Fewest Switches Surface Hopping in Liouville Space - The Journal of

Subscriber access provided by - Access paid by the | UCSB Libraries

Letter

Fewest Switches Surface Hopping in Liouville Space Linjun Wang, Andrew E. Sifain, and Oleg V. Prezhdo J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.5b01502 • Publication Date (Web): 08 Sep 2015 Downloaded from http://pubs.acs.org on September 9, 2015

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry Letters is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry Letters

Fewest Switches Surface Hopping in Liouville Space Linjun Wang,*,† Andrew E. Sifain,‡ and Oleg V. Prezhdo*,† †

Department of Chemistry, University of Southern California, Los Angeles, CA 90089-1062,

USA ‡

Department of Physics and Astronomy, University of Southern California, Los Angeles, CA

90089-0485, USA

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.

Keywords: Liouville space, surface hopping, quantum coherence and interference, tunneling, superexchange

ACS Paragon Plus Environment

1


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 22

TOC


2

Page 3 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Numerous processes of interest in modern physics, chemistry, and biology require consideration of quantum dynamics. However, a fully quantum description is generally forbidden due to computational cost. Frequently, only a small portion of particles under investigation, such as electrons and excitons, must be treated quantum mechanically, while the remaining degrees of freedom, typically nuclei, can be considered classically. Motivated by this fact, mixed quantum-classical methods have become the most popular choice for studying timedependent phenomena including charge transport,1-5 photoinduced electron transfer,6 photochemistry,7 exciton relaxation,8 singlet fission,9 and bioluminescence.10 A series of quantum-classical techniques have been developed over the last few decades.1114

Tully's fewest switches surface hopping (FSSH)15 is the most popular option. FSSH is

straightforward to implement, accurate and efficient. It is also easily compatible with modern electronic structure methods. FSSH is widely adopted as the platform to introduce modifications as needed. It has been corrected for decoherence effects soon after its invention.16 More strategies to incorporate decoherence have been presented,17,

18

although proper handling of

decoherence in realistic systems is still not finally settled. A QM/MM-like flexible surface hopping has been proposed, grasping all charge transport regimes in organic crystals.3, 19 Selfconsistency checks20 have been recently added, solving the trivial crossing problem arising due to high state density in large systems. The fewest switches concept has been generalized to gross population flow between states.21 FSSH has been reformulated in the second quantization language to include entanglement within the trajectory ensemble.22 Standard FSSH is carried out in Hilbert space. Fifty years ago, Fano coined the concept of Liouville space, which is a direct product of two Hilbert spaces.23 This space has been extensively applied by Zwanzig24 and Redfield25 in the quantum theory of relaxation and by


3


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 22

Mukamel in nonlinear optical spectroscopy.26 Quantum dynamics in Liouville space is described by a time-dependent density vector, which is formally equivalent to the time-dependent wave function in Hilbert space. However, the resulting dynamic pathways are completely different.27 In Hilbert space, surface hops happen between directly coupled states. Population transfer is expressed in a more elaborate way in Liouville space: Changes in quantum coherence depend on magnitudes of quantum populations and vice versa. Coherences play an extremely important role in quantum dynamics.28, 29 Liouville space allows us to treat populations and coherences on equal footing, leading to dramatic, qualitative improvements in accuracy, and allowing FSSH to treat many important phenomena. Quantum-mechanical mapping between the Hilbert and Liouville spaces is known in quantum optics26 and computing.30 While such mapping is straightforward in pure quantum mechanics, it is neither obvious nor unique in a quantum-classical description. Quantumclassical theories raise multiple questions, such as the quantum backreaction,31-33 answers to which require extensive research and testing. Further, quantum-classical theories are much more broadly applicable and carry significant impact in a wide spectrum of fields, including nanoscale, condensed phase, soft matter and gas phase systems in physics, chemistry, biology, materials and related disciplines. This letter reports a novel method for description of quantum dynamics. For the first time, the most practical semiclassical technique is formulated in Liouville space. The quantitative and qualitative advantages are demonstrated with two standard models, representing quantum interference, tunneling and superexchange – phenomena that are essential for many areas of physics, including quantum transport, scattering, Auger processes, quantum computing and optics. The method retains the advantages of the traditional technique, including simple


4

Page 5 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


formalism, internal consistency, ease of implementation, and low computational cost. Currently formulated for FSSH, the Liouville representation can be applied easily to other surface hopping methods. The time-dependent wave function, ψ (t ) , is expressed as a linear combination of a set of orthogonal basis states in Hilbert space,

{ i }, ψ (t ) = ∑ i ci (t ) i .

