Analysis of the Trajectory Surface Hopping Method from the Markov State Model Perspective

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Journal of the Physical Society of Japan 84, 094002 (2015) http://dx.doi.org/10.7566/JPSJ.84.094002

Alexey V. Akimov1, Dhara Trivedi2, Linjun Wang1, and Oleg V. Prezhdo1+

1Department of Chemistry, University of Southern California, Los Angeles, CA 90089, U.S.A. 2Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, U.S.A. (Received May 20, 2015; accepted July 22, 2015; published online August 18, 2015) We analyze the applicability of the seminal fewest switches surface hopping (FSSH) method of Tully to modeling quantum transitions between electronic states that are not coupled directly, in the processes such as Auger recombination. We address the known deficiency of the method to describe such transitions by introducing an alternative definition for the surface hopping probabilities, as derived from the Markov state model perspective. We show that the resulting transition probabilities simplify to the quantum state populations derived from the time-dependent Schrödinger equation, reducing to the rapidly switching surface hopping approach of Tully and Preston. The resulting surface hopping scheme is simple and appeals to the fundamentals of quantum mechanics. The computational approach is similar to the FSSH method of Tully, yet it leads to a notably different performance. We demonstrate that the method is particularly accurate when applied to superexchange modeling. We further show improved accuracy of the method, when applied to one of the standard test problems. Finally, we adapt the derived scheme to atomistic simulation, combine it with the time-domain density functional theory, and show that it provides the Auger energy transfer timescales which are in good agreement with experiment, significantly improving upon other considered techniques.

estimated in FSSH theory because the transitions through 1. Introduction such intermediate states are inhibited by the probabilities of Theory of molecular dynamics with quantum transitions is overcoming the high-energy barrier. Simulations of non- an actively developing field of computational chemistry.1–14) adiabatic processes using FSSH are usually performed in the The interest in this subject is constantly stimulated by the adiabatic basis, in which FSSH shows reasonable accuracy. need for accurate and computationally tractable simulation However, the use of diabatic states (e.g., donor and acceptor methodologies for modeling processes involving coupled states) may be advantageous, both in terms of computational electron–nuclear dynamics in contemporary materials. In efforts and for interpretation of the problem physics. Diabatic particular, such techniques are required for studying photo- states are used regularly in phenomenological models of induced charge transfer in photovoltaic and photocatalytic charge and energy transfer because they have well-defined materials,15–20) long-distance energy transfer in biological physical meaning. Superexchange effects are traditionally photosynthetic complexes, or photo- and electro-stimulated described in a diabatic representation. An adiabatic picture mechanical response in nanoscale systems.20,21) can lead to spurious effects arising from artificial delocaliza- A great body of methods for modeling molecular dynamics tion of an adiabatic state between donor and acceptor sites with quantum transitions has been developed over the last that are very distant from each other. Finally, the adiabatic several decades. We refer the reader to specialized reviews representation leads to numerical problems with calculation discussing some of these methods.22–27) Among the variety of non-adiabatic coupling.45,46) Under the conditions when of techniques, Tully’s fewest switches surface hopping diabatic states are strongly localized and many pairs of states (FSSH)1) remains one of the most utilized methods, due to are decoupled, the accuracy of the original FSSH method is its conceptual and practical simplicity, good accuracy, and often unacceptable when diabatic states are used, motivating high computational efficiency. The FSSH method is not a development of the alternative formulations.28,47) panacea, and there are examples where it breaks down. In Recently, the problem of superexchange has been particular, the superexchange problem28) requires a more addressed by Wang et al.28) who proposed a global flux advanced treatment. surface hopping (GFSH) algorithm. In it, hopping between The need for consideration of superexchange effects is a pair of states is determined by a relative change of state motivated in applications by excitonic, many-particle proc- population during the infinitesimal time slice. The method esses, such as Auger recombination and energy transfer,29–35) avoids computing the hopping probabilities based on one- singlet fission36–39) or Raman scattering. The superexchange particle properties (such as non-adiabatic couplings between mechanism is also important for tunneling and conduc- one-particle states), thus solving the superexchange problem tivity in molecular electronics.40–44) In all these processes, phenomenologically. In the alternative second-quantized transitions between excitonic states can involve more than surface hopping (SQUASH) approach,47) the superexchange one simultaneous single-particle transition. A direct consid- problem was solved by extending the quantum dynamics to eration of the dynamics of such excitations is prohibited by the space of second-quantized states representing coupled the Slater rules, and only sequential mechanisms are viable. trajectories. The method introduces many-particle states, The problem of superexchange arises when there is at least allowing a single transition between a pair of such states to one pair of states that are not coupled to each other directly. capture a many-body transition. The states can be either diabatic or adiabatic. The transition Although the developed techniques provide notable between these states is mediated by an intermediate high- improvement of the accuracy in modeling the superexchange lying state. The overall transition rates are strongly under- problem, they have certain limitations. Specifically, the 094002-1 ©2015 The Physical Society of Japan J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by Univ of Southern California on 11/07/19

