Ultrafast Electronic Relaxation through a Conical Intersection:Nonadiabatic Dynamics Disentangled through an OscillatorStrength-Based Diabatization FrameworkGregory R. Medders,*,† Ethan C. Alguire,† Amber Jain,† and Joseph E. Subotnik†,‡

†Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States‡Stanford PULSE Institute, SLAC National Accelerator Laboratory, Menlo Park, California 94025, United States

*S Supporting Information

ABSTRACT: We employ surface hopping trajectories to model the short-timedynamics of gas-phase and partially solvated 4-(N,N-dimethylamino)benzonitrile(DMABN), a dual fluorescent molecule that is known to undergo a nonadiabatictransition through a conical intersection. To compare theory vs time-resolvedfluorescence measurements, we calculate the mixed quantum−classical density matrixand the ensemble averaged transition dipole moment. We introduce a diabatizationscheme based on the oscillator strength to convert the TDDFT adiabatic states intodiabatic states of La and Lb character. Somewhat surprisingly, we find that the rate ofrelaxation reported by emission to the ground state is almost 50% slower than theadiabatic population relaxation. Although our calculated adiabatic rates are largelyconsistent with previous theoretical calculations and no obvious effects ofdecoherence are seen, the diabatization procedure introduced here enables anexplicit picture of dynamics in the branching plane, raising tantalizing questionsabout geometric phase effects in systems with dozens of atoms.

■ INTRODUCTION

From enabling protein imaging to controlling the efficiency ofsolar cell materials, the energetics and dynamics of electroni-cally excited states play a key role in the development of newmaterials and biological assays.1−4 For example, the sensitivityof fluorescent probes to their environment has enabledcharacterization of protein structure and dynamics.5 Forsystems in which dual fluorescence has been observed,obtaining a molecular-level interpretation of the electronicand nuclear dynamics that control fluorescence has provennontrivial from both an experimental and a theoreticalperspective.6 From a theoretical standpoint, a description ofthe electronic dynamics that influence fluorescence yield andline shape is complicated by several factors; while fluorescentmolecules typically emit from their first excited state, they areoften initially excited to a higher excited stated from which theyrelax.7 If the excited states come close to one another, rapidtransitions can occur between electronic states. In this case, themotion of nuclei and electrons can no longer be treated asdecoupled, leading to a breakdown of the Born−Oppenheimer(BO) approximation.To circumvent the breakdown of the BO approximation,

several approaches to simulating the nonadiabatic moleculardynamics (NAMD) of large systems have been developed.8−13

Perhaps the most widely used of such algorithms, Tully’s fewestswitches surface hopping (FSSH), involves classical treatmentof the nuclear degrees of freedom, which are propagated alongan adiabatic potential energy surface subject to stochastic hops

between the PESs.8 FSSH has been successfully applied to awide range of systems,14−22 and in this contribution, we use itto study the population relaxation of the dual fluorescentmolecule 4-(N,N-dimethylamino)benzonitrile (DMABN). Inits gas-phase minimum-energy geometry, DMABN has twolow-lying electronically excited singlet states that are separatedfrom higher excited states by >1 eV. These two excited statesare centered at 4.32 and 4.90 eV above the ground-stateminimum-energy geometry based on MRCIS(8,7)+P/ANO-DZ calculations,23 which is in reasonable agreement withexperimentally determined absorption maxima of 4.13 and 4.57eV.24,25 In the Franck−Condon region, the electron density ofthe first excited state, S1, is “locally excited” (LE) within thebenzyl ring, resulting in a transition with relatively low oscillatorstrength. In contrast, the second excited state shifts electrondensity from the amine to the benzonitrile, resulting in anintramolecular charge transfer (ICT) state with a largeoscillator strength.23,26−30 In analogy with the electronicstructure of cata-condensed aromatic molecules,31 the LEstate is often described as Lb-like and the ICT state as La-like inthe Franck−Condon region. While DMABN fluoresces fromthe LE state in the gas phase and in nonpolar solvents, the ICTstate is preferentially stabilized in polar environments, enablingdual fluorescence from both the LE and ICT states on S1. The

Received: December 1, 2016Revised: January 15, 2017Published: January 18, 2017

Article

pubs.acs.org/JPCA

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precise nature of the emitting ICT state has been the subject ofa great deal of experimental and computational re-search.6,28,32−38

Now, even though the fluorescence of DMABN has beenlong established,39 conflicting mechanisms for the relaxationfrom the S2-La state to the S1-La and S1-Lb states have beenproposed. Upon photoexcitation with 270 nm light in the gasphase at 400 K, DMABN has been found by nonresonantmultiphoton dissociative ionization techniques to absorb intothe second excited (S2-La) state and undergo subsequentnonadiabatic relaxation to S1.

