Crossing Classified and Corrected Fewest Switches Surface Hopping

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Spectroscopy and Photochemistry; General Theory

Crossing Classified and Corrected Fewest Switches Surface Hopping Jing Qiu, Xin Bai, and Linjun Wang J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b01902 • Publication Date (Web): 16 Jul 2018 Downloaded from http://pubs.acs.org on July 16, 2018

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The Journal of Physical Chemistry Letters

Crossing Classified and Corrected Fewest Switches Surface Hopping Jing Qiu, Xin Bai, and Linjun Wang*

Department of Chemistry, Zhejiang University, Hangzhou 310027, China

ABSTRACT: In the traditional fewest switches surface hopping (FSSH), trivial crossings between uncoupled or weakly-coupled states have highly peaked nonadiabatic couplings and thus are difficult to deal with in the preferred, adiabatic representation. Here, we classify surface crossings into four general types and propose a parameter-free crossing corrected FSSH (CC-FSSH) algorithm, which could treat multiple trivial crossings within a time interval. As examples, Holstein Hamiltonians with different parameters are adopted to mimic electron dynamics in tens to hundreds of molecules, which suffer from severe trivial crossing problems. Using existed surface hopping approaches as references, we show that CC-FSSH exhibits significantly fast time interval convergence and weak system size dependence. In all cases, a reliable description is achieved with a large time interval of 1 fs. With simple formalism and the ability to describe complex surface crossings, CC-FSSH could potentially simulate general nonadiabatic dynamics in nanoscale materials with a high efficiency.

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Tully's fewest switches surface hopping (FSSH) algorithm1 for mixed quantum-classical dynamics has received growing attention in the past years.2-5 It is based on the hypothesis that nonadiabatic dynamics can be approximately described by an ensemble of independent semiclassical trajectories. For each trajectory, the classical particles move on an active potential energy surface (PES) through Newtonian equations, and the wave function of the quantum subsystem is propagated according to the time-dependent Schrödinger equation. Based on the flow of population flux between quantum states, the active PES is updated via stochastic surface hops. Due to the simple formalism, good balance between efficiency and reliability, and great compatibility with ab initio electronic structure methods, FSSH and its variant approaches have been extensively utilized to study a variety of dynamical processes in different research fields.6-12 It is well known that FSSH is representation dependent, and better results are normally obtained in the adiabatic representation.13-14 For states with zero or very weak diabatic couplings, surface crossings between the corresponding adiabatic PESs are associated with hopping probabilities close to unity, and thus are usually called trivial crossings.15 In these cases, the nonadiabatic couplings change rapidly with time around the crossing points, and an accurate description of the wave function propagation and surface hops generally requires an extremely small time interval, resulting in the trivial crossing problem of the traditional FSSH algorithm. With the increase of time interval or system size, the trivial crossing problem is amplified due to the larger probability to encounter trivial crossings within a time step. In addition, recent studies have shown that widely used decoherence correction approaches further enhances the difficulty to treat trivial crossings.16 To

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study nonadiabatic dynamics in general systems with surface hopping, a clean and efficient treatment of surface crossings is essential and becomes a subject of great interest.4 In literature, different strategies have been proposed to deal with trivial crossings. One may identify trivial crossings through detecting unphysical discontinuities of the wave function15 or the potential energies17 between the corresponding adiabatic states and make manual surface hops. Effective nonadiabatic couplings18 and hopping probabilities19 have been also suggested, and a norm-preserving interpolation of the adiabatic electronic wave function can provide good approximations.20 For large systems with localized quantum states, we can choose only the important subsystem for surface hopping and most of the trivial crossings are avoided naturally.21 As trivial crossings require much smaller time intervals than normal avoided crossings, adaptive time intervals are helpful, and have been utilized in different surface hopping techniques.15,21,22 Another alternative to FSSH is the global flux surface hopping, which computes the hopping probabilities based on quantum populations instead of nonadiabatic couplings.23 The most widely used strategy to deal with trivial crossings nowadays is perhaps the local diabatization approach of FSSH (LD-FSSH) proposed by Granucci and co-workers,24,25 which has been implemented in several nonadiabatic dynamics programs (e.g. Newton-X,26 SHARC27 and DFTbaby28). There, the locally diabatic representation is adopted to propagate the wave function and calculate the hopping probabilities. Recently, Wang and Prezhdo proposed a self-consistent FSSH (SC-FSSH) approach,29 which aims at solving the trivial crossing problem in the preferred, adiabatic representation. SC-FSSH introduces automatic corrections to the problematic hopping probabilities with the assumption that at most one trivial crossing is encountered during a time step. When multiple trivial crossings are present in a 3



