J. Chem. Phys. 150, 204124 (2019); https://doi.org/10.1063/1.5095810 150, 204124

© 2019 Author(s).

Ehrenfest and classical path dynamics withdecoherence and detailed balanceCite as: J. Chem. Phys. 150, 204124 (2019); https://doi.org/10.1063/1.5095810Submitted: 13 March 2019 . Accepted: 09 May 2019 . Published Online: 31 May 2019

Parmeet Nijjar, Joanna Jankowska , and Oleg V. Prezhdo

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Paper published as part of the special topic on Dynamics of Open Quantum Systems

Note: This paper is part of a JCP Special Topic on Dynamics of Open Quantum Systems.

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Ehrenfest and classical path dynamicswith decoherence and detailed balance

Cite as: J. Chem. Phys. 150, 204124 (2019); doi: 10.1063/1.5095810Submitted: 13 March 2019 • Accepted: 9 May 2019 •Published Online: 31 May 2019

Parmeet Nijjar,1 Joanna Jankowska,1,2 and Oleg V. Prezhdo1,a)

AFFILIATIONS1Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA2Faculty of Chemistry, University of Warsaw, Warsaw, 02-093, Poland

Note: This paper is part of a JCP Special Topic on Dynamics of Open Quantum Systems.a)Author to whom correspondence should be addressed: prezhdo@usc.edu

ABSTRACTWe present a semiclassical approach for nonadiabatic molecular dynamics based on the Ehrenfest method with corrections for decoher-ence and detailed balance. Decoherence is described via a coherence penalty functional that drives dynamics away from regions in Hilbertspace characterized by large values of coherences. Detailed balance is incorporated by modification of the off-diagonal matrix elements witha quantum correction factor used in semiclassical approximations to quantum time-correlation functions. Both decoherence and detailedbalance corrections introduce nonlinear terms to the Schrödinger equation. At the same time, the simplicity of fully deterministic dynamicsand a single trajectory for each initial condition is preserved. In contrast, surface hopping is stochastic and requires averaging over multi-ple realization of the stochastic process for each initial condition. The Ehrenfest-decoherence-detailed-balance (Ehrenfest-DDB) method isadapted to the classical path approximation and ab initio time-dependent density functional theory and applied to an experimentally studiednanoscale system consisting of a fluorophore molecule and an scanning tunneling microscopy tip and undergoing current-induced chargeinjection, cooling, and recombination. Ehrenfest-DDB produces time scales that are similar to those obtained with decoherence induced sur-face hopping, which is a popular nonadiabatic molecular dynamics technique applied to condensed matter. At long times, Ehrenfest-DDBdynamics slows down considerably because the detailed balance correction makes off-diagonal elements go to zero on approach to Boltzmannequilibrium. The Ehrenfest-DDB technique provides efficient means to study quantum dynamics in large systems.

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I. INTRODUCTION

Quantum nonadiabatic (NA) processes, such as electron andproton transport,1 exciton relaxation,2,3 dissociation,4,5 recombina-tion,6,7 and energy transfer,8–11 are widespread in nature.12–16 Apurely quantum description of large systems, typically involved insuch processes, is too complex and time consuming, and thus,approximate semiclassical treatments are unavoidable. Such phe-nomena can generally be modeled by representing dynamics of afew key quantum particles coupled to classical degrees of freedom.Mixed quantum-classical dynamics (MQCD) methods, in whichelectrons are treated at the quantum level and nuclei are treated clas-sically, have become the most appropriate approach to model theseprocesses.

Fewest switches surface hopping (FSSH)17 is one of the mostpopular NA molecular dynamics (MD) techniques. In this approach,

the dynamics of classical particles evolving on a potential energy sur-face is described by Newtonian equations, while quantum particlesare propagated according to the time-dependent (TD) Schrödingerequation (SE). NA transitions (hops) among coupled electronicstates incorporate feedback between electronic and nuclear subsys-tems. The FSSH probabilities are designed to minimize the numberof state switches. Once a surface hop is assigned, the nuclear veloci-ties are adjusted in the direction of the NA coupling vector so as toconserve the total energy of the system. If the nuclear kinetic energyis less than the energy required for the quantum transition, the hop isrejected. Such velocity rescaling and hop rejection produce detailedbalance between the upward and downward transitions, approxi-mately reproducing the Boltzmann distribution of the excited statepopulations.18,19 An ensemble of surface hopping trajectories is sim-ulated independently, and the population of each quantum state isgiven by the fraction of trajectories that are assigned to that state.

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In the absence of velocity rescaling, FSSH probabilities satisfy inter-nal consistency, i.e., the fraction of trajectories on a given state atany time equals the quantum population of that state obtained bythe time-dependent Schrödinger equation. Individual trajectoriesalways propagate on a pure state (eigenstate) of the system. Owing tothese features, the FSSH method has been successful in many appli-cations, making it immensely popular for modeling NA processes.Its conceptual simplicity and ease of implementation have helpedFSSH retain its popularity.

The popularity of FSSH has also brought forward its limita-tions, as demonstrated by the numerous types of corrections andmodifications to the standard FSSH algorithm. A detailed discus-sion of such methods can be found in Ref. 20. Since FSSH trajecto-ries hop between potential energy surfaces, it is not straightforwardto apply FSSH to systems with a continuum of states, and a dis-cretization of the continuum is required for FSSH dynamics.21 Inlarge scale systems, such as extended polyatomic molecules, molec-ular crystals, or assemblies of quantum dots, there may exist elec-tronic states localized in different parts of a system and coupledvery weakly.17 Such noninteracting states may have similar energyand cross during dynamics, leading to trivial crossings in whichthe NA couplings behave as delta functions of time. Very smalltime steps are required for identification of such crossings, whichis extremely time-consuming.22 Furthermore, FSSH is known to givemore accurate results in the adiabatic representation rather than dia-batic representation because the velocity adjustment process differsstrongly in the two representations. The representation dependenceof the results may be viewed as an undesirable property.23 All MQCDmethods are intrinsically limited by the classical mechanical descrip-tion of the nuclear motion. Many nuclear quantum effects, such astunneling and zero-point vibrational energy that are important forlow temperature dynamics, cannot be easily described by classicaltrajectories.24,25 Last but not least, loss of quantum coherence in theelectronic subsystem induced by coupling to nuclear motions has tobe addressed.26–33