40−42 From polarization-resolvedtransient ionization measurements, three fast time scales ofrelaxation were obtained and assigned to τ1 = 5 fs, relaxation inthe Franck−Condon region of S2-La; τ2 = 68 fs, transition fromS2-La into both S1-La and S1-Lb via conical intersection (CI);and τ3 = 1 ps, equilibration within S1 into the Lb state.

41 Thispicture, while consistent with previous experimental measure-ments24,39,43,44 and (static) theoretical characterization of theexcited-state landscape, has been called into question by tworecent NAMD simulations of the excited-state relaxation ofDMABN. Kochman et al., using FSSH trajectories at the MP2/ADC(2)/cc-pVDZ level, found ultrafast relaxation (τ = 8.5 fs)from S2 → S1 via a CI.45 At around the same time, FSSHsimulations of Du et al. at the CAM-B3LYP/6-31G* levelfound a similar time scale of 5.9 fs for transit through the CI.46

One difficulty in the interpretation of surface hoppingcalculations is that most published studies report only adiabaticpopulations, rather than spectroscopic signals or electronicexpectation values; as we will show below, this simplificationcan lead to reasonably-sized errors in predicted rate constants.Besides the aforementioned problem of interpretation,

another possible origin of the discrepancy may arise fromtreatment of the overcoherence problem of FSSH.47−49 In themixed quantum−classical framework of FSSH, the evolution ofthe quantum subsystem is perfectly coherent. In a simplisticsense, this causes individual FSSH trajectories to maintain thememory of previous excursions into regions of strongnonadiabatic coupling; for example, it was shown that when atrajectory exits and then subsequently re-enters the couplingregion, the overcoherence biases the trajectory toward hopping,resulting in artificially high rates of population transfer and longtime oscillations in the adiabatic populations.48,50 Numerousdecoherence corrections have been developed to account forthis effect;48,50−53 however, most rely on either empiricallymotivated or adjustable parameters to determine under whatconditions the coherence of the quantum amplitudes should bereset. In the previous studies of DMABN, for example,decoherence effects were treated empirically46 or neglected.45

To address decoherence, we employ a recently introducedversion of A-FSSH. Although a comparison of the FSSHalgorithm with the quantum−classical Liouville equation10,54−58proves that the true decoherence rate cannot be correctlydescribed within the noninteracting trajectory approximation ofFSSH,59 the A-FSSH correction estimates a minimal rate ofwave packets separating onto different surfaces in a parameter-free way60−62 such that accuracy is almost always improved.21

With this background in mind, our goals for this paper are torevisit the short-time DMABN dynamics and (i) makepredictions for the ultrafast emission of DMABN followingphotoexcitation to its La state using the density matrixformulation of the A-FSSH (rather than simply using theadiabatic populations to connect DMABN dynamics with anexternal light field); (ii) verify whether or not “overcoherence”

problems have impacted previous results; and (iii) investigatesolvent effects on the short-time dynamics of DMABN. As willbe discussed later, note that in this paper we will not addressdynamics beyond 100 fs.

■ THEORYDynamics. In order to minimize computational time, we

use a recently developed version of the A-FSSH algorithm thatexhibits improved efficiency without significantly compromisingaccuracy.63 This algorithm, described in ref 63, builds onstandard implementations of surface hopping in two ways:(i) In its original form, FSSH required calculation of the

derivative coupling between all (pairwise) combinations ofstates in order to propagate the quantum amplitudes anddetermine the hopping probability. While not prohibitive incost for two states, the number of derivative couplings grows as

n( )Q2 , where nQ is the number of FSSH states, and their

calculation rapidly becomes the bottleneck as the number ofFSSH electronic states increases. As was recognized byHammes-Schiffer and Tully,64 the explicit calculation ofderivative coupling can be avoided by expressing the effect ofclassical motion on the quantum subsystem through overlap ofthe electronic states at different times. In the new A-FSSHimplementation described in ref 63, a key improvement wasenabled by the norm-preserving interpolation approach ofMeek and Levine,65 which was invoked and generalized tocalculate the time-derivative coupling such that it is wellbehaved near the crossing of arbitrarily many electronic states.(ii) Formally, the A-FSSH algorithm requires propagation of