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short period of time, a smaller time interval is needed to isolate the trivial crossings with each other. In this work, we classify surface crossings into four general types, and present a parameter-free crossing corrected FSSH (CC-FSSH) algorithm, which substantially improves the efficiency to deal with multiple surface crossings within a time interval. We use SC-FSSH and LD-FSSH as references and investigate electron dynamics in a series of Holstein models.30-32 CC-FSSH exhibits significantly high performance in time interval convergence and system size dependence. Within the mixed quantum-classical description, the system Hamiltonian is generally expressed as

Hˆ (r; R ) , where r and R are coordinates of the quantum and classical degrees of freedom, respectively. At any given geometry, R , one may solve the time-independent Schrödinger equation,

Hˆ (r; R ) φi (r; R ) = Ei (R ) φi (r; R ) , to obtain all the eigenstates, φi (r; R ) , and eigenenergies,

Ei (R) . In the adiabatic representation, the time-dependent wave function of the quantum subsystem is expressed as a linear expansion of the corresponding eigenstates, ψ (r, t ) = ∑i wi (t ) φi (r; R(t )) , where wi (t ) are the wave function coefficients. In the traditional FSSH algorithm,1 an ensemble of independent trajectories is generated to simulate the nonadiabatic dynamics. For each trajectory, the initial R , R& , ψ (r ) , and active PES are set according to the problem under investigation. In the following dynamics, the classical particles move on the active PES following Newtonian equations. The wave function is propagated through the time-dependent Schrödinger equation, ∂ ψ (r, t ) / ∂t = Hˆ (r; R(t )) ψ (r, t ) / ih , which yields

& (t ) ⋅ d (R(t )) . w& i (t ) = wi (t ) Ei (R(t )) / ih − ∑ w j (t )R ij

(1)

j

4


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Here, d ij ( R ) = φi (r; R ) ∇ R φ j (r; R )

is the nonadiabatic coupling vector. According to Eq. (1), the 2

quantum population of each state i is obtained through ai (t ) = wi (t ) , and its time derivative is

analytically expressed as a sum over flux contributions from different pathways, a&i (t ) = ∑ j ≠i bij (t ) ,

& (t ) ⋅ d (R(t ))] . Within a time step dt, the FSSH probability of where bij (t ) = −2 Re[wi* (t )w j (t )R ij switching from the active state φa (r; R (t ))

to another state φi ( ≠ a ) (r; R (t + dt ))

gi =

−dt ⋅ bai (t ) wa (t )

2

is calculated as1

.

(2)

By design, ga = 0 and the other hopping probabilities are reset to zero when negative. A uniform random number, ξ , is generated and a hop to surface j is assigned if

∑

j −1 i =1

gi < ξ ≤

∑

j i =1

g i . The

hop is successful only when the velocities can be adjusted along the nonadiabatic coupling vector daj (R) for energy conservation.