J. Phys. Soc. Jpn. 84, 094002 (2015) A. V. Akimov et al.

GFSH approach relies on an ad hoc scheme for splitting total MSSH method to the classical path approximation (CPA), probability to leave a state i into probabilities of doing this which neglects electron back-reaction on nuclei, and apply via different channels i ! j. As it is pointed out by Wang it to compute Auger-type energy transfer dynamics in et al.,28) there are, in principle, many possibilities of such atomistic quantum dots (QDs). We conclude that the method splitting. Different splitting schemes define different hopping can be useful for a number of systems and processes, where probabilities, which may affect the results of surface hopping the standard FSSH may have problems. We also discuss simulations. Thus, the question of uniqueness, physical limitations of the proposed approach. meaning and rigor of the definition of surface hopping probabilities remains open, presenting a conceptual chal- 2. Theory lenge. On the other hand, SQUASH47) introduces coupled 2.1 Analysis of the FSSH trajectories that may exchange energy. The number of N- One of the basic elements of Tully’s FSSH method is the particle states describing N coupled trajectories increases definition of hopping probabilities. The method is derived the dimensionality of the problem very rapidly, making from the following assumption of hopping out of a single the computations costly. The method also relies on the state. Assume that the number of trajectories in the current conceptually challenging prescription of the energy exchange state i at time t is NiðtÞ. If at the later time t þ dt this number in this ensemble-based approach to quantum dynamics. decreases to Niðt þ dtÞ, with Niðt þ dtÞ