Ehrenfest dynamics is another primary MQCD method. Here,classical particles evolve on a mean potential energy surface whichis averaged over all quantum states and weighted by the correspond-ing populations. Ehrenfest dynamics is fully deterministic and hencemore time-efficient than FSSH which is based on a stochastic algo-rithm and requires averaging over multiple realizations of the ran-dom process to exhibit good statistics. Additional sampling may beneeded for rare events.29 Although the Ehrenfest technique is simpleand easy to implement, the mean-field approximation is only ade-quate either when the quantum and classical subsystems are weaklycoupled to each other or when the classical subsystem reacts sim-ilarly to all quantum states included in the active space. Anotherserious limitation of the Ehrenfest approach is its inability to collapsethe system’s wavefunction on a pure state. After passing through aregion of strong electronic coupling, the trajectory evolves with awavefunction that is a combination of nearby adiabatic eigenfunc-tions even in regions of negligible coupling between the states.30

However, this deficiency of the Ehrenfest method can be resolved ifthe results of the Ehrenfest dynamics are interpreted as the averagevalues of the properties and their distributions. Moreover, by treat-ing vibrational motions classically, both the Ehrenfest MD and theFSSH techniques maintain coherence in the quantum (electronic)subsystem and do not account for decoherence that occurs when

the quantum subsystem is coupled to a quantum vibrational bath.31

Many decoherence correction schemes for FSSH as well as Ehren-fest dynamics have been proposed.32,33 Additionally, the standardEhrenfest method does not satisfy detailed balance between tran-sitions downward and upward in energy, and therefore, it cannotgenerate thermodynamic equilibrium between quantum and classi-cal subsystems. In particular, the Ehrenfest method does not pro-duce the Boltzmann equilibrium populations of the quantum statesat a finite temperature.3,34 A proper theory that accounts for deco-herence and detailed balance is of fundamental importance to accu-rately simulate processes happening in complex condensed mattersystems.

In this work, we propose a modified Ehrenfest method withcorrections to account for both decoherence and detailed balance.Decoherence is introduced via the coherence penalty functional(CPF) method developed by Akimov et al.33 The CPF methoddynamically penalizes development of coherences during the evo-lution of quantum degrees of freedom via an additional term inthe classically mapped Hamiltonian, thus preserving the overallHamiltonian structure of the equations of motion. The detailedbalance correction is applied to the coupling matrix of the Hamil-tonian, as proposed by Bastida et al.35 for simulating vibrationalenergy transfer between quantum systems and classical baths.We have introduced the two corrections to the Ehrenfest tech-nique, applied the classical path approximation, implemented theapproach within the PYXAID software package,36,37 and studiedNA dynamics in a hybrid molecule/metal-tip system to analyze theperformance.

II. THEORY: FORMULATION OF THE EHRENFESTMETHOD WITH DECOHERENCE AND DETAILEDBALANCE CORRECTIONS

We begin with a derivation of the Ehrenfest technique. Then,we present the decoherence and detailed balance corrections tothe Ehrenfest technique and formulate equations of motions thatinclude both corrections. After that, we describe the optical responsetheory formalism to compute the decoherence rates and providedetails of the simulation of charge injection, relaxation, and recom-bination in a molecule/metal-tip system studied experimentally38,39

by scanning tunneling microscopy (STM).

A. The Ehrenfest methodConsider the time-dependent (TD) Schrödinger equation (SE)

ih∂Ψ(r3n, R3N , t)

∂t= H(r3n, R3N , t)Ψ(r3n, R3N , t), (1)

where r3n and R3N are coordinates of n electrons and N nuclei,respectively. For simplicity of notation, the electronic and nucleardegrees of freedom will be denoted by r and R, respectively. Thenonrelativistic molecular Hamiltonian is defined as

H(r,R, t) = −N∑J=1

12MJ

∇2J −

n∑j=1

12∇

2i +∑

J<K

ZJZK

∣Ð→RJ −

Ð→RK ∣

+∑j<k

1∣Ð→rj −Ð→rk ∣

−∑J,j

ZJ

∣Ð→RJ −

Ð→rj ∣, (2)

J. Chem. Phys. 150, 204124 (2019); doi: 10.1063/1.5095810 150, 204124-2

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where MJ and ZJ are the mass and the charge of the Jth nucleus,respectively. Combining all the terms except the nuclear kineticenergy term defines the electronic Hamiltonian operator

H(r,R, t) = −N∑J=1

12MJ

∇2J + Hel(r,R, t). (3)

Next, consider the Born-Oppenheimer separation ansatz forthe wavefunction Ψ(r, R, t). The total wavefunction Ψ(r, R, t) isdescribed by a linear combination of products of electronic wave-functions for different adiabatic states, Φi(r;R(t)), and nuclear wave-functions, χi(t,R(t)), associated with the electronic states

Ψ(r,R, t) =∑iχi(t,R(t))Φi(r;R(t)). (4)

The adiabatic Born-Oppenheimer representation is based on the factthat the mass of the nuclei is much larger than the mass of theelectrons, and the electrons move rapidly in the nuclear field.

