Decoherence Allows Model Reduction in Nonadiabatic Dynamics

Article pubs.acs.org/JPCA

Decoherence Allows Model Reduction in Nonadiabatic Dynamics Simulations Dhara J. Trivedi† and Oleg V. Prezhdo*,‡ †

Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, United States Department of Chemistry, University of Southern California, Los Angeles, California 90089, United States

‡

ABSTRACT: A nonadiabatic (NA) molecular dynamics (MD) simulation requires calculation of NA coupling matrix elements, the number of which scales as a square of the number of basis states. The basis size can be huge in studies of nanoscale materials, and calculation of the NA couplings can present a significant bottleneck. A quantum-classical approximation, NAMD overestimates coherence in the quantum, electronic subsystem, requiring decoherence correction. Generally, decoherence times decrease with increasing energy separation between pairs of states forming coherent superpositions. Since rapid decoherence stops quantum dynamics, one expects that decoherence-corrected NAMD can eliminate the need for calculation of NA couplings between energetically distant states, notably reducing the computational cost. Considering several types of dynamics in a semiconductor quantum dot, we demonstrate that indeed, decoherence allows one to reduce the number of needed NA coupling matrix elements. If the energy levels are spaced closer than 0.1 eV, one obtains good results while including only three nearest-neighbor couplings, and in some cases even with just the first nearest-neighbor coupling scheme. If the energy levels are spaced by about 0.4 eV, the nearest-neighbor model fails, while three or more nearest-neighbor schemes also provide good results. In comparison, the results of NAMD simulation without decoherence vary continuously with changes in the number of NA couplings. Thus, decoherence effects induced by coupling to a quantum-mechanical environment not only provide the physical mechanism for NAMD trajectory branding and improve the accuracy of NAMD simulations, but also afford significant computational savings. attention.26,27 The simulations of photoinduced dynamics in large molecular systems involving coupled electronic excited states require an “on-the-fly”28,29 calculation of energies, forces and nonadiabatic couplings (NACs). While the number of energies and forces is a linear function of the number of the electronic states involved in the dynamics, the number of the NACs scales quadratically. As the number of states increases, the NAC calculation start to pose a computational bottleneck. For example, multiple exciton generation even in small quantum dots (QDs) involves hundreds of thousands of double exciton states.30,31 Therefore, it is highly desirable to decrease the number of NAC matrix elements needed for accurate description of NAMD. Alternative schemes have been proposed in order to reduce the computational cost associated with evaluating the NAC terms.32 Although such schemes introduce error in the propagation of the electronic wave function, they help to reduce the computational cost. A recent study has shown that the standard SH technique requires evaluation of all or most NAC terms.27 It has long been recognized and appreciated33 that MQCD overestimates the magnitude of coherence effects in the

1. INTRODUCTION Mixed quantum-classical dynamics (MQCD) methodologies provide an intuitive approach to time-domain simulations of physical and chemical processes in condensed phases.1−9 Only a small subset of particles, such as electrons, holes and protons, are dealt with quantum mechanically in MQCD. The remaining parts of the system, e.g., heavy atoms, are treated classically. MQCD simulations of the nonadiabatic molecular dynamics (NAMD) provide an atomistic time-domain description of complex ultrafast photophysical and photochemical processes, including charge and energy transfer, and nonradiative relaxation.10−18 They generate a comprehensive perspective on the elastic and inelastic electron−phonon scattering dynamics of photoexcited charge carriers in nanoscale materials, directly mimicking numerous time-resolved experiments, and offering insights into the fundamental mechanisms underlying key processes in optics,19 photovoltaics,20,21 electronics,22,23 and related fields. A number of programs are capable of performing NAMD, employing the Ehrenfest24 and surface hopping (SH)25 methods. The stability of these methods with respect to changes in parameters and the number of trajectories has been extensively analyzed for simple models, while the robustness of the method applied to larger systems has received less © XXXX American Chemical Society