J. Phys. Soc. Jpn. 84, 094002 (2015) A. V. Akimov et al.

FSSH prescription, including hop rejection, velocity rescal- The elements of the matrix Kðt; t þ dtÞ are non-negative, so ing, propagation of nuclei and electronic amplitudes. The are the elements of the matrix T. Thus, the ad hoc schemes only distinction from the FSSH enters via the way the surface for setting negative transition probabilities to zero, used hopping probabilities are defined. To derive these rates, we in other surface hopping algorithms, including FSSH and follow the kinetic considerations discussed below. Consider GFSH, are not needed. the number of trajectories, Niðt þ dtÞ, in state i at time t þ dt. Finally, the expression Eq. (7) can be simplified using the It can be computed based on the numbers of trajectories, definitions in Eqs. (5) and (6): fNjðtÞg, in all states at time t, and the “rates” of transition ajjðt þ dtÞaiiðtÞ k t; t dt X between states i and j, f j!ið þ Þg: 2 X annðtÞ N t dt k t; t dt N t : 4 Kji n ajjðt þ dtÞaiiðtÞ ið þ Þ¼ j!ið þ Þ jð Þ ð aÞ Ti j ¼ X ¼ ¼ X ! X a t dt a t j Kki kkXð þ Þ iið Þ akkðt þ dtÞaiiðtÞ k a2 t k It can be seen from the dimensionality analysis that the k nnð Þ “rates” should be interpreted as the fraction of trajectories n transferred from the source to target states during a given ajjðt þ dtÞ ajjðt þ dtÞ ¼ X ¼ ¼ ajjðt þ dtÞ: ð8Þ time interval. Thus, it is more accurate to think of the akkðt þ dtÞ 1 quantities fkj!iðt; t þ dtÞg as the matrix elements of an k effective transition operator that describes the flow of We now see that MSSH defines the surface hopping population within a given time slice. The introduced “rates” probabilities as the quantum populations of the target states at are not local in time, unlike the quantities a_iiðtÞ=aiiðtÞ used to the next time step. The result is unexpectedly simple and define the hopping probabilities in FSSH. Instead, the “rates” reflects the well-known concept of quantum mechanics — carry integral information about the dynamics during for the that the quantum amplitudes aiiðtÞ have the meaning of the time slice ½t; t þ dt. This information may implicitly include probability to find the system in a given state i at time t. Note many-particle transitions. that, in general, the probability to find the system in a specific Normalization to the total number of trajectories trans- state and the probability to hop to a given state from another forms Eq. (4a) to the equation for quantum populations: arbitrary state are different quantities. For example, in both X FSSH and GFSH, the hopping probability depends on the a t dt k t; t dt a t ; 4 iið þ Þ¼ j!ið þ Þ jjð Þ ð bÞ current quantum state. In MSSH, the hopping probability j depends only on the population of the final state. Moreover, and in the matrix form: this probability does not explicitly depend on non-adiabatic coupling between the source and target states. Thus, a aðt þ dtÞ¼Kðt; t þ dtÞaðtÞ: ð4cÞ transition is possible even when the non-adiabatic coupling T Here, aðtÞ¼ða00ðtÞ a11ðtÞ aMMðtÞÞ is the population between these states is zero. The transition is accommodated state-vector, M is the number of states, Kðt; t þ dtÞ is the by the population flow via alternative channels, as implicitly matrix representation of the population transfer operator encoded in the diagonal elements of the propagated density discussed above. Note that the element ði; jÞ of the matrix matrix, aðt þ dtÞ. defines the rate of the “reverse” transition, that is, the We note that the result, Eq. (8), was utilized in one of the transition from the state j to the state i: first surface hopping algorithms by Tully and Preston.48) However, there are two main distinctions with our approach K k : 5 ij ¼ j!i ð Þ we would like to point out. First, at the technical level, the Now the question is how to define the population transfer MSSH hopping probability is defined differently from the operator such that the surface hopping populations are one used by Tully and Preston.48) In their formulation the consistent with the populations obtained from the TD-SE, surface hopping probabilities are defined as the TS-SE initial Eq. (3). Using vector algebra, one can see that the matrix populations of the target states, Ti!j ¼ ajjðtÞ, while MSSH Kðt; t þ dtÞ is an outer product of the source and target uses final populations of the target states Ti!j ¼ ajjðt þ dtÞ. population state-vectors: Second, the derivation of the Tully–Preston and MSSH ff aðt þ dtÞaTðtÞ aðt þ dtÞaTðtÞ methods are conceptually di erent. In our approach we KðtÞ¼ ¼ : ð6Þ start by explicitly requesting the higher-order effects to be aTðtÞaðtÞ a t 2 k ð Þk included and employ the general Markov state model Note that all elements of the matrix Kðt; t þ dtÞ are non- framework. The frequent switches surface hopping method negative, because all elements of the population vectors are of Tully and Preston relied on the intuitive probabilistic non-negative. Thus, there is no additional complication asso- interpretation of quantum populations, without rigorous ciated with negative hopping probabilities, unlike in FSSH. derivation. We are now in position to define the Markov state model. It is defined by the transition probability matrix, T, the 3. Applications elements of which are related to the elements of the “rate” In this section, we present the results of our calculations on matrix Kðt; t þ dtÞ by the following equation: model and atomistic systems. To demonstrate the capabilities of MSSH with regard to the superexchange, we apply the ki!j Kji Ti j ¼ X ¼ X : ð7Þ ! k K method to the three-state superexchange problem of Wang i!k ki et al.28) Next, to show the performance of the method with k k the standard test models and to illustrate the advantages of 094002-3 ©2015 The Physical Society of Japan J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by Univ of Southern California on 11/07/19

J. Phys. Soc. Jpn. 84, 094002 (2015) A. V. Akimov et al.

(a) (b)

Fig. 1. (Color online) A three-state superexchange model. Schematic representation of the model Hamiltonian (a). Probabilities of transmission on (b) ﬁrst, (c) second, and (d) third states. Performance of the MSSH method is compared with the results of FSSH and exact quantum simulations.

