Fewest Switches Surface Hopping in Liouville Space † ‡ † Linjun Wang,*, Andrew E

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 ρ̂()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. The vertical orange dashed lines in panels B−D express the critical kinetic − | | ≫ 2 energy, which is equal to V22 V11 at x 5. E = k /2m is the initial kinetic energy. transition time, the Boltzmann approximation achieves the FSSH is formally similar, the fraction of trajectories on state | ⟩⟩ detailed balance. ij , Pij, also equals to the corresponding population, pij. In the Several additional definitions are needed. First, we define the absence of hop rejection,15 the fraction of trajectories on state i, | ⟩⟩ energy for each ij state, Eij.Ifi = j, Eij equals the energy of the Pi, can be obtained through ≠ ith state in Hilbert space, Vii. When i j, we follow the analysis of refs 36 and 37 and adopt Eij =(Vii + Vjj)/2. In principle, only Piii=+PPP∑ ()/2ij + ji Hilbert states have well-defined energies. If a coherence state is ji≠ (17) regarded as a superposition of two Hilbert states, its energy Σ Σ Σ ρ ρ Σ ρ ρ ρ should be an average over both states, perhaps weighted by the which equals j Pij = j pij = j ij ji = j ii jj = ii considering corresponding amplitudes. Placing the energy of the Liouville Pij = Pji. Hence, Liouville space FSSH holds the same level of space coherence state in the middle of the two corresponding internal consistency as Hilbert space FSSH. Note that rigorous Hilbert states is an approximate yet rational way to mimic the internal consistency cannot be realized when running each 17 role of coherence in quantum dynamics. (See the numerical trajectory independently. 15 demonstrations later.) The same convention has been adopted First, we examine Tully’s dual avoided crossing model, 36,37 fi in the previous works. Second, the fraction of trajectories which is de ned in the diabatic representation: V11(x)=0, | ⟩⟩ − −0.28x2 −0.06x2 populating coherence states in Liouville space, that is, ij with V22(x)= 0.1e + 0.05, V12(x)=V21(x) = 0.015e i ≠ j, needs to be converted back to Hilbert space populations (see Figure 1A). The mass of the x degree of freedom, m,is for data analysis. Energy conservation is essential. When the 2000 au, which is close to the mass of a hydrogen atom. total energy, that is, classical kinetic energy (Ek) plus Eij, is more Initially, a Gaussian wave packet is prepared on the negative x | ⟩⟩ than both Vii and Vjj, half of the trajectories on ij contribute side of state 1, and the corresponding width is set to be 20/k, to |i⟩ and the other half contribute to |j⟩, to be consistent with with k being the initial momentum. This model exhibits two the energy convention. If the total energy is larger than Vii but avoided crossings, where the quantum interference causes | ⟩⟩ | ⟩ 15 smaller than Vjj, the trajectories on ij fully contribute to i in Stueckelberg oscillations in the transmission probabilities. Hilbert space or vice versa. When decoherence occurs rapidly − − For such a two-level system, there are two basis states and and is properly considered,16 18,38 41 no additional treatment only one channel of population transfer in Hilbert space. The is needed. evolution of the wave function is represented by a single-sided The internal consistency,15 which ensures that the fraction of Feynman diagram27 (see Figure 2). In contrast, there exist four trajectories on each potential energy surface is equivalent to the basis states and population transfer channels in Liouville space. corresponding quantum probability, is a key characteristic of The wave function propagation is expressed by a double sided the standard Hilbert space FSSH. Because the Liouville space Feynman diagram. Because (i) the coherence states (|12⟩⟩ and

3829 DOI: 10.1021/acs.jpclett.5b01502 J. Phys. Chem. Lett. 2015, 6, 3827−3833 The Journal of Physical Chemistry Letters Letter

