Letter pubs.acs.org/JPCL Fewest Switches Surface Hopping in Liouville Space † ‡ † Linjun Wang,*, Andrew E. Sifain, and Oleg V. Prezhdo*, † Department of Chemistry, University of Southern California, 3620 McClintock Avenue, Los Angeles, California 90089-1062, United States ‡ Department of Physics and Astronomy, University of Southern California, 825 Bloom Walk, Los Angeles, California 90089-0485, United States ABSTRACT: The novel approach to nonadiabatic quantum dynamics greatly increases the accuracy of the most popular semiclassical technique while maintaining its simplicity and efficiency. Unlike the standard Tully surface hopping in Hilbert space, which deals with population flow, the new strategy in Liouville space puts population and coherence on equal footing. Dual avoided crossing and energy transfer models show that the accuracy is improved in both diabatic and adiabatic representations and that Liouville space simulation converges faster with the number of trajectories than Hilbert space simulation. The constructed master equation accurately captures superexchange, tunneling, and quantum interference. These effects are essential for charge, phonon and energy transport and scattering, exciton fission and fusion, quantum optics and computing, and many other areas of physics and chemistry. umerous processes of interest in modern physics, extensively applied by Zwanzig24 and Redfield25 in the quantum N chemistry, and biology require consideration of quantum theory of relaxation and by Mukamel in nonlinear optical dynamics; however, a fully quantum description is generally spectroscopy.26 Quantum dynamics in Liouville space is forbidden due to computational cost. Frequently, only a small described by a time-dependent density vector, which is formally portion of particles under investigation, such as electrons and equivalent to the time-dependent wave function in Hilbert excitons, must be treated quantum mechanically, while the space; however, the resulting dynamic pathways are completely remaining degrees of freedom, typically nuclei, can be different.27 In Hilbert space, surface hops happen between considered classically. Motivated by this fact, mixed quantum- directly coupled states. Population transfer is expressed in a classical methods have become the most popular choice for more elaborate way in Liouville space: Changes in quantum studying time-dependent phenomena including charge trans- coherence depend on magnitudes of quantum populations and − port,1 5 photoinduced electron transfer,6 photochemistry,7 vice versa. Coherences play an extremely important role in 28,29 exciton relaxation,8 singlet fission,9 and bioluminescence.10 quantum dynamics. Liouville space allows us to treat A series of quantum-classical techniques have been populations and coherences on equal footing, leading to − developed over the last few decades.11 14 Tully’s fewest dramatic, qualitative improvements in accuracy and allowing switches surface hopping (FSSH)15 is the most popular option. FSSH to treat many important phenomena. ffi Quantum-mechanical mapping between the Hilbert and Downloaded via UNIV OF SOUTHERN CALIFORNIA on November 8, 2019 at 00:08:19 (UTC). FSSH is straightforward to implement, accurate, and e cient. It See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. 26 is also easily compatible with modern electronic structure Liouville spaces is known in quantum optics and comput- 30 methods. FSSH is widely adopted as the platform to introduce ing. While such mapping is straightforward in pure quantum modifications as needed. It has been corrected for decoherence mechanics, it is neither obvious nor unique in a quantum- effects soon after its invention.16 More strategies to incorporate classical description. Quantum-classical theories raise multiple 31−33 decoherence have been presented,17,18 although proper questions, such as the quantum backreaction, answers to handling of decoherence in realistic systems is still not finally which require extensive research and testing. Furthermore, settled. A QM/MM-like flexible surface hopping has been quantum-classical theories are much more broadly applicable fi fi proposed, grasping all charge transport regimes in organic and carry signi cant impact in a wide spectrum of elds, crystals.3,19 Self-consistency checks20 have been recently added, including nanoscale, condensed phase, soft matter, and gas- solving the trivial crossing problem arising due to high state phase systems in physics, chemistry, biology, materials, and density in large systems. The fewest switches concept has been related disciplines. generalized to gross population flow between states.21 FSSH This letter reports a novel method for description of fi has been reformulated in the second quantization language to quantum dynamics. For the rst time, the most practical include entanglement within the trajectory ensemble.22 Standard FSSH is carried out in Hilbert space. Fifty years Received: July 14, 2015 ago, Fano coined the concept of Liouville space, which is a Accepted: September 8, 2015 direct product of two