(1)

Its time evolution follows the Schrödinger equation, d ψ (t ) 1 = Hˆ ψ (t ) , dt ih

(2)

where Hˆ is the system Hamiltonian. Alternatively, one can describe the quantum system using the density operator, ρˆ . For a pure state, it reads

ρˆ (t ) = ψ (t ) ψ (t ) .

(3)

The dynamics follows the Liouville equation,

d ρˆ (t ) 1 ˆ =  H , ρˆ  . dt ih

(4)

Consider a two-level system as an illustration. The Hamiltonian is written as

V V  Hˆ =  11 12  .  V21 V22 

(5)

Eq. (4) becomes

d  ρ11  dt  ρ 21

ρ12  1  −V21 ρ12 + V12 ρ 21 −V12 ρ11 + (V11 -V22 ) ρ12 + V12 ρ 22  =  , ρ22  ih  V21ρ11 + (V22 − V11 ) ρ21 -V21 ρ 22 V21 ρ12 − V12 ρ 21 

(6)

and can be reformatted as


5


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 22

−V21 V12 0   ρ11   ρ11   0      0 V12   ρ12  d  ρ12  1  −V12 V11 -V22 = . 0 V22 − V11 -V21   ρ 21  dt  ρ 21  ih  V21      V21 −V12 0   ρ 22   ρ 22   0

(7)

If we define

ρ = ∑ ij ρij ij , where, ij

≡ i

(8)

j , Eq. (7) can be further simplified as d ρ dt

=

1 ˆˆ L ρ . ih

ˆ Here, the superoperator Lˆ connects all elements of ρ

(9)

.

Tully's FSSH is carried out in Hilbert space. The time derivative of the population of state

i , ρii , is expressed as15 d ρii = ∑ bij , dt j ≠i

(10)

where

bij = 2h −1 Im ( ρij* H ij ) .

(11)

In FSSH, the switching probability from state i to another state j within a time interval ∆t is gij =

−∆tbij

ρii

.

(12)

Drawing upon the similarity between the Hilbert and Liouville formulations, Eqs. (1-2) and (8-9), we define the population of state ij

2

in Liouville space as pij ≡ ρij . Trρ 2 = 1 always


6

Page 7 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


holds for pure states, hence,

∑

ij

pij = 1 .

{ ij }

quantum system in Liouville space, similar to dpij dt

form a complete, orthonormal basis for a

{ i } in Hilbert space. Analogously, we obtain

∑

=

bij ,kl ,

(13)

k ≠ i or l ≠ j

where bij ,kl = 2h -1 δ ik Im( ρij ρ kl* H jl ) + δ jl Im( ρij* ρ kl H ik )  .

(14)

Similar with Eq. (12), the surface hopping probability in Liouville space is given by

gij ,kl =

−∆tbij ,kl pij

.

(15)

Eqs. (11) and (14) are derived for time-independent basis in the diabatic representation. In general, one needs to replace Hˆ with Hˆ − ihv ⋅ d , where v ⋅ d is the nonadiabatic coupling,

( v ⋅ d )ij =

i

d j . dt

(16)

When a surface hop is assigned, velocities need to be adjusted to conserve the total quantumclassical energy. In the adiabatic representation of Hilbert space FSSH, velocity rescaling is normally performed in the direction of the nonadiabatic coupling vector.15 If a surface hop happens from ij

to kl

in Liouville space, one projects dik onto d jl to define the scaling

direction. The situations is less clear in the diabatic representation, since the nonadiabatic coupling is zero in this case. In principle, one needs to find the direction along which the surface hopping probabilities based on Eqs. (12) and (15) are maximized. Velocity rescaling is straightforward for one-dimensional models as there is only one classical degree of freedom. Quantum-classical energy conservation achieved by velocity rescaling is critical to achieve the detailed balance between transitions up and down in energy and the quantum-classical