MSSH over the MF method, we apply MSSH to a modified stable and symplectic integration. The time-step is set to double avoided crossing (m-DAC) problem.1,49) Finally, a 200=p a.u., where p is the initial momentum. Each trajectory CPA version of the MSSH algorithm is developed and is propagated for 2000 steps with properties of interest applied to compute the timescales of Auger relaxation in a printed every 2 steps. The propagation of each individual small QD. trajectory is stopped whenever the trajectory leaves the interest interval defined as ½15; 20. The exact solution is 3.1 A three-state superexchange problem obtained via integration of the TD-SE on a grid numerically, The model consists of three states, with the energy as explained in the previous work.28) alignment and coupling shown schematically in Fig. 1(a). The results obtained with the MSSH method coincide with The direct coupling between states 1 and 2 is set to zero. The the exact solution to a great degree of accuracy, Fig. 1. On Hamiltonian is defined by the matrix elements in the diabatic the contrary, FSSH reproduces only the high-energy results. representation (diabatic energy levels and diabatic cou- However, the high incident momentum region is essentially plings): E1ðxÞ¼0, E2ðxÞ¼0:005, E3ðxÞ¼0:01, V12ðxÞ¼ classical, with negligible quantum effects. In this region, 2 V21ðxÞ¼0, V13ðxÞ¼V31ðxÞ¼0:001 expðx =2Þ, V23ðxÞ¼ any semiclassical theory, including mean-field, as well as 2 V32ðxÞ¼0:01 expðx =2Þ. The simulations are performed in classical propagation would give the same accuracy as the the diabatic representation. The atomic units of length and fully quantum description. Thus, we judge the quality of the energy are used. The mass of the particle is set to 2000 a.u. MSSH based on its capability to capture low-energy effects. The initial momenta and positions of the classical particle are The main difference between MSSH and FSSH is observed set to the defined values for all trajectories, with no additional in this region of the initial momentum. For the initial spread, to exclude any related effects. The initial position is momentum in the range of 4–6 a.u. FSSH simply does not set to −15 a.u., so that trajectory starts far away from the allow a transition to state 2. The transition would require strong coupling region. The initial momentum varies over the overcoming a high-energy barrier to the intermediate state 3. interval from 1 to 50 a.u. with the 0.25 step in scattering This cannot be accomplished when the nuclear kinetic energy calculations, or is set to a specified value in the trajectory is low. In contrast, MSSH makes the surface hopping analysis calculations. The ensemble of 2500 trajectories is decision solely on the basis of the TD-SE-derived popula- propagated, to generate statistically-meaningful and smooth tions. The hop rejection criterion based on the energy distributions of electronic populations at all times. Equations conservation is verified after each attempted transition, of motion for the electronic and nuclear degrees of freedom similar to FSSH. However, the transition is no longer are integrated using split-operator technique, which leads to mediated by the high-lying state 3 and can proceed directly. 094002-4 ©2015 The Physical Society of Japan J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by Univ of Southern California on 11/07/19

J. Phys. Soc. Jpn. 84, 094002 (2015) A. V. Akimov et al.

(a) (b)

Fig. 2. (Color online) Evolution of the ensemble-averaged TD-SE populations (SE) and TSH-based populations (SH) computed with FSSH (a, c) and MSSH (b, d) methods. The active state (istate) evolution for a representative individual trajectory is shown by dotted line. The evolutions for two initial momenta are shown: p ¼ 5 a.u. (a, b) and p ¼ 10 a.u. (c, d).

