Chemical Physics 481 (2016) 77–83 Contents lists available at ScienceDirect Chemical Physics journal homepage: www.elsevier.com/locate/chemphys Surface hopping from the perspective of quantum–classical Liouville dynamics Raymond Kapral Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, ON M5S 3H6, Canada article info abstract Article history: Fewest-switches surface hopping is studied in the context of quantum–classical Liouville dynamics. Both Available online 20 May 2016 approaches are mixed quantum–classical theories that provide a way to describe and simulate the nona- diabatic quantum dynamics of many-body systems. Starting from a surface-hopping solution of the quan- Keywords: tum–classical Liouville equation, it is shown how fewest-switches dynamics can be obtained by dropping Nonadiabatic dynamics terms that are responsible for decoherence and restricting the nuclear momentum changes that accom- Fewest-switches surface hopping pany electronic transitions to those events that occur between population states. The analysis provides Quantum–classical Liouville equation information on some of the elements that are essential for the construction of accurate and computation- ally tractable algorithms for nonadiabatic processes. Ó 2016 Elsevier B.V. All rights reserved. 1. Introduction ber of hops consistent with the electronic populations. While this method has known shortcomings it is simple to use and often The Born–Oppenheimer approximation [1] figures prominently yields reasonable results. in studies of quantum structure and dynamics. It relies on a scale More specifically, the equations for the electronic density separation that is controlled by a small parameter gauged by the matrix elements governing the dynamics are as follows: The diag- ratio m=M of the light m to heavy M masses of different con- onal density matrix elements satisfy stituents of the system. This approximation forms the basis for most of electronic structure theory and is also used in adiabatic d mm Pt Ãma0 q ðXt; tÞ¼2R Á dma0 ðRtÞq ðXt; tÞ ; ð1Þ quantum dynamics where nuclei move on single Born–Oppen- dt M heimer surfaces. Although the Born–Oppenheimer approximation has wide utility, it does break down and this breakdown signals while the off-diagonal elements evolve by the fact that quantum nuclear motion can no longer be described d ma0 ma0 Pt ba0 as motion on a single electronic state. Nonadiabatic dynamics is q ðX ; tÞ¼ixma0ðR Þq ðX ; tÞ Á dmbðR Þq ðX ; tÞ dt t t t M t t important for the description of many excited-state physical pro- Pt à mb cesses. Quantum dynamical methods that account for the break- À Á da0bðR Þq ðX ; tÞ: ð2Þ M t t down of the Born–Oppenheimer approximation must then be used to follow the time evolution of the system [2]. Fewest- In this equation dabðRÞ is the nonadiabatic coupling matrix element, switches surface hopping [3] is one of the most widely used dabðRÞ¼ha; RjrRjb; Ri and ja; Ri, denotes the a adiabatic eigenstate. schemes for this purpose. More generally, the basic elements of The summation convention was used above and will be used the fewest-switches algorithm often enter into molecular dynam- throughout the paper except where summations are written in full ics methods that involve quantum transitions [4]. for clarity. In fewest-switches surface hopping the nuclei are assumed fol- Transitions between adiabatic states occur probabilistically ¼ð ; Þ ¼ð ; ; ...; Þ low stochastic trajectories Xt Rt Pt , with R R1 R2 RN and with a transition rate selected so that the fraction of trajectories ¼ð ; ; ...; Þ P P1 P2 PN denoting the N nuclear positions and momenta, in a given adiabatic state corresponds to the electronic population respectively [5]. Trajectory evolution takes place on single adia- of that state. Energy is conserved along the stochastic trajectories batic surfaces with stochastic ‘‘hops” to other surfaces that occur and to ensure that this is the case whenever a nonadiabatic transi- with probabilities that are constructed to lead to the fewest num- tion causes the system to change its state the nuclear momenta are adjusted to compensate for the energy change in the quantum E-mail address: [email protected] transition. For example, if a transition from state a to state b occurs http://dx.doi.org/10.1016/j.chemphys.2016.05.016 0301-0104/Ó 2016 Elsevier B.V. All rights reserved. 