7


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 22

equilibrium in the long time limit.34 For multi-dimensional realistic systems, one can adopt the classical path approximation,35 and replace velocity rescaling by multiplying the surface hopping probabilities upward in energy with the Boltzmann factor. Valid in the case of rapid redistribution of energy within the classical subsystem, compared to the transition time, the Boltzmann approximation achieves the detailed balance. Several additional definitions are needed. First, we define the energy for each ij

state,

Eij . If i = j , Eij equals to the energy of the i-th state in Hilbert space, Vii . When i ≠ j , we

follow the analysis of refs 36, 37 and adopt Eij = (Vii + V jj ) / 2 . In principle, only Hilbert states have well-defined energies. If a coherence state is regarded as a superposition of two Hilbert states, its energy should be an average over both states, perhaps weighted by the corresponding amplitudes. Placing the energy of the Liouville space coherence state in the middle of the two corresponding Hilbert states is an approximate yet rational way to mimic the role of coherence in quantum dynamics (see the numerical demonstrations below). The same convention has been adopted in the earlier works.36, 37 Second, the fraction of trajectories populating coherence states in Liouville space, i.e., ij

with i ≠ j , needs to be converted back to Hilbert space populations for data

analysis. Energy conservation is essential. When the total energy, i.e., classical kinetic energy ( Ek ) plus Eij , is more than both Vii and V jj , half of the trajectories on ij

contribute to i and

the other half contribute to j , to be consistent with the energy convention. If the total energy is larger than Vii but smaller than V jj , the trajectories on ij

fully contribute to i in Hilbert

space, or vice versa. When decoherence occurs rapidly and is properly considered,16-18, 38-41 no additional treatment is needed.


8

Page 9 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The internal consistency,15 which ensures that the fraction of trajectories on each potential energy surface is equivalent to the corresponding quantum probability, is a key characteristic of the standard Hilbert space FSSH. Because the Liouville space FSSH is formally similar, the fraction of trajectories on state ij , Pij , also equals to the corresponding population, pij . In the absence of hop rejection,15 the fraction of trajectories on state i , Pi , can be obtained through Pi = Pii + ∑ j ≠i ( Pij + Pji ) / 2 ,

which equals to

∑

j

(17)

Pij = ∑ j pij = ∑ j ρij ρ ji = ∑ j ρii ρ jj = ρii considering Pij = Pji . Hence,

Liouville space FSSH holds the same level of internal consistency as Hilbert space FSSH. Note that rigorous internal consistency cannot be realized when running each trajectory independently.17


9


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 22

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 (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 (B-D) express the critical kinetic energy, which is equal to V22 − V11 at x >> 5 . E = k 2 / 2m is the initial kinetic energy. First, we examine Tully's dual avoided crossing model,15 which is defined in the diabatic 2

2

representation: V11 ( x) = 0 , V22 ( x) = −0.1e −0.28 x + 0.05 , V12 ( x) = V21 ( x) = 0.015e −0.06 x (see Figure 1A). The mass of the x degree of freedom, m , is 2000 a.u., which is close to the mass of a hydrogen atom. Initially, a Gaussian wave packet is prepared on the negative x side of state 1, and the corresponding width is set to be 20 / k with k being the initial momentum. This model exhibits two avoided crossings, where the quantum interference causes Stueckelberg oscillations in the transmission probabilities.15


10

Page 11 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 2. Comparison of Hilbert and Liouville spaces for a two-level system. Basis states and surface hopping channels are shown in the upper two panels. A representative surface hopping channel is highlighted with dark circles. It is chosen for a detailed investigation of the energy landscape and further analysis. The lower two panels demonstrate the energy landscape and Feynman diagrams. The wavy arrows depict interaction with the classical particles at specific time points. For such a two-level system, there are two basis states and only one channel of population transfer in Hilbert space. The evolution of the wave function is represented by a single sided Feynman diagram27 (see Figure 2). In contrast, there exists four basis states and population transfer channels in Liouville space. The wave function propagation is expressed by a double sided Feynman diagram. Because (i) the coherence states ( 12 the energy gap of the two population states ( 11

and 21 ) are in the middle of

and 22 ), and (ii) 11


and 22

are

11


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

coupled via the 12

and 21

Page 12 of 22

states according to Eqs. (14-15), coherence states play an

important role in the Liouville space FSSH. When V22 − V11 is larger than Ek , the surface hop 1 → 2 is completely forbidden in the Hilbert space FSSH due to violation of energy conservation. The situation changes significantly if (V22 − V11 ) / 2 ≤ Ek . Surface hops to the coherence states become energetically allowed in Liouville space. It is well known15,