Therefore, one observes a notable population transfer to (both FSSH and MSSH) states that the SH populations must state 2 in the intermediate energy region. reach the SE populations, in the limit of sufficiently large The transmission on state 3 as a function of the initial number of trajectories. There can be two sources of deviation incident momentum [Fig. 1(d)] shows a slight deviation of of SH populations from the corresponding SE values: a) due the MSSH curve with respect to the exact results. The to the wavepacket branching; b) due to intrinsic properties of deviation is larger in the region of low initial momenta. The the TSH algorithm. The branching affects the results in the high-energy results are very close to the exact and FSSH following way. In Figs. 2 and 4, we plot SH and SE along a data. The total transmission probability on state 3 is very single trajectory, but other trajectories may take alternative small, an order of magnitude smaller than the transmission on paths, resulting in distinction of the SH and SE populations state 2. Thus, the better agreement between the FSSH and computed along the path chosen for plotting. The trajectory exact data compared to the MSSH performance is not branching is absent in the superexchange model, because the indicative of the overall superiority of FSSH over MSSH. It potential energy surfaces are flat and the diabatic basis is can be attributed to technical sides, such as choice of initial used. Thus, the results in Fig. 2 reflect only the intrinsic conditions, statistical properties of the trajectory ensemble, properties of the TSH algorithm. Hence all trajectories will etc. One can observe a small, but systematic tendency of the take the same path. Although the evolution of populations MSSH to overestimate transition probabilities slightly. Still, implied their dependence on time, we plot the populations as MSSH provides qualitative and quantitative improvement the function of positions, to facilitate the analysis with the over FSSH for a number of systems, including the present model Hamiltonian in mind. In particular, the position x ¼ 0 superexchange model. correspond to the maximum of the off-diagonal (coupling) We also illustrate the distinctions between the two methods Hamiltonian matrix element. by considering the evolution of the initial state population First, we analyze the behavior of the populations in the and active state index (denoted istate) for a typical trajectory low incident momentum region ( p ¼ 5 a.u.) — Figs. 2(a) (Fig. 2). We compute the ensemble-averaged populations of and 2(b). One can clearly observe that the SH populations two types — those that come from the fully coherent TD-SE computed with FSSH do not follow the corresponding SE (denoted SE) and those computed via the surface hopping populations. The active state remains unchanged. In other algorithms (denoted SH). These populations are computed words, no population transfer to other states occurs, as we along the particular trajectory shown in figures. The main have observed in Fig. 1. On the contrary, the SH populations assumption used to derive the surface hopping probabilities computed with MSSH follow the corresponding SE values. 094002-5 ©2015 The Physical Society of Japan J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by Univ of Southern California on 11/07/19

J. Phys. Soc. Jpn. 84, 094002 (2015) A. V. Akimov et al.

(a)

(b) (c)

Fig. 3. (Color online) Comparison of the survival probabilities on states 1 and 2 computed with MSSH, FSSH, MF and exact numerical propagation for the modiﬁed DAC model. (a) Adiabatic potential energy surfaces of the model; (b, c) transmission probabilities on the upper state 2 computed with MSSH, FSSH, and exact propagation (b) and with MSSH, MF, and exact propagation (c).

One can also observe multiple switches of the active state B ¼ 0:028, increasing the separation between the avoided (dotted line). This behavior is consistent with non-negligible crossing regions and allowing stronger interference patterns population transfer probabilities observed at this value of to develop. The simulations are performed in the adiabatic incident momentum. representation, as in the original work.1) The adiabatic Second, we extend the above analysis to the larger incident potential energy profiles are shown in Fig. 3(a). momentum value — p ¼ 10 a.u. As it follows from Fig. 1, The scattering probabilities computed with MSSH and there are no important superexchange effects in this region of FSSH are compared with the reference exact results in kinetic energies: both FSSH and MSSH perform well. This Figs. 3(b) and 3(c). The exact numerical solution was conclusion is indeed supported by the time evolution of reported in the earlier work.49) In general, the qualitative populations shown in Figs. 2(c) and 2(d). In both cases, the behavior of the MSSH solution agrees with both FSSH and SH populations coincide with the corresponding SE counter- exact results. Notably, the MSSH solution is closer to the parts. There is no surface hop for the trajectory depicted in fully quantum result than FSSH, as demonstrated by the good Fig. 2(c), but it occurs for other trajectories in the ensemble. agreement of the peak positions in Fig. 3(b). The position of Surface hops are still more frequent in the MSSH algorithm, the MSSH peaks is in the middle between the positions of the but are less frequent than for p ¼ 5 a.u. momentum. FSSH and exact peaks, leaning toward the exact solution. The recently developed GFSH method28) is also capable This effect is especially pronounced for the low incident of handling the superexchange problem. However, some momentum region. As we noted earlier, the quantum effects disagreement between the GFSH and exact results remains. become less important at higher energies. As a consequence, MSSH shows much closer agreement with the exact results. all methods produce similar results in this limit. By construction, the MSSH method is based on frequent 3.2 Modified double avoided crossing problem surface hops. As discussed by Tully,1) such frequent hops A double avoided crossing (DAC) model was originally make the MSSH method similar to the MF=Ehrenfest proposed by Tully.1) The model is formulated in terms approach. The similarity originates from the observation that of matrix elements of the diabatic Hamiltonian as the effective potential energy surface on which a system 2 E1ðxÞ¼0, E2ðxÞ¼A expðBx ÞþE0, V12ðxÞ¼V21ðxÞ¼ evolves in SH with frequent hops is, on average, close to the C expðDx2Þ, with the parameters chosen to be A ¼ 0:10, wavefunction-weighted potential energy surface used in the B ¼ 0:28, E0 ¼ 0:05, C ¼ 0:015, and D ¼ 0:06. We have MF dynamics. The similarity of the frequent switches surface employed a modified version of this problem in one of our hopping with the MF theory was one of the reasons why the previous studies.49) Our modification sets the parameter approach was abandoned, and why the FSSH method gained 094002-6 ©2015 The Physical Society of Japan J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by Univ of Southern California on 11/07/19