nQ2ncl degrees of freedom, where ncl and nQ are the number of

classical and quantum degrees of freedom, respectively. Forsystems whose gradient and derivative coupling evaluation istrivial (e.g., model Hamiltonians), the A-FSSH calculation canbecome a computational bottleneck. In our new implementa-tion, we calculate only the diagonal moments of the auxiliarydecoherence variables, decreasing the number of degrees offreedom to nQncl and allowing the decoherence correction to becomputed with only minimal overhead.In the end, simulations of DMABN are 28 times faster using

the new A-FSSH implementation63 relative to the originalimplementation62 (for two states when including the differ-ences in the time step needed to maintain energyconservation). Timings demonstrating the performance of thenew A-FSSH implementation for systems with many electronicstates are provided in Figure S7.

Diabatization. While the wave functions resulting fromelectronic structure calculations are the adiabatic states thatdiagonalize the electronic Hamiltonian, it is not always the casethat experiments probe the adiabatic states of the system. Forinstance, Marcus theory is most natural in a localized, diabaticbasis. Formally, a diabatic basis is an electronic representationin which the first-order derivative coupling

= ⟨Φ |∇ |Φ ⟩d R R( ) ( )IJ I JR

R[ ]

(1)

is zero. Such a basis does not exist in general, and the best onecan hope for is a quasi-diabatic basis whose derivative couplingis vanishingly small.66 The choice of such diabatic states (andthe associated criteria used to generate them) is obviously notunique, and many definitions of diabatic states have beenemployed.67−79 Because the electronic characters of La and Lbare distinguished primarily in terms of their differing oscillatorstrengths, a physically motivated route for constructing the

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diabatic states of DMABN (i.e., La and Lb) is to build a matrixcontaining the dot product of the transition dipole momentsbetween each adiabatic state (S1, S2) and the ground state

μ μ μ μ

μ μ μ μ=

· ·

· ·

⎛⎝⎜⎜

⎞⎠⎟⎟D

01 01 01 02

02 01 02 02 (2)

Subsequent diagonalization of D results in two states withmaximally different transition dipole moment magnitudes, andthe eigenvectors of D can be interpreted as the rotation matrixthat converts the adiabatic states into high and low oscillatorstrength diabatic states. As will be shown later, this diabaticrepresentation naturally allows for a description of fluorescencein molecules with electronic states of differing oscillatorstrength.To demonstrate that this diabatization produces excited

states of expected electronic character (i.e., La and Lb), we willcalculate and compare attachment and detachment densities inthe adiabatic and diabatic representations.80 In this work, theexcited states of DMABN are modeled through TDDFT in theTamm−Dankoff approximation using the long-range correctedhybrid density functional ωB97X in the cc-pVDZ basis,81,82

which is similar to the functional employed in ref 46. Nosymmetry was employed in the calculation of the electronicstates. The electronic character of the adiabatic states wasassigned at the minimum-energy geometry of the ground state(GS0), where the coupling between the excited states isrelatively small. Here, the adiabatic states S1 and S2 exhibit theexpected low and high oscillator strength and are assigned as Lband La, respectively (see Figures S1 and S2 in the SupportingInformation). Diagonalization of eq 2 at GS0 preserves thecharacter of the adiabatic states, with a mixing angle (θ,obtained by interpreting the eigenvectors of eq 2 as a 2 × 2rotation matrix) of 0.1°. In regions of strong electroniccoupling, the La/Lb character of the adiabatic states is stronglymixed. Shown in Figure 1 are the attachment densities in theadiabatic and diabatic representations for a configuration in thevicinity of the S2/S1 CI; in this case, the diabatic states definedthrough the transition moments maintain the La/Lb characterobserved near GS0 through mixtures of the adiabatic states (θ =43.0°). Note that, in its twisted conformation, the transition

dipole moments of the first two states of DMABN becomesimilar in magnitude, causing a breakdown of oscillatorstrength-based property diabatization. To avoid this issue, werestrict our analysis of the diabatic properties to the first 100 fswhen the molecule has not yet twisted.