10-fold improvement in accuracy. This model shows that the Liouville space FSSH achieves higher accuracy than the standard Hilbert space FSSH, while maintaining simplicity and reliability. It is worth noting that the interference effect may play an even more important role in other systems, and several novel surface hopping methods have shown significant improvement − in this case.42 46 Liouville space representation can be combined with these methods, for instance, phase-corrected surface hopping,42,43 to improve the performance further. We move further to the more complex three-level super- 21 fi exchange model de ned in atomic units as V11(x)=0,V22(x) −x2/2 = 0.01, V33(x) = 0.005, V12(x)=V21(x) = 0.001e , V23(x)= −x2/2 V32(x) = 0.01e , and V13(x)=V31(x) = 0. As illustrated in Figure 3A, diabatic states 1 and 3 are coupled indirectly through the intermediate state 2 with a higher energy. The other parameters are chosen to be the same as those in the dual avoided crossing model previously discussed. − We start the dynamics from state 1. When V22 V11 > Ek > − fl V33 V11, a certain amount of population can ow from state 1 to state 3 because they are indirectly coupled, representing the superexchange mechanism of population transfer; however, the Figure 2. Comparison of Hilbert and Liouville spaces for a two-level hop 1 → 2 is forbidden in the Hilbert space FSSH due to system. Basis states and surface-hopping channels are shown in the violation of energy conservation, and thus there is no net upper two panels. A representative surface hopping channel is population flux from state 1 to state 3. The superexchange highlighted with dark circles. It is chosen for a detailed investigation dynamics is totally misrepresented. The situation is completely of the energy landscape and further analysis. The lower two panels different in Liouville space, which allows additional population demonstrate the energy landscape and Feynman diagrams. The wavy transfer channels (See Figure 4). Coherence states act as arrows depict interaction with the classical particles at specific time | ⟩⟩ | ⟩⟩ points. bridges connecting states 11 and 33 with smaller energy barriers. A typical example is highlighted in Figure 4, where the channel |11⟩⟩ → |21⟩⟩ → |31⟩⟩ → |32⟩⟩ → |33⟩⟩ involves a − |21⟩⟩) are in the middle of the energy gap of the two global energy barrier of (V22 + V33)/2 V11 = 0.0075 au population states (|11⟩⟩ and |22⟩⟩) and (ii) |11⟩⟩ and |22⟩⟩ are instead of 0.01 au in Hilbert space. Furthermore, trajectories fi coupled via the |12⟩⟩ and |21⟩⟩ states according to eqs 14 and occupying coherence states also contribute to the nal | ⟩⟩ → 15, coherence states play an important role in the Liouville population of state 3. For instance, the channel 11 − |21⟩⟩ → |31⟩⟩ has an overall energy barrier of only (V − V )/ space FSSH. When V22 V11 is larger than Ek, the surface hop 22 11 1 → 2 is completely forbidden in the Hilbert space FSSH due 2 = 0.005 au and thus plays an even more important role in the to violation of energy conservation. The situation changes superexchange dynamics. fi − ≤ Figure 3B−D compare transmissions on different states signi cantly if (V22 V11)/2 Ek. Surface hops to the coherence states become energetically allowed in Liouville obtained with Hilbert and Liouville spaces. The superexchange space. model can be analyzed in detail by dividing all investigated 15,42,43 fi It is well known that Hilbert space FSSH in the momenta into three regimes. In the rst regime, Ek is less than − adiabatic representation gives unrealistically large reflection on V33 V11. Both methods agree perfectly with the quantum 21 the lower state for energy lower than V − V (i.e., log (E)< standard. There, the particle remains on state 1 after crossing 22 11 e − −3; see Figure 1B). Such inaccuracy is strongly related to the interaction zone. In the second regime, V22 V11 > Ek > V33 − fi quantum interference. Simply switching to Liouville space V11. The exact solution shows signi cant transmission on results in a much better agreement with the exact solution. As state 3, illustrating superexchange. The transmission is blocked expected, no transmission on the upper state is observed when in the Hilbert space FSSH because the transition probability − loge(E)< 3 (see Figure 1D) because energy conservation directly from state 1 and state 3 is zero, and the indirect, cannot be fulfilled. FSSH reproduces quantum interference and superexchange pathway is classically disallowed. In contrast, the the peak positions. At high energies, the Liouville space Liouville space FSSH has multiple surface hopping channels to transmission is almost identical to the Hilbert space results. At state 3 and can reproduce nearly all of the transmission. (See low energies, the Liouville space FSSH improves upon the Figure 3D.) In the adiabatic representation, the results of Hilbert space FSSH in both diabatic and adiabatic representa- Hilbert space FSSH are already quite close to the exact tions. In Figure 1C, the peak intensity in the former case is solution. Nevertheless, Liouville space FSSH still shows enhanced, while the peak has been shifted to the right position improvement, indicating that surface hopping in Liouville in the latter case, getting closer to the quantum standard in space is generally superior to the standard strategy in Hilbert fi − both cases. Figure 1B shows an arti cially high peak for the space. In the third regime, where Ek > V22 V11, both methods Hilbert adiabatic representation, overestimating the exact give similar results again. Now, all channels are energetically answer 10-fold. By switching to Liouville space, while remaining enabled. The Liouville space FSSH also gives a better in the adiabatic representation, one reduces the peak intensity description for the “M-shaped” transmission on state 2. (See to within 50% of the exact answer, thus achieving more than Figure 3C.) This model demonstrated that the new method

