Nonadiabatic Molecular Dynamics for Thousand Atom Systems: ATight-Binding Approach toward PYXAIDSougata Pal,† Dhara J. Trivedi,∥ Alexey V. Akimov,§ Balint Aradi,‡ Thomas Frauenheim,‡

and Oleg V. Prezhdo*,†

†Department of Chemistry, University of Southern California, Los Angeles, California 90089, United States∥Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, United States§Department of Chemistry, University at Buffalo, The State University of New York, Buffalo, New York 14260-3000, United States‡Bremen Center for Computational Materials Science, Universitat Bremen, Otto-Hahn-Alle 1, 28359 Bremen, Germany

ABSTRACT: Excited state dynamics at the nanoscale requirestreatment of systems involving hundreds and thousands ofatoms. In the majority of cases, depending on the processunder investigation, the electronic structure component of thecalculation constitutes the computation bottleneck. Wedeveloped an efficient approach for simulating nonadiabaticmolecular dynamics (NA-MD) of large systems in theframework of the self-consistent charge density functionaltight binding (SCC-DFTB) method. SCC-DFTB is combinedwith the fewest switches surface hopping (FSSH) anddecoherence induced surface hopping (DISH) techniques forNA-MD. The approach is implemented within the Pythonextension for the ab initio dynamics (PYXAID) simulationpackage, which is an open source NA-MD program designed tohandle nanoscale materials. The accuracy of the developed approach is tested with ab initio DFT and experimental data, byconsidering intraband electron and hole relaxation, and nonradiative electron−hole recombination in a CdSe quantum dot andthe (10,5) semiconducting carbon nanotube. The technique is capable of treating accurately and efficiently excitation dynamics inlarge, realistic nanoscale materials, employing modest computational resources.

1. INTRODUCTION

Nanoscale materials, composed of hundreds and thousands ofatoms, bridge the gap between molecules and bulk. Theyexhibit a variety of interesting and novel phenomena, which arebeing explored for technological and medical applications. Forinstance, nanoscale systems enable one to efficiently harvestalternative energy sources, using devices such as photo-voltaic1−3 and photocatalytic3,4 cells. The applications stimulateextensive research on charge and energy transfer in hybridnanoscale materials and interfaces,5 because these processesdetermine efficiencies of solar cells and other devices.Absorption of a photon creates an electronically excited state,which subsequently dissociates into a pair of spatially separatedcharges. Charge separation occurs at interfaces5−9 andcompetes with energy losses due to thermal relaxation ofexcited electrons and holes and electron−hole recombination.In order to understand and control the competition betweenthe alternative dynamical pathways, with the goal of improvingsolar cell parameters, such as voltage, current, and photon-to-electron conversion yield, one needs to investigate the highlynonequilibrium nature of the excited state dynamics at theatomistic level of detail.10 Atomistic description is required inorder to account for the intrinsically statistical origin of

nanoscale materials, involving relatively broad distributions ofsizes, surfaces, and chiralities, nonstoichiometric and spatiallyvarying compositions, etc. Perhaps even more importantly,nanoscale materials are synthesized by chemists and involve,intentionally or unintentionally, defects, ligands, dopants, andother atom-size features that often govern material’s propertiesand should be well understood.Numerous experimental11−17 and theoretical18−31 efforts are

dedicated to studies of nonequilibrium, excited state dynamicson the nanoscale. Ultrafast, time-resolved spectroscopy32,33 is aparticularly valuable experimental tool that helps characterizingthe excited states dynamics in many materials. It allows one toobtain the time scale of each process and to determine therelationship between favorable carrier generation and transportand nonproductive energy losses. The analysis of these dataleads to a thorough understanding of the origins of a particulardevice performance. Theoretical studies parallel the exper-imental characterization, since they interpret experimentalresults, establish mechanisms of the processes under inves-tigation, and provide insights into many important details,

Received: December 28, 2015Published: March 8, 2016

Article

pubs.acs.org/JCTC

© 2016 American Chemical Society 1436 DOI: 10.1021/acs.jctc.5b01231J. Chem. Theory Comput. 2016, 12, 1436−1448

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ultimately stimulating generalizations and the development ofessential design principles. Arguably, nonadiabatic (NA)molecular dynamics (MD)34,35constitutes the most efficientmeans for modeling nonequilibrium excited state phenomena.Various theoretical groups perform NA-MD simulations ofatomic, molecular, nanoscale, and bulk systems in order tocapture photoinduced phenomena, such as electron−holeseparation, energy exchange and recombination,36−46 chargeand energy transfer and transport,47−50 singlet fission,28,51,52

multiple exciton generation,22−24,53,54 photochemistry, etc.55−57

Still, theoretical studies are few in comparison to thecorresponding experimental investigations, partially due tocomputational limitations on system size and simulation time.58

More efficient computational NA-MD approaches are, there-fore, required.NA-MD is often used in practice in its quantum-classical

formulation. Within the quantum-classical framework, themotion of nuclei is treated classically, whereas the electronicevolution is propagated quantum mechanically. NA-MDgeneralizes classical ground-state MD to include NA transitionsbetween quantum (electronic) states. NA-MD simulationsinvolve two components: an algorithm for the electron−nucleardynamics, including the possibility of NA transitions, and amethod for electronic structure calculations, which providesnuclear forces and NA couplings.59,60 Typically, the electronicstructure is computed “on the fly”61−66 and provides input tostochastic trajectory surface hopping (TSH)34,67 algorithms.Many electronic structure methods are used for this purpose,including ab initio “frozen ionic bond”,68 ab initio configurationinteraction,69 time-dependent density functional theory (TD-DFT),70−74 and semiempirical methods.55,75−77 TD-DFT isparticularly popular in this regard, since it provides a goodcompromise between accuracy and efficiency. DFT is transfer-rable and applicable to the majority systems. It can treat mostof the relevant processes and can take into account the fullcomplexity of a material, including composition, shape, defects,dopants, surface, ligands, and environment. Still, TD-DFTsimulations are limited to about hundred atoms systems.Rooted in DFT and parametrized against it, the SCC-DFTB iscapable of treating much larger systems.A number of programs featuring the NA-MD capability have

been developed over the past decade.78−87 Most of them relyon computationally expensive electronic structure methods and,thus, are applicable to rather small systems. The codes typicallyimplement the most basic NA-MD techniques, such as theEhrenfest88−93 and standard TSH34,66 methods. These NA-MDapproaches capture only some of the processes involved inoperation of nanoscale devices. For instance, the Ehrenfestmethod mistreats electron-vibrational energy exchange and failsto equilibrate the system to Boltzmann distribution. Thestandard TSH cannot handle properly superexchange andmany-particle transitions.94,95 Both the Ehrenfest and standardSH disregard decoherence effects, leading to discrepancies incomputed and experimentally measured time scales forelectronic dynamics by orders of magnitude.The Prezhdo group has developed a spectrum of advanced

NA-MD techniques94−100 designed for studying the interplaybetween productive processes, such as charge and energytransfer, and energy losses to heat to due electron-vibrationalrelaxation and charge recombination. The focus is on nanoscaleand condensed phase systems. Important practical advancesinclude decoherence induced surface hopping (DISH),96 globalflux surface hopping (GFSH),94 and classical path approx-

imation (CPA) for SH.97 DISH incorporates decoherenceeffects in a natural and efficient way. GFSH treats super-exchange and many-particle Auger-type phenomena. CPA-SHgreatly increases simulation efficiency. The utility of these andrelated methods has been demonstrated with differentnonequilibrium phenomena occurring in a variety of nanoscalesystems.101−105 Recently, the Prezhdo group released an opensource package for simulation of NA dynamics on thenanoscale.103,104 Named PYXAID for PYthon eXtension forAb Initio Dynamics, it interfaces the Kohn−Sham (KS)106

formulation of NA-MD70,107 with Quantum Espresso108 andVienna Ab initio Simulation Package (VASP),109 which areused as DFT drivers for adiabatic electronic structure and MDcalculations. PYXAID implements advanced techniques forintegration of the time-dependent Schrodinger equation (TD-SE), and a number of basic and more advanced NA-MDfunctionalities, including fewest-switches surface hopping(FSSH),67 DISH,96 multielectron adiabatic representation ofthe time-dependent KS (TD-KS) equations, and directsimulation of photoexcitation via explicit light-matter inter-action. The CPA implemented in PYXAID achieves consid-erable computational savings. With the exception of processesinvolving hundreds of thousands of electronic states, such asmultiple exciton generation,53,101,102,110,111 the computationalbottleneck of PYXAID is determined by the scaling of theelectronic structure computations.Realistic representation of many nanoscale systems requires

hundreds and thousands of atoms, extending beyond thecapabilities of the existing NA-MD codes. The self-consistentcharge density functional tight binding method (SCC-DFTB)112−118 is capable of increasing the system size by anorder of magnitude. Parameterized against DFT, the SCC-DFTB provides an accurate quantum-mechanical description ofgeometries, vibrational frequencies, reaction energies, and otherproperties115 of very large systems at a modest computationaleffort.In this article, we describe a technique for NA-MD

simulation of many atom systems by combining SCC-DFTB112−118 with PYXAID.103,104 The approach is illustratedwith charge-phonon relaxation in a CdSe quantum dot119,120

(QD) and nonradiative electron−hole recombination a semi-conducting carbon nanotube (CNT),121,122 including bothinelastic and elastic electron−phonon scattering effects. Theresults obtained from NA-MD simulations based on the SCC-DFTB formalism (DFTB-NA-MD) are then tested against thatof ab initio DFT based NA-MD (DFT-NA-MD) and arecompared with experimental data.The paper is organized as follows. Section 2 describes the

computational method. Namely, subsection 2.1 presents thebasic formalism of the electronic structure calculation byDFTB, while subsection 2.2 outlines the working principles ofthe NA-MD simulation as implemented in PYXAID. Section 3reports the QD and CNT simulations, demonstrating andvalidating the approach. The paper concludes with a summaryand outlook.