Simulation Details. Given a method of reliably generatingthe La and Lb diabatic states (namely, through eq 2), weperformed A-FSSH simulations using the newly proposedalgorithm63 and analyzed the trajectories to study thepopulation transfer dynamics of DMABN. To enablecomparison with experiment,41 ground-state AIMD simulationsof DMABN modeled with ωB97X/cc-pVDZ were performed at400 K using the Nose−Hoover chains thermostat.83 Config-urations and velocities were extracted from the constant-temperature simulations at ∼100 fs intervals to serve as initialconditions for the A-FSSH simulations. No attempt to accountfor nuclear quantum effects was made in the choice of the initialconditions. Each of the 200 A-FSSH trajectories was initializedon the state with the highest oscillator strength, which in allcases was S2, and was propagated for 100 fs using a time step of21 au (∼0.5 fs).

■ RESULTSThe relaxation of the excited-state population from S2 to S1 ispresented in Figure 2. The dashed lines indicate thepopulations in the adiabatic representation of the mixedquantum−classical density matrix, while solid lines indicatethe populations in the diabatic representation.84 The adiabaticpopulation relaxes quickly with a rate of τ = 11 ± 1 fs.Decoherence is found to minimally affect the dynamics (seeFigure S3 for a comparison of FSSH and A-FSSH trajectories).Because DMABN is operating in the normal (as opposed toinverted) regime, the dynamics primarily involve rapidrelaxation downhill from the strongly absorbing S2 state toS1. After the first hop, few recrossings occur because the excitedstates quickly separate in energy, which results in theinsensitivity of DMABN dynamics to decoherence effects.In the diabatic representation, conversion from La to Lb is

somewhat slower than in the adiabatic representation, with atime scale of 15 ± 1 fs. Therefore, while essentially alltrajectories have relaxed into S1 within 50 fs, some La character

Figure 1. Excited-state character of the first two excited singlets of DMABN for a configuration in the vicinity of the CI. The mixing angle for thisconfiguration is 43.0°. The attachment densities of the S1 and S2 states are shown. In the diabatic representation of the excited states, the S1 and S2excitation amplitudes are mixed to restore the distinct La and Lb states. The corresponding detachment densities, as well as the excitation densities ofa configuration far from the region of coupling, are presented in Figures S1 and S2 in the Supporting Information.

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remains up to 100 fs. To understand the somewhat slowerrelaxation of the diabatic states, it is useful to examine thebehavior of a representative surface hopping trajectory. Roughly80% of the trajectories undergo a single hop from S2 to S1during the course of a simulation. In Figure 3(a), the excited-state energies of one such trajectory are plotted relative to theenergy at GS0. The active adiabatic state (indicated by opensquares) is initially S2-La. At 5 fs, the molecule hops betweenadiabatic states (dashed lines) to S1. In the diabaticrepresentation, however, no hop has taken place because thediabatic character of S1 and S2 also switches at 5 fs. At 14 fs, theadiabatic states flip diabatic character again; this time, becauseno hop takes place, the molecule is now evolving on S1-Lb.After 14 fs, the molecule leaves the region of strong coupling,and the adiabatic states separate in energy, with no further hopstaking place.

In this particular trajectory, the change in diabatic characterlags behind the hop between adiabatic states. To assess this, thetime of the final hop from S2 to S1 (thop) and the time of thecrossing of diabatic states that restores the Lb character to S1(tcross) were determined for each of the trajectories. Specifically,tcross was obtained by constructing the list of times at which theenergetic ordering of the diabatic states changed from S1-La toS1-Lb; because all trajectories in the A-FSSH simulations endedin the S1-Lb state, the final hop to S1 was located, and the list ofcrossings was searched to find the closest diabatic crossing thatrestored the S1-La character. The distribution of these timedelays, Δt = tcross − thop, is plotted in panel (b) of Figure 3.Negative delay times indicate that the diabatic character of S1changed from La to Lb before the hop occurred, while positivedelays indicate that the diabatic character changes after the hop.The vast majority of trajectories undergo restoration of S1-Lbcharacter after the final hop, with an average delay of 5 fs that isconsistent with the longer time scale of the populationrelaxation in the diabatic representation. The majority of thehops, therefore, occur from S2-La to S1-La and are followed by asubsequent conversion from S1-La to S1-Lb. Because thecrossing of diabatic states happens in a region of strongcoupling (the average adiabatic energy gap at the point ofdiabatic crossing is 0.24 ± 0.01 eV), the fact that thedistribution is peaked away from Δt = 0 fs indicates that themolecule is either in a region of extended coupling after thehop or that the molecule exits and re-enters the coupling regionin order to change its diabatic character. That being said, due tothe similarities of the population dynamics with and withoutdecoherence, however, we can infer that such recrossings occuron time scales that are faster than the decoherence rate (seeFigure S5).