3830 DOI: 10.1021/acs.jpclett.5b01502 J. Phys. Chem. Lett. 2015, 6, 3827−3833 The Journal of Physical Chemistry Letters Letter

Figure 3. Superexchange model. (A) Diabatic potential energies and interstate couplings. Transmission on (B) ﬁrst, (C) second, and (D) third state obtained from exact quantum dynamics,21 Hilbert and Liouville space FSSH in diabatic and adiabatic representations. The details of the transmission peak on state 3 are highlighted in the inset of panel D. The energies of the coherence states are shown as dashed horizontal lines. The vertical blue (red) dashed lines express the critical momenta, which give the kinetic energies equaling to the energy gap between the third (second) and ﬁrst states.

quantum electronics, charge and energy transfer, Auger phenomena, photochemistry, and related ﬁelds. Finally, we investigate convergence of the surface hopping results. As shown in Figure 5, the standard deviation of

Figure 5. Standard deviations of transmission on (A) upper state in the dual avoided crossing model with k = 31 and (B) state 3 in the superexchange model with k = 5, as functions of the number of Figure 4. Comparison of Hilbert and Liouville spaces for the three- trajectories. The results from Hilbert and Liouville space FSSH in level superexchange model with the same information as in Figure 2. diabatic and adiabatic representations are shown. In general, Liouville space results converge faster with the number of trajectories than provides a dramatic, qualitative improvement over the standard Hilbert space results. Note in panel B that there is no population in technique for superexchange and tunneling, common in state 3 for Hilbert space FSSH in diabatic representation, Figure 3D.

3831 DOI: 10.1021/acs.jpclett.5b01502 J. Phys. Chem. Lett. 2015, 6, 3827−3833 The Journal of Physical Chemistry Letters Letter quantum transmission in both dual avoided crossing and nonadiabatic quantum dynamics. The novel Liouville space superexchange models decays rapidly as a function of the version maintains all advantages of the standard, Hilbert space number of trajectories. One thousand trajectories are required formulation, including internal consistency, ease of implemen- to achieve a 99% accuracy (standard deviation around 0.01). In tation, and low computational cost. Liouville space results both cases, Liouville space FSSH requires fewer trajectories converge faster with the number of FSSH trajectories than than Hilbert space FSSH to achieve the same statistical Hilbert space data. By treating populations and coherences on convergence. Considering the fact that Liouville and Hilbert equal footing, the method allows us to gain a comprehensive space FSSH need similar efforts to obtain the surface hopping understanding of complex nonequilibrium quantum-mechanical probabilities, one concludes that Liouville space FSSH is processes. While the standard Tully’s FSSH technique exhibits statistically more accurate. qualitative problems in dealing with superexchange and The quantum-classical Liouville equation (QCLE) method of tunneling, the developed method provides a satisfactory Martens47 and Kapral and Ciccotti37 related closely to description of these phenomena. The reported advance opens quantum-classical Lie brackets36 starts with the full quantum up a new direction in semiclassical treatment of nonadiabatic Liouville equation and takes the semiclassical limit through a dynamics and enables one to treat efficiently a broad spectrum partial Wigner transform over the nuclear degrees of freedom. of phenomena that are of great importance in multiple areas of The resulting equations of motion account for the time physics and chemistry, including charge and energy transport, evolution of the coupled electronic and nuclear subsystems. electron−phonon scattering, multiple exciton generation and QCLE can be simulated by an ensemble of trajectories annihilation, nonlinear optics, and quantum computing. undergoing momentum jumps.48 Recently, Subotnik et al. demonstrated that Tully’s FSSH algorithm approximately obeys ■ AUTHOR INFORMATION QCLE, provided that several conditions are satisfied.49 In Corresponding Authors particular, Hilbert space FSSH can be viewed as a method for *L.W.: E-mail: [email protected]. approximate propagation of quantum populations while * neglecting the evolution of coherences. Liouville space FSSH O.V.P.: E-mail: [email protected]. is more closely related to QCLE. By treating populations and Notes coherences on equal footing while maintaining FSSH simplicity The authors declare no competing financial interest. and independent trajectory approximation, the new approach bridges the QCLE and FSSH worlds and achieves a good ■ ACKNOWLEDGMENTS balance between accuracy and efficiency. Frustrated hops, which violate the quantum-classical energy This work is supported by U.S. National Science Foundation, conservation, require special consideration. On the one hand, grant CHE-1300118. they provide a mechanism to achieve detailed balance between transitions upward and downward in energy, leading to ■ REFERENCES thermodynamic equilibrium.34 On the other hand, they violate (1) Stafström, S. 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3833 DOI: 10.1021/acs.jpclett.5b01502 J. Phys. Chem. Lett. 2015, 6, 3827−3833