J. Phys. Soc. Jpn. 84, 094002 (2015) A. V. Akimov et al.

(a) (b)

Fig. 4. (Color online) Evolution of the ensemble-averaged TD-SE populations (SE) and TSH-based populations (SH) computed with FSSH (a, c) and MSSH (b, d) methods. The active state (istate) evolution for a representative individual trajectory is shown by dotted line. The evolutions for two initial momenta are shown: p ¼ 10 a.u. (a, b) and p ¼ 25 a.u. (c, d). more popularity. In this work we demonstrate that this for trajectory analysis. Namely, we analyze trajectories with conclusion is not always true. Quite the opposite, the MSSH initial p ¼ 10 a.u. [Figs. 4(a) and 4(b)] and p ¼ 25 a.u. method performs much better than the MF technique. To [Figs. 4(c) and 4(d)]. In case when the initial momentum demonstrate this result, we utilize the modified DAC is small, the FSSH results show the trajectories turning (m-DAC) model defined above. twice — approximately at the points of strong nonadiabatic The scattering probabilities for the m-DAC model problem coupling, x ¼5 a.u. The turning of the trajectory is the computed with the MSSH and MF methods are shown in reason why Fig. 4(a) shows a multi-valued function. For Fig. 3(c). The exact solution is also shown as a reference. larger initial momentum we do not observe spatial oscil- The transition probabilities computed with MSSH and MF lations — the trajectory simply passes through the regions of are very close to each other when the initial momentum is strong coupling, without reflecting. sufficiently high (e.g., >20 a.u.). This agreement can be In the FSSH case [Figs. 4(a) and 4(c)], one can also observe attributed to the almost classical regime of the dynamics. As two surface hops (0 ! 1) and (1 ! 0) per the representative the nuclear kinetic energy decreases, quantum effects become trajectory. The hops occur around the crossing points, as dominating. In the intermediate energy region (15

J. Phys. Soc. Jpn. 84, 094002 (2015) A. V. Akimov et al.

branching than FSSH, making it more suitable for higher nuclear kinetic energy problems. The latter condition is also a prerequisite of the classical path approximation (CPA) to hold. Thus, when the CPA is applicable, it is likely that the MSSH can also provide good level of description of nonadiabatic eﬀects. In the next section, we report the test study of Auger relaxation process in atomistic model computed with the CPA + MSSH hybrid methodology.