33-36

Otherwise, a frustrated hop occurs, and the system remains on the

current active PES. These steps are repeated until a certain predefined criterion is achieved. SC-FSSH differs with FSSH in two major aspects.29 Firstly, the wave function propagation is carried out with the representation transformation technique. If a set of orthogonal diabatic bases,

{χ

j

(r )

} , are defined, the adiabatic states and the wave function are expressed accordingly as

φ i ( r ; R ) = ∑ j p ji ( R ) χ j ( r )

and ψ (r , t ) = ∑ j c j (t ) χ j (r ) , respectively. At each time step, the

wave function propagation can be solved in the diabatic representation through c&i (t ) = ∑ c j (t ) χ i (r ) Hˆ (r; R (t )) χ j (r ) / ih ,

(3)

j

and converted to the adiabatic representation through wi (t ) = ∑ j p*ji (R (t ))c j (t ) . Thereby, instead of solving Eq. (1), the time-dependent wave function in the adiabatic representation is obtained without calculating the nonadiabatic couplings explicitly. Note that Eq. (3) can be solved in any diabatic bases,16 including the locally diabatic bases proposed by Granucci and co-authors,24-25 which are

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very helpful for realistic applications. Secondly, the problematic hopping probabilities are corrected self-consistently. The total probability for surface hops out of the active surface a, by definition, is 2

∑g

i

=

wa (t ) − wa (t + dt ) wa (t )

i

2

2

(4)

Eq. (4) may differ from the sum calculated by Eq. (2). SC-FSSH assumes that the error is due to one single pathway a→ j, and the corresponding hopping probability, g j , is thus corrected as29

g j = ∑ gi − ∑ gi i

(5)

i≠ j

Here, the first and second terms on the right-hand side are obtained from Eqs. (4) and (2), respectively. In the original SC-FSSH algorithm, j is identified as the adiabatic state which is closest in energy to the active state.29 This is a natural choice for trivial crossings because they happen between states that are very close in energy. Apparently, SC-FSSH deals with at most one trivial crossing at a time.

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Figure 1. Schematic representation of (A) type 1, (B) type 2, (C) type 3, and (D) type 4 surface crossings during a time interval. The adiabatic PESs are represented by solid lines and the adiabatic states at both time t and t + dt are shown as indicated (e.g., the active state is a at time t). The Roman numerals represent trivial crossings with active or non-active PESs. Different number of trivial crossings may exist and only representative trivial crossings are shown to guide the eyes.

In principle, surface crossings within a time interval can be classified into four general types (see Figure 1), including purely avoided crossings (type 1), trivial crossings with the active PES (type 2), trivial crossings with non-active PESs (type 3), and trivial crossings with both active and non-active PESs (type 4). We propose the CC-FSSH method to treat all types of surface crossings properly. A step-by-step outline of our algorithm is as follows. 1. As in standard FSSH, we initialize the coordinates and velocities of the classical particles, set the wave function of the quantum subsystem, and select the active PES at time zero. 2. At each time step t, the classical particles move on the active PES by solving Newtonian equations for a time interval dt. The wave function is propagated in the diabatic bases through Eq. (3), and further converted to the adiabatic bases by representation transformations as in SC-FSSH. The nonadiabatic coupling vectors are calculated with the Hellmann-Feynman theorem.29 3. We calculate the overlap between the active state a at time t and every adiabatic state i at time t + dt, S ai = φa (r; R (t )) φi (r; R (t + dt )) , and find the state j with the maximum overlap, i.e.,

S aj = max i S ai . If j is identical to a, we suppose the active PES is not associated with any trivial crossing within the time interval and the crossing belongs to type 1 (Figure 1A) or type 3 (Figure

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1C); Otherwise, trivial crossing with the active PES is detected, and the case is of type 2 (Figure 1B) or type 4 (Figure 1D). 4. We evaluate the hopping probabilities based on the type of crossing. For types 1 and 3, the hopping probabilities are calculated through Eq. (2) as in standard FSSH. For types 2 and 4, the hopping probability g j is redefined by Eq. (5), while the other hopping probabilities gi ( ≠ j ) are still obtained through Eq. (2). Here, j is detected by step 3. 5. As in standard FSSH, a uniform random number, ξ ∈[0,1] , is generated, and a surface hop to state b ≠ a is assigned when