Received: June 19, 2015

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DOI: 10.1021/acs.jpca.5b05869 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A quantum subsystem. An equivalent subsystem coupled to a quantum-mechanical rather than classical environment loses coherence quite rapidly, typically on a femtosecond time scale.34−36 Decoherence effects within a subsystem can be captured properly by considering quantum dynamics of open systems, which extends the unitary evolution of the Schrödinger equation for the wave function into nonunitary stochastic Schrödinger equation,37 or, alternatively, which generalizes the quantum Liouville−von Neumann equation for the density matrix to a quantum Fokker−Planck equation.38 A classical environment in an MQCD simulation does not modify the Schrödinger or von Neumann equation, but rather provides a classical external field, and therefore, cannot properly describe loss of coherence. It has been shown that decoherence effects have a strong influence on the calculated time scales, as exemplified by the quantum Zeno effect in the limit of infinitely fast decoherence.39−41 Further, decoherence provides a physical mechanism for trajectory branching in MQCD, leading naturally to SH algorithms.37,42−47 Inclusion of decoherence strengthens quantum-classical correlations, in the context of trajectory surface hopping simulations, and can increase in the internal stability.48,49 Generally, decoherence effects become more important as the energy gaps between pairs of states forming coherent superpositions grow. Since decoherence slows down quantum dynamics in general, rapid decoherence between energetically distant states shuts off populations transfer, suggesting that NAC between such states can be ignored. Combined with the fact that the magnitude of NAC typically decreases with increasing state separation,50 decoherence effects can be used to reduce the number of NAC evaluations and to increase the computational efficiency of NAMD. In this paper, we investigate the possibility of reducing the number of NAC terms in a NAMD simulation that incorporated decoherence effects. We compare performance of several NAC reduction schemes within fewest switches surface hopping (FSSH), which is the most popular NAMD approach, and within decoherence induced surface hopping (DISH), which uses decoherence as a SH mechanism. Focusing on a CdSe QD, we consider three different models, including a dense manifold of quantum states, a sparse state manifold, and a combination of dense and sparse states. Confirming earlier results, we find that SH without decoherence requires full consideration of NAC matrix elements. At the same time, we show that indeed, decoherence effects allow one to construct reduced models of NAMD. In the case of a dense state manifold, it is already sufficient to include three nearestneighbor NACs. Even the nearest neighbor scheme can be quite accurate for some properties. As the energy gaps between quantum states grow, one requires a more complete NAC model. Overall, methods such as DISH, which include decoherence effects, can scale better with number of states than the standard approaches, such as FSSH. The demonstration that decoherence effects can be used to reduce the number of NAC matrix elements provides an important tool for improving the efficiency of NAMD simulations of large systems.

provides a rigorous description of quantum dynamics. Here, Ψ(r, R, t) is the full wave function, and Η̂(r, R, t) is the total Hamiltonian of the system. They depend on the electronic (r) and nuclear (R) coordinates. A quantum-classical approximation greatly reduces computational requirements on the solution of the TDSE by treating nuclei as classical particles. This leads to elimination of the terms in the Hamiltonian operator associated with the nuclear degrees of freedom and introduction of parametric dependence of the electronic Hamiltonian on classical nuclear coordinates. In the Born− Oppenheimer approximation, one can write the total wave function as a product of the electronic, ψ(r; R(t)), and nuclear, χ(R(t)), components, Ψ(r , R , t ) = ψ (r ; R(t ))χ (R(t ))

Nuclei evolve according to classical equations of motion in NAMD, and the classical paths are determined by a quantum force, which can be computed as an adiabatic Hellmann− Feynman force or mean-field Ehrenfest force. The electronic wave functions are obtained as solutions of the electronic TDSE, iℏ

∂Ψ(r , R , t ) = Η̂(r , R , t )Ψ(r , R , t ) ∂t

∂ψ (r ; R(t )) = Η̂el(r ; R(t ))ψ (r ; R(t )) ∂t

(3)

where Η̂el is the electronic Hamiltonian of the system. It can be further decomposed into 2

ℏ Η̂el = − 2

1 2 ∇i + Vel(r ; R(t )) mi

∑ i

(4)

Here, Vel(r; R(t)) includes the electron−nuclear, electron− electron, and nuclear−nuclear interactions. It is evaluated for given nuclear positions, neglecting nuclear quantum-effects. The first term in eq 4 is the electron kinetic energy. The nuclear kinetic energy is ignored, as justified by the small electron-to-nucleus mass ratio. The electronic wave function can be expanded on the basis of adiabatic wave functions, ψ (r ; R(t )) =