3.3 Atomistic simulation of Auger relaxation in quantum dots To emphasize the strength of the MSSH method and to show its applicability to atomistic system, we have studied the decay of hot electrons in colloidal CdSe QDs. The experimentally observed relaxation time is in the sub- picosecond range,50–52) indicating absence of the so-called (a) phonon bottleneck. The Auger process should play a key role, as explained below. Our atomistic simulations reveal the importance of the Auger process in the electron–phonon relaxation dynamics. We have carried out a simulation of the nonadiabatic photoinduced dynamics in the Cd33Se33 QD [Fig. 5(a)]. The Cd33Se33 QD is a “magic”-size cluster with the diameter of 1.3 nm. It has been shown experimentally to be very stable,53) making it an excellent model for the electronic structure studies of CdSe QDs.15,28,54) We firstly run quantum-mechanical molecular dynamics (MD) using the Vienna Ab initio Simulation Package (VASP)55,56) followed by nonadiabatic molecular dynamics (NA-MD) simulations performed using the PYthon eXtension for Ab Initio Dynamics (PYXAID).57,58) The QD structure is initially optimized using the conjugate-gradient algorithm until the force convergence threshold of 0.001 eV=Å is reached. Subsequently, the (b) system is thermalized by running a 10 ps ab initio MD simulation in the canonical ensemble. The collisional Fig. 5. (Color online) (a) The optimized structure of the Cd33Se33 QD. (b) The time evolution of the energy levels of the Cd33Se33 QD for the first thermostat as implemented in the VASP program maintains 3.5 ps of the MD simulation at 300 K. the average temperature. The velocities are rescaled every 4 fs to match the target phonon temperature of 300 K. The possibility for the individual trajectories to take The thermalized structures are utilized as input for the alternative paths is enhanced in the low-momentum region NA-MD calculations. Starting from these structures, 10 ps due to increased fraction of frustrated hops and possibilities MD trajectories are computed in the microcanonical of reflection. As the initial momentum of the classical ensemble. All MD calculations utilize the velocity Verlet59) particle increases (p ¼ 25 a.u.), the frustrated hops become algorithm with the time step of 1 fs to propagate nuclear less frequent, leading to less significant wavepacket bran- coordinates. The Kohn–Sham energies and orbitals computed ching — more trajectories follow similar paths. This effect is at each time-step are utilized to obtain the NA couplings and illustrated in Figs. 4(c) and 4(d). One can see much closer to form the NA-MD Hamiltonian.57,60) This time-dependent agreement of SH and SE population along the chosen path, in information is stored and used to perform the NA-MD contrast to low-momentum region [Figs. 4(a) and 4(b)]. A simulations with the help of the PYXAID package.57,58) The small distinction is still present: the FSSH trajectories show CPA is applied to all three semiclassical methods — MSSH, some branching after the second avoided crossing region, FSSH, and GFSH — to achieve considerable computational whereas all MSSH trajectories follow similar paths in all savings. coordinate interval ½15; 10. All quantum calculations employ density functional theory We conclude that MSSH approach is applicable in both (DFT) with the PBE exchange–correlation functional61,62) diabatic and adiabatic representations. Since FSSH works best as implemented in the VASP program.55,56) To accelerate in the adiabatic representations, it is generally challenging calculations and to account for the effects of core electrons, to improve its accuracy further. Yet, as the m-DAC model the projector-augmented-wave (PAW) pseudopotentials55,56) demonstrates, MSSH provides a notable improvement over for Cd and Se are employed. Valence electrons are described FSSH in its capability to predict position of the population using plane waves with the energy cutoff of 280 eV. To avoid transfer peaks. We also conclude that MSSH performs much interaction of the QD with its image, a sufficient vacuum better than simple MF, despite a very strong similarity of the is added around the QD, resulting in a 25 25 25 Å two methods. Finally, we observe that MSSH shows less simulation box. Only the gamma-point of the Brillouin zone 094002-8 ©2015 The Physical Society of Japan J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by Univ of Southern California on 11/07/19

