Decoherence-induced surface hopping Cite as: J. Chem. Phys. 137, 22A545 (2012); https://doi.org/10.1063/1.4757100 Submitted: 08 June 2012 . Accepted: 19 September 2012 . Published Online: 15 October 2012 Heather M. Jaeger, Sean Fischer, and Oleg V. Prezhdo ARTICLES YOU MAY BE INTERESTED IN Molecular dynamics with electronic transitions The Journal of Chemical Physics 93, 1061 (1990); https://doi.org/10.1063/1.459170 Perspective: Nonadiabatic dynamics theory The Journal of Chemical Physics 137, 22A301 (2012); https://doi.org/10.1063/1.4757762 Critical appraisal of the fewest switches algorithm for surface hopping The Journal of Chemical Physics 126, 134114 (2007); https://doi.org/10.1063/1.2715585 J. Chem. Phys. 137, 22A545 (2012); https://doi.org/10.1063/1.4757100 137, 22A545 © 2012 American Institute of Physics. THE JOURNAL OF CHEMICAL PHYSICS 137, 22A545 (2012) Decoherence-induced surface hopping Heather M. Jaeger,1 Sean Fischer,2 and Oleg V. Prezhdo1 1Department of Chemistry, University of Rochester, Rochester, New York 14627-0216, USA 2Department of Chemistry, University of Washington, Seattle, Washington 98195-1700, USA (Received 8 June 2012; accepted 19 September 2012; published online 15 October 2012) A simple surface hopping method for nonadiabatic molecular dynamics is developed. The method derives from a stochastic modeling of the time-dependent Schrödinger and master equations for open systems and accounts simultaneously for quantum mechanical branching in the otherwise classical (nuclear) degrees of freedom and loss of coherence within the quantum (electronic) subsystem due to coupling to nuclei. Electronic dynamics in the Hilbert space takes the form of a unitary evolution, intermittent with stochastic decoherence events that are manifested as a localization toward (adia- batic) basis states. Classical particles evolve along a single potential energy surface and can switch surfaces only at the decoherence events. Thus, decoherence provides physical justification of sur- face hopping, obviating the need for ad hoc surface hopping rules. The method is tested with model problems, showing good agreement with the exact quantum mechanical results and providing an improvement over the most popular surface hopping technique. The method is implemented within real-time time-dependent density functional theory formulated in the Kohn-Sham representation and is applied to carbon nanotubes and graphene nanoribbons. The calculated time scales of non-radiative quenching of luminescence in these systems agree with the experimental data and earlier calculations. © 2012 American Institute of Physics.[http://dx.doi.org/10.1063/1.4757100] I. INTRODUCTION fluctuating system states in contact with a bath. Concomi- Mixed quantum-classical methodologies provide an tantly, through the quantum backreaction, the macroscopic intuitive approach to real-time dynamic simulations of bath follows an array of (non-classical) trajectories associ- physical and chemical processes in condensed phases.1 ated with individual components of the quantum state. Di- Quantum-classical simulations of solution chemistry,2–4 sur- vergence among the bath trajectories leads to a loss of phase face chemistry,5–9 and excitons in polymers10 and nano-sized relationship among the states of the total system. When the structures11–13 prove valuable toward understanding quan- divergence is slow relative to transitions within the quantum tum systems in a macroscopic environment. Classical equa- subsystem, standard quantum-classical approaches model dy- tions of motion are applied to a large number of degrees namic evolution quite well. However, when the opposite is of freedom, avoiding intractable quantum mechanical ba- true and the divergence is fast relative to the rates of tran- sis set calculations. Dynamics of a small quantum subsys- sitions, quantum-classical approaches must be corrected for tem can be treated utilizing a reduced description of the decoherence. Coherence loss is facilitated by a lack of phase- total system. Quantum-classical models are widely used de- space structure,24 and furthermore, becomes an irreversible spite fundamental obstacles in the interaction between quan- phenomenon in systems with many degrees of freedom. Mod- tum and classical regimes.14, 15 The Ehrenfest method16–18 ap- eling the dynamic evolution of condensed phase systems proximates quantum-classical coupling through expectation requires a theoretical account of decoherence as either an in- values of quantum variables. An improved treatment of the herent quantum effect of the bath or a phenomenological con- quantum-classical interaction introduces the quantum backre- sequence of the system-bath interaction. action through a distribution of classical variables individu- The