2. THEORETICAL METHOD

2.1. Electronic Structure Calculations and the DFTB+Code. The SCC-DFTB method112−118 is implemented withinthe DFTB+ code.112,123It is used to calculate the electronicstructure for the system of interest on-the-fly. The foremostidea of the SCC-DFTB method is to derive the total energy

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from the second order Taylor series expansion of the KS-DFTtotal energy functional around a given reference density ρ0(r)

∫ ∫

∫ ∫ ∫

∑ ψ ψδδρδρ

δρδρ

ρ ρρ ρ ρ

= ⟨ | | ⟩ +| − ′|

+′| ′

−′

| − ′| ′ + − +

ρ

⎛⎝⎜

⎞⎠⎟E n H

r rE

r rdrdr E V dr V

12

1

12

[ ] [ ]

i

occ

i i ixc

xc xc NN

02

0 00 0 0

0

(1)

where H0 in the first term of the above equation is the KSHamiltonian at the reference density, ψi are the KS orbitals, andni are the occupation numbers. The second term in eq 1explicitly depends on the interactions between chargefluctuations with the Coulomb and exchange-correlationintegral kernel. It can be decomposed into atom-centeredmonopole contributions113 of the form

∑ γΔ Δαβ

α β αβq q12 (2)

Here, Δqα and Δqβ denote atomic net Mulliken charges ofatoms α and β. The function γαβ

114 interpolates between a pureCoulomb interaction for large interatomic distance and anelement-specific constant in the atomic limit, which is directlyrelated to chemical hardness113 of the atomic species. The thirdterm represents the Hartree Coulomb interaction. EXC, VXC, andVNN represent the exchange-correlation energy, the correspond-ing potential, and the nucleus−nucleus repulsion terms,respectively.In SCC-DFTB, the KS orbitals, ψi are expanded in linear

combinations of the atomic orbitals (LCAO)124of the form

ϕ = ∑ ∑μ ζ ζζ+ − ′( )c r e Y( )i

l i rlm

rr. These are obtained variation-

ally by solving the atomic KS equation with an additionalharmonic confining potential, (r/r0)

2

ρ ϕ ε ϕ− ∇ + + =μ μα

μα

μα

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥v

rr

12

( )eff2

0

2

(3)

The introduction of the additional potential term124,125 intothe equation forces the atomic wave functions to avoid areas faraway from the nucleus resulting in compressed electron densityrather than that of the free atom. The approach gives betterelectronic structure results for solids, molecules, and clusters.117

Once the basis set is defined, the matrix elements of thezeroth-order Hamiltonian can be calculated within the twocenter approximation

ε μ

ϕ ρ ρ ϕ α β =

=

⟨ | − ∇ + + | ⟩ ≠μν

μα

μα α β

νβ

⎧⎨⎪⎪

⎩⎪⎪

H

if v

v if

otherwise

12

[ ]

0

eff0 2

0 0

(4)

where vef f is the effective KS potential (external, Hartree plusexchange-correlation) corresponding to the reference electrondensity ρ0. The latter is given as a superposition of the electrondensities of the individual neutral atoms, α and β. The diagonalterms of the matrix elements are the DFT atomic eigenvalues.All off-diagonal terms are obtained via DFT calculations for anα−β dimer. With this approximation, the SCC-DFTB totalenergy can be expressed as follows

= +E E EtotalDFTB

elecDFTB

rep (5)

where the second term is represented as

∫ ∫ ∫ρ ρρ ρ ρ= −

′| − ′|

+ − +Er r

E V dr V12

[ ] [ ]rep xc xc NN0 0

0 0 0

(6)

The energy Erep in eq 6 can be approximated as a sum ofshort-range two-body potentials i.e., = ∑αβ

αβE V R( )rep rep . It can

be obtained by the difference of the total energy resulting froma self-consistent DFT calculation and the electronic part of theSCC-DFTB as a function of internuclear distance of chosenreference systems. The SCC-DFTB electronic energy is thengiven by

∑ ∑ ϕ ϕ γ μ α ν β= ⟨ | | ⟩+ Δ Δ ∀ ∈ ∈αβ μ ν

μ ν μ ν αβα β

⎛⎝⎜⎜

⎞⎠⎟⎟E a a H q q

12

, ,elecDFTB

,

0

(7)

where aμ and aν are the combination coefficients used toexpand the KS wave function. As the KS wave function hasbeen written as a LCAO ansatz, thus, for a given set of atomicpositions, a trial set of combination coefficients can be taken toconstruct a SCC-DFTB secular equation of the form

∑ ε μ α ν β− = ∀ ∈ ∈μν μν μνa H S( ) 0, ,i

i(8)

where aμν and εi represent the coefficient of the KS molecularorbital and its corresponding orbital energy, respectively. Hμν

and Sμν are the Hamiltonian and overlap matrices given by

∑ γ γ= + Δ +μν μν μνω

ωαω βωH H S q

12

( )0

(9)

ϕ ϕ μ α ν β= ⟨ | | ⟩ ∀ ∈ ∈μν μ νH H ,0 0(10)

ϕ ϕ μ α ν β= ⟨ | ⟩ ∀ ∈ ∈μν μ νS , (11)

Thus, eq 8 can be solved self-consistently with respect to thecharge fluctuation Δq, in conjunction with eq 7 to find theoptimal set of the linear combination coefficients.116,126

The computational efficiency of the SCC-DFTB approach isbased on the fact that all necessary Hamiltonian and overlapmatrix elements for each pair of atoms in a system versusdistance only need to be calculated once, and stored in aseparate file, known as the Slater-Koster table.

2.2. Nonadiabatic Molecular Dynamics Simulationand the PYXAID Code. “On-the-fly” NA-MD simulationdeveloped in the present work requires calculation of adiabaticelectronic state energies, NA couplings, dki, and forces acting onthe nuclei. These are calculated in the framework of the SCC-DFTB method.65,127,128

The time-dependent wave function of the system isrepresented in the basis of the KS orbitals as

∑ψ ϕ| ⟩ = | ⟩=

r R t c t r R t( ; ( )) ( ) ( ; ( ))k

N

k k0 (12)

where ck(t) is the time-dependent expansion coefficient, and |ϕk(r;R(t))⟩ is the adiabatic wave function representing theelectronic state k, which is parametrically dependent on theclassical nuclear trajectory R(t). N is the size of the electronicbasis set used in temporal evolution. In the NA-MD simulation,

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nuclear degrees of freedom are treated classically, and nucleartrajectories, R(t), are obtained by solving the classical Newton’sequations of motion using quantum-mechanical forces. Byusing the ansatz, eq 12, time evolution of the coefficients ck(t)along a given classical trajectory can be obtained by solving thetime-dependent Schrodinger equation

∑ ε δℏ = − ℏ=

idc t

dti d c t

( )( ) ( )k

i

N

k ki ki i0 (13)

where

ϕϕ

=∂

∂d r R t

r R t

t( , ( ))

( , ( ))ki k

i

(14)

Here, εk is the energy of the adiabatic state k, and dki is the NAcoupling between states k and i.The NA coupling arises because electronic wave functions

depend parametrically on nuclear coordinates. It reflects theinelastic electron-vibrational interaction. The coupling iscalculated numerically using the overlap of orbitals k and i atsequential time steps129

ϕ ϕ

ϕ ϕ

ϕ ϕ

ϕ ϕ

= − ℏ⟨ |∇ | ⟩

=− ℏ⟨ ∂∂

⟩

≈ − ℏΔ

⟨ | + Δ ⟩

−⟨ + Δ | ⟩

d i r R t r R tdR t

dt

i r R tt

r R t

it

r R t r R t t

r R t t r R t

( , ( )) ( , ( ))( )

( , ( )) ( , ( ))

2( ( , ( )) ( , ( ))

( , ( )) ( , ( )) )

ki k R i

k i

k i

k i (15)

where Δt is the time step used for the integration of theclassical Newton’s equation of motion.The numerical solution of eq 13 yields the time-dependent

amplitudes of the adiabatic states, ck(t). The amplitudes areutilized to calculate the hopping probabilities needed forswitching the trajectory between the electronic states.Classical-mechanical prescription of nuclear dynamics in

response to changes in electron density is known as thequantum backreaction problem.97 NA-MD provides a general-ization of the original SH approach.130 Correlations betweenthe nuclear motion and electronic states are built in using SHtechniques. SH can be interpreted as a master equation, inwhich transition rates are nonperturbative and evolve in time.Tully’s FSSH algorithm34 (also called molecular dynamics withquantum transitions or MDQT) prescribes a probability forhopping between electronic states. FSSH minimizes thenumber of surface hops, while maintaining consistency withthe Schrodinger equation.34 Moreover, it satisfies approximatelythe detailed balance between transitions upward and downwardin energy,67 as required for proper description of electron-vibrational energy exchange and relaxation to thermodynamicequilibrium.131

The probability of hopping from state k to i within the timestep dt is given in FSSH by34

=dPbc

dtkiki

ki (16)

where

ϕ ϕ= * = ℏ ⟨ | | ⟩ −−c c t c t b a H c d( ) ( ) and 2 Im( ) 2Re( )ki k i ki ki k i ki ki1

(17)

If the calculated dPki is negative, the hopping probability isset to zero. A hop from state k to state i can occur only whenthe electronic occupation of state k decreases and theoccupation of state i increases, minimizing the number ofhops. To conserve the total electron−nuclear energy after ahop, the original FSSH technique34 rescales the nuclearvelocities along the direction of the NA coupling. If a NAtransition to a higher energy electronic state is predicted by eq16, while the kinetic energy available in the nuclear coordinatesalong the direction of the derivative coupling vector isinsufficient to accommodate the increase in the electronicenergy, the hop is rejected.Despite the great success of FSSH, absence of the

decoherence effects and the problem due to frustrated hoppingevents, which prohibit modeling of superexchange andmultiparticle phenomena, have hindered the range of FSSHapplications. Advanced SH approaches, such as DISH96 andGFSH,94 have been developed and implemented in thePYXAID code by the Prezhdo group in order to addressthese issues. By combining PYXAID with DFTB, we haveintegrated the advanced techniques for simulating NA-MDwithin the tight binding approximation (DFTB-NA-MD).