Using the representation of the total wavefunction given inEq. (4), inserting it into the TD-SE, Eq. (1), and projecting it onto thecorresponding electronic states, one obtains the dynamics of nuclearwavepackets correlated with different electronic states

ih∂χi(t,R(t))

∂t=∑

j

⎧⎪⎪⎨⎪⎪⎩

(Tnucl + Ej(t,R))δij

− h2 d(1)ij

M∇ − h2 d

(2)ij

2M

⎫⎪⎪⎬⎪⎪⎭

χi(t,R(t)), (5)

where d(n)ij are the n-th order nonadiabatic couplings between elec-tronic states i and j, defined as

d(1)ij = ⟨Φi(r;R(t))|∇Φj(r;R(t))⟩, (6a)

d(2)ij = ⟨Φi(r;R(t))|∇2Φj(r;R(t))⟩. (6b)

The quantities Ei(t,R) are the adiabatic potential energy surfaces,corresponding to the electronic states Φi(r;R(t)). They are the eigen-values of the stationary electronic SE, parametrically dependent onthe nuclear coordinates R(t),

Hel(r, t;R)Φi(r, t;R(t)) = Ei(t,R)Φi(r, t;R(t)). (7)

The diagonal parts of the term −h2 d(2)ij

2M in Eq. (5) can be consideredas a correction to the adiabatic energies

Ei(R) = Ei(R) − h2 d(2)ij

2M. (8)

The off-diagonal terms representing the second order NA couplingbetween electronic states are often neglected under the assump-tions that adiabatic wavefunctions are slowly varying with respect tonuclear coordinates.40–42 However, these terms may become impor-tant in some cases, and one must be careful when using this ansatz.37

After applying these assumptions, Eq. (5) simplifies to

ih∂χi(t,R(t))

∂t=∑

j

⎧⎪⎪⎨⎪⎪⎩

(Tnucl + Ej(t,R))δij − h2 d(1)ij

M∇

⎫⎪⎪⎬⎪⎪⎭

χj(t,R(t)).

(9)

The next approximation involves treating the nuclear degreesof freedom classically, using the following standard quantum-classical correspondence rules:

Tnucl ≡ −h2

2M∇

2→

P2

2M, (10)

−h2 d(1)ij

M∇→ −ihd(1)ij

PM

. (11)

The solution of the TD-SE is expressed on a finite set of adiabaticbasis functions defined as the eigenfunctions of the stationary SE[Eq. (7)], {Φi}, i = 0, . . ., Nb − 1, where Nb is the basis size

Ψ(r,R, t) =∑Nb−1i=0 ci(t)Φi(r;R(t)). (12)

The notation Φi(r; R(t)) indicates functional dependence of the elec-tronic wavefunctions, Φi, on the electronic coordinates, r, and itsparametric dependence on the nuclear coordinates, R. It should benoted that with this definition, there exists no explicit functionaldependence of Ψ(r, R, t) on the nuclear positions R.

Using the ansatz [Eq. (12) together with Eqs. (10) and (11)],one converts the TD-SE [Eq. (9)] into the equations of motion forthe amplitudes, ci(t),

ih∂ci∂t

=∑Nb−1j=0 (Eiδij − ihd(1)ij

PM

)cj. (13)

The solution of Eq. (13) describes the evolution of the electronicdegrees of freedom. The first term represents the evolution of thecoefficient of an eigenfunction of the Hamiltonian, and the secondterm couples the electronic states with each other via the nuclearkinetic energy and the nonadiabatic coupling.

In the Ehrenfest method, the nuclei move on the averageelectronic potential energy surface, subject to the mean-field force43

Fmf = −⟨Ψ(r,R, t)|∇RHel(r, t;R)∣Ψ(r,R, t)⟩

= −⟨∑i ci(t)Φi(r;R(t))|∇RHel(r, t;R)|∑j cj(t)Φj(r;R(t))⟩

= ⟨∑i c∗

i ciΦi|∇RHel|Φi⟩ − ⟨∑ i, jj≠i

c∗i cjΦi∣∇RHel∣Φj⟩

=∑i∣ci∣2Fi +∑

i,jc∗i cj(Ei − Ej)d(1)ij ,

(14)where

Fi = −⟨Φi∣dHel

dR∣Φi⟩ (15)

is the Hellmann-Feynman force corresponding to the adiabatic elec-tronic state Φi. The mean-field force is computed on the basis ofthe contribution of each adiabatic state to the time-dependent elec-tronic wavefunction. The first term couples the populations of theelectronic states |ci|2 to the nuclear trajectory, while the secondterm with the coefficients c∗i ci includes interferences between thesestates. The system of coupled differential equations (13) and (14)constitutes the Ehrenfest method. The trajectories generated in theEhrenfest dynamics are fully deterministic and continuous.

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B. Coherence penalty functionalAs noted above, the Ehrenfest dynamics does not account for

decoherence that would be induced within the electronic subsystemif the nuclei were treated quantum mechanically. Following the workof Akimov, Long, and Prezhdo,33 we add the decoherence correctionto the dynamics via the CPF method as follows.

In the variables qi = Re(ci) and pi = Im(ci), the semiclassicalTD-SE,

ih∂ci∂t

=∑j[Ei(R(t))δi,j − ih

PM

d(1)ij ]cj(t), (16)

becomes equivalent to the following Hamiltonian equations ofmotion:

pi = −∂H∂qi

, (17a)

qi =∂H∂pi

, (17b)

with the Hamiltonian given by

H =∑i

Ei2h

(q2i + p2

i ) −PM∑i,j

d(1)ij piqj. (18)

The following augmented Hamiltonian to account for decoherenceeffects has been proposed:

H = H +∑i, ji≠j

λij(q2i + p2

i )(q2j + p2

j ), (19)

where the term (q2i + p2

i )(q2j + p2

j ) = ∣c∗i cj∣2 is simply a square of the

magnitude of coherence between the pair of states i and j. This termpenalizes coherence development between pairs of states. The con-stant λij determines the magnitude of the penalty for each pair ofstates. It is interpreted as the decoherence rate and is calculated asdescribed in Sec. II D. Each term (q2

i + p2i )(q2

j + p2j ) creates an addi-

tional potential that biases system’s dynamics to choose its evolutionpathways such that they minimize the developed coherences (pro-vided λij > 0). The sum in the right-hand side of Eq. (19) is called theCPF.