∑

b −1 i =1

g i < ξ ≤ ∑ i =1 g i . If b ≠ j, we calculate the overlap between state b

b at time t and every adiabatic state i at time t + dt, Sbi = φb (r; R (t )) φi (r; R (t + dt )) , and find the state k with the maximum overlap, i.e., Sbk = maxi Sbi . The type of crossing is assigned as

1 (if j = a and k = b ) 2 (if j ≠ a and (b = j or k = b ≠ j ))  . type =  3 (if j = a and k ≠ b) 4 (if j ≠ a and b ≠ j and k ≠ b)

(6)

6. For types 1 and 3, if the velocities of the classical particles can be adjusted along the nonadiabatic coupling vector for energy conservation, the new active state at time t + dt is reassigned to k; Otherwise, the system remains on the active state a. For types 2 and 4 with b ≠ j, if the velocities can be adjusted along the nonadiabatic coupling vector for energy conservation, the new active state at time t + dt is reassigned to k; Otherwise, the new active state is reassigned to j. For type 2 with b = j, the hop is accepted immediately without considering the energy conservation. 7. Decoherence correction is implemented. 8. Return to step 2 and repeat the steps until a certain predefined criterion is achieved. A certain number of trajectories are used to simulate nonadiabatic dynamics. 8


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Compared with SC-FSSH, CC-FSSH has three major improvements. First, a general classification of surface crossings is achieved by state tracking, and all types of surface crossings are corrected by CC-FSSH. In contrast, SC-FSSH cannot describe type 3 and type 4 crossings, where trivial crossings happen with non-active PESs. Note that state tracking has been previously utilized to detect unphysical discontinuities of the PESs.15,17 Here, state tracking is used to make surface hopping classifications and ensure a proper treatment of each crossing. Second, CC-FSSH selects the state which overlaps most with the active state for hopping probability correction, while SC-FSSH applies the hopping probability correction to the state which is closest in energy to the active state. Thereby, CC-FSSH gives more proper hopping probabilities when multiple trivial crossings happen with the active PES in type 2 and type 4 cases (see Figures 1B and 1D). Third, energy conservation is not considered when a surface hop occurs between trivial crossing states (i.e., b = j) to eliminate numerical instability of the corresponding nonadiabatic couplings. This is reasonable because the energy difference between the corresponding states is normally negligible. In fact, the nonadiabatic coupling is significant only at the crossing point in this case, and the direction for energy conservation is hardly well defined with a large time interval. As a result, CC-FSSH gives more appropriate hopping probabilities and can deal with complex multiple trivial crossings in a time interval, overcoming the main drawback of SC-FSSH. To assess the performance of CC-FSSH, we adopt the Holstein Hamiltonian30-32 for one-dimensional molecular stacks with open boundary condition,

1 H = ∑α xk | k >< k | +∑−τ (| k >< k +1| + | k +1 >< k |) + ∑ (Kxk2 + mvk2 ) . k k k 2

(7)

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The model system contains N molecular sites with equal spacing of L between neighboring molecules. Each molecule k is associate with an electronic orbital |k> and one vibrational degree of freedom

xk (the corresponding velocity, force constant, and mass are vk, K and m, respectively). -τ

is the electronic coupling between orbitals in adjacent molecules, and α is the local electron-phonon coupling. Similar Hamiltonians have been widely adopted to study charge transport in organics.16,21,37-42 Following previous studies, the parameters are chosen as α = 3500 cm-1/Å, K = 14500 amu/ps2, m = 250 amu, and L = 5 Å, which are typical for organic materials. Surface hopping simulation of charge transport has been detailed described in previous studies.16,42 We only briefly discuss about it below. For each surface hopping trajectory m, the charge is initially localized at the center of the stack and further relaxed to the bottom of the energy band.16 At temperature T, the initial nuclear coordinates and velocities follow Boltzmann distributions with a variance of kBT/K and kBT/m, respectively.38 The nuclear dynamics is described by the Langevin 2 2 equation,21 md xk / dt = −Vk′ − γ mvk + ξ , where Vk′ is the gradient of the active PES a, γ is the

friction coefficient characterizing system-bath interaction, ξ is a Gaussian random force with standard deviation (2γmkBT/dt)1/2, and kB is the Boltzmann constant. The electronic wave function is expressed as a linear expansion of the diabatic bases