J. Phys. Soc. Jpn. 84, 094002 (2015) A. V. Akimov et al.

geometries are selected randomly from the initially computed MD trajectory. The computed averaged population of the 1Se state is presented in Fig. 6(b), along with the fitted exponential curves. The fits are used to extract the timescale for the electron relaxation. The standard FSSH1) and the recently developed GFSH28) methods are used as references to understand the performance of the MSSH method reported in this work. MSSH exhibits significantly faster dynamics than the other two surface hopping techniques. The 0.7 ps time scale given by (a) the MSSH is an order of magnitude smaller than the 8.2 ps time scale obtained through FSSH. The two-photon photo- emission50) and transient absorption spectroscopy51,52) measurements have shown sub-picosecond timescales for the electron relaxation from 1Pe to 1Se. In particular, the work of Sippel et al.50) reports the 0.22 ps relaxation timescale. The MSSH timescale agrees much better with the experiment than the FSSH and GFSH data. GFSH produces Auger dynamics that is faster than the one obtained in the FSSH method. Still, the GFSH dynamics is too slow compared with MSSH and the experiments. The MSSH approach provides the best description of the Auger-type process, involving a simultaneous transition of two particles.

(b) 4. Conclusions We have presented the analysis of the trajectory surface Fig. 6. (Color online) (a) Schematic representation of the Auger relaxation hopping algorithms from the Markov state model perspec- process: the electron in the initially prepared 1Pe state relaxes down to the 1Se state by exchanging energy with the hole; (b) Time evolution of the tive. Using the Markov state model we designed the MSSH electron population in the 1Se state computed with the FSSH, GFSH, and method for accounting for many-particle transitions and MSSH methods. higher-order effects in semiclassical nonadiabatic dynamics. One of such effects is superexchange, which is commonly is used for integration because this is a finite system. Further considered to constitute one of the key charge and energy computational details can be found in the work of Trivedi transfer mechanisms, and which arises when a diabatic basis et al.,63) in which the Auger process was studied using the is used in simulations. GFSH approach.28) We show that the general MSSH approach reduces to the The electronic energy levels of the CdSe QD are shown in frequent switches surface hopping method of Tully and Fig. 5(b). The asymmetry of the density of states (DOS) Preston, but it is derived from different perspectives and may favors the Auger relaxation channel. That is the photoexcited be generalized beyond the Tully–Preston version. Unlike electron overcomes the larger energy barrier between the 1Pe Tully and Preston, who used the interpretation of the and 1Se states by transferring its energy to the hole, as shown quantum mechanical probabilities to define surface hopping schematically in Fig. 6(a). In turn, the hole rapidly transfers probabilities, our theoretical analysis is based on a general its energy to phonons, by relaxing down the manifold of master equation approach. In our construction, we explicitly dense energy levels, Figs. 5(b) and 6(a). The calculated mean require the many-particle quantum transitions to be feasi- energy gap between the 1Pe and 1Se states of the electron is ble — the point which is missing in the frequent switching 0.4 eV. It agrees reasonably well with the experimental surface hopping scheme. Although the original Tully–Preston observations of 0.1–0.3 eV for CdSe QDs.50) The slightly approach was considered inferior to the FSSH formulation, larger value arises because the investigated QD is smaller and was eventually abandoned, it can address the conceptual than experimental QDs. Smaller size emphasizes the quantum shortcoming of the standard first-order FSSH technique, confinement (particle-in-a-box) effect that increases the namely its inability to account properly for superexchange- energy level separation. Because of the large gaps between type processes. This problem is also resolved with the one-electron orbital energies, the many-electron states presented MSSH model. involved in the Auger energy transfer are separated by the We demonstrate with two model systems (superexchange gaps of comparable magnitude.63) Thus, the conditions and double avoided crossing) that the approach leads to a favoring the emergence of superexchange are met most of good agreement with the exact quantum solutions. When the time, and it is expected that MSSH can work well. the MSSH algorithm is applied to model Auger relaxation To study the Auger relaxation dynamics, we compute the dynamics in a CdSe QD, it produces results that are in good population of the 1Se state, into which the photogenerated agreement with available experimental measurements. On the electron relaxes due to Auger-type energy exchange with the contrary, the dynamics generated with FSSH and GFSH is hole [Fig. 6(a)]. To obtain reliable statistics, we average the characterized by too slow relaxation time scales. Given the results over 500 stochastic realizations of the surface hopping conceptual and computational simplicity, the MSSH method algorithms and over 30 starting geometries. The starting and its CPA-based modification can be regarded as useful 094002-9 ©2015 The Physical Society of Japan J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by Univ of Southern California on 11/07/19

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