When the system accumulates coherence between the pairof states i and j, the point in the phase space of the dynami-cal system is located uphill the bias potential. The gradient of theeffective energy surface acts in the opposite direction and movesthe phase space point away from the regions of large coherences.In this way, the system avoids developing large coherences. Suchavoidance of the regions with large coherences in turn affectsthe average populations of the electronic states as functions oftime, by changing the rates of transitions between them. Fur-ther details and illustrations of the CPF method can be found inRef. 33.

C. Detailed balance and thermodynamic equilibriumNext, the detailed balance correction is applied based on the

work of Bastida et al.35 They show that the coupling matrix elementsbetween the basis states modified with the quantum correctionfactor,2

−ihRdqckj = −ihRdkj(2

1 + ehωkj/KT)

12, (20)

result in transition rates that obey the detailed balance condition.Here, d(1)ij is represented by dij.

In order to maintain the symmetry of the coupling matrix,keeping the overall Hamiltonian Hermitian, the following modifi-cation is made:

dsqcjk = dsqckj = ρkdqcjk − ρjd

qckj , k > j, (21a)

dsqcjj = djj, j, k = 1, . . . ,N�, (21b)

where ρk = |ck| is the absolute value of the wavefunction coefficient.35

Overall, the following scaling for the off-diagonal Hamiltonianmatrix elements was proposed:

Hij = Hij ⋅ ρj

¿ÁÁÁÀ

2

1 + exp( ∆EkBT

)−Hji ⋅ ρi

¿ÁÁÁÀ

2

1 + exp(−∆EkBT)

, (22)

where ∆E = Ei − Ej.The Hamiltonian considered in the original approach was real-

valued.35 In contrast, the NA coupling is imaginary. In such a case,hermiticity requires that Hij = −Hji. In order to maintain the NAHamiltonian Hermitian, we modified the original scaling to thefollowing form:

Hij → Hij : Im[Hij]

= Im[Hij] ⋅

RRRRRRRRRRRRRRR

ρj

¿ÁÁÁÀ

2

1 + exp( ∆EkBT

)− ρi

¿ÁÁÁÀ

2

1 + exp(−∆EkBT)

RRRRRRRRRRRRRRR

, (23a)

Re[Hij] = Re[Hij]. (23b)

Such scaling preserves the Hermitian character of the Hamilto-nian and ensures that its off-diagonal elements vanish when statepopulations reach the Boltzmann distribution.

The final form of the Hamiltonian, Hij, for the proposed Ehren-fest method with the decoherence and detailed balance correctionsis given by

Im[Hij] = Im[Hij] ⋅

RRRRRRRRRRRRRRR

ρj

¿ÁÁÁÀ

2

1 + exp( ∆EkBT

)− ρi

¿ÁÁÁÀ

2

1 + exp(−∆EkBT)

RRRRRRRRRRRRRRR

,

(24a)

Re[Hij] = Re[Hij] +∑i, ji≠j

λij(q2i + p2

i )(q2j + p2

j ). (24b)

The resulting method combining both corrections is calledEhrenfest-DDB (decoherence detailed balance).

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D. The classical path approximationThe original Ehrenfest approach and Ehrenfest-DBB require

evaluation of the force that contains contributions from manyelectronic states [Eq. (14)]. The need to compute excited state prop-erties can significantly increase the computational cost, especiallywithin ab initio electronic structure implementations. Akimov andPrezhdo36,37 pointed out that nuclear trajectories in many nanoscaleand condensed matter systems differ little between ground andexcited adiabatic potential energy surfaces and are driven by ther-mal nuclear fluctuations. This observation allowed them to apply theclassical path approximation to FSSH36 and decoherence inducedsurface hopping (DISH).37 Here, we apply the classical path approx-imation to Ehrenest-DBB and analyze its performance in applica-tion to the system exhibiting current-induced molecular lumines-cence.38,39 Specifically, the electronic evolution is obtained exactly asdescribed above for the Ehrenfest-DBB method, while the nucleartrajectory is propagated using the ground state Hellmann-Feynmanforce [Eq. (15)] instead of the mean-field force [Eq. (14)]. The classi-cal path approximation allows us to utilize the optical response the-ory to evaluate the decoherence time since both approaches utilizeequilibrated trajectories.

E. Pure-dephasing/decoherence timeWe estimate the decoherence time as the pure-dephasing time

for pairs of initial and final states using the optical response theoryformalism.35,44 Fluctuations in the energy gap between the electronicstates due to nuclear motions are characterized by the autocorrela-tion function (ACF)

C(t) = ⟨∆E(t)∆E(0)⟩T . (25)

The brackets indicate canonical averaging. Fourier transformationof the ACF characterizes the phonon modes that couple to the elec-tronic subsystem. The resulting spectrum is known as the influencespectrum, or spectral density. The pure-dephasing function is com-puted using the second-order cumulant expansion of the opticalresponse function35,36

Dcum(t) = exp(−g(t)), (26)

where g(t) is

g(t) = ∫t

0dτ1 ∫

τ1

0dτ2C(τ2). (27)

TABLE I. Pure-dephasing time (fs) (lower triangle) and time-averaged absolutevalue of nonadiabatic coupling (meV) (upper triangle) between the pairs of statesconsidered in the application.

S0 S1 S2 S3 S4

S0 . . . 0.91 0.73 0.35 0.17S1 5.2 . . . 34.99 7.22 2.25S2 4.2 38.1 . . . 20.90 3.85S3 4.4 31.5 84.6 . . . 3.63S4 4.4 23.9 61.9 36.1 . . .