{ k } , and the time evolution is solved through

Eq. (3) and converted to the adiabatic bases at each time step. Both nuclear dynamics and wave function propagation are solved with the fourth-order Runge-Kutta algorithm.21 The hopping probabilities are calculated, and surface hops are carried out accordingly. Decoherence corrections should be considered, and the decoherence time is widely used in literature.43-50 Recently, we have adopted the force-based decoherence time proposed by Schwartz and co-workers46 and the 10


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energy-based decoherence strategy proposed originally by Truhlar and co-workers45 and latter simplified by Granucci and co-workers.47 Both strategies have shown similar results for charge transport,16 and thus we only adopt the simpler energy-based decoherence approach in this study. At each time step, the decoherence time is calculated through τ i = h(1 + C / Ekin )/ | Ei − Ea | ,47 where Ea and Ei are potential energies of the active and the i-th PESs, respectively. Ekin is the total nuclear kinetic energy, and C is set to be 0.1 Hartree. The wave function coefficients are corrected by wi′ = wi exp( − dt / τ i ) (∀i ≠ a ) and wa′ = wa (1 − ∑ i ≠ a wi′ )1/ 2 / wa 2

to conserve the total population.

Based on the time-dependent wave function of active state,

φa( m ) (t ) , the mean-squared

displacement (MSD) reads

MSD(t ) =

2 1 M  ( m) 2 ( m) ( m) ( m) , φ t r φ t − φ t r φ t ( ) ( ) ( ) ( ) ∑ a a a a  M m=1 

(8)

where M is the number of trajectories, and the elements can be computed with k r l = δkl kL and

k r2 l =δkl k2L2 . In this study, we assess the performance of different surface hopping strategies based on the MSD at t = 2 ps.16 We adopt different electronic couplings to mimic different materials. The temperature is fixed as T = 300 K, and the friction coefficient is set as γ = 100 ps-1.

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Figure 2. Distribution of the four types of surface crossings for (A) different dt with N = 9 and (B) different N with dt = 1 fs by CC-FSSH. τ = 10 cm-1 is adopted. For each calculation, at least ten thousand trajectories are carried out and each trajectory lasts 2 ps.

In Figure 2, we show the distribution of the four types of surface crossings detected by CC-FSSH. As a representative example, the Holstein model with a small electronic coupling of τ = 10 cm-1 is investigated. Different time intervals (i.e., dt = 0.1, 0.2, 0.5, and 1.0 fs) and system sizes (i.e., N = 9, 13, 17, 21, and 25) are considered. In all cases, over 98% of the surface crossings are type 1 avoided crossings that can be described by standard FSSH. The rest are trivial crossings, and the majority are of type 2, whose distribution is two orders of magnitude larger than those of type 3 and type 4. Apparently, the amounts of type 2, 3 and 4 crossings all increase significantly with both the time interval and the system size. These trivial crossings can easily induce artificial long-range charge transfer and lead to overestimation of the MSD. In principle, the MSD is converged only when all trivial crossings are treated properly, and thus the time interval convergence and the system size dependence can be used to benchmark the performance of different surface hopping strategies.

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Figure 3. The MSD at 2 ps as a function of the time interval, dt, for (A) τ = 10 cm-1 with N = 9, (B) τ = 50 cm-1 with N = 17, (C) τ = 200 cm-1 with N = 45 and (D) τ = 800 cm-1 with N = 101 by SC-FSSH,16

LD-FSSH,

mSC-FSSH,

CC-FSSH,

SC-FSSH-RD,16

LD-FSSH-RD,

and

mSC-FSSH-RD methods. To eliminate statistical error as much as possible, the number of trajectories varies between one hundred thousand and one million. The smaller the electronic coupling, the larger the number of trajectories.