■ DISCUSSION

Time-Resolved Fluorescence and Solvent Effects. Thearguments above have important consequences for interpretingexperimental time-resolved spectra that probe excited-statedynamics. Due to the complexity of modeling the nonresonantmultiphoton ionization employed in refs 40−42, we have notattempted to directly reproduce these experiments. Rather,

Figure 2. Dynamics of the excited-state relaxation. Excited-statepopulations in the adiabatic and diabatic representations (left y-axis)and the ensemble-averaged transition dipole moment (right y-axis, eq5) are presented as a function of simulation time following excitationto S2. Note that the diabatic populations decay almost 50% slower thanthe adiabatic populations (matching the decay of the ensemble-averaged transition dipole moment).

Figure 3. (a) Total energies of the adiabatic and diabatic states of a single trajectory shifted by the energy of the ground-state equilibrium geometry.The simulation time following the excitation to S2 is plotted on the x-axis. The active adiabatic state of the A-FSSH simulation is indicated by theopen squares. Note that while the molecule hops between adiabatic states at 5 fs, the active surface maintains La diabatic character until 14 fs. Ahistogram of this lag time between adiabatic and diabatic transitions (Δt) accumulated over the 200 trajectories is presented in panel (b).

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because a key factor of DMABN is the molecule’s dualfluorescence, we have considered how the straightforwardadiabatic rates obtained from NAMD simulations may differfrom the emission that might be experimentally observed. Inparticular, ultrafast measurements of the dynamics of DMABNin acetonitrile (MeCN) have recently been performed usingtime-resolved fluorescence.37 From these measurements, aninstrument-limited ultrafast signal of <30 fs was observed acrossa spectral window encompassing fluorescence from both the Lbstate as well as the La state. While it has long been known thatpolar solvents such as MeCN alter the shape of S1 by stabilizingthe highly polar La state (commonly referred to as the ICTstate outside of the Franck−Condon region, as mentioned inthe Introduction) and bringing it to a similar energy as the Lbstate,38 fluorescence measurements provide compelling evi-dence of an ultrafast nonadiabatic process occurring at a ratesimilar to what has been found theoretically in the gasphase.45,46,85

To provide initial exploration of how solvent impacts theultrafast dynamics of DMABN, A-FSSH simulations wereperformed for DMABN interacting with a single MeCNmolecule. A total of 200 trajectories were calculated beginningfrom initial conditions corresponding to 180 K, with the lowertemperature being necessary to prevent the bimolecularcomplex from dissociating. The DMABN adiabatic populationdynamics appear to be largely unaffected by the MeCNmolecule, with slight differences being attributed to thedifference in temperature (see Figure S7). To investigate whythe short-time population dynamics are unaffected by theinteractions with MeCN, the derivative coupling between S1and S2 was examined for all hops and hop attempts that wereencountered during the A-FSSH simulations. As shown inFigure 4, an average of >99.5% of the total derivative coupling

was localized on DMABN while only <0.5% of the derivativecoupling was contributed by the MeCN molecule. Thus, whilethe long-time dynamics and populations are expected to beinfluenced by interactions with the polar MeCN, the non-adiabatic dynamics appear to be unaffected by MeCN due to S1and S2 being localized on DMABN. While we can by no meansquantify conclusively the influence of the solvation on theshort-time dynamics based on these simulations using only asingle solvent molecule, these results suggest that gas-phase

DMABN may also have an ultrafast process that is comparableto nonadiabatic relaxation found in ref 37. Indeed, thisconclusion that short-time dynamics are unaffected by solventeffects is supported by transient absorption measurements.34

Now, because the experiments in ref 37. involve fluorescenceand therefore probe the transition moment between the excitedand ground electronic states, some care is required to compareour surface hopping calculations with the experimentallydetermined rates. For simplicity, suppose we probe a stimulatedemission signal back to the ground state with an instantaneouselectric field in the x-direction (which could be compared tothe time-resolved fluorescence measurements of ref 37). Insuch a case, let |Ψ⟩ be the time-dependent nuclear−electronicexcited-state wave function, |Ψ⟩ = |χ1⟩|Φ1⟩ + |χ2⟩|Φ2⟩, where |χ⟩and |Φ⟩ indicate nuclear and electronic wave functions,respectively. Using Fermi’s Golden Rule, the relevant time-delayed signal will be proportional to α = ∑N |⟨χ0