78 R. Kapral / Chemical Physics 481 (2016) 77–83 the momenta of the nuclei along the direction of the nonadiabatic where the Hellmann–Feynman forces are FaðRÞ¼@EaðRÞ=@R. The FS coupling vector are adjusted by P ! P þ DPab, with operator, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 1 @ à P 1 @ FS ^ ^ ^ 2 ^ ^ D ¼ ð Á Þ ð Á Þ þ D À ð Á Þ; ð Þ J aa0;bb0 ¼dab Á þ DEab da0b0 À da0b0 Á þ DEa0b0 dab; Pab dab sgn P dab P dab 2 EabM dab P dab 3 M 2 @P M 2 @P to conserve energy. Here the energy gap is DEab ¼ Ea À Eb. For ð6Þ upward transitions it may happen that there is insufficient energy in the nuclear degrees of freedom to insure energy conservation. couples the dynamics on the individual and mean adiabatic sur- L 0 In this case the transition rule needs to be modified, usually by set- faces. The last line of Eq. (4) defines the QCL operator, i aa0;bb . ting the transition probability to zero. A few features of this equation are worth noting. The classical This algorithm captures many of the important physical fea- evolution operators iLaa0 describe adiabatic evolution on either sin- a ¼ a0 a – a0 tures of nonadiabatic dynamics and is easy to implement in com- gle ( ) surfaces or on the mean of two surfaces when . putations, thus justifying its widespread use. It is not without No approximation is made to obtain such evolution on the mean defects. Since there is no mechanism for the decay of the off-diag- of two surfaces for off-diagonal elements; it follows naturally from onal density matrix elements, it cannot describe the effects of the representation of the QCLE in the adiabatic basis. The coupling J 0 decoherence on nonadiabatic processes. A considerable amount term aa0;bb not only involves nonadiabatic coupling matrix ele- of effort has been devoted to modification of fewest-switches sur- ments, dabðRÞ, but also derivatives with respect to the nuclear face hopping to introduce decoherence into the scheme [6–15]. momenta. This term accounts for part of the influence of the nona- The aim of this article is to determine the conditions under diabatic quantum electronic dynamics on the nuclei. This impor- which quantum–classical Liouville dynamics [16] can be reduced tant coupling adds complexity to the equation of motion and its to fewest-switches surface hopping. The quantum–classical Liou- exact treatment precludes a simple description of the nuclear ville equation provides a basis for the derivation of various quan- evolution. tum–classical methods [17]. While solutions to this equation can One may attempt to solve this equation by any convenient be obtained by a variety of methods, solutions may also be method and considerable effort and schemes have been devised obtained by a surface-hopping algorithm [18–20], and it is in the with the aim of obtaining accurate yet computationally tractable context of the approximations to this surface-hopping dynamics solutions. [22–33] Since the goal of this paper is to explore connec- that we shall consider fewest-switches surface hopping. In partic- tions to fewest-switches surface hopping (FSSH), we consider ular, it will be shown that by dropping terms that account for the approximate solutions that are based on surface-hopping trajecto- effects of decoherence and modifying how nonadiabatic transitions ries. It is useful to observe that while the QCLE conserves energy, and the nuclear momentum changes that accompany them are nothing is implied about conservation of energy in any single tra- treated, one can arrive fewest-switches surface hopping. jectory that might be used in solutions to this equation. The main text begins in Section 2 with a brief outline and crit- ical discussion of the features of the quantum–classical Liouville 2.1. Surface-hopping solution of the QCLE equation in the adiabatic basis and its solution by a surface-hop- ping algorithm. This sets the stage for the analysis in Section 3 that The basis for the surface-hopping solution was described some allows one to see in some detail the approximations to the dynamics time ago [21] and the details of the algorithms and their applica- that lead to fewest-switches surface hopping. The last section of the tions to various problems have been discussed previously paper discusses how the results of this study may provide ingredi- [18–20]. Nevertheless, it is useful to present a brief account of this ents for the construction of new surface-hopping algorithms. solution scheme in order to contrast it with FSSH in the next section, and to point to some of its features that are often 2. Quantum–classical Liouville dynamics in the adiabatic basis overlooked. In general terms the surface-hopping method is a stochastic algorithm for the solution of the QCLE that relies on Since surface-hopping methods are often formulated in the adi- Monte Carlo sampling of diagonal and off-diagonal electronic abatic basis, it is instructive to discuss the dynamical picture that states and accounts for nuclear momentum changes when transi- emerges when the quantum–classical Liouville equation (QCLE) is tions occur. In its usual implementation only one basic approxima- expressed in this basis. The partially Wigner transformed Hamilto- tion is made: the momentum-jump approximation [21,34,16,17]. ^ This approximation, outlined below, replaces the infinitesimal nian, HW , for the system can be written as the sum of the nuclear 2 nuclear momentum changes contained in the J aa0;bb0 coupling term kinetic energy, P =2M, and the remainder of the electronic, nuclear and coupling terms contained in the operator by finite momentum changes.