3. RESULTS AND DISCUSSIONSTo validate the accuracy of the developed DFTB-NA-MDapproach, herein we compare the DFTB-NA-MD method withab initio DFT-NA-MD and experiments for the relaxationdynamics of hot carriers in two low dimensional nanostructuredmaterials. In particular, we study electron and hole relaxation ina zero-dimensional nanoscale cluster, known as QDs, andelectron−hole recombination involving both inelastic andelastic electron−phonon scattering in a one-dimensionalsingle-walled carbon nanotube (SWNT). Their potential inmodern nanotechnology, in particular photovoltaic and photo-cata ly t ic appl icat ions , has received great at ten-tion.5,26,119−122,132−135 The electronic structure of low dimen-sional systems is drastically different from the electronicstructure of the corresponding bulk materials. Investigation ofthe dynamical processes in such systems provides solidvalidation of our method, which is designed for modeling ofnanoscale objects. The considered dynamical processes carryboth fundamental and practical importance. In this work, wehave investigated two different systems: the Cd33Se33 QD and(10,5) SWNT. They are described in sections 3.1 and 3.2,respectively.

3.1. Intraband Charge Relaxation in a CdSe QuantumDot. CdSe gives rise to one of the most investigated types ofsemiconductor QDs. The hot carrier relaxation dynamics inthese systems has been extensively studied, both experimen-tally120 and theoretically.119 With this motivation, we havechosen the Cd33Se33 QD as a model system to study theintraband electron and hole relaxation dynamics by using bothDFTB-NA-MD and DFT-NA-MD simulation techniques. TheCd33Se33 QD is derived from wurtzite bulk structure and is a“magic” size cluster with diameter of 1.3 nm. It is one of thesmallest stable CdSe QD with crystalline-like core.133,136,137

This makes the Cd33Se33 cluster an excellent model forquantum-mechanical studies of electronic properties of CdSeQDs.The ab initio DFT calculation has been carried out with a

plane-wave basis, as incorporated in VASP.109 The valenceelectrons are described with the PBE exchange-correlationfunctional138 and a converged plane-wave basis. The core

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electrons are accounted for with the projector-augmented-wave(PAW) pseudopotentials.139 A periodically replicated cubic cellwith at least 8 Å of vacuum between the QD replicas is used asa molecular model of the nonperiodic cluster. DFTBcalculations have been performed by using the SCC-DFTB112−118 scheme as implemented in the DFTB+code112,123 using the recently developed parameters, whichhave been tested extensively for a variety of Cd-chalcogenidesystems.140 Similar simulation strategies were adopted for bothDFT and DFTB. Starting with the initially optimized groundstate cluster, at 0 K, we heated the system to 300 K withrepeated velocity rescaling. Microcanonical trajectories weregenerated for 3 ps using the Verlet algorithm141 with 1 fs timestep and Hellman-Feynman forces. At each snapshot, theenergies of the KS molecular orbitals and the correspondingstate-to-state NA couplings were calculated at both the DFTBand DFT levels. This time dependent information was storedand used in the NA-MD simulations, performed with CPA-FSSH.In order to make a general analysis of the DFTB and DFT

calculations, we first compare the geometric parameters of theQD obtained from both methods. Surface reconstructionprovides a typical test, since it determines the interionicinteractions and the force tolerance on individual ions. TheDOS plot obtained by DFTB of this bare QD is shown inFigure 1 (a) with the optimized structure given in the inset. It isnoteworthy that, for the stoichiometric cluster such Cd33Se33,the surface relaxation affects all surface atoms similarly, andthus all surface Cd−Se bonds are approximately equal. Thesurface reconstruction predicted by the DFTB and DFT issimilar, even though it is known that DFTB overbinds.142 TheDFTB DOS is consistent with DOS from the DFT calculationsas shown in Figure 1 (b). The calculated zero temperature bandgap obtained with DFTB is slightly larger than the onecomputed at the DFT level: ca. 1.84 eV and ca. 1.6 eV,respectively. A common feature of all semilocal exchange-correlation functionals, such as PBE used in the present work,ab initio DFT tends to underestimate the band gap.143,144 Onthe contrary, the DFTB-derived band structure is not affectedin the same way as in LDA or GGA-PBE calculations. This is aconsequence of the small basis set employed. As a result, theband gap obtained in DFTB is much closer to theexperimentally determined value.145

The DOS shows that the HOMO is dominated by the Seatoms, while the isolated LUMO state is mostly composed of

Cd-localized atomic orbitals. The projected DOS (PDOS) ofboth DFT and DFTB calculations shows that there is a notablecontribution of the Se atom electron to the total electrondensity of the valence band edge, while the p orbitals of Cdatoms contribute to the total electron density of the conductionband edge of the CdSe QD. The DOS plot also shows that forboth cases the state structure in the valence band (VB) is lesspronounced as compared to the conduction band (CB), sincethe states are closer in energy in the VB. This fact agrees withthe effective-mass theory, which uses a higher effective mass forholes than for electrons. Our results show that the geometricrelaxation and the electronic structure obtained from DFTBagree with those of the DFT calculation, which is computa-tionally much more demanding.Next, we compare the average absolute values of the NA

couplings between representative states computed by DFT andDFTB, Table 1.We considered first four energy states in both

VB (HOMO to HOMO−3) and CB (LUMO to LUMO+3).The NA couplings calculated by the two methods range from1.13 to 13.61 meV. In both VB and CB, the NA coupling ishigher between two neighboring energy levels than betweenmore distant levels. Also, the coupling generally increases withan increase in the energy level. The NA couplings obtained byDFTB and DFT follow similar trends and are consistent witheach other. The agreement between the DOS and NA

Figure 1. Projected density of states of the Cd33Se33 quantum dot calculated by (a) DFTB and (b) DFT. The optimized geometry is shown in theinset obtained by (a) DFTB and (b) DFT. Blue and red balls represent Cd and Se atoms, respectively. The zero of energy is set equal to the middleof the HOMO−LUMO gap.

Table 1. Average Absolute Values of Nonadiabatic Couplingbetween Representative Pairs of States in the Cd33Se33Quantum Dot Computed by DFTB and VASP

⟨|NAC|⟩, meV DFTB DFT

(HOMO)-(HOMO−1) 8.86 5.57(HOMO)-(HOMO−2) 1.97 2.25(HOMO)-(HOMO−3) 1.13 1.56(HOMO−1)-(HOMO−2) 9.85 8.55(HOMO−1)-(HOMO−3) 3.01 2.91(HOMO−2)-(HOMO−3) 10.61 9.40(LUMO+1)-(LUMO) 5.72 3.42(LUMO+2)-(LUMO) 1.70 1.71(LUMO+3)-(LUMO) 1.29 1.36(LUMO+2)-(LUMO+1) 10.41 7.43(LUMO+3)-(LUMO+1) 3.87 4.00(LUMO+3)-(LUMO+2) 13.61 8.79