∑ ci(t )φi(r ; R(t ))

(5)

i

obtained as eigenstates of the electronic Hamiltonian. A wide variety of methods have been developed to calculate the electronic energies, densities, and wave functions, while balancing the desired accuracy with computational cost. The less expensive methods range from semiempirical to various formulations of density functional theory (DFT).51−58 We employ ab initio DFT and its time-dependent generalization.59 The adiabatic Kohn-sham (KS) orbitals, obtained from timeindependent DFT are used as the basis in the time-dependent DFT (TDDFT) simulation.57 Substituting a basis expansion into the TDSE results in a set of coupled equations of the expansion coefficients: iℏ

2. THEORETICAL BACKGROUND The time-dependent Schrödinger equation (TDSE), ıℏ

(2)

∂ci(t ) = ∂t

∑ cj(t )(ϵjδij − iℏd ij·Ṙ ) j

(6)

Here, ϵj is the energy of the adiabatic KS orbital, and dij Ṙ is the NAC. The NAC is calculated numerically10 as the overlap of the adiabatic KS orbitals φi and φj at sequential MD time steps Δt:

(1) B


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dR = dt

φi

DISH, as compared to decoherence corrected FSSH, is that branching appears directly due to decoherence rather than with the help of an ad hoc algorithm, which has to be corrected for decoherence effects. The original idea of decoherence-driven trajectory branching was implemented within the stochastic mean-field approach.37 DISH extends the TDSE through a stochastic adaption of the time-evolution operator of quantum system coupled to a quantum environment, and provides a mechanism for surface hopping that models diverging evolutions.42 The calculations were performed with the Vienna Ab initio Simulation Package (VASP).75 The PBE functional76 with projector-augmented-wave (PAW) pseudopotentials77 were employed in a converged plane wave basis. The simulations were performed in a periodically replicated cubic cell with at least 8 A of vacuum between the QD replicas. Following geometry optimization at 0K, we heated the QD to 300 K with repeated velocity rescaling. Then, we generated a 3 ps microcanonical MD trajectory using the Verlet algorithm with the 1 fs time step and Hellman-Feynman forces. At each time step, we computed NACs. Along with KS orbital energies, this time-dependent information was stored and used to perform NAMD within the recently developed PYthon eXtension for Ab Initio Dynamics (PYXAID) program.72,78

∂ φ ∂t j

1 (⟨φi(t )|φj(t + Δt )⟩ − ⟨φi(t + Δt )|φj(t )⟩) 2Δt

(7)

The NAC describes electron−phonon interactions that arise from dependence of the adiabatic KS orbitals on the nuclei coordinates, R(t). The above equations establish evolution of the electrons subject to the classical field of nuclei. Defining the evolution of the nuclei constitutes the so-called quantum backreaction problem, which prescribes how the quantum subsystem influences the classical subsystem.60−67 NAMD provides a generalization of ordinary molecular dynamics to include transitions between electronic states. The branching of the overall nuclear wave function into wavepackets, correlated with different electronic states, is a quantum mechanical phenomenon. Correlations between the nuclear motion and electronic states are built into MQCD using SH techniques.25,42,68,69 SH can be interpreted as a master equation in which transition rates are nonperturbative and evolve in time. FSSH minimizes the number of surface hops by prescribing a probability for hopping between the electronic states that is based on the change in the electronic state populations. The probability of hopping from state i to j during the time step Δt is given in FSSH by,25 ⎧ bji ·Δt ⎫ , 0⎬ gij(t , Δt ) = max⎨ ⎩ aii(t ) ⎭

3. RESULTS AND DISCUSSION The study is performed with the Cd33Se33 “magic” size cluster. Known experimentally,79 the cluster is 1.3 nm in diameter and is one of the smallest stable CdSe QDs that support a crystalline-like core.80,81 These properties make Cd33Se33 an excellent model for quantum-mechanical and quantumdynamical studies of semiconductor QDs. While smaller stable clusters with bulk topology are known, for instance Pb16Se16,82−84 the properties of CdSe are more representative of semiconductor QDs. For instance, CdSe has a significantly higher density of valence band (VB) than conduction band (CB) states, while the band structure of PbSe is unusually symmetrical. The density of states (DOS) of Cd33Se33 is shown in Figure 1. The optimized structure of the system is given as insert. Surface reconstruction results in saturation of unsaturated valencies and removal of associated localized surface states. Consequently, this opens the energy gap between the highest occupied (HOMO) and lowest unoccupied (LUMO) Kohn−