In Figure 3, we make a systematic assessment of the time interval convergence. We consider a series of model systems with τ = 10, 50, 200, 800 cm-1, ranging from weak to strong electronic couplings. In all cases, CC-FSSH shows a remarkable performance. Namely, reliable results can be achieved even with a large time interval of dt = 1 fs, indicating that almost all trivial crossings are 13



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captured and properly described by CC-FSSH. Note that a larger dt = 2 fs gives slightly overestimated results. As a comparison, recent studies have shown that the original SC-FSSH needs a small time interval of dt = 0.002 fs to obtain converged results for the same systems.16 In SC-FSSH, only type 1 and 2 crossings are considered and at most one trivial crossing is treated at a time. According to Figure 2A, a small dt could greatly eliminate type 3 and type 4 crossings as well as multiple trivial crossings with the active PES of type 2, and thus makes SC-FSSH robust. SC-FSSH selects the state which is closest in energy to the active state for hopping probability correction. Also, we can apply the hopping probability correction to the state that overlaps most with the active state with SC-FSSH as in CC-FSSH. This modified SC-FSSH (mSC-FSSH) shows much faster time interval convergence than SC-FSSH, and converged results can be achieved with about dt = 0.02 fs (see Figure 3). Besides, the LD-FSSH24,25 approach is also adopted as reference. At each time t, LD-FSSH constructs a locally diabatic basis based on the adiabatic states of time t - dt, propagates the wave function without referring to the nonadiabatic coupling terms, and defines the hopping probabilities through representation transformation of the diabatic propagator. For the present study, the performance of LD-FSSH lies between that of SC-FSSH and mSC-FSSH (see Figure 3). These three approaches differ mostly in the definition of hopping probabilities. They all show slower time interval convergence than CC-FSSH, implying the importance of the additional surface classification and correction proposed in this study. Recently, we found that the adopted decoherence correction approach tends to quickly collapse the wave function to a wrong state when a trivial crossing is not captured, and the accelerated artificial long-range charge transfer greatly increases the difficulty to treat trivial crossings in surface 14


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hopping.16 A simple restricted decoherence (RD) strategy has been proposed to weaken this decoherence enhanced trivial crossing problem. Namely, we make decoherence only when the population of the active state exceeds a critical population, Pc, which indicates whether an improper surface hop has happened. It has been shown that Pc = 0.0001 is a reasonable parameter in general cases.16 This strategy has been combined with SC-FSSH and can be easily extended to other surface hopping strategies, e.g., mSC-FSSH and LD-FSSH, discussed above. The resulting SC-FSSH-RD, mSC-FSSH-RD, and LD-FSSH-RD approaches are also studied. From Figure 3, it is easy to find that the time interval convergence is improved significantly and the time interval for convergence is magnified to about 0.1 fs when restricted decoherence is implemented. Restricted decoherence reduces the negative effects of decoherence corrections when the charge has already hopped to a wrong PES.16 In contrast, the present CC-FSSH provides a much more robust solution through a complete classification of the surface crossings and a proper description of each type of crossing. Surface hop to wrong PESs with long-range charge transfer character does not happen in CC-FSSH. Especially, this is achieved without introducing any empirically derived parameter.

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Figure 4. The MSD at 2 ps as a function of the number of molecules, N, for τ = 10 cm-1 by LD-FSSH, SC-FSSH, mSC-FSSH, LD-FSSH-RD, SC-FSSH-RD, mSC-FSSH-RD, and CC-FSSH approaches. Different time intervals have been chosen for different approaches as indicated.