42, 43

that Hilbert space FSSH in the adiabatic representation gives

unrealistically large reflection on the lower state for energy lower than V22 − V11 (i.e., log e ( E ) < −3 , see Figure 1B). Such inaccuracy is strongly related to quantum interference. Simply switching to Liouville space results in a much better agreement with the exact solution. As expected, no transmission on the upper state is observed when log e ( E ) < −3 (see Figure 1D) because energy conservation cannot be fulfilled. FSSH reproduces quantum interference and the peak positions. At high energies, the Liouville space transmission is almost identical to the Hilbert space results. At low energies, the Liouville space FSSH improves upon the Hilbert space FSSH in both diabatic and adiabatic representations. The peak intensity in the former case is enhanced, while the peak has been shifted to the right position in the latter case, getting closer to the quantum standard in both cases. Figure 1B shows an artificially high peak for the Hilbert adiabatic representation, overestimating the exact answer 10-fold. By switching to Liouville space, while remaining in the adiabatic representation, one reduces the peak intensity to within 50% of the exact answer, thus achieving a 20-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.


12

Page 13 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


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 phasecorrected surface hopping,42, 43 in order to improve the performance further.

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 (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.


13


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 22

We move further to the more complex three-level superexchange model21 defined in atomic units as: V11 ( x) = 0 , V22 ( x) = 0.01 , V33 ( x) = 0.005 , V12 ( x) = V21 ( x) = 0.001e− x

V23 ( x) = V32 ( x) = 0.01e − x

2

/2

2

/2

,

, 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 discussed above.

Figure 4. Comparison of Hilbert and Liouville spaces for the three-level superexchange model with the same information as in Figure 2.


14

Page 15 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


We start the dynamics from state 1. When V22 − V11 > Ek > V33 − V11 , a certain amount of population can flow from state 1 to state 3 because they are indirectly coupled, representing the superexchange mechanism of population transfer. However, the hop 1 → 2 is forbidden in the Hilbert space FSSH due to violation of energy conservation, and thus there is no net population flux from state 1 to state 3. The superexchange dynamics is totally misrepresented. The situation is completely different in Liouville space, which allows additional population transfer channels (see Figure 4). Coherence states act as 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 global energy barrier of (V22 + V33 ) / 2 − V11

= 0.0075 a.u. instead of 0.01 a.u. in Hilbert space. Further, trajectories occupying coherence states also contribute to the final population of state 3. For instance, the channel 11 → 21 → 31

has an overall energy barrier of only (V22 − V11 ) / 2 = 0.005 a.u., and thus

plays an even more important role in the superexchange dynamics. Figures 3B-D compare transmissions on different states obtained with Hilbert and Liouville spaces. The superexchange model can be analyzed in detail by dividing all investigated momenta into three regimes. In the first regime, Ek is less than V33 − V11 . Both methods agree perfectly with the quantum standard.21 There, the particle remains on state 1 after crossing the interaction zone. In the second regime, V22 − V11 > Ek > V33 − V11 . The exact solution shows significant transmission on state 3, illustrating superexchange. The transmission is blocked in the Hilbert space FSSH, because the transition probability directly from state 1 and state 3 is zero, and the indirect, superexchange pathway is classically disallowed. In contrast, the Liouville space FSSH has multiple surface hopping channels to state 3 and can reproduce nearly all of the


15


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 22

transmission (see Figure 3D). In the adiabatic representation, the results of Hilbert space FSSH are already quite close to the exact solution. Nevertheless, Liouville space FSSH still shows improvement, indicating that surface hopping in Liouville space is generally superior to the standard strategy in Hilbert space. In the third regime, where Ek > V22 − V11 , both methods give similar results again. Now, all channels are energetically enabled. The Liouville space FSSH also gives a better description for the "M-shaped" transmission on state 2 (see Figure 3C). This model demonstrated that the new method provides a dramatic, qualitative improvement over the standard technique for superexchange and tunneling, common in quantum electronics, charge and energy transfer, Auger phenomena, photochemistry and related fields.