(8) −1

where aji(t) = cj*(t)ci(t) and bji = 2ℏ Im(aji⟨φj|H|φi⟩) − 2Re(ajidji). A hop from state i to state j can occur only if the electronic occupation of state i decreases and the electronic occupation of state j increases, minimizing the number of hops. By contrast, the original SH technique70 uses the standard quantum mechanical probabilities to hop, and allows transitions even if state populations do not change. The total electron−nuclear energy is conserved by velocity rescaling,10,25 normally performed in the direction of the NAC vector, dji. If the velocity component is too small in this direction to reflect the increase in the electronic energy, the hop is rejected. This phenomenon leads to the detailed balance between upward and downward transitions.71 In order to further simplify the implementation of FSSH, it is assumed that the energy exchanged between the electronic and vibrational degrees of freedom during a hop is rapidly redistributed among all of the vibrational modes. The velocity rescaling and hop rejection procedure is replaced by multiplying the probability for transitions upward in energy by the Boltzmann factor.13,72 This allows for the use of a ground state nuclear trajectory to determine the time-dependent potential that drives the electronic dynamics. This simplification of the original FSSH technique provides great computational savings, eliminating the need to run multiple excited state trajectories. By treating nuclei classically, the original FSSH scheme excludes coherence loss that occurs if the electronic subsystem is coupled to quantum nuclei.73,74 Decoherence can be viewed as an environment induced destruction of quantum state superpositions. The effect of decoherence on the evolution of quantum subsystem is implemented here using the DISH approach.42 Originally, Rossky and co-workers introduced the decoherence correction to FSSH63 by working with the Liouville-von Neumann equation. A conceptual advantage of

Figure 1. Density of states in Cd33Se33. The optimized geometry is shown in the inset. C


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partial coupling (PC) schemes. PC1 includes NAC only between nearest neighbor states. PC3 and PC5 include three and five nearest-neighbor NACs, respectively. The CC, PC1, PC3 and PC5 NAC schemes are applied to the full intraband relaxation, dense manifold, and sparse manifold models (Figure 2) to obtain the rates of decay of the initial electronic state population and electronic energy (Figures 3−5). Figure 6 reports values of NACs, while Figure 7

Sham orbitals of the CdSe cluster. The calculated zero temperature energy gap is about 1.6 eV. This value is underestimateda typical feature of most semilocal DFT functionals, such as PBE used in our calculations. The problem is attributed to incomplete elimination of the self-interaction energy by these functionals. Nonlocal corrections are routinely used as remedy to the problem, for instance, in hybrid functionals that include a portion of the exact Hartree−Fock exchange.81 However, hybrid functionals require significant additional computational efforts. The VB DOS is higher than the CB DOS in agreement with the effective-mass theory,85 which uses a higher effective mass for holes than for electrons. The LUMO is well separated energetically from the rest of the CB manifold, giving rise to the so-called phonon bottleneck to the electron relaxation, observed experimentally,86 and rationalized by ab initio time-domain simulation.41 We focus on intraband relaxation of the excited electron in the CdSe QD and consider three scenarios, Figure 2. First, we

Figure 3. Decay of (a) population and (b) energy of the initially excited state for complete intraband relaxation at 300 K, obtained with FSSH and DISH using complete coupling (CC), coupling with the first five nearest neighbors (PC5), coupling with the first three nearest neighbors (PC3), and coupling with nearest neighbors (PC1). Figure 2. Schematics of (a) complete intranband relaxation, (b) dense manifold relaxation, and (c) sparse manifold relaxation via nonadiabatic vibronic dynamics. Schemes (a) and (b) employ 15 excited states, from LUMO to LUMO+14, spanning about 1.3 eV of energy. Scheme (c) has 7 excited states distributed evenly in energy with gaps around 0.4 eV, spanning 2.3 eV of energy. The LUMO to LUMO+1 gap is close to 0.4 eV (Figure 1).