Hilbert spaces.23 This space has been Published: September 8, 2015 © 2015 American Chemical Society 3827 DOI: 10.1021/acs.jpclett.5b01502 J. Phys. Chem. Lett. 2015, 6, 3827−3833 The Journal of Physical Chemistry Letters Letter semiclassical technique is formulated in Liouville space. The Tully’s FSSH is carried out in Hilbert space. The time | ⟩ ρ 15 quantitative and qualitative advantages are demonstrated with derivative of the population of state i , ii, is expressed as two standard models, representing quantum interference, − dρii tunneling, and superexchange phenomena that are essential = ∑ b dt ij for many areas of physics, including quantum transport, ji≠ (10) scattering, Auger processes, quantum computing, and optics. The method retains the advantages of the traditional technique, where including simple formalism, internal consistency, ease of bH=ℏ2Im()−1 ρ* implementation, and low computational cost. Currently ij ij ij (11) formulated for FSSH, the Liouville representation can be In FSSH, the switching probability from state i to another state applied easily to other surface hopping methods. j within a time interval Δt is The time-dependent wave function, |ψ(t)⟩, is expressed as a linear combination of a set of orthogonal basis states in Hilbert −Δtbij | ⟩ = space, { i } gij ρii (12) |ψ()tcti⟩=∑ () |⟩ i Drawing upon the similarity between the Hilbert and i (1) Liouville formulations, eqs 1 and 2 and 8 and 9,wedefine ̈ | ⟩⟩ ≡ |ρ |2 ρ2 Its time evolution follows the Schrodinger equation the population of state ij in Liouville space as pij ij .Tr Σ | ⟩⟩ d()|⟩ψ t 1 = 1 always holds for pure states, and hence ijpij =1.{ij } = Ht̂|ψ()⟩ forms a complete, orthonormal basis for a quantum system in (2) dtiℏ Liouville space, similar to {|i⟩} in Hilbert space. Analogously, where Ĥis the system Hamiltonian. Alternatively, one can we obtain describe the quantum system using the density operator, ρ̂. For dpij a pure state, it reads = ∑ b dt ij, kl ρ()̂ ttt=|ψψ () ⟩⟨ ()| (3) ki≠≠or lj (13) The dynamics follows the Liouville equation where d()ρ̂ t 1 bHH=ℏ2‐1 [δρρδρρ Im(* )+ Im(* )] = [,]Ĥ ρ̂ ij, kl ik ij kl jl jl ij kl ik (14) dtiℏ (4) Similar to eq 12, the surface hopping probability in Liouville Consider a two-level system as an illustration. The space is given by Hamiltonian is written as ⎛ ⎞ −Δtbij, kl VV11 12 g = Ĥ = ⎜ ⎟ ij, kl p ij (15) ⎝VV21 22⎠ (5) Equations 11 and 14 are derived for time-independent basis in Equation 4 becomes the diabatic representation. In general, one needs to replace Ĥ ⎛ρρ⎞ ̂ d 11 12 1 with H − iℏv·d, where v·d is the nonadiabatic coupling ⎜ ⎟ = ⎝ρρ⎠ ℏ dti21 22 d ⎛ ⎞ (vd·=⟨||⟩) i j −+VVρρ −+− V ρ() VV ij ⎜ 21 12 12 21 12 11 11 22 ⎟ dt (16) ⎜ ρρ+ V ⎟ ⎜ 12 12 22 ⎟ ⎜ ⎟ When a surface hop is assigned, velocities need to be adjusted ⎝VVVVρρρρρ+−() − VV − ⎠ 21 11 22 11 21 21 22 21 12 12 21 (6) to conserve the total quantum-classical energy. In the adiabatic representation of Hilbert space FSSH, velocity rescaling is and can be reformatted as normally performed in the direction of the nonadiabatic ⎛ ρ ⎞ ⎛ 00− VV ⎞⎛ ρ ⎞ coupling vector.15 If a surface hop happens from |ij⟩⟩ to |kl⟩⟩ in ⎜ 11⎟ ⎜ 21 12 ⎟⎜ 11⎟ fi ρ ρ Liouville space, one projects dik onto djl to de ne the scaling d ⎜ 12⎟ 1 ⎜−−VV12 11 V 220 V 12 ⎟⎜ 12⎟ = ⎜ ⎟ direction. The situation is less clear in the diabatic ⎜ ρ ⎟ ⎜ ρ ⎟ dti⎜ 21⎟ ℏ ⎜ VVVV210 22−− 11 21⎟⎜ 21⎟ representation because the nonadiabatic coupling is zero in ⎜ ⎟ ⎜ ⎟⎜ ⎟ this case. In principle, one needs to find the direction along ⎝ρ ⎠ ⎝ − ⎠⎝ρ ⎠ 22 00VV21 12 22 which the surface hopping probabilities based on eqs 12 and 15 (7) are maximized. Velocity rescaling is straightforward for 1D If we define models because there is only one classical degree of freedom. Quantum-classical energy conservation achieved by velocity |ρρ⟩⟩ =∑ ij |ij ⟩⟩ rescaling is critical to achieve the detailed balance between ij (8) transitions up and down in energy and the quantum-classical equilibrium in the long time limit.34 For multidimensional where |ij⟩⟩ ≡ |i⟩⟨j|, eq 7 can be further simplified as realistic systems, one can adopt the classical path approx- d 1 imation,35 and replace velocity rescaling by multiplying the |⟩⟩=ρρL̂|⟩⟩̂ dtiℏ (9) surface hopping probabilities upward in energy with the Boltzmann factor. Valid in the case of rapid redistribution of Here the superoperator L̂̂ connects all elements of |ρ⟩⟩. energy within the classical subsystem, compared with the 3828 DOI: 10.1021/acs.jpclett.5b01502 J. Phys. Chem. Lett. 2015, 6, 3827−3833 The Journal of Physical Chemistry Letters Letter Figure 1. Dual avoided crossing model. (A) Diabatic potential energies and interstate couplings. (B) Reflection on the lower state. Transmission on (C) the lower state and (D) the upper state. The energy of the coherence state |12⟩⟩ is shown as a dashed blue line in panel A. Open circles are exact quantum mechanical results,15,42 red (blue) lines are FSSH results in Hilbert (Liouville) space with diabatic representation, and purple (green) lines are Hilbert (Liouville) space FSSH results with adiabatic representation.