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 trajectories. The results from Hilbert and Liouville space FSSH in diabatic and adiabatic representations are shown. Generally, Liouville space results converge faster with the


16

Page 17 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


number of trajectories than Hilbert space results. Note in (B) that there is no population in state 3 for Hilbert space FSSH in diabatic representation, Figure 3D. Finally, we investigate convergence of the surface hopping results. As shown in Figure 5, the standard deviation of quantum transmission in both dual avoided crossing and superexchange models decays exponentially as a function of the number of trajectories. One thousand trajectories are required to achieve a 99% accuracy (standard deviation around 0.01). In both cases, Liouville space FSSH requires fewer trajectories than Hilbert space FSSH to achieve the same statistical convergence. Considering the fact that Liouville and Hilbert space FSSH need similar efforts to obtain the surface hopping probabilities, one concludes that Liouville space FSSH is statistically more accurate. The quantum-classical Liouville equation (QCLE) method of Martens,47 Kapral and Ciccotti,37 related closely to quantum-classical Lie brackets,36 starts with the full quantum Liouville equation, and takes the semiclassical limit through a partial Wigner transform over the nuclear degrees of freedom. The resulting equations of motion accounts for the time evolution of the coupled electronic and nuclear subsystems. QCLE can be simulated by an ensemble of trajectories undergoing momentum jumps.48 Recently, Subotnik et al. demonstrated that Tully's FSSH algorithm approximately obeys QCLE, provided that several conditions are satisfied.49 In particular, Hilbert space FSSH can be viewed as a method for approximate propagation of quantum populations, while neglecting the evolution of coherences. Liouville space FSSH is more closely related to QCLE. By treating populations and coherences on equal footing, while maintaining FSSH simplicity and independent trajectory approximation, the new approach bridges the QCLE and FSSH worlds and achieves a good balance between accuracy and efficiency.


17


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 22

Frustrated hops, which violate the quantum-classical energy conservation, require special consideration. On the one hand, they provide a mechanism to achieve detailed balance between transitions upward and downward in energy, leading to thermodynamic equilibrium.34 On the other hand, they violate internal consistency between the quantum mechanical and statistical surface hopping probabilities, and prevent superexchange processes. The superexchange problem has been recently addressed by global flux surface hopping, which redefines probability flux and introduces more reliable surface hopping probabilities,21 and second-quantized surface hopping, which uses an ensemble of correlated trajectories.22 Liouville space FSSH introduces coherence states into the surface hopping framework, and could be combined with these methods to enhance the performance further. Quantum decoherence effects can be very important and have been taken into account previously.50-52 The present technique treats the population and coherence terms on equal footing, potentially allowing for a more straightforward description and interpretation of decoherence. Providing a novel foundation for surface hopping, the developed approach can be combined with the existing decoherence ideas.16-18, 38-41 This will be done in future work. The concept of surface hopping cannot be applied directly to a continuous spectrum of states. In such cases, one can either discretize the states, or resort to other approaches developed for Hilbert space, such as combined mean-field/surface hopping,53, 54 quantum trajectory meanfield approach for nonadiabatic molecular dynamics,55 and Bohmian quantum-classical dynamics.33, 56 Similar ideas can be applied to surface hopping in Liouville space. In conclusion, by drawing analogy between the Hilbert and Liouville representations of quantum mechanics, we have proposed a new family of semiclassical techniques for nonadiabatic quantum dynamics. The novel Liouville space version maintains all advantages of


18

Page 19 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the standard, Hilbert space formulation, including internal consistency, ease of implementation, and low computational cost. Liouville space results converge faster with the number of FSSH trajectories than Hilbert space data. By treating populations and coherences on equal footing, the method allows us to gain a comprehensive understanding of complex non-equilibrium quantummechanical processes. While the standard Tully's FSSH technique exhibits qualitative problems in dealing with superexchange and tunneling, the developed method provides a satisfactory description of these phenomena. The reported advance opens up a new direction in semiclassical treatment of nonadiabatic dynamics and enables one to treat efficiently a broad spectrum of phenomena that are of great importance in multiple areas of physics and chemistry, including charge and energy transport, electron-phonon scattering, multiple exciton generation and annihilation, nonlinear optics and quantum computing. AUTHOR INFORMATION

Corresponding Authors *E-mail: [email protected]. *E-mail: [email protected].

Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work is supported by U.S. National Science Foundation, grant CHE-1300118.


19


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 22

REFERENCES (1) Stafström, S., Electron Localization and the Transition from Adiabatic to Nonadiabatic Charge Transport in Organic Conductors. Chemical Society Reviews 2010, 39, 2484-2499. (2) Troisi, A., Charge Transport in High Mobility Molecular Semiconductors: Classical Models and New Theories. Chemical Society Reviews 2011, 40, 2347-2358. (3) Wang, L. J.; Beljonne, D., Flexible Surface Hopping Approach to Model the Crossover from Hopping to Band-Like Transport in Organic Crystals. The Journal of Physical Chemistry Letters 2013, 4, 1888-1894. (4) Ren, J.; Vukmirović, N.; Wang, L.-W., Nonadiabatic Molecular Dynamics Simulation for Carrier Transport in a Pentathiophene Butyric Acid Monolayer. Physical Review B 2013, 87, 205117. (5) Wang, L. J.; Prezhdo, O. V.; Beljonne, D., Mixed Quantum-Classical Dynamics for Charge Transport in Organics. Physical Chemistry Chemical Physics 2015, 17, 12395-12406. (6) Hazra, A.; Soudackov, A. V.; Hammes-Schiffer, S., Isotope Effects on the Nonequilibrium Dynamics of Ultrafast Photoinduced Proton-Coupled Electron Transfer Reactions in Solution. The Journal of Physical Chemistry Letters 2011, 2, 36-40. (7) Weingart, O.; Lan, Z.; Koslowski, A.; Thiel, W., Chiral Pathways and Periodic Decay in Cis-Azobenzene Photodynamics. The Journal of Physical Chemistry Letters 2011, 2, 1506-1509. (8) Gao, X.; Peng, Q.; Niu, Y.; Wang, D.; Shuai, Z., Theoretical Insight into the Aggregation Induced Emission Phenomena of Diphenyldibenzofulvene: A Nonadiabatic Molecular Dynamics Study. Physical Chemistry Chemical Physics 2012, 14, 14207-14216. (9) Wang, L. J.; Olivier, Y.; Prezhdo, O. V.; Beljonne, D., Maximizing Singlet Fission by Intermolecular Packing. The Journal of Physical Chemistry Letters 2014, 5, 3345-3353. (10) Yue, L.; Lan, Z.; Liu, Y.-J., The Theoretical Estimation of the Bioluminescent Efficiency of the Firefly Via a Nonadiabatic Molecular Dynamics Simulation. The Journal of Physical Chemistry Letters 2015, 6, 540-548. (11) Herman, M. F., Dynamics by Semiclassical Methods. Annual Review of Physical Chemistry 1994, 45, 83-111. (12) Drukker, K., Basics of Surface Hopping in Mixed Quantum/Classical Simulations. Journal of Computational Physics 1999, 153, 225-272. (13) Tully, J. C., Perspective: Nonadiabatic Dynamics Theory. The Journal of Chemical Physics 2012, 137, 22A301. (14) Wang, L. J.; Long, R.; Prezhdo, O. V., Time-Domain Ab Initio Modeling of Photoinduced Dynamics at Nanoscale Interfaces. Annual Review of Physical Chemistry 2015, 66, 549-579. (15) Tully, J. C., Molecular Dynamics with Electronic Transitions. The Journal of Chemical Physics 1990, 93, 1061-1071. (16) Bittner, E. R.; Rossky, P. J., Quantum Decoherence in Mixed Quantum-Classical Systems: Nonadiabatic Processes. The Journal of Chemical Physics 1995, 103, 8130-8143. (17) Granucci, G.; Persico, M., Critical Appraisal of the Fewest Switches Algorithm for Surface Hopping. J. Chem. Phys. 2007, 126, 134114. (18) Jasper, A. W.; Truhlar, D. G., Non-Born-Oppenheimer Molecular Dynamics of Na...Fh Photodissociation. J. Chem. Phys. 2007, 127, 194306.