(N)Φ0|μx|Ψ⟩|2,

where N indexes the nuclear eigenstates (|χ0(N)⟩) on the ground

state and μx is the dipole operator in the x-direction. This sumcan be rewritten as

ρα = DTr (3)

where ρ = |Ψ⟩⟨Ψ| and

∑ μ χ χ μ

μ μ

= |Φ ⟩⟨Φ |

= |Φ ⟩ ⟨Φ |

D

IN

x N N x

x x

0 0( )

0 0( )

0 N 0 (4)

Here, IN is the resolution of identity over the nuclear (N) space.Equation 3 can be re-expressed through a partial Wignertransformation over the nuclear degrees of freedom as

∫ ∫∑α ρ= R P R P D R Pd d ( , ) ( , )ij

ij jiW W

(5)

where the trace over the electronic states (i and j) has beenwritten out explicitly. Here, ρij

W(R,P) is the adiabatic mixedquantum−classical density matrix obtained (approximately)from the surface hopping trajectories and Dji

W(R,P) is the partialWigner transform of the dipole operator. The role of Dji

W(R,P)can be most easily seen in a diabatic basis (Φa,Φb). In thisrepresentation, the transformed operator is, to within anormalization constant

∫

∫

μ μ

μ μ

= Δ − Δ Φ Φ Φ Φ + Δ

= Δ − Δ + Δ

Δ·

Δ·

D R P R R

R R

( , ) d e2

; ;2

d e2 2

P x x

P x x

abW i

a 0 0 b

ia0 0b

(6)

where the electronic integral over the dipole operator isperformed in the first line, resulting in the transition dipole μa0

x

between the ground diabatic state Φ0 and one of the twoexcited diabatic states Φa. Because this transition dipole isdiagonal in the nuclear position basis, we can further simplifythe expression

Figure 4. (a) Snapshot from A-FSSH simulations of DMABN with anMeCN molecule. (b) Percent contribution to the total derivativecoupling from each atom. Derivative couplings are averaged over the392 hops and hop attempts encountered in the simulations ofDMABN with MeCN. The contribution from the hydrogen atoms onthe benzene ring are all less than 5% and are not shown. Thecontributions on the methyl groups include both the derivativecoupling on the methyl hydrogens and the carbon atom.

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∫

∫

μ μ

μ δ μ

μ μ

= Δ − Δ − Δ + Δ + Δ

= Δ − Δ Δ + Δ

=

Δ·

Δ·

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

D R P R R R R

R R

R R

( , ) d e2 2 2 2

d e2

( )2

( ) ( )

P x x

P x x

x x

abW i

a0 0b

ia0 0b

a0 0b

(7)

More generally, if we average over all spatial directionsisotropically, we simply replace μx · μx by μ · μ and obtainDab

W(R,P) = ∑γ=x,y,z μa0γ (R)μ0b

γ (R) . After rotating back into theadiabatic basis, this result motivates the diabatization criterionin eq 2 and provides a straightforward way to evaluate eq 5 via amixed quantum−classical density matrix interpretation of thesurface hopping algorithm. Using such an interpretation, thetime-dependent stimulated emission (α) is plotted in Figure 2.Interestingly, emission-based measurements of the excited-staterelaxation of DMABN are predicted to yield a decay on theorder of 15 fs, rather than 11 fs. Specifically, due to the differingelectronic character of the excited states, proper treatment ofthe spectroscopic observables results in a change of nearly 50%in the predicted decay relative to the rate obtained from theadiabatic populations. Moreover, the agreement between α(t)and the diabatic dynamics suggests that our diabatic states(from eq 2) are meaningful for understanding relaxationthrough CIs involving states of differing transition dipolecharacter.Dynamics in the CI. So far, we have analyzed the dynamics

of DMABN relaxation through a CI in both adiabatic anddiabatic representations. In a diabatic representation that wouldbe probed by stimulated emission, we observe a single rate ofdecay at 15 fs, which is in between the experimental ratesreported for DMABN in the gas phase (5 and 68 fs). However,given that the interaction with MeCN does not appear to affectthe nonadiabatic rate, our 15 fs rate is consistent with the sub-30 fs rate observed through time-resolved fluorescence ofDMABN in MeCN, leading us to suggest that the 5 fsexperimental signal observed in the gas phase likelycorresponds to the nonadiabatic transition.Now, regarding the experimental signal at 68 fs, our

