB-SH satisfies detailed balance across a wide range of diabatic electronic coupling strengths provided that the velocities are adjusted along the direction of the NACV to satisfy total energy conservation upon a surface hop. This criterion produces the right fraction of energy-forbidden (frustrated) hops, which is essential for correct population of excited states, especially when diabatic couplings are on the order of the thermal energy or larger, as in organic semiconductors and DNA. Furthermore, we find that FOB-SH is internally consistent, that is, the electronic surface population matches the average quantum amplitudes, but only in the limit of small diabatic couplings. For large diabatic couplings, inconsistencies are observed as the decrease in excited state population due to frustrated hops is not matched by a corresponding decrease in quantum amplitudes. The derivation provided here for the NACV should be generally applicable to any electronic structure approach where the electronic Hamiltonian is constructed in a diabatic electronic state basis.

Flickering Polarons Extending over Ten Nanometres Mediate Charge Transport in High‐Mobility Organic Crystals

“…Hence, although it is extremely useful and important to have analytic theories, it is paramount to develop also numerical schemes that give insight into the actual dynamics and that seamlessly bridge the gap between different mechanistic regimes. With this purpose in mind, we have developed an efficient fully atomistic non-adiabatic molecular dynamics approach, denoted fragment orbital-based surface hopping (FOB-SH), [29][30][31][32] which allows us to propagate the coupled charge-nuclear motion in realtime for condensed phase systems. The methodology is based on a DFT-parametrized tight-binding representation of the electronic Hamiltonian (updated on-the-fly) to naturally incorporate local and non-local electron-phonon couplings, thus encompassing in a non-perturbative manner a broad range of possible transport mechanisms.…”

“…We verified that the average polaron size and shape is insensitive to the choice of the initial carrier wavefunction while the time for relaxation to this state may, of course, vary. This is because FOB-SH maintains detailed balance in the long-time limit to a good approximation [30,32] ensuring that after initial relaxation, the equilibrium (Boltzmann) populations of electronic band states are reached, independently on the initial starting point. It should be noted that the finite size of the polaron is due to the thermal diagonal and off-diagonal disorder in the electronic Hamiltonian Equation (1).…”

Quasi-Diabatic Propagation Scheme for Direct Simulation of Proton-Coupled Electron Transfer Reaction

“…Note that a similar violation of the internal consistency has been found in FSSH. [82][83][84] Resolving this issue in future could potentially improve the SQC approach. Nevertheless, the above calculations demonstrate that the QD propagation scheme indeed provides opportunities to assess the performance of approximate diabatic dynamics approaches, with test cases beyond simple diabatic model systems, and at the same time, completely avoid any additional efforts to reformulate these methods back to the adiabatic representation.…”

Strong Influence of Decoherence Corrections and Momentum Rescaling in Surface Hopping Dynamics of Transition Metal Complexes

“…In the Ep h and Ep g schemes, the momentum is rescaled along the h αλ and g αλ directions, respectively. Based on formal arguments 9,65,66 and numerical results on simple few-state model systems, 31,67 a number of authors have concluded that Ep h is the most rigorous option. Here we shall investigate whether this conclusion holds for a larger system containing more states and many degrees of freedom and how strong the effect is.…”

“…It has been argued that frustrated hops are required to maintain quantum detailed balance, i.e., the statistical ratio of up and down transitions between different energy surfaces. 31,32,67 Only by rejecting upward hops is it possible to assure that the lower energy state has increased population in agreement with the Boltzmann distribution. 31,32 In the case of a frustrated hop the active state does not change.…”