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couplings leads to agreement in the electron and hole relaxationdynamics.To study the intraband relaxation dynamics of excited

electron and hole in the Cd33Se33 QD we prepare initialexcitations by creating a hole approximately 1 eV below theHOMO level and an electron approximately 1 eV above theLUMO level. The dynamics of the electronic excitation isrestricted to the dense mainfold of unoccupied states above theLUMO, excluding the latter. The LUMO is energetically well-separated from the rest of the CB, giving rise to the so-calledphonon bottleneck to the electron relaxation.119 Simulation ofsuch slow relaxation processes requires incorporation ofdecoherence effects, as exemplified in the following section.The quasi-continuous DOS in the VB suggests ultrafastrelaxation of holes without a phonon bottleneck.Figure 2 presents the time evolution of the excited electron

computed at the DFTB and DFT levels of theory, panels (a)

and (b), respectively. The figures show a three-dimensional plotof the state populations as a function of energy and time. Inboth instances, the initial photoexcitation peak at around 0.8 eVvanishes and reappears in the final states. Although the carriersvisit multiple intermediate states during relaxation, none ofthem play any special role. Analogous results computed for holerelaxation are summarized in Figure 3. Figures 2 and 3 indicatethat the dynamics of electron and hole obtained with DFTB-NA-MD and DFT-NA-MD are very similar, validating theformer method. To obtain further insight into the relaxation

dynamics of electrons and holes, we compute population-weighted energy of the evolving electron and hole (Figure 4).The zero of energy is chosen to be at the average energy ofLUMO+1 for electron and HOMO for hole. Black and redcurves are obtained from DFTB-NA-MD and DFT-NA-MDsimulations, respectively. The decay curves predicted by bothsimulation techniques are composed of two parts. The short-time Gaussian decay gradually switches to exponentialdynamics. The observed nonexponential relaxation componentis in agreement with the strongly non-Lorentzian line shapesobserved experimentally.146 The exponential componentbecomes important at lower energies for the electrons and athigher energies for the holes. The results indicate that DFTB-NA-MD and DFT-NA-MD predict comparable electron andhole transfer dynamics and the energy decay times.We characterize the phonon modes participating in the

intraband relaxation of electron and hole by computing Fouriertransforms (FT) of the energy gaps between the initial and finalstates of the relaxation processes. Parts (a) and (c) of Figure 5refer to electron, while parts (b) and (d) represent hole. Theseinfluence spectra are computed using both DFT (panels a, b)and DFTB (panels c, d). Two phonon frequencies are mainlyassociated with electron relaxation, one below 100 cm−1 andanother around 200 cm−1. The peak below 100 cm−1 is lessintense in DFT than DFTB. The difference can be attributed toa difference in the extent of surface relaxation between the twomethods. Both DFTB and DFT predict that the hole couples tothe phonons with a wide range of frequencies spreading over400 cm−1. The corresponding spectra are broader than thosefor the electron relaxation. The overall appearance of the FTspectra for both electron and hole relaxation is qualitativelyconsistent between DFTB and DFT, although notablequantitative differences are seen in this case.We conclude that the newly developed DFTB-NA-MD

formalism performs well in characterizing the charge-phononinteraction and energy exchange during intraband relaxation ofelectrons and holes in the CdSe QD. The results are inagreement with DFT-NA-MD simulations. The band gap inparticular is more accurate in DFTB and DFT. At the sametime, the DFTB-NA-MD scheme is computationally muchmore efficient that the ab initio method.

3.2. Electron−Hole Recombination in a CarbonNanotube. As a further validation and robustness test of theDFTB-NA-MD scheme, we applied it to study the electron−hole recombination across the fundamental bank gap in asemiconducting SWNT. We compared the simulation resultswith the experimental time scales for both inelastic and elasticelectron−phonon scattering in this case. The carrier relaxationdynamics in SWNT is well investigated both theoret-ically121,147,148 and experimentally.122,132,135,149−151 Reproduc-ing the exciton lifetime of a semiconducting SWNT is a goodbenchmark for the new implementation of the NA-MDmethodologies. Exciton life times are notably shorter inSWNTs compared to other semiconductor materials. Experi-ments produce electron−hole recombination times on theorder of 100 ps.122,132 Motivated by these experiments, wecompute the recombination time in the (10,5) chiral SWNT,using the DFTB-NA-MD approach.The electronic structure of SWNT is strongly dependent

upon the chirality of the tube; in particular, the (10,5) SWNT isfound to be semiconducting with the experimental band gap of0.86 eV.152 Figure 6 depicts the DFTB calculated DOS of the(10,5) SWNT at 0 K along with its optimized geometry. The

Figure 2. Intraband relaxation of electron energy at 300 K computedwith FSSH using (a) DFTB and (b) ab initio DFT.

Figure 3. Intraband relaxation of hole energy at 300 K computed withFSSH using (a) DFTB and (b) ab initio DFT.

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simulations are performed in a periodically replicated cell with30 Å of vacuum between SWNT replicas in the X- and Y-directions. The tube is periodic in the Z-direction with theoptimized cell length of 11.33 Å. The optimized tube diameteris 10.30 Å, and the total number of carbon atoms in thesimulation cell is 140. Ab initio DFT-NA-MD simulations arerather challenging in this case. The DFTB calculations wereperformed using the DFTB+ code112,123 with a well testedparameter set developed for carbon based systems.114 The k-point sampling was done with the (1 × 1 × 256) Monkhorst−Pack k-point grid. The calculated DOS agrees well with thepreviously reported theoretical results.153 The calculated energygap between the VB maximum (VBM) and CB minimum(CBM) at the gamma point is 0.69 eV.Electron−hole recombination across a large energy gap

occurs much more slowly than quantum transitions betweennearly degenerate states. Further, quantum transitions acrosslarge gaps are slower than loss of quantum coherence, requiringan explicit treatment of decoherence effects in the quantum-

Figure 4. Evolution of (a) electron and (b) hole energy during intraband relaxation at 300 K computed with FSSH using DFTB and DFT.

Figure 5. Fourier transform of the energy gap between the initial and final states of (a), (c) electron and (b), (d) hole relaxation computed using (a),(b) DFT and (c), (d) DFTB.

Figure 6. Density of states of the (10,5) single wall carbon nanotube at0 K calculated by DFTB. The optimized geometry of the unit cell isshown in the inset. The zero of energy is set equal to the middle of theband gap.

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classical NA-MD simulation. Hence, we used the DISHmethod.96

Figure 7 shows the decay of population of the lowest energyexcitation in the (10,5) SWNT. The relaxation follows

exponential decay, P(t) = exp(−t/τ1). Using the short-time,first-order Taylor expansion, we obtained the electron−holerecombination time of 206 ps, which is comparable with theexperimental result of Huang and Krauss.122

To elucidate the atomic motions that are responsible for theenergy loss, we have identified the phonon modes that couplewith the electronic subsystem. Figure 8 presents FT of the

VBM-CBM energy gap in the (10,5) SWNT. The FT spectrumindicates that the electronic subsystem couples to the G-typelongitudinal optical (LO) phonon. The computed frequency isaround 1800 cm−1 and is overestimated compared to theexperimental value of 1600 cm−1. Malola et al.154 showed thatgamma-point calculation by the DFTB method overestimatesthe frequency of the high-energy G-mode in SWNTs. Theoverestimation occurs due to the incomplete atomic basis setused and the approximate description of part of the electron−electron interactions by pair potentials.Electron−phonon interactions give rise to both elastic and

inelastic scattering. The energy released during the electronictransition of from CBM to VBM is accommodated via inelasticprocess leading to energy exchange between the electronic andvibrational subsystems. Elastic electron−phonon interactiondestroys coherence between the CBM and VBM states. Thecoherence is formed during both nonradiative and radiative

transitions. In the latter case, within the framework of theoptical response theory, elastic electron−phonon scattering isknown as pure-dephasing.155 In particular, its inversedetermines the line width of single particle luminescence.The pure-dephasing time is inversely related to the

magnitude of the energy gap fluctuation: the smaller thefluctuation of the gap, the larger the dephasing time.156 Thepure-dephasing time is also related to the memory of theenergy gap fluctuation. Using the optical response theory, onecomputes the gap autocorrelation function (ACF)

= ⟨Δ Δ ⟩C t E t E( ) ( ) (0) T (18)

where the bracket indicates canonical averaging. The ACFnormalized by its initial value

=⟨Δ Δ ⟩

⟨Δ ⟩C t

E t EE

( )( ) (0)

(0)normT

T2

(19)

is shown in Figure 9 (a). The decay is very slow and indicateslong memory in the gap fluctuation. The extremely long decay

of the ACF arises because the electronic subsystem couplesonly to the G-mode (Figure 8), and because this phonon is veryharmonic.111

The pure-dephasing function is computed using the second-order cumulant expansion of the optical response function as155

= −D t g t( ) exp( ( ))cumu (20)

where g(t) is

∫ ∫τ τ τ=τ

g t d d C( ) ( )t

01

02 2

1

(21)

Figure 7. Electron−hole recombination dynamics in the (10, 5)carbon nanotube at 300 K, computed using DFTB electronic structureand DISH nonadiabatic molecular dynamics.

Figure 8. Fourier transform of the band gap, characterizing thephonon modes promoting electron−hole recombinationin the (10,5)carbon nanotube at 300 K.

Figure 9. (a) Autocorrelation and (b) pure-dephasing functions of theband gap in the (10,5) carbon nanotube at 300 K.

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Fitting eq 20 with a Gaussian gives the pure-dephasing time.The pure-dephasing function is plotted in Figure 9 (b). Thecalculated dephasing time is approximately 42 fs. Its inverse,obtained using the time-energy Heisenberg uncertaintyrelationship, is equal to 23.7 meV. Corresponding to theroom temperature line width of single molecule luminescence,this value is in good agreement with the experimentallyobserved line width of 23 meV.157

4. CONCLUSIONSConcluding, we have developed and implemented a novelmethodology for studying NA charge and energy transferdynamics in large systems. The tight-binding model providesthe electronic structure at a very good compromise betweenaccuracy and computational demand. Electronic states, andthereafter NA couplings between pairs of these states,calculated within the tight-binding scheme have been utilizedfor studying excited state dynamics explicitly in the time-domain. We have tested the method by considering excitationdynamics in zero- and one-dimensional materials: a CdSe QDand a SWNT. The relaxation dynamics of the hot chargecarriers, the time scale of the electron−hole recombination, andthe elastic electron−phonon scattering time match well withthe time-domain ab initio DFT calculations and theexperimental results. The test studies justify the accuracy andefficiency of the approach, which is expected to perform wellfor other large systems and to require modest computerresources.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: prezhdo@usc.edu.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThe authors are grateful to Parmeet Ninjar for comments onthe manuscript. A.V.A. acknowledges financial support from theUniversity at Buffalo, The State University of New York startuppackage. O.V.P. acknowledges financial support from the U.S.National Science Foundation, grant CHE-1530854. S.P.acknowledges financial support from SERB-DST, New Delhi,Govt. of India through the project ref. No. CS-085/2014.