The pure-dephasing functions are fitted by a Gaussian, exp[−0.5(t/τ)2], to extract the pure-dephasing time constant, τ. The fittingresults are presented in Table I. Table I also contains the nona-diabatic couplings between pairs of adjacent electronic states. Thedecoherence rate, λij, that enters Eq. (24b) is computed as the inverseof the pure-dephasing time, τ.

F. Simulation detailsIn order to test the proposed method, we have modeled an

electric-field induced charge and energy transfer in the systemcomposed of a silver tip represented by a Ag20 tetrahedral clus-ter and a Zn(II)-etioporphyrin I molecule, as shown in Fig. 1(a).This test-system has been chosen for its complex electronic struc-ture, including energetically well-separated molecular states anddensely packed levels of the metal tip. The system choice has beenmotivated by the experiments showing current-induced molecularluminescence.38,39

We employed the plane-wave density functional theoryapproach, as implemented in the QUANTUM ESPRESSO pack-age,45 to describe the electronic properties of the system. ThePerdew-Burke-Ernzerhof (PBE)46 exchange-correlation functionalwith norm-conserving pseudopotential and scalar-relativistic cor-rection was used. The kinetic energy cutoff was set to 60 Ry forgeometry optimization and molecular dynamics and to 800 Ryfor charge density calculations. The Grimme’s density functionaldispersion correction version 2 (DFT-D2) was included.47,48 Themolecule-tip distance was set to 6 Å. Following the geometry opti-mization, a 1 ps long thermalization dynamics was performed at300 K using the Andersen thermostat. Ground state MD was thenperformed to generate a 1 ps microcanonical trajectory with a classi-cal time step of 1 fs to provide initial conditions for the excited statedynamics.

The nonadiabatic dynamics were performed at 300 K usingthe Ehrenfest-DDB method as well as decoherence induced surfacehopping (DISH).32 We have performed the dynamics with DISHfor comparison of the results of the two methods. DISH is a pop-ular surface hopping method for nonadiabatic molecular dynam-ics (NAMD) that includes decoherence effects and satisfies detailedbalance. Both methods have been implemented in the PYXAIDpackage36,46 and use the classical path approximation. Motivated bythe fact that the nuclear dynamics in the system under investiga-tion involves vibrational motions near the equilibrium geometry,which are sampled sufficiently accurately during the 1 ps MD tra-jectory, we repeated the NA Hamiltonian multiple 1000 times inorder to study long-time dynamics, up to 1 ns. This approxima-tion cannot sample slow motions of the molecule with respect tothe tip; however, it represents well faster nuclear motions that con-tribute most to the NA coupling. Furthermore, in the experimentmotivating the study,38,39 the porphyrin molecule resides on a sub-strate, and the location of the Ag electrode tip is controlled by theSTM setup such that the tip can be moved relative to the molecule,providing spatially resolved data and eliminating slow large-scalemotions.

For each studied physical process [as described in Fig. 1(b)]and method (Ehrenfest-DDB and DISH), 100 initial conditions weresampled from the 1ps MD trajectory and used as initial condi-tions for NAMD. The NA dynamics were initiated by instantly

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FIG. 1. (a) Top view on the tip-porphyrinsystem. (b) The studied nonadiabaticprocesses and charge densities of thecorresponding states.

promoting the system to a particular excited state. Additionally forthe DISH scheme, 100 realizations of the stochastic surface hop-ping process were sampled for each initial condition. The popula-tions of different electronic states in Ehrenfest-DDB were computedas squares of the coefficients of expansion of the time-dependentwavefunction in the adiabatic basis [Eq. (12)]. In DISH, the statepopulations were obtained as fractions of trajectories in each adia-batic state. The atomic structure and orbital visualizations have beenprepared with the XCRYSDEN software.49

III. RESULTS: CHARGE INJECTION, COOLING, ANDRECOMBINATION IN A METALLIC-TIP/PORPHYRINSYSTEM.

The optimized structure of the system under investigation38,39

is shown in Fig. 1(a), while Fig. 1(b) summarizes the different elec-tronic processes studied with the two methods, Ehrenfest-DDB andDISH.32 First, the electron injection from the tip to the moleculeis analyzed in a stepwise manner, with transition S4 → S3 (ini-tial phase of the electron transfer) and the complete electroninjection process (transitions from S4 through S2) studied sepa-rately. Then, the combined electron injection and cooling processes(transitions from S4 through S1) are investigated. Finally, the fulldynamics (S4 through S0) and charge recombination within themolecule (transition S1 → S0) are modeled with the two methods.Table I reports the pure-dephasing times and the average abso-lute NA coupling for all pairs of states. Besides summarizing thekey electronic processes, Fig. 1(b) also reveals the nature of rele-vant electronic states. The molecular highest occupied molecularorbital (HOMO) representing state S0 is localized on the metalcenter and complexing nitrogen atoms of the porphyrin molecule.The lowest unoccupied molecular orbital (LUMO) and LUMO+1,representing states S1 and S2, are composed of the π carbonorbitals of the porphyrin. LUMO+2 (S3) is delocalized over thetip and the porphyrin, demonstrating strong top-molecule inter-action in this state. LUMO+3 (S4) is fully localized on the silvertip.

Figure 2(a) shows phonon-induced fluctuations in the ener-gies of all electronic states during 500 fs of the MD trajectory, whileFig. 2(b) zooms onto the excited states and shows the energy gapsbetween nearest neighbor states. The gap between states S1 andS2 fluctuates most among the excited states, often reaching below0.025 eV, which corresponds to kBT at room temperature. TheNA coupling is also largest for this pair of states [Fig. 2(c)]. Thisis both because the energy gap between these states reaches smallvalues [Fig. 2(b)] and since the states are localized on and cou-ple to high frequency modes of the porphyrin molecule [Figs. 1(b)and 2(e)]. Because most electronic transitions couple to one ortwo phonon modes, Fig. 2(e), with the exception of the S2 → S1transition, the ACF functions of the energy gap fluctuations decayslowly [Fig. 2(d)]. The initial ACF value, corresponding to the aver-age gap fluctuation squared [Eq. (25)] is by far the largest for theS1 → S0 transition. The large initial ACF value rationalizes36

why the pure-dephasing function [Eq. (26)] decays fastest for theS1→ S0 transition.