In Figure 4, we further study the system size dependence. We focus on the system with τ = 10 cm-1, where the trivial crossing problem is maximized among all investigated systems in Figure 3. Based on the time interval convergence, different dt has been chosen for different surface hopping approaches. For SC-FSSH and LD-FSSH with dt = 0.01 fs, the MSD starts to diverge when the system size is enlarged to about 21. In comparison, mSC-FSSH gives better size dependence due to its better hopping probabilities when multiple trivial crossings of type 2 are present. Again, restricted decoherence is helpful when combined with different surface hopping approaches, although it does not solve the problem completely. For instance, the performance of mSC-FSSH-RD with dt = 0.1 fs is already similar to that of mSC-FSSH with dt = 0.01 fs. Impressively, almost size independent results have been obtained by CC-FSSH even with a large time interval of dt = 1 fs, surpassing all other approaches with much smaller time intervals. Note that the energy-based decoherence approach adopted here is related to the total nuclear kinetic energy which changes with the system size. As a result, there exists a weak intrinsic system size dependence in Figure 4. From Figures 3 and 4, it is easy to find that good size dependence requires a much more accurate description of trivial crossings and is more difficult to achieve than the time interval convergence. The outstanding performance of CC-FSSH significantly increases the efficiency and reliability to simulate nonadiabatic dynamics especially in large systems.

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Figure 5. Statistical error of the MSD at 2 ps with different number trajectories for τ = 10, 50, 200, and 800 cm-1 with dt = 1 fs. Ten thousand realizations have been used to evaluate the error for each data point. The dashed line is shown to guide the eyes.

For realistic applications, there exist two general strategies. One may build model Hamiltonians, obtain the parameters from ab initio calculations, and run surface hopping simulations based on the model Hamiltonian.42,51 In this case, it is straightforward to implement CC-FSSH, and a huge number of trajectories can be generated as we do in Figure 3 and Figure 4. One can also calculate electronic structures on the fly and run ab initio nonadiabatic dynamics simulations directly. Due to the high computational cost of ab initio electronic structure calculations, the number of trajectories is normally limited. In Figure 5, we show the relative error of CC-FSSH for different number of trajectories. The error is defined as the standard deviation of MSD at 2 ps divided by the average value. In general, the error decreases rapidly with the number of trajectories, and slower dynamics with smaller electronic couplings require more trajectories to get the same statistical accuracy. For 17



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all the investigated systems, 2,000 trajectories are enough to achieve a 90% accuracy at 300 K. Note that a variant algorithm of SC-FSSH has been recently proposed by Blumberger and co-workers, and charge transport in one-dimensional chains of ethylene-like molecules has been studied.52 From this point of view, the present CC-FSSH approach is also very promising for on-the-fly nonadiabatic dynamics simulations in real materials. We may take advantage of different diabatic bases, such as the locally diabatic states or the quasi-diabatic molecular/atomic orbitals. The relevant studies are currently under way. Finally, we point out that all surface hopping approaches investigated in this study converge to the same result when an extremely small dt is used. The only difference is how large the time interval can be used to get the converged result. Thereby, the present CC-FSSH can only be regarded as a new version of FSSH with a much higher numerical efficiency. Because a large time interval can already be adopted, the use of adaptive time intervals is not of key importance within CC-FSSH. FSSH is an ad hoc method, and various corrections have been introduced to increase its reliability.4 Thereby, CC-FSSH also provides a fundamental platform to introduce modifications or approximations as needed and deserves a systematic study. In summary, we have proposed a novel CC-FSSH approach for surface hopping, which could describe complex surface crossings with a high density of electronic states. We have shown that surface crossings can be generally classified into four types, and a proper description of each type of crossing allows us to get converged results with a large time interval of 1 fs. Especially, a weak size dependence has been obtained with CC-FSSH, implying that the trivial crossing problem is no longer a major bottleneck of surface hopping simulations. Compared with other surface hopping approaches 18


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with trivial crossing corrections, CC-FSSH has shown significant advantage when studying large systems. CC-FSSH is free of parameters and highly efficient, and thus can be a powerful approach for general nonadiabatic dynamics simulations of electrons and excitons. AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]. Notes The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

L.W. acknowledges support from the “Thousand Young Talents Plan” of China, the “Hundred Talents Plan” of Zhejiang University, the Fundamental Research Funds for the Central Universities (Grant No. 2017QNA3010), and the National Natural Science Foundation of China (Grant No. 21703202).

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TOC figure 49x49mm (300 x 300 DPI)


Crossing Classified and Corrected Fewest Switches Surface Hopping

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