importance of decoherence in condensed-phase ally correlated to quantum basis states.19 Renowned surface chemical reactions has long been appreciated.25, 26 Reaction hopping methods20, 21 stochastically sample the dynamic evo- rates carry an explicit dependence on decoherence.27, 28 The lution of this quasi-classical distribution. As the system size environment acts as a source of friction that damps nonadi- grows, however, the quantum subsystem responds with in- abatic effects and, accordingly, reaction probabilities. Mini- creasing measure to subtle quantum effects in the classically mizing environment-induced decoherence has become an ide- treated degrees of freedom. alized goal of quantum control reaction mechanisms,29 where A quantum mechanical wave packet of the total sys- quantum control relies on the existence and manipulation tem, accounting for all quantum effects of the bath, splits of electronic coherences. In addition to coherence-controlled into uncorrelated branches and loses coherence over time. chemistry,30–32 the rates and mechanisms of energy transfer The resulting decoherence can be viewed as an environment- within large molecular aggregates,33–37 polymers,38 and or- induced destruction of superpositioned quantum states of the ganic chromophores39 depend on decoherence. The quantum microscopic system.22, 23 In more detail, the microscopic sys- Zeno effect is an extreme example of such dependence, in tem evolves quantum mechanically as a superposition of which quantum transitions cease in the limit of infinitely fast 0021-9606/2012/137(22)/22A545/14/$30.00137, 22A545-1 © 2012 American Institute of Physics 22A545-2 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012) decoherence.40–42 In quantum dots, nano-wires, and nano- puted as an adiabatic Hellmann-Feynman force or a mean- tubes, exciton dephasing43–45 is pervasive as the wave packet field force.54, 55 Alternatively, the Pechukas method56 offers evolves nonadiabatically throughout a large number of quan- a self-consistent solution for the nuclear path under the in- tum states. fluence of quantum transitions. As coherences are quenched, Modifications of quantum-classical surface hopping ap- quantum transition probabilities approach zero, and the “best” proaches treat the effect of decoherence on the evolution classical path is determined by the Hellmann-Feynman force of the quantum subsystem. Decoherence corrections have associated with a single adiabatic state. Environment-induced been designed and applied to electronic46 and vibrational47, 48 decoherence leads to a conceptually and computationally sim- excited state dynamics. Interestingly, inclusion of decoher- ple procedure for switching potential energy surfaces. ence strengthens the quantum-classical correspondence, and, Introducing decoherence at the microscopic level for in the context of trajectory surface hopping simulations, is condensed phases can be achieved by an approximate so- most readily seen as an increase in the internal stability.49, 50 lution to the Nakajima-Zwanzig generalized quantum mas- Many decoherence-corrected approaches utilize information ter equation,57–62 quantum correction factors to classical about the excited state potential surfaces to estimate the ef- reaction rates,63–65 numerical formulations of path inte- fect of trajectory divergence on the transition probability.51 gral methods,66–68 or phenomenological equations of motion A computationally simple approach assumes the classical for the reduced density matrix.69–72 Phenomenological ap- environment establishes an open system in which diffu- proaches aim to generate quantum Langevin-like dynamics, sive, coherence-destroying, dynamics can be described by in which the dynamics become dissipative due to a “friction” Markovian evolution.52 Combining wave packet collapse with generated by quantum fluctuations in the environment. We the time-dependent Schrödinger equation (TDSE) provides a follow a distinct, yet parallel, procedure toward modeling de- complete description of electron dynamics subject to quantum coherence by applying a dissipative extension to the TDSE. decoherence. Previous work on the stochastic Schrödinger Drawing from the methodology of stochastic Schrödinger equation along a mean-field trajectory53 sets the stage for the and master equations,73–78 solutions to the modified equa- current work, in which decoherence induces stochastic sur- tions of motion are found through a computationally efficient, face hopping. “quantum-jump” procedure. Hence, decoherence results in Decoherence-induced surface hopping (DISH) relies on a surface hopping, and, in DISH, quantum transitions only oc- simple picture of quantum state diffusion, that is applicable to cur at the time of decoherence. large,