Analysis of the Trajectory Surface Hopping Method from the Markov State Model Perspective Alexey V. Akimov 1 , Dhara Trivedi 2 , Linjun Wang 1 , and Oleg V. Prezhdo 1+ 1 Department of Chemistry, University of Southern California, Los Angeles, CA 90089, U.S.A. 2 Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, U.S.A. (Received May 20, 2015; accepted July 22, 2015; published online August 18, 2015) We analyze the applicability of the seminal fewest switches surface hopping (FSSH) method of Tully to modeling quantum transitions between electronic states that are not coupled directly, in the processes such as Auger recombination. We address the known deficiency of the method to describe such transitions by introducing an alternative definition for the surface hopping probabilities, as derived from the Markov state model perspective. We show that the resulting transition probabilities simplify to the quantum state populations derived from the time-dependent Schrödinger equation, reducing to the rapidly switching surface hopping approach of Tully and Preston. The resulting surface hopping scheme is simple and appeals to the fundamentals of quantum mechanics. The computational approach is similar to the FSSH method of Tully, yet it leads to a notably different performance. We demonstrate that the method is particularly accurate when applied to superexchange modeling. We further show improved accuracy of the method, when applied to one of the standard test problems. Finally, we adapt the derived scheme to atomistic simulation, combine it with the time-domain density functional theory, and show that it provides the Auger energy transfer timescales which are in good agreement with experiment, significantly improving upon other considered techniques. 1. Introduction Theory of molecular dynamics with quantum transitions is an actively developing field of computational chemistry. 1–14) The interest in this subject is constantly stimulated by the need for accurate and computationally tractable simulation methodologies for modeling processes involving coupled electron–nuclear dynamics in contemporary materials. In particular, such techniques are required for studying photo- induced charge transfer in photovoltaic and photocatalytic materials, 15–20) long-distance energy transfer in biological photosynthetic complexes, or photo- and electro-stimulated mechanical response in nanoscale systems. 20,21) A great body of methods for modeling molecular dynamics with quantum transitions has been developed over the last several decades. We refer the reader to specialized reviews discussing some of these methods. 22–27) Among the variety of techniques, Tully’s fewest switches surface hopping (FSSH) 1) remains one of the most utilized methods, due to its conceptual and practical simplicity, good accuracy, and high computational efficiency. The FSSH method is not a panacea, and there are examples where it breaks down. In particular, the superexchange problem 28) requires a more advanced treatment. The need for consideration of superexchange effects is motivated in applications by excitonic, many-particle proc- esses, such as Auger recombination and energy transfer, 29–35) singlet fission 36–39) or Raman scattering. The superexchange mechanism is also important for tunneling and conduc- tivity in molecular electronics. 40–44) In all these processes, transitions between excitonic states can involve more than one simultaneous single-particle transition. A direct consid- eration of the dynamics of such excitations is prohibited by the Slater rules, and only sequential mechanisms are viable. The problem of superexchange arises when there is at least one pair of states that are not coupled to each other directly. The states can be either diabatic or adiabatic. The transition between these states is mediated by an intermediate high- lying state. The overall transition rates are strongly under- estimated in FSSH theory because the transitions through such intermediate states are inhibited by the probabilities of overcoming the high-energy barrier. Simulations of non- adiabatic processes using FSSH are usually performed in the adiabatic basis, in which FSSH shows reasonable accuracy. However, the use of diabatic states (e.g., donor and acceptor states) may be advantageous, both in terms of computational efforts and for interpretation of the problem physics. Diabatic states are used regularly in phenomenological models of charge and energy transfer because they have well-defined physical meaning. Superexchange effects are traditionally described in a diabatic representation. An adiabatic picture can lead to spurious effects arising from artificial delocaliza- tion of an adiabatic state between donor and acceptor sites that are very distant from each other. Finally, the adiabatic representation leads to numerical problems with calculation of non-adiabatic coupling. 45,46) Under the conditions when diabatic states are strongly localized and many pairs of states are decoupled, the accuracy of the original FSSH method is often unacceptable when diabatic states are used, motivating development of the alternative formulations. 28,47) Recently, the problem of superexchange has been addressed by Wang et al. 28) who proposed a global flux surface hopping (GFSH) algorithm. In it, hopping between a pair of states is determined by a relative change of state population during the infinitesimal time slice. The method avoids computing the hopping probabilities based on one- particle properties (such as non-adiabatic couplings between one-particle states), thus solving the superexchange problem phenomenologically. In the alternative second-quantized surface hopping (SQUASH) approach, 47) the superexchange problem was solved by extending the quantum dynamics to the space of second-quantized states representing coupled trajectories. The method introduces many-particle states, allowing a single transition between a pair of such states to capture a many-body transition. Although the developed techniques provide notable improvement of the accuracy in modeling the superexchange problem, they have certain limitations. Specifically, the Journal of the Physical Society of Japan 84, 094002 (2015) http://dx.doi.org/10.7566/JPSJ.84.094002 094002-1 © 2015 The Physical Society of Japan J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by Univ of Southern California on 11/07/19