20

Page 21 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(19) Wang, L. J.; Beljonne, D., Charge Transport in Organic Semiconductors: Assessment of the Mean Field Theory in the Hopping Regime. The Journal of Chemical Physics 2013, 139, 064316. (20) Wang, L. J.; Prezhdo, O. V., A Simple Solution to the Trivial Crossing Problem in Surface Hopping. The Journal of Physical Chemistry Letters 2014, 5, 713-719. (21) Wang, L. J.; Trivedi, D.; Prezhdo, O. V., Global Flux Surface Hopping Approach for Mixed Quantum-Classical Dynamics. Journal of Chemical Theory and Computation 2014, 10, 3598-3605. (22) Akimov, A. V.; Prezhdo, O. V., Second-Quantized Surface Hopping. Physical Review Letters 2014, 113, 153003. (23) Fano, U., Pressure Broadening as a Prototype of Relaxation. Physical Review 1963, 131, 259-268. (24) Zwanzig, R., On the Identity of Three Generalized Master Equations. Physica 1964, 30, 1109-1123. (25) Redfield, A. G., The Theory of Relaxation Processes. In Advances in Magnetic and Optical Resonance, John, S. W., Ed. Academic Press: 1965; Vol. 1, pp 1-32. (26) Mukamel, S., Principles of Nonlinear Optical Spectroscopy. Oxford University Press: New York, 1995. (27) Skourtis, S. S.; Mukamel, S., Superexchange Versus Sequential Long Range Electron Transfer; Density Matrix Pathways in Liouville Space. Chemical Physics 1995, 197, 367-388. (28) Dani, K. M.; Tignon, J.; Breit, M.; Chemla, D. S.; Kavousanaki, E. G.; Perakis, I. E., Ultrafast Dynamics of Coherences in a Quantum Hall System. Physical Review Letters 2006, 97, 057401. (29) Robertson, A.; Galitski, V. M.; Refael, G., Dynamic Stimulation of Quantum Coherence in Systems of Lattice Bosons. Physical Review Letters 2011, 106, 165701. (30) Aspuru-Guzik, A.; Dutoi, A. D.; Love, P. J.; Head-Gordon, M., Simulated Quantum Computation of Molecular Energies. Science 2005, 309, 1704-1707. (31) Anderson, A., Quantum Backreaction on "Classical" Variables. Phys. Rev. Lett. 1995, 74, 621. (32) Diósi, L.; Halliwell, J. J., Coupling Classical and Quantum Variables Using Continuous Quantum Measurement Theory. Phys. Rev. Lett. 1998, 81, 2846-2849. (33) Prezhdo, O. V.; Brooksby, C., Quantum Backreaction through the Bohmian Particle. Phys. Rev. Lett. 2001, 86, 3215-3219. (34) Parandekar, P. V.; Tully, J. C., Mixed Quantum-Classical Equilibrium. The Journal of Chemical Physics 2005, 122, 094102. (35) Akimov, A. V.; Prezhdo, O. V., The Pyxaid Program for Non-Adiabatic Molecular Dynamics in Condensed Matter Systems. Journal of Chemical Theory and Computation 2013, 9, 4959-4972. (36) Prezhdo, O. V.; Kisil, V. V., Mixing Quantum and Classical Mechanics. Physical Review A 1997, 56, 162-175. (37) Kapral, R.; Ciccotti, G., Mixed Quantum-Classical Dynamics. The Journal of Chemical Physics 1999, 110, 8919-8929. (38) Prezhdo, O. V., Mean Field Approximation for the Stochastic Schrödinger Equation. The Journal of Chemical Physics 1999, 111, 8366-8377. (39) Hack, M. D.; Truhlar, D. G., A Natural Decay of Mixing Algorithm for Non-Born– Oppenheimer Trajectories. The Journal of Chemical Physics 2001, 114, 9305-9314.