study the relaxation involving all CB states. Second, we restrict the dynamics to the dense state manifold, excluding the LUMO, which is separated energetically from the rest of the CB, Figure 1. Finally, we construct a sparse manifold model by retaining few CB orbitals such that they are separated by approximately 0.4 eV, which is the gap between LUMO and LUMO+1. The dense manifold model provides a good representation of excitation dynamics in nanoscale materials, such as QDs. The sparse manifold model is typical of molecular systems, which exhibit large energy gaps between electronic states. We report two sets of data. On one hand, we present timedependent population of the initially excited state. Such data correspond to pump−probe experiments that trace the signal at a fixed photoexcitation energy, for instance, by monitoring signal bleach and recovery. On the other hand, we show the evolution of the excited electron energy, which decreases due to coupling to vibrations. Energy losses are detrimental to many applications, in which nonradiative relaxation competes with productive processes, such as charge separation. The ratio of the rates of energy losses and productive processes determines the efficiency of a device. In order to investigate the possibility of reducing the number of NAC matrix elements needed to obtain reliable results, we compare the complete coupling (CC) description with three

Figure 4. Same as Figure 3, for the dense state manifold defined in Figure 2.

characterizes pure-dephasing times. Tables 1 and 2 present the time scales of decay of the initial state population and energy, respectively. The time scales are obtained by exponential fits of the data shown in Figures 3−5. The results are obtained with both the decoherence free FSSH approach,25 and the DISH method42 that uses decoherence as the physical mechanism of SH. Note that including decoherence in DISH slows down the dynamics relative to decoherence free FSSH, as should be expected.35,36,78 As demonstrated by the data, DISH allows significant reduction in the number of needed NACs, while the D


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Figure 5. Same as Figure 3, for the sparse state manifold defined in Figure 2.

Figure 7. Same as Figure 6, for the pure-dephasing (decoherence) time.

Table 1. Time Scales (ps) of Relaxation of Population of the Initially Excited Statea model

intraband relaxation

dense manifold

sparse manifold

FSSH_CC FSSH_PC5 FSSH_PC3 FSSH_PC1 DISH_CC DISH_PC5 DISH_PC3 DISH_PC1

0.33 0.40 0.46 0.79 0.97 1.03 1.08 1.20

0.33 0.40 0.45 0.80 0.98 1.04 1.09 1.19

2.58 2.65 2.91 9.80 26.0 27.2 28.8 55.6

a

The data are presented for NAMD with (DISH) and without (FSSH) decoherence. CC stands for complete coupling. PCN refers to partial coupling schemes, with N denoting the number of nearest neighbor couplings. Intraband, dense and sparse manifold relaxations are illustrated in Figure 2.

Table 2. Time Scales (ps) of Relaxation of the Excess Electronic Energya

Figure 6. Canonically averaged root-mean-squared value of the NAC for each state involved in complete intraband relaxation for different coupling schemes: complete coupling (CC), coupling with first five nearest neighbors (PC5), coupling with first three nearest neighbors (PC3), and coupling with nearest neighbor (PC1). The NAC increases with orbital energy and is largest between nearest neighbor states.

FSSH results are much more sensitive to reduction of the number of NAC matrix elements. Focusing on the numerical values presented in Tables 1 and 2, we observe that FSSH shows much more variation with the number of NAC matrix elements than DISH. The variation is larger for the energy relaxation (Table 2) than for the decay of the initial state population (Table 1). For instance, the FSSH decay of the initial state population in the dense manifold model changes from 0.33 ps for the CC scheme to 0.80 ps for PC1, almost by a factor of 3. The corresponding variation is insignificant in DISH, 0.98 and 1.19 ps, respectively. With both FSSH and DISH, the initial state population in the sparse manifold model show little variance for the CC, PC5, and PC3 schemes, while PC1 gives notably different results. The relaxation of energy in the dense manifold model differs between the CC and PC1 schemes by an order of magnitude in FSSH, 1.28 ps vs 12.7 ps. The corresponding difference in

a

model

intraband relaxation

dense manifold

sparse manifold

FSSH_CC FSSH_PC5 FSSH_PC3 FSSH_PC1 DISH_CC DISH_PC5 DISH_PC3 DISH_PC1

0.76 2.32 4.61 12.2 10.5 18.7 22.3 30.5

1.28 2.56 4.71 12.7 11.9 20.0 24.3 32.4

4.94 5.77 7.00 54.7 88.5 83.2 95.6 234

The notation is the same as in Table 1.