simulations cannot make any new predictions. Historically,this rate was assigned to passage through a CI based on theassumption that torsional motion was one of the modes of thebranching plane (and the main nuclear degree of freedom thatgives anisotropy).40−42 However, higher-level electronicstructure calculations have found that the twisting motion isnot in the S1/S2 branching space.23,86 Furthermore, previousFSSH simulations45,46 and ab initio multiple spawning (AIMS)calculations85 predict that the nonadiabatic relaxation throughthe CI occurs much more quickly than the torsional motion.Indeed, in the recent work using AIMS, Martinez et al. foundthat the time scale of torsional motion is in good agreementwith τ2 (68 fs). However, at this point, it is important to pointout that the methods employed in the previous FSSHsimulations (i.e., CAM-B3LYP46 and MP2/ADC(2)45), therecently reported AIMS simulations (using ω-PBE85), and thepresent calculations (ω-B97X) all significantly overstabilize thetwisted S1-ICT state relative to the S1-LE state.23,38 That beingsaid, the separation of time scales (i.e., 5 and 68 fs) suggeststhat the shorter-time dynamics may still be reliable.Finally, before we conclude, we note that attosecond

spectroscopy has the potential to become a powerfulexperimental tool for visualizing nonadiabatic dynamics.87,88

To that extent, there is one more set of physical effects that wehave not yet discussed and that deserves mention, namely,geometric phase effects.89,90 In the BO picture, the electronicwave functions (Φj(r;R)) are parametric functions of thenuclear coordinates R whose solutions (i.e., the adiabaticelectronic states) satisfy the time-independent Schrodingerequation. When the nuclear coordinates are varied to encircle aCI, the sign of the electronic wave function must change due tothe geometric (or Berry) phase. In order for the total (nuclear+ electronic) wave function Ψ(r,R,t) = ∑jΦj(r;R)χj(R,t) toremain single-valued, the nuclear wave function (χj(R,t)) mustalso be double-valued to account for the sign of the electronicwave function. The phase of the nuclear wave function cancause trajectories that evolve in opposite directions around a CIto destructively interfere. In the case of dynamical processesproceeding on a single adiabatic surface around an energeticallyinaccessible CI, this interference can suppress the rate ofreaction.91 Recent analysis of the continuity of the totalmolecular wave function near CIs suggests that mixedquantum−classical methods such as FSSH should not includethe second-order derivative coupling or diagonal BO correction(DBOC, ⟨Φi(R)|∇R2Φi(R)⟩r).

92 Furthermore, in excited-statepopulation relaxation, Izmaylov and co-workers93 have arguedthat the suppression of reactivity caused by the geometric phaseis compensated by neglecting the DBOC, which is divergent atthe CI,94 such that FSSH actually performs fairly well near CIs(at least for model systems). There is no guarantee, however,that such an error cancelation can be generally relied upon togive accurate short-time dynamics for realistic, multidimen-sional ab initio Hamiltonians.With that in mind, in the context of the DMABN molecule,

the time evolution of the adiabatic-to-diabatic mixing angle canbe used to gain further insight into possible geometric phaseeffects. When the mixing angle, θ, is nπ (n ∈ {..., −1, 0, 1, ...}),the excited states have S1-Lb/S2-La character, while the excited-

state character is inverted at θ = π+n(2 1)2

. The mixing angles

for the swarm of A-FSSH trajectories are presented in Figure 5.

Because the diabatic surfaces cross at θ = π+n(2 1)4

, the rate at

which the molecule passes through the coupling region can beinferred from the slope of the rotation angle at that point. Thetrajectory shown in Figure 3 is highlighted in red in Figure 5. Inthis case, the molecule begins in S2-La, crosses to S1-La from 5to 14 fs (θ = π