21


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 22

(40) Bedard-Hearn, M. J.; Larsen, R. E.; Schwartz, B. J., Mean-Field Dynamics with Stochastic Decoherence (Mf-Sd): A New Algorithm for Nonadiabatic Mixed Quantum/Classical Molecular-Dynamics Simulations with Nuclear-Induced Decoherence. The Journal of Chemical Physics 2005, 123, 234106. (41) Prezhdo, O. V.; Rossky, P. J., Relationship between Quantum Decoherence Times and Solvation Dynamics in Condensed Phase Chemical Systems. Physical Review Letters 1998, 81, 5294-5297. (42) Gorshkov, V. N.; Tretiak, S.; Mozyrsky, D., Semiclassical Monte-Carlo Approach for Modelling Non-Adiabatic Dynamics in Extended Molecules. Nat Commun 2013, 4, 2144. (43) Shenvi, N.; Subotnik, J. E.; Yang, W., Phase-Corrected Surface Hopping: Correcting the Phase Evolution of the Electronic Wavefunction. The Journal of Chemical Physics 2011, 135, 024101. (44) Subotnik, J. E., Fewest-Switches Surface Hopping and Decoherence in Multiple Dimensions. The Journal of Physical Chemistry A 2011, 115, 12083-12096. (45) Subotnik, J. E.; Shenvi, N., A New Approach to Decoherence and Momentum Rescaling in the Surface Hopping Algorithm. The Journal of Chemical Physics 2011, 134, 024105. (46) White, A. J.; Gorshkov, V. N.; Tretiak, S.; Mozyrsky, D., Non-Adiabatic Molecular Dynamics by Accelerated Semiclassical Monte Carlo. The Journal of Chemical Physics 2015, 143, 014115. (47) Martens, C. C.; Fang, J.-Y., Semiclassical-Limit Molecular Dynamics on Multiple Electronic Surfaces. The Journal of Chemical Physics 1997, 106, 4918-4930. (48) Kapral, R., Progress in the Theory of Mixed Quantum-Classical Dynamics. Annual Review of Physical Chemistry 2006, 57, 129-157. (49) Subotnik, J. E.; Ouyang, W.; Landry, B. R., Can We Derive Tully's Surface-Hopping Algorithm from the Semiclassical Quantum Liouville Equation? Almost, but Only with Decoherence. The Journal of Chemical Physics 2013, 139, 214107. (50) Zhu, C.; Nangia, S.; Jasper, A. W.; Truhlar, D. G., Coherent Switching with Decay of Mixing: An Improved Treatment of Electronic Coherence for Non-Born-Oppenheimer Trajectories. J. Chem. Phys. 2004, 121, 7658-7670. (51) Zhu, C.; Jasper, A. W.; Truhlar, D. G., Non-Born-Oppenheimer Liouville-Von Neumann Dynamics. Evolution of a Subsystem Controlled by Linear and Population-Driven Decay of Mixing with Decoherence and Coherent Switching. Journal of Chemical Theory and Computation 2005, 1, 527-540. (52) Zheng, J.; Meana-Pañeda, R.; Truhlar, D. G., Including Tunneling in Non-BornOppenheimer Simulations. J. Phys. Chem. Lett. 2014, 5, 2039-2043. (53) Sholl, D. S.; Tully, J. C., A Generalized Surface Hopping Method. J. Chem. Phys. 1998, 109, 7702-7710. (54) Prezhdo, O. V.; Rossky, P. J., Mean-Field Molecular Dynamics with Surface Hopping. J. Chem. Phys. 1997, 107, 825-834. (55) Feng, W.; Xu, L.; Li, X.-Q.; Fang, W.; Yan, Y., Nonadiabatic Molecular Dynamics Simulation: An Approach Based on Quantum Measurement Picture. AIP Advances 2014, 4, 077131. (56) Gindensperger, E.; Meier, C.; Beswick, J. A., Mixing Quantum and Classical Dynamics Using Bohmian Trajectories. J. Chem. Phys. 2000, 113, 9369.


22

Fewest Switches Surface Hopping in Liouville Space - The Journal of

Recommend Documents