Figures 3–8 show the evolution of excited state populations forvarious scenarios differing in the initially excited state and num-ber of states involved. Figures 3–8 are constructed the same way.Namely, the Ehrenfest-DDB data are shown in the upper row, andthe DISH results are given in the lower row. The panels in theleft column present the data up to 1 ns, while the panels in theright column show the same results for the first 100 ps of theevolution.

The dynamics for the electron injection process was startedwith the tip/porphyrin system excited to state S4. Figure 3 focuseson states S3 and S4. The population relaxation is very sim-ilar for Ehrenfest-DDB and DISH, with Ehrenfest-DDB show-ing slightly slower dynamics. This result indicates that Ehrenfest-DDB can provide reasonable relaxation times. At long times,approaching the thermodynamic limit, the relaxation slows downsharply for Ehrenfest-DDB, and the populations reach finite steadystate values. In comparison, the DISH relaxation continues untilall electronic population relaxes from the initial state S4 tostate S3.

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FIG. 2. (a) Phonon-driven evolution ofthe electronic state energies, S0–S4.(b) Energy gaps between excited states,S1–S4. (c) Absolute value of nonadia-batic coupling. (d) Unnormalized auto-correlation functions. (e) Influence spec-tra. (f) Pure-dephasing functions fortransitions between adjacent electronicstates. The energy gap between statesS1 and S2 drops below kBT = 0.025 eVat room temperature [part (b)]. Transi-tion between this pair of states exhibitsthe largest nonadiabatic coupling [part(c)] and couples to high frequency vibra-tions [part (e)], arising from the porphyrinmolecule that supports these states. Theenergy gap fluctuation is largest for theS0–S1 state pair [part (d)] which exhibitsfastest dephasing [part (f)].

The following analysis rationalizes the observed result. Notethat detailed balance is achieved in the modified Ehrenfest method35

by multiplying the coupling matrix element by a correction thatdecays to zero when the system approaches Boltzmann equilibrium,

i.e., the closer the system is to equilibrium, the smaller the modi-fied coupling, and the slower the dynamics. Specifically, considera 2-level system with energies of states k and j equal to E and 0,respectively. For simplification, assume E

kBT= 1.

FIG. 3. Dynamics of electronic state pop-ulations displayed up to 1 ns (left panels)and focused on the first 100 ps (rightpanels) calculated within the Ehrenfest-DDB method (crosses) and the DISHmethod (solid lines) for the partial elec-tron injection process (S4 → S3). Bothmethods produce similar time scales,with Ehrenfest-DBB marginally slowerthan DISH. Whereas the DISH methodshows almost complete relaxation, theEhrenfest-DDB scheme stops at about6% short of complete relaxation.

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FIG. 4. Dynamics of electronic state pop-ulations displayed up to 1 ns (left panels)and focused on the first 100 ps (rightpanels) calculated within the Ehrenfest-DDB method (crosses) and the DISHscheme (solid lines) for the completeelectron injection process (S4 → S3→ S2). Although the initial relaxation rateis marginally slower with Ehrenfest-DDBas compared to DISH, electronic popu-lations equilibrate at a similar state withboth methods.

At equilibrium, the Boltzmann quantum populations of the twostates are

ρ2k =

11 + e

, (28a)

ρ2j =

e1 + e

, (28b)

where e is Euler’s number. Let the populations ρ2k and ρ2

j differ fromtheir equilibrium values by x such that

ρk =√

11 + e

+ x, (29a)

ρj =√ e

1 + e− x. (29b)

Plugging the values from Eq. (29) into Eq. (24a), we obtain

Im[Hjk] = Im[Hjk] ⋅

RRRRRRRRRRR

√2e

1 + e⎛

⎝(√

1 + x(1 + e)) −⎛

⎝

√

1 − x(1 + ee

)⎞

⎠

⎞

⎠

RRRRRRRRRRR

.

(30)

To estimate the value of the coupling between the two states whenx approaches 0, we Taylor expand the two functions of x at x = 0 inthe above equation and obtain

(√

1 + x(1 + e)) −⎛

⎝

√

1 − x(1 + ee

)⎞

⎠=

(1 + e)2

2ex + O(x3

). (31)

Therefore,

Im[Hjk]∝ Im[Hjk] ⋅ ∣cx∣, (32)

where c is a constant.Equation (32) demonstrates that the detailed balance correc-

tion makes the coupling go to zero as the system approachesequilibrium. For this reason, the original Ehrenfest modified fordetailed balance method35 and Ehrenfest-DDB require infinite timeto achieve true Boltzmann distribution of populations. In prac-tice, one usually requires fairly short-time dynamics in order toobtain the characteristic time scale. For example, the 100 ps data inFig. 3 are sufficient to perform an exponential fitting and deducethe quantum transition time. Although surface hopping meth-ods such as DISH38 can achieve the thermodynamic equilibriummore closely than the current version of Ehrenfest-DDB, they aremore computationally expensive since they involve stochastic sam-pling that has to be repeated multiple times for each initial con-dition. In comparison, Ehrenfest-DDB needs only one trajectoryfor each initial condition. Furthermore, Ehrenfest-DDB requires

FIG. 5. Dynamics of electronic state populations displayed up to 1 ns (left panels) and focused on the first 100 ps (right panels) calculated within the Ehrenfest-DDB scheme(crosses) and the DISH scheme (solid lines) for the combined injection and cooling processes (S4→ S1). The initial dynamics are similar and slightly slower with Ehrenfest-DDB compared to DISH. At the long-time limit, the populations of states S1 and S2 reach different values. The different behavior of the two methods can be attributed to thefacts that the energy gap between these states drops below kBT [Fig. 2(b)] and that the nonadiabatic coupling is large [Fig. 2(c)].