■ REFERENCES(1) Hagfeldt, A.; Gratzel, M. Molecular photovoltaics. Acc. Chem. Res.2000, 33 (5), 269−277.(2) Anderson, N. A.; Lian, T. Ultrafast electron transfer at themolecule-semiconductor nanoparticle interface. Annu. Rev. Phys. Chem.2005, 56, 491−519.(3) Zhao, W.; Ma, W.; Chen, C.; Zhao, J.; Shuai, Z. Efficientdegradation of toxic organic pollutants with Ni2O3/TiO2‑x Bx undervisible irradiation. J. Am. Chem. Soc. 2004, 126 (15), 4782−4783.(4) Anfuso, C. L.; Snoeberger, R. C., III; Ricks, A. M.; Liu, W.; Xiao,D.; Batista, V. S.; Lian, T. Covalent attachment of a rhenium bipyridylCO2 reduction catalyst to rutile TiO2. J. Am. Chem. Soc. 2011, 133(18), 6922−6925.(5) Akimov, A. V.; Neukirch, A. J.; Prezhdo, O. V. Theoreticalinsights into photoinduced charge transfer and catalysis at oxideinterfaces. Chem. Rev. 2013, 113 (6), 4496−4565.(6) Ardo, S.; Meyer, G. J. Photodriven heterogeneous charge transferwith transition-metal compounds anchored to TiO2 semiconductorsurfaces. Chem. Soc. Rev. 2009, 38 (1), 115−164.

(7) Gratzel, M. Solar energy conversion by dye-sensitized photo-voltaic cells. Inorg. Chem. 2005, 44 (20), 6841−6851.(8) Liu, F.; Meyer, G. J. A nuclear isotope effect for interfacialelectron transfer: Excited-state electron injection from Ru amminecompounds to nanocrystalline TiO2. J. Am. Chem. Soc. 2005, 127 (3),824−825.(9) McConnell, I.; Li, G.; Brudvig, G. W. Energy conversion innatural and artificial photosynthesis. Chem. Biol. 2010, 17 (5), 434−447.(10) Wang, L.; Long, R.; Trivedi, D.; Prezhdo, O. V. Time-DomainAb Initio Modeling of Charge and Exciton Dynamics in Nanomateri-als. In Green Processes for Nanotechnology; Springer InternationalPublishing: Switzerland, 2015; pp 353−392.(11) Barber, G. D.; Hoertz, P. G.; Lee, S.-H. A.; Abrams, N. M.;Mikulca, J.; Mallouk, T. E.; Liska, P.; Zakeeruddin, S. M.; Gratzel, M.;Ho-Baillie, A. Utilization of Direct and Diffuse Sunlight in a Dye-Sensitized Solar Cell - Silicon Photovoltaic Hybrid ConcentratorSystem. J. Phys. Chem. Lett. 2011, 2 (6), 581−585.(12) Barea, E. M.; Shalom, M.; Gimenez, S.; Hod, I.; Mora-Sero, I.;Zaban, A.; Bisquert, J. Design of injection and recombination inquantum dot sensitized solar cells. J. Am. Chem. Soc. 2010, 132 (19),6834−6839.(13) Chan, W.-L.; Tritsch, J. R.; Zhu, X.-Y. Harvesting singlet fissionfor solar energy conversion: one-versus two-electron transfer from thequantum mechanical superposition. J. Am. Chem. Soc. 2012, 134 (44),18295−18302.(14) Johnson, J. C.; Nozik, A. J.; Michl, J. High triplet yield fromsinglet fission in a thin film of 1, 3-diphenylisobenzofuran. J. Am.Chem. Soc. 2010, 132 (46), 16302−16303.(15) Kim, D. R.; Lee, C. H.; Rao, P. M.; Cho, I. S.; Zheng, X. HybridSi microwire and planar solar cells: passivation and characterization.Nano Lett. 2011, 11 (7), 2704−2708.(16) Musser, A. J.; Al-Hashimi, M.; Maiuri, M.; Brida, D.; Heeney,M.; Cerullo, G.; Friend, R. H.; Clark, J. Activated singlet exciton fissionin a semiconducting polymer. J. Am. Chem. Soc. 2013, 135 (34),12747−12754.(17) Zhu, J.; Hsu, C.-M.; Yu, Z.; Fan, S.; Cui, Y. Nanodome solarcells with efficient light management and self-cleaning. Nano Lett.2010, 10 (6), 1979−1984.(18) Akimov, A. V.; Prezhdo, O. V. Nonadiabatic dynamics of chargetransfer and singlet fission at the pentacene/C60 interface. J. Am.Chem. Soc. 2014, 136 (4), 1599−1608.(19) Difley, S.; Van Voorhis, T. Exciton/charge-transfer electroniccouplings in organic semiconductors. J. Chem. Theory Comput. 2011, 7(3), 594−601.(20) Fischer, S. A.; Duncan, W. R.; Prezhdo, O. V. Ab initiononadiabatic molecular dynamics of wet-electrons on the tio2 surface.J. Am. Chem. Soc. 2009, 131 (42), 15483−15491.(21) Fischer, S. A.; Isborn, C. M.; Prezhdo, O. V. Excited states andoptical absorption of small semiconducting clusters: Dopants, defectsand charging. Chem. Sci. 2011, 2 (3), 400−406.(22) Fischer, S. A.; Madrid, A. B.; Isborn, C. M.; Prezhdo, O. V.Multiple exciton generation in small si clusters: a high-level, Ab initiostudy. J. Phys. Chem. Lett. 2010, 1 (1), 232−237.(23) Hyeon-Deuk, K.; Prezhdo, O. V. Time-domain ab initio study ofauger and phonon-assisted auger processes in a semiconductorquantum dot. Nano Lett. 2011, 11 (4), 1845−1850.(24) Isborn, C. M.; Kilina, S. V.; Li, X.; Prezhdo, O. V. Generation ofmultiple excitons in PbSe and CdSe quantum dots by directphotoexcitation: first-principles calculations on small PbSe and CdSeclusters. J. Phys. Chem. C 2008, 112 (47), 18291−18294.(25) Kilin, D. S.; Pereversev, Y. V.; Prezhdo, O. V. Electron-nuclearcorrelations for photo-induced dynamics in molecular dimers. J. Chem.Phys. 2004, 120 (23), 11209−11223.(26) Kilina, S. V.; Craig, C. F.; Kilin, D. S.; Prezhdo, O. V. Ab initiotime-domain study of phonon-assisted relaxation of charge carriers in aPbSe quantum dot. J. Phys. Chem. C 2007, 111 (12), 4871−4878.(27) Lee, J.; Vandewal, K.; Yost, S. R.; Bahlke, M. E.; Goris, L.; Baldo,M. A.; Manca, J. V.; Voorhis, T. V. Charge transfer state versus hot

Journal of Chemical Theory and Computation Article

DOI: 10.1021/acs.jctc.5b01231J. Chem. Theory Comput. 2016, 12, 1436−1448

1444

exciton dissociation in polymer-fullerene blended solar cells. J. Am.Chem. Soc. 2010, 132 (34), 11878−11880.(28) Paci, I.; Johnson, J. C.; Chen, X.; Rana, G.; Popovic, D.; David,D. E.; Nozik, A. J.; Ratner, M. A.; Michl, J. Singlet fission for dye-sensitized solar cells: Can a suitable sensitizer be found? J. Am. Chem.Soc. 2006, 128 (51), 16546−16553.(29) Sanchez-de-Armas, R. o.; Oviedo Lopez, J.; San-Miguel, M. A.;Sanz, J. F.; Ordejon, P.; Pruneda, M. Real-time TD-DFT simulationsin dye sensitized solar cells: the electronic absorption spectrum ofAlizarin supported on TiO2 nanoclusters. J. Chem. Theory Comput.2010, 6 (9), 2856−2865.(30) Xiao, D.; Martini, L. A.; Snoeberger, R. C., III; Crabtree, R. H.;Batista, V. S. Inverse design and synthesis of acac-Coumarin anchorsfor robust TiO2 sensitization. J. Am. Chem. Soc. 2011, 133 (23), 9014−9022.(31) Yost, S. R.; Wang, L.-P.; Van Voorhis, T. Molecular insight intothe energy levels at the organic donor/acceptor interface: A quantummechanics/molecular mechanics study. J. Phys. Chem. C 2011, 115(29), 14431−14436.(32) Furube, A.; Katoh, R.; Yoshihara, T.; Hara, K.; Murata, S.;Arakawa, H.; Tachiya, M. Ultrafast direct and indirect electron-injection processes in a photoexcited dye-sensitized nanocrystallinezinc oxide film: the importance of exciplex intermediates at the surface.J. Phys. Chem. B 2004, 108 (33), 12583−12592.(33) Furube, A.; Du, L.; Hara, K.; Katoh, R.; Tachiya, M. Ultrafastplasmon-induced electron transfer from gold nanodots into TiO2