DISH is only a factor of 3: 11.9 ps versus 32.4 ps. Similarly to the initial state population decay, the energy relaxation times agree between the CC, PC5, and PC3 schemes applied to the sparse manifold model, while the PC1 scheme shows significant deviations, with both FSSH and DISH. Just as with the dense manifold, here the difference between CC and PC1 is a factor of 10 in FSSH, and a factor of 3 in DISH. The reported results lead to the overall conclusion that reduction in the number of NAC matrix elements is much more justified in DISH than FSSH. It is dangerous to apply the nearest neighbor scheme PC1 with sparse states manifold, since E


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decoherence time can be understood qualitatively. The NAC depends on overlap of wave functions at different timesteps. Such overlap generally decreases for wave functions with a larger difference in the number of nodes. The pure-dephasing (decoherence) time depends on fluctuation in the energy gap between a given pair of states. More distant states have larger energy gaps, and typically, larger gap fluctuations. Because quantum transition rates are more sensitive to the coherence time than the NAC, methods including decoherence effects, such as DISH, are more suitable for reduction of NAC to a few nearest neighbors.

one may miss an important NAC matrix element. Including several nearest neighbors gives good results, especially with DISH. The energy relaxation time scale is much more sensitive to elimination of distant NAC matrix elements than decay of the initial state population. For practical purposes, one can employ the PC3 scheme with DISH and expect to obtain results that differ from the CC scheme by a factor of 2 or less. The possibility of model reduction in NAMD arises due to the general properties of NAC matrix elements and decoherence times. Figure 6 shows root-mean-squared values of the NAC between a given state and all other states in a particular scheme, averaged over the MD trajectory. Similarly, Figure 7 shows pure-dephasing (decoherence) times between a given state and all other states involved in the four coupling schemes. NAC grows with energy. More importantly, NACs between nearest neighbors are significantly larger than NACs between distant states. This fact forms the basis for model reduction within FSSH. The behavior of pure-dephasing times gives additional motivation of model reduction within DISH. It is important to note that the frequency of quantum transitions depends on the decoherence time much more strongly than on the NAC. As follows from Fermi’s golden rule,87,88 the transition rate is a quadratic function of NAC and a Gaussian function of the decoherence time. Just as NAC, the puredephasing time is larger for nearest neighbor states than for distant states. Longer coherence favors quantum transitions.50 As a result, the nearest neighbor pathways dominate the dynamics much more strongly in DISH than FSSH. Qualitatively, once can argue that NACs are larger for nearest neighbor states because these states have more “similar” wave functions, resulting in larger overlaps in the NAC expression (eq 7). For instance, distant wave functions have very different nodal structure. A shift in the nodes induced by a nuclear displacement will give both positive and negative contributions, which will cancel each other. Decoherence time decreases with increasing fluctuation of the energy gap between a given pair of states.34 It is natural to expect that larger energy gaps between more distant states also have larger fluctuations.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support of the U.S. National Science Foundation, Grant CHE-1300118, is gratefully acknowledged.

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REFERENCES

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4. CONCLUSIONS To recapitulate, we have demonstrated that decoherence effects, which improve accuracy of MQCD and provide the physical basis for trajectory branching, also help to reduce the computational effort. In particular, a major computational bottleneck in NAMD of large systems, computation of NAC matrix elements, which scales as the square of the number of states, can be reduced to a few nearest neighbors, provided that a NAMD includes decoherence. Denser state manifolds allow one to consider fewer nearest neighbors. This is a particularly favorable feature, since dynamics involving denser state manifolds require more basis states. It is not advisable to restrict the calculation to just the first nearest neighbor, since one may easily miss an important dynamical pathway. Investigation of electron−phonon energy relaxation over a broad energy range requires a more complete NAC scheme than study of decay of the initial state population. Both NAC and decoherence time exhibit the properties needed for reduction of the computational effort. They both decrease for pairs of more distant states. Larger NAC and longer coherence times accelerate quantum dynamics in general. Therefore, decrease in the NAC and coherence time with state separation reduces the importance of quantum transitions between distant states. This feature of the NAC and F


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Decoherence Allows Model Reduction in Nonadiabatic Dynamics

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