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FIG. 6. Dynamics of electronic state pop-ulations displayed up to 1 ns (left panels)and focused on the first 100 ps (rightpanels) calculated within the Ehrenfest-DDB scheme (crosses) and the DISHscheme (solid lines) just for the pair ofstates S1 and S2 that exhibit differentlong-time limits during the S4 → S1dynamics shown in Fig. 5. The differ-ences arise because the S1–S2 energygap drops below kBT [Fig. 2(b)] and thenonadiabatic coupling is large [Fig. 2(c)].

only propagation of the Schrödinger equation (with the modi-fied Hamiltonian), while DISH and related surface hopping meth-ods require both propagation of the Schrödinger equation as wellas additional steps, making surface hopping more algorithmicallycomplex.

Following the above two state dynamics, a slightly more com-plex system was considered. The dynamics for the complete chargeinjection process [Fig. 1(b)] including states S4, S3, and S2 wasstarted with the system excited to state S4. Figure 4 shows thechange in the electronic population of the three states vs time. TheEhrenfest-DDB and DISH schemes arrive at similar distributions ofpopulations of the three states at long time (left column of Fig. 4).On the shorter time scale (right column of Fig. 4), the dynamicsis slightly slower for Ehrenfest-DDB than for DISH. Overall, thebehavior of the two methods in the current example involving statesS2–S4 is similar to the previous example involving states S3–S4(Fig. 3).

As the next step, we added the S1 state to the above set of statesand studied the dynamics of the electron injection followed by thecooling process, including states S1, S2, S3, and S4. The system wasexcited to state S4 at the initial time for both Ehrenfest-DDB andDISH. The evolution of the state populations is shown in the upperrow of Fig. 5 for Ehrenfest-DDB and in the lower row of Fig. 5 forDISH. Ehrenfest-DDB produces different long-time limits for popu-lations of states S1 and S2 compared to DISH. The difference can beattributed to the energy gap and NA coupling values for the S1–S2

state pair [Figs. 2(b) and 2(c)]. In particular, the energy gap fre-quently drops below 0.025 eV, which corresponds to kBT at roomtemperature. The fluctuation of the energy gap from less than kBTto nearly 10 times kBT makes the concept of equilibrium popula-tions ill-defined. Furthermore, the NA coupling between states S1and S2 reaches over 20 kBT, strongly mixing these states, and onceagain, making equilibrium populations ill-defined. The early timedynamics is slower in Ehrenfest-DDB than in DISH, similarly to theprevious examples.

Figure 6 shows the results of simulations involving states S1and S2 only, with the initial population in S2. In DISH, the popu-lations of S1 and S2 are similar, whether or not the higher energystates S3 and S4 are included. The difference in the S1 and S2 pop-ulations between Figs. 5 and 6 is more pronounced in Ehrenfest-DDB, indicating that population equilibration is a more complexprocess showing nontrivial dependence on fluctuations in the energygap, NA coupling, and number of states. The Ehrenfest-DDB equa-tions of motion are nonlinear, with nonlinearities arising in bothdetailed balance and decoherence terms. Therefore, the dependenceof the ensuing dynamics on the parameters of the time-dependentHamiltonian can be rather involved and is hard to analyze ana-lytically, beyond simple arguments such as those presented inEqs. (28)–(32).

Next, we performed dynamics involving all states from S0through S4, describing the complete set of processes from theexcitation, to the electron injection, to the cooling, and to the

FIG. 7. Dynamics of electronic state populations displayed up to 1 ns (left panels) and focused on the first 100 ps (right panels) calculated within the Ehrenfest-DDB scheme(crosses) and the DISH scheme (solid lines) for the complete set of processes, including charge injection, cooling, and recombination (S4→ S0). The initial dynamics aresimilar and slightly slower in Ehrenfest-DDB compared to DISH. At long times, the Ehrenfest-DDB dynamics slows down considerably, with the largest differences observedfor the populations of states S0, S1, and S2. See Figs. 5 and 6 for additional insights.

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FIG. 8. Dynamics of electronic state populations displayed up to 1 ns (left panels) and focused on the first 100 ps (right panels) calculated within the Ehrenfest-DDB scheme(crosses) and the DISH scheme (solid lines) for the recombination process (S1 → S0). Both DISH and Ehrenfest-DDB require more than 1 ns to reach equilibrium, owingto the large energy gap [Fig. 2(a)] and small nonadiabatic coupling [Fig. 2(c) and Table I] between the states. Ehrenfest-DDB approaches equilibrium faster with two states(S1→ S0) than with five states (S4→ S0) (Fig. 7).

recombination. As before (Figs. 3–5), all of the electronic popu-lation was assigned to state S4 at the initial time. The results areshown in Fig. 7. The major difference from the previous simula-tions is that the ground state S0 is strongly separated in energy fromthe excited states by about 1 eV [Fig. 2(a)]. Furthermore, the NAcoupling between S0 and all excited states is at least an order ofmagnitude smaller (0.2–0.9 meV) than the NA coupling betweenthe excited states (2–35 meV) (Table I), and pure-dephasing is anorder of magnitude shorter (4–5 fs vs 25–85 fs). The large energygap, and small NA couplings and pure-dephasing times lead to veryslow population of the ground state S0. Even the 1 ns simulations arenot sufficient to reach equilibrium populations in both Ehrenfest-DDB and DISH. The 100 ps data (right panels in Fig. 7) demon-strate that the ground state S0 population reaches about 20% in bothmethods, indicating that they predict similar charge recombinationtimes. Transiently, the populations of states S1–S3 rise and then fallin DISH, while in Ehrenfest-DBB, they reach steady-states. This isbecause the dynamics within the energetically closely spaced S1–S4manifold slows down after 10 ps in Ehrenfest-DBB (Fig. 5), and therate of transitions from S4 into S1–S3 becomes similar to the rateof transitions from S1 to S0 (Fig. 8). At the long time (left panelsof Fig. 7), the Ehrenfest-DDB dynamics slows down considerablyeven though the population of S0 continues to grow, as required bythermal equilibration.