nanoparticles. J. Am. Chem. Soc. 2007, 129 (48), 14852−14853.(34) Tully, J. C. Molecular dynamics with electronic transitions. J.Chem. Phys. 1990, 93 (2), 1061−1071.(35) Tully, J. C. Perspective: Nonadiabatic dynamics theory. J. Chem.Phys. 2012, 137 (22), 22A301.(36) Abuabara, S. G.; Rego, L. G.; Batista, V. S. Influence of thermalfluctuations on interfacial electron transfer in functionalized TiO2

semiconductors. J. Am. Chem. Soc. 2005, 127 (51), 18234−18242.(37) Duncan, W. R.; Craig, C. F.; Prezhdo, O. V. Time-domain abinitio study of charge relaxation and recombination in dye-sensitizedTiO2. J. Am. Chem. Soc. 2007, 129 (27), 8528−8543.(38) Duncan, W. R.; Prezhdo, O. V. Theoretical studies ofphotoinduced electron transfer in dye-sensitized TiO2. Annu. Rev.Phys. Chem. 2007, 58, 143−184.(39) Jakubikova, E.; Snoeberger, R. C., III; Batista, V. S.; Martin, R.L.; Batista, E. R. Interfacial Electron Transfer in TiO2 SurfacesSensitized with Ru (II)- Polypyridine Complexes†. J. Phys. Chem. A2009, 113 (45), 12532−12540.(40) Kilin, D. S.; Micha, D. A. Surface photovoltage at nanostructureson Si surfaces: ab initio results. J. Phys. Chem. C 2009, 113 (9), 3530−3542.(41) Kilin, D. S.; Micha, D. A. Modeling the Photovoltage of DopedSi Surfaces. J. Phys. Chem. C 2011, 115 (3), 770−775.(42) Labat, F.; Ciofini, I.; Hratchian, H. P.; Frisch, M.; Raghavachari,K.; Adamo, C. First principles modeling of eosin-loaded ZnO films: astep toward the understanding of dye-sensitized solar cell perform-ances. J. Am. Chem. Soc. 2009, 131 (40), 14290−14298.(43) Labat, F. d. r.; Le Bahers, T.; Ciofini, I.; Adamo, C. First-principles modeling of dye-sensitized solar cells: challenges andperspectives. Acc. Chem. Res. 2012, 45 (8), 1268−1277.(44) Maggio, E.; Martsinovich, N.; Troisi, A. Theoretical study ofcharge recombination at the TiO2-electrolyte interface in dyesensitised solar cells. J. Chem. Phys. 2012, 137 (22), 22A508.(45) Maggio, E.; Martsinovich, N.; Troisi, A. Evaluating chargerecombination rate in dye-sensitized solar cells from electronicstructure calculations. J. Phys. Chem. C 2012, 116 (14), 7638−7649.(46) Rego, L. G.; Batista, V. S. Quantum dynamics simulations ofinterfacial electron transfer in sensitized TiO2 semiconductors. J. Am.Chem. Soc. 2003, 125 (26), 7989−7997.(47) Fernandez-Alberti, S.; Kleiman, V. D.; Tretiak, S.; Roitberg, A.E. Nonadiabatic Molecular Dynamics Simulations of the EnergyTransfer between Building Blocks in a Phenylene EthynyleneDendrimer†. J. Phys. Chem. A 2009, 113 (26), 7535−7542.

(48) Fernandez-Alberti, S.; Kleiman, V. D.; Tretiak, S.; Roitberg, A.E. Unidirectional Energy Transfer in Conjugated Molecules: TheCrucial Role of High-Frequency CC Bonds. J. Phys. Chem. Lett.2010, 1 (18), 2699−2704.(49) Nelson, T.; Fernandez-Alberti, S.; Chernyak, V.; Roitberg, A. E.;Tretiak, S. Nonadiabatic excited-state molecular dynamics modeling ofphotoinduced dynamics in conjugated molecules. J. Phys. Chem. B2011, 115 (18), 5402−5414.(50) Wu, C.; Malinin, S. V.; Tretiak, S.; Chernyak, V. Y. MultiscaleModeling of Electronic Excitations in Branched Conjugated MoleculesUsing an Exciton Scattering Approach. Phys. Rev. Lett. 2008, 100 (5),057405.(51) Greyson, E. C.; Vura-Weis, J.; Michl, J.; Ratner, M. A.Maximizing Singlet Fission in Organic Dimers: Theoretical Inves-tigation of Triplet Yield in the Regime of Localized Excitation and FastCoherent Electron Transfer†. J. Phys. Chem. B 2010, 114 (45),14168−14177.(52) Smith, M. B.; Michl, J. Singlet fission. Chem. Rev. 2010, 110(11), 6891−6936.(53) Hyeon-Deuk, K.; Prezhdo, O. V. Multiple exciton generationand recombination dynamics in small Si and CdSe quantum dots: Anab initio time-domain study. ACS Nano 2012, 6 (2), 1239−1250.(54) Isborn, C. M.; Li, X. Modeling the doubly excited state withtime-dependent Hartree−Fock and density functional theories. J.Chem. Phys. 2008, 129 (20), 204107.(55) Fabiano, E.; Keal, T.; Thiel, W. Implementation of surfacehopping molecular dynamics using semiempirical methods. Chem.Phys. 2008, 349 (1), 334−347.(56) Kuhlman, T. S.; Glover, W. J.; Mori, T.; Møller, K. B.; Martínez,T. J. Between ethylene and polyenes-the non-adiabatic dynamics of cis-dienes. Faraday Discuss. 2012, 157, 193−212.(57) Tao, H.; Allison, T.; Wright, T.; Stooke, A.; Khurmi, C.; VanTilborg, J.; Liu, Y.; Falcone, R.; Belkacem, A.; Martinez, T. Ultrafastinternal conversion in ethylene. I. The excited state lifetime. J. Chem.Phys. 2011, 134 (24), 244306.(58) Akimov, A. V.; Prezhdo, O. V. Large-Scale Computations inChemistry: A Bird’s Eye View of a Vibrant Field. Chem. Rev. 2015, 115(12), 5797.(59) Barbatti, M. Nonadiabatic dynamics with trajectory surfacehopping method. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2011, 1 (4),620−633.(60) Drukker, K. Basics of surface hopping in mixed quantum/classical simulations. J. Comput. Phys. 1999, 153 (2), 225−272.(61) de Carvalho, F. F.; Bouduban, M. E.; Curchod, B. F.; Tavernelli,I. Nonadiabatic molecular dynamics based on trajectories. Entropy2014, 16 (1), 62−85.(62) Kapral, R. Progress in the theory of mixed quantum-classicaldynamics. Annu. Rev. Phys. Chem. 2006, 57, 129−157.(63) Kapral, R.; Ciccotti, G. Mixed quantum-classical dynamics. J.Chem. Phys. 1999, 110 (18), 8919−8929.(64) Marx, D.; Hutter, J. Ab initio molecular dynamics: basic theory andadvanced methods; Cambridge University Press: New York, 2009.(65) Mitric, R.; Werner, U.; Wohlgemuth, M.; Seifert, G.; Bonacic-Koutecky, V. Nonadiabatic Dynamics within Time-Dependent DensityFunctional Tight Binding Method. J. Phys. Chem. A 2009, 113 (45),12700−12705.(66) Tully, J. Mixed quantum−classical dynamics. Faraday Discuss.1998, 110, 407−419.(67) Parandekar, P. V.; Tully, J. C. Mixed quantum-classicalequilibrium. J. Chem. Phys. 2005, 122 (9), 094102.(68) Hartmann, M.; Pittner, J.; Bonacic-Koutecky, V. Ab initiononadiabatic dynamics involving conical intersection combined withWigner distribution approach to ultrafast spectroscopy illustrated onNa3F2 cluster. J. Chem. Phys. 2001, 114 (5), 2123−2136.(69) Mitric, R.; Bonacic-Koutecky, V.; Pittner, J.; Lischka, H. Abinitio nonadiabatic dynamics study of ultrafast radiationless decay overconical intersections illustrated on the Na3F cluster. J. Chem. Phys.2006, 125 (2), 024303.