2), and finally converts back to S1-Lb (θ = π) after

14 fs. The sharp slope of θ(t) at the crossing points of the redtrajectory indicates that the molecule exists only briefly in thecoupling region. The adiabatic and diabatic energies for atrajectory that persists in the region of strong coupling areshown in Figure S4 and are highlighted in purple in Figure 5; inthis case, the extended nature of the coupling is manifest in theflat slope of θ(t ≈ 10 fs).The product branching through the CI can be obtained by

analysis of the time dependence of the electronic diabatizationangle. As shown in Figure 3, the majority of trajectories proceedfrom S2-La through S1-La and finally to S1-Lb. This is reflected inthe diabatization angle trajectories; all trajectories begin with adiabatization angle of nearly 0 and after 25 fs have almost allrelaxed into the S1-Lb state (the adiabatic identity is inferredfrom the adiabatic density in Figure 2). During this time, sometrajectories circle around the CI, acquiring an electronic phaseof ±2π (i.e., a diabatic angle of ±π). Because the nuclear wavepacket splits apart, with one component traveling around the

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CI clockwise and the other component counterclockwise, onemight expect that the phases would destructively interfere,decreasing the rate of population transfer. Because FSSH isunable to describe the geometric phase, this effect is missingfrom these simulations. Even though mounting evidencesuggests that FSSH dynamics should yield accurate relaxationrates through CIs, one need not necessarily conclude that thesegeometric phase effects cannot be observed experimentally witha collection of short enough pulses. To truly understand short-time DMABN dynamics, the next step forward is clearly to runmore explicit wave function calculations, whereby one caninvestigate such dynamics more rigorously and makepredictions for nonlinear spectroscopic signals. Because of itsmodest size, straightforward electronic structure, and rapiddescent into a CI, DMABN should be a vigorous test case forsuch future quantum dynamical simulations.

■ CONCLUSIONSWe have simulated ultrafast nonadiabatic population relaxationfrom S2 to S1 using A-FSSH simulations of the dual fluorescentmolecule DMABN. Our results are largely consistent withprevious theoretical calculations,45,46 though we have gainedseveral new pieces of insight. First, decoherence is found to notplay an important role in propagating these dynamics due tothe rapid excited-state relaxation and subsequent energeticseparation of the first two excited states. Second, we ransimulations of DMABN in the presence of a MECN moleculeand found that the ultrafast relaxation was largely unaffected bythe solvent, in agreement with time-resolved fluorescencemeasurements of DMABN in MeCN. Third, we introduced aphysically motivated diabatization procedure that is capable ofconverting the TDDFT excited (adiabatic) states into excited(diabatic) states of La and Lb character (which are diagonal withrespect to the rotationally averaged transition dipole operator).Transforming the mixed quantum−classical density matrix intothis diabatic representation, the population relaxation from La

to Lb was found to be somewhat slower than the adiabatic statesand consistent with the decay of the ensemble-averagedtransition moment; specifically, our calculations predict adecay in the emission on the order of 15 fs (rather than 11fs). Fourth and finally, we investigated the dynamics of thediabatization angle for all relaxation events through the CI. Wefound that 16% of the trajectories that we simulated fullycircled the CI clockwise, 18% circled counterclockwise, and theremainder did not circle the CI at all. From these numbers andthe fact that the majority of the electronic relaxation occurswithin 30 fs according to our TDDFT functional, wehypothesize that if geometric phase effects are important inlarger molecules, the 21 atom molecule DMABN will be anideal test case for higher-level calculations, both for theelectronic structure and the nuclear dynamics. It remains anopen question as to whether or not we can predict anobservable to prove such effects with nonlinear spectroscopy.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.jpca.6b12120.

Attachment and detachment densities for DMABN indifferent configurations, a comparison of adiabatic anddiabatic population dynamics using FSSH and A-FSSH,excited-state energies for the trajectory highlighted inpurple in Figure 5, an examination of decoherence times,computational scaling of the new A-FSSH algorithmversus the number of FSSH states, and the temperaturedependence of the adiabatic population dynamics forDMABN with and without MeCN (PDF)

■ AUTHOR INFORMATIONCorresponding Author*E-mail: gmedders@sas.upenn.edu.ORCIDGregory R. Medders: 0000-0001-8988-4401NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work was supported by NSF CAREER Grant CHE-1150851. J.E.S. gratefully acknowledges support from theStanford PULSE Institute and a John Simon GuggenheimMemorial fellowship.

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Figure 5. Adiabatic-to-diabatic rotation angle versus time followingexcitation for the swarm of A-FSSH trajectories. An angle of nπ, wheren is an integer, corresponds to the S1 state having Lb character and S2having La, while angles of π+n(2 1)

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