Figure 8 focuses on charge recombination due to nonradia-tive relaxation from the first excited state S1 to the ground stateS0. The short time dynamics is similar between Ehrenfest-DDBand DISH. Judging by the 100 ps data (right panels in Fig. 8),Ehrenfest-DDB even shows faster relaxation than DISH. The popu-lation of the ground state reaches 40% in Ehrenfest-DDB and slightlyless than 30% in DISH. The ground state population reaches 50%at about the same time with the two methods, while the longertime decay is significantly slower in Ehrenfest-DDB (left panels ofFig. 8).

The current work demonstrates that both detailed balanceand decoherence can be introduced into the Ehrenfest method,making it applicable for simulation of large realistic systems andlong time scales. At the same time, further studies are requiredto investigate how the detailed balance modification of the Ehren-fest method affects the quantum transition rates and approach to

equilibrium, with and without inclusion of decoherence. Thereexist several quantum correction factors that account for detailedbalance.32 Bastida et al.35 employed Q(ω) = 2/[1 + exp(−βhω)],obtained by assuming that the real part of the quantum correla-tion function is equal to the classical correlation function. Otherchoices include Q(ω) = βhω/[1 − exp(−βhω)], corresponding tothe harmonic approximation, Q(ω) = βhω, as well as more com-plicated versions.32 Each quantum correction will produce a dif-ferent scaling of the coupling matrix elements, leading to differentrates of quantum transitions. Furthermore, Kleinekathöfer and co-workers50 pointed out that the original approach does not reproducethe high-temperature limit and proposed an additional normaliza-tion. They divided the scaled coupling by the difference ρi − ρj, whichrestores the high-temperature limit of equally populated states,but causes divergence as the system approaches this limit and ρi− ρj → 0. A more detailed analysis and further tests are needed toorder to establish which type of correction works best for a particularclass of problems.

IV. CONCLUSIONSWe have presented the Ehrenfest-DDB method for NAMD

simulations that accounts for decoherence effects and detailed bal-ance between transitions upward and downward in energy. Bothfeatures are essential for modeling excited state dynamics in largesystems on long time scales. The methodology is based on themixed quantum-classical Ehrenfest approach that couples quantummechanical expectation values to classical variables. This simplestquantum-classical approximation is rooted in the Ehrenfest theoremand can be derived in different ways.21,35,44,50 Decoherence effectsare missing in the Ehrenfest method because the “bath” is treatedclassically, and the evolution of the quantum subsystem remains uni-tary even when coupled to an environment. The Ehrenfest approachcannot describe quantum-classical thermalization because it is amean-field method and lacks correlations of classical trajectorieswith different quantum states. Decoherence effects are incorporatedinto the Ehrenfest approach via an additional term that penalizesdevelopment of coherences during the evolution of the quantumdegrees of freedom. In the classically mapped Hamiltonian, thecoherence penalty term adds an energy penalty to the total energy.33

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The decoherence time scale is estimated as the pure-dephasing timeof the optical response theory.51,52 Detailed balance is achieved byscaling the off-diagonal matrix elements of the Hamiltonian witha quantum correction factor known in semiclassical approxima-tions to quantum time-correlation functions.53 The detailed balancecorrection is a function of temperature and energy gap betweena given pair of states. Both modifications modify the Schrödingerequation, making it nonlinear. At the same time, the key feature ofthe Ehrenfest method, which describes quantum-classical dynamicsfully deterministically and with wavefunctions, is preserved. In con-trast, other approaches such as surface hopping or master equationseither introduce stochastic processes (hops) that require extensivesampling or deal with N2 dimensional density matrices rather thanN dimensional wavefunctions. Thus, the Ehrenfest-DDB method iscomputationally more efficient than the alternative techniques. Theefficiency is increased further by the classical path approximation,which allows one to use a precomputed nuclear trajectory to drivethe electron dynamics. It is important to note that surface hoppingalgorithms are still needed to represent properly dynamics outcomeswhen nuclear trajectories vary strongly between different adiabaticpotential energy surfaces.

We illustrated the Ehrenfest-DDB method by performing cal-culations on a hybrid molecule-tip system studied experimentally.The system undergoes several important processes that are typical ofmodern applications of nanoscale materials, including charge injec-tion, relaxation, and recombination. The Ehrenfest-DDB results arecompared to calculations performed using DISH, which is a pop-ular surface hopping technique with decoherence and detailed bal-ance effects. The Ehrenfest-DDB approach gives time scales that aresimilar and slightly longer than those of DISH. In the long-timelimit, the Ehrenfest-DDB dynamics slow down considerably, due tothe main feature of the detailed balance correction that drives theoff-diagonal matrix elements of the Hamiltonian to zero as the sys-tem approaches Boltzmann equilibrium. Nevertheless, correct timescales can be obtained with relatively short simulations. The currentversion of Ehrenfest-DDB uses one of the several possible quan-tum correction factors responsible for detailed balance,54 followingthe original publication.35 The performance of other detailed bal-ance corrections50,55 deserves further studies that are planned in nearfuture.

The Ehrenfest-DDB method, being an extension of the Ehren-fest dynamics, is easy to implement and computationally efficient.Avoiding the need to generate ensembles of stochastic trajectories,it is capable of describing thermal relaxation and loss of quan-tum coherence, making it appropriate for calculations on largecondensed matter and nanoscale systems.

ACKNOWLEDGMENTSThe research was supported by the U.S. National Science Foun-

dation, Award No. CHE-1900510.

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