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DOI: 10.1021/acs.jctc.5b01231J. Chem. Theory Comput. 2016, 12, 1436−1448

1445

(70) Craig, C. F.; Duncan, W. R.; Prezhdo, O. V. Trajectory surfacehopping in the time-dependent Kohn-Sham approach for electron-nuclear dynamics. Phys. Rev. Lett. 2005, 95 (16), 163001.(71) Mitric, R.; Werner, U.; Bonacic-Koutecky, V. Nonadiabaticdynamics and simulation of time resolved photoelectron spectra withintime-dependent density functional theory: Ultrafast photoswitching inbenzylideneaniline. J. Chem. Phys. 2008, 129 (16), 164118.(72) Tapavicza, E.; Tavernelli, I.; Rothlisberger, U. Trajectory surfacehopping within linear response time-dependent density-functionaltheory. Phys. Rev. Lett. 2007, 98 (2), 023001.(73) Tapavicza, E.; Tavernelli, I.; Rothlisberger, U.; Filippi, C.;Casida, M. E. Mixed time-dependent density-functional theory/classical trajectory surface hopping study of oxirane photochemistry.J. Chem. Phys. 2008, 129 (12), 124108.(74) Werner, U.; Mitric, R.; Suzuki, T.; Bonacic-Koutecky, V.Nonadiabatic dynamics within the time dependent density functionaltheory: Ultrafast photodynamics in pyrazine. Chem. Phys. 2008, 349(1), 319−324.(75) Fabiano, E.; Thiel, W. Nonradiative deexcitation dynamics of9H-adenine: An OM2 surface hopping study. J. Phys. Chem. A 2008,112 (30), 6859−6863.(76) Granucci, G.; Persico, M.; Toniolo, A. Direct semiclassicalsimulation of photochemical processes with semiempirical wavefunctions. J. Chem. Phys. 2001, 114 (24), 10608−10615.(77) Lan, Z.; Fabiano, E.; Thiel, W. Photoinduced nonadiabaticdynamics of pyrimidine nucleobases: on-the-fly surface-hopping studywith semiempirical methods. J. Phys. Chem. B 2009, 113 (11), 3548−3555.(78) CPMD; IBM Corp.: Armonk, NY, 1990−2008; MPI furFestkorperforschung Stuttgart: Stuttgart, Germany, 1997−2001.http://www.cpmd.org (accessed 15 June 2015).(79) Andrade, X.; Alberdi-Rodriguez, J.; Strubbe, D. A.; Oliveira, M.J.; Nogueira, F.; Castro, A.; Muguerza, J.; Arruabarrena, A.; Louie, S.G.; Aspuru-Guzik, A. Time-dependent density-functional theory inmassively parallel computer architectures: the octopus project. J. Phys.:Condens. Matter 2012, 24 (23), 233202.(80) Barbatti, M.; Granucci, G.; Persico, M.; Ruckenbauer, M.;Vazdar, M.; Eckert-Maksic, M.; Lischka, H. The on-the-fly surface-hopping program system Newton-X: Application to ab initiosimulation of the nonadiabatic photodynamics of benchmark systems.J. Photochem. Photobiol., A 2007, 190 (2), 228−240.(81) Barbatti, M. G. G.; Lischka, H.; Ruckenbauer, M. Newton-X: apackage for Newtonian dynamics close to the crossing seam. www.univie.ac.at/newtonx (accessed 30 June 2015).(82) Castro, A.; Appel, H.; Oliveira, M.; Rozzi, C. A.; Andrade, X.;Lorenzen, F.; Marques, M. A.; Gross, E.; Rubio, A. octopus: a tool forthe application of time-dependent density functional theory. Phys.Status Solidi B 2006, 243 (11), 2465−2488.(83) Doltsinis, N. L.; Marx, D. Nonadiabatic car-parrinello moleculardynamics. Phys. Rev. Lett. 2002, 88 (16), 166402.(84) Levine, B. G.; Coe, J. D.; Virshup, A. M.; Martinez, T. J.Implementation of ab initio multiple spawning in the Molpro quantumchemistry package. Chem. Phys. 2008, 347 (1), 3−16.(85) Marques, M. A.; Castro, A.; Bertsch, G. F.; Rubio, A. Octopus: afirst-principles tool for excited electron−ion dynamics. Comput. Phys.Commun. 2003, 151 (1), 60−78.(86) Thiel, W. MNDO program; Mulheim, Germany, 2007.(87) Werner, H. J.; Knowles, P. J.; Knizia, G.; Manby, F. R.; Schutz,M.; Celani, P.; Korona, T.; Lindh, R.; Mitrushenkov, A.; Rauhut, G.;Shamasundar, K. R.; Adler, T. B.; Amos, R. D.; Bernhardsson, A.;Berning, A.; Cooper, D. L.; Deegan, M. J. O.; Dobbyn, A. J.; Eckert, F.;Goll, E.; Hampel, C.; Hesselmann, A.; Hetzer, G.; Hrenar, T.; Jansen,G.; Koppl, C.; Liu, Y.; Lloyd, A. W.; Mata, R. A.; May, A. J.;McNicholas, S. J.; Meyer, W.; Mura, M. E.; Nicklass, A.; O’Neill, D. P.;Palmieri, P.; Peng, D.; Pfluger, K.; Pitzer, R.; Reiher, M.; Shiozaki, T.;Stoll, H.; Stone, A. J.; Tarroni, R.; Thorsteinsson, T.; Wang, M.MOLPRO, version 2012.1, a package of ab initio programs; CardiffUniversity, UK, 2012. See http://www.molpro.net/ (accessed Mar 1,2016).

(88) Ehrenfest, P. Adiabatische Transformationen in der Quanten-theorie und ihre Behandlung durch Niels Bohr. Naturwissenschaften1923, 11 (27), 543−550.(89) Ehrenfest, P. Bemerkung uber die angenaherte Gultigkeit derklassischen Mechanik innerhalb der Quantenmechanik. Eur. Phys. J. A1927, 45 (7−8), 455−457.(90) Fischer, S. A.; Chapman, C. T.; Li, X. Surface hopping withEhrenfest excited potential. J. Chem. Phys. 2011, 135 (14), 144102.(91) Kab, G. Mean field Ehrenfest quantum/classical simulation ofvibrational energy relaxation in a simple liquid. Phys. Rev. E: Stat. Phys.,Plasmas, Fluids, Relat. Interdiscip. Top. 2002, 66 (4), 046117.(92) Li, X.; Tully, J. C.; Schlegel, H. B.; Frisch, M. J. Ab initioEhrenfest dynamics. J. Chem. Phys. 2005, 123 (8), 084106.(93) Subotnik, J. E. Augmented Ehrenfest dynamics yields a rate forsurface hopping. J. Chem. Phys. 2010, 132 (13), 134112.(94) Wang, L.; Trivedi, D.; Prezhdo, O. V. Global flux surfacehopping approach for mixed quantum-classical dynamics. J. Chem.Theory Comput. 2014, 10 (9), 3598−3605.(95) Akimov, A. V.; Prezhdo, O. V. Second-quantized surfacehopping. Phys. Rev. Lett. 2014, 113 (15), 153003.(96) Jaeger, H. M.; Fischer, S.; Prezhdo, O. V. Decoherence-inducedsurface hopping. J. Chem. Phys. 2012, 137 (22), 22A545.(97) Prezhdo, O. V.; Duncan, W. R.; Prezhdo, V. V. Photoinducedelectron dynamics at the chromophore−semiconductor interface: Atime-domain ab initio perspective. Prog. Surf. Sci. 2009, 84 (1), 30−68.(98) Wang, L.; Prezhdo, O. V. A simple solution to the trivialcrossing problem in surface hopping. J. Phys. Chem. Lett. 2014, 5 (4),713−719.(99) Akimov, A. V.; Prezhdo, O. V. Formulation of quantizedHamiltonian dynamics in terms of natural variables. J. Chem. Phys.2012, 137 (22), 224115.(100) Akimov, A. V.; Long, R.; Prezhdo, O. V. Coherence penaltyfunctional: A simple method for adding decoherence in Ehrenfestdynamics. J. Chem. Phys. 2014, 140 (19), 194107.(101) Hyeon-Deuk, K.; Prezhdo, O. V. Photoexcited electron andhole dynamics in semiconductor quantum dots: phonon-inducedrelaxation, dephasing, multiple exciton generation and recombination.J. Phys.: Condens. Matter 2012, 24 (36), 363201.(102) Neukirch, A. J.; Hyeon-Deuk, K.; Prezhdo, O. V. Time-domainab initio modeling of excitation dynamics in quantum dots. Coord.Chem. Rev. 2014, 263-264, 161−181.(103) Akimov, A. V.; Prezhdo, O. V. The PYXAID program for non-adiabatic molecular dynamics in condensed matter systems. J. Chem.Theory Comput. 2013, 9 (11), 4959−4972.(104) Akimov, A. V.; Prezhdo, O. V. Advanced capabilities of thePYXAID program: integration schemes, decoherence effects, multi-excitonic states, and field-matter interaction. J. Chem. Theory Comput.2014, 10 (2), 789−804.(105) Duncan, W. R.; Stier, W. M.; Prezhdo, O. V. Ab InitioNonadiabatic Molecular Dynamics of the Ultrafast Electron Injectionacross the Alizarin-TiO2 Interface. J. Am. Chem. Soc. 2005, 127 (21),7941−7951.(106) Kohn, W.; Sham, L. J. Self-consistent equations includingexchange and correlation effects. Phys. Rev. 1965, 140 (4A), A1133.(107) Fischer, S. A.; Habenicht, B. F.; Madrid, A. B.; Duncan, W. R.;Prezhdo, O. V. Regarding the validity of the time-dependent Kohn−Sham approach for electron-nuclear dynamics via trajectory surfacehopping. J. Chem. Phys. 2011, 134 (2), 024102.(108) Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.;Cavazzoni, C.; Ceresoli, D.; Chiarotti, G. L.; Cococcioni, M.; Dabo, I.QUANTUM ESPRESSO: a modular and open-source software projectfor quantum simulations of materials. J. Phys.: Condens. Matter 2009,21 (39), 395502.(109) Kresse, G.; Furthmuller, J. Efficient iterative schemes for abinitio total-energy calculations using a plane-wave basis set. Phys. Rev.B: Condens. Matter Mater. Phys. 1996, 54 (16), 11169.(110) Jaeger, H. M.; Hyeon-Deuk, K.; Prezhdo, O. V. Excitonmultiplication from first principles. Acc. Chem. Res. 2013, 46 (6),1280−1289.

Journal of Chemical Theory and Computation Article

DOI: 10.1021/acs.jctc.5b01231J. Chem. Theory Comput. 2016, 12, 1436−1448

1446

(111) Madrid, A. B.; Hyeon-Deuk, K.; Habenicht, B. F.; Prezhdo, O.V. Phonon-induced dephasing of excitons in semiconductor quantumdots: Multiple exciton generation, fission, and luminescence. ACSNano 2009, 3 (9), 2487−2494.(112) Aradi, B.; Hourahine, B.; Frauenheim, T. DFTB+, a sparsematrix-based implementation of the DFTB method. J. Phys. Chem. A2007, 111 (26), 5678−5684.(113) Elstner, M. SCC-DFTB: What is the proper degree of self-consistency? J. Phys. Chem. A 2007, 111 (26), 5614−5621.(114) Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.;Frauenheim, T.; Suhai, S.; Seifert, G. Self-consistent-charge density-functional tight-binding method for simulations of complex materialsproperties. Phys. Rev. B: Condens. Matter Mater. Phys. 1998, 58 (11),7260.(115) Kruger, T.; Elstner, M.; Schiffels, P.; Frauenheim, T. Validationof the density-functional based tight-binding approximation methodfor the calculation of reaction energies and other data. J. Chem. Phys.2005, 122 (11), 114110.(116) Niehaus, T. A.; Suhai, S.; Della Sala, F.; Lugli, P.; Elstner, M.;Seifert, G.; Frauenheim, T. Tight-binding approach to time-dependentdensity-functional response theory. Phys. Rev. B: Condens. MatterMater. Phys. 2001, 63 (8), 085108.(117) Porezag, D.; Frauenheim, T.; Kohler, T.; Seifert, G.; Kaschner,R. Construction of tight-binding-like potentials on the basis of density-functional theory: Application to carbon. Phys. Rev. B: Condens. MatterMater. Phys. 1995, 51 (19), 12947.(118) Seifert, G. Tight-binding density functional theory: anapproximate Kohn-Sham DFT scheme. J. Phys. Chem. A 2007, 111(26), 5609−5613.(119) Trivedi, D. J.; Wang, L.; Prezhdo, O. V. Auger-MediatedElectron Relaxation Is Robust to Deep Hole Traps: Time-Domain AbInitio Study of CdSe Quantum Dots. Nano Lett. 2015, 15 (3), 2086−2091.(120) Sippel, P.; Albrecht, W.; Mitoraj, D.; Eichberger, R.;Hannappel, T.; Vanmaekelbergh, D. Two-photon photoemissionstudy of competing Auger and surface-mediated relaxation of hotelectrons in CdSe quantum dot solids. Nano Lett. 2013, 13 (4), 1655−1661.(121) Habenicht, B. F.; Craig, C. F.; Prezhdo, O. V. Time-domain abinitio simulation of electron and hole relaxation dynamics in a single-wall semiconducting carbon nanotube. Phys. Rev. Lett. 2006, 96 (18),187401.(122) Huang, L.; Krauss, T. D. Quantized bimolecular Augerrecombination of excitons in single-walled carbon nanotubes. Phys.Rev. Lett. 2006, 96 (5), 057407.(123) Frauenheim, T.; Seifert, G.; Elstner, M.; Niehaus, T.; Kohler,C.; Amkreutz, M.; Sternberg, M.; Hajnal, Z.; Di Carlo, A.; Suhai, S.Atomistic simulations of complex materials: ground-state and excited-state properties. J. Phys.: Condens. Matter 2002, 14 (11), 3015.(124) Eschrig, H.; Bergert, I. An optimized LCAO version for bandstructure calculations application to copper. Phys. Status Solidi B 1978,90 (2), 621−628.(125) Eschrig, H. Optimized LCAO method and the electronic structureof extended systems, 1st ed.; Akademie Verlag and Springer: Berlin,Germany, 1989; 221 pages.(126) Oliveira, A. F.; Seifert, G.; Heine, T.; Duarte, H. A. Density-functional based tight-binding: an approximate DFT method. J. Braz.Chem. Soc. 2009, 20 (7), 1193−1205.(127) Gao, X.; Geng, H.; Peng, Q.; Ren, J.; Yi, Y.; Wang, D.; Shuai, Z.Nonadiabatic Molecular Dynamics Modeling of the Intrachain ChargeTransport in Conjugated Diketopyrrolo-pyrrole Polymers. J. Phys.Chem. C 2014, 118 (13), 6631−6640.(128) Niehaus, T. A.; Heringer, D.; Torralva, B.; Frauenheim, T.Importance of electronic self-consistency in the TDDFT basedtreatment of nonadiabatic molecular dynamics. Eur. Phys. J. D 2005,35 (3), 467−477.(129) Hammes-Schiffer, S.; Tully, J. C. Proton transfer in solution:Molecular dynamics with quantum transitions. J. Chem. Phys. 1994,101 (6), 4657−4667.

(130) Tully, J. C.; Preston, R. K. Trajectory surface hoppingapproach to nonadiabatic molecular collisions: the reaction of H+ withD2. J. Chem. Phys. 1971, 55 (2), 562−572.(131) Prezhdo, O. V. Mean field approximation for the stochasticSchrodinger equation. J. Chem. Phys. 1999, 111 (18), 8366−8377.(132) Carlson, L. J.; Krauss, T. D. Photophysics of individual single-walled carbon nanotubes. Acc. Chem. Res. 2008, 41 (2), 235−243.(133) Kilina, S.; Ivanov, S.; Tretiak, S. Effect of surface ligands onoptical and electronic spectra of semiconductor nanoclusters. J. Am.Chem. Soc. 2009, 131 (22), 7717−7726.(134) Kilina, S. V.; Kilin, D. S.; Prezhdo, O. V. Breaking the phononbottleneck in PbSe and CdSe quantum dots: Time-domain densityfunctional theory of charge carrier relaxation. ACS Nano 2009, 3 (1),93−99.(135) Ma, Y.-Z.; Stenger, J.; Zimmermann, J.; Bachilo, S. M.; Smalley,R. E.; Weisman, R. B.; Fleming, G. R. Ultrafast carrier dynamics insingle-walled carbon nanotubes probed by femtosecond spectroscopy.J. Chem. Phys. 2004, 120 (7), 3368−3373.(136) Kasuya, A.; Sivamohan, R.; Barnakov, Y. A.; Dmitruk, I. M.;Nirasawa, T.; Romanyuk, V. R.; Kumar, V.; Mamykin, S. V.; Tohji, K.;Jeyadevan, B. Ultra-stable nanoparticles of CdSe revealed from massspectrometry. Nat. Mater. 2004, 3 (2), 99−102.(137) Puzder, A.; Williamson, A.; Gygi, F.; Galli, G. Self-healing ofCdSe nanocrystals: first-principles calculations. Phys. Rev. Lett. 2004,92 (21), 217401.(138) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradientapproximation made simple. Phys. Rev. Lett. 1996, 77 (18), 3865.(139) Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to theprojector augmented-wave method. Phys. Rev. B: Condens. MatterMater. Phys. 1999, 59 (3), 1758.(140) Sarkar, S.; Pal, S.; Sarkar, P.; Rosa, A.; Frauenheim, T. Self-Consistent-Charge Density-Functional Tight-Binding Parameters forCd−X (X= S, Se, Te) Compounds and Their Interaction with H, O,C, and N. J. Chem. Theory Comput. 2011, 7 (7), 2262−2276.(141) Verlet, L. Computer″ experiments″ on classical fluids. I.Thermodynamical properties of Lennard-Jones molecules. Phys. Rev.1967, 159 (1), 98.(142) Trani, F.; Barone, V. Silicon Nanocrystal Functionalization:Analytic Fitting of DFTB Parameters. J. Chem. Theory Comput. 2011, 7(3), 713−719.(143) Godby, R.; Schluter, M.; Sham, L. Self-energy operators andexchange-correlation potentials in semiconductors. Phys. Rev. B:Condens. Matter Mater. Phys. 1988, 37 (17), 10159.(144) Gygi, F.; Baldereschi, A. Quasiparticle energies in semi-conductors: self-energy correction to the local-density approximation.Phys. Rev. Lett. 1989, 62 (18), 2160.(145) Moreira, N. H.; Dolgonos, G.; Aradi, B.; da Rosa, A. L.;Frauenheim, T. Toward an accurate density-functional tight-bindingdescription of zinc-containing compounds. J. Chem. Theory Comput.2009, 5 (3), 605−614.(146) Califano, M.; Zunger, A.; Franceschetti, A. Efficient inverseAuger recombination at threshold in CdSe nanocrystals. Nano Lett.2004, 4 (3), 525−531.(147) Habenicht, B. F.; Kamisaka, H.; Yamashita, K.; Prezhdo, O. V.Ab initio study of vibrational dephasing of electronic excitations insemiconducting carbon nanotubes. Nano Lett. 2007, 7 (11), 3260−3265.(148) Habenicht, B. F.; Prezhdo, O. V. Nonradiative quenching offluorescence in a semiconducting carbon nanotube: A time-domain abinitio study. Phys. Rev. Lett. 2008, 100 (19), 197402.(149) Lanzani, G.; Cerullo, G.; Gambetta, A.; Manzoni, C.; Menna,E.; Meneghetti, M. Exciton relaxation in single wall carbon nanotubes.Synth. Met. 2005, 155 (2), 246−249.(150) Manzoni, C.; Gambetta, A.; Menna, E.; Meneghetti, M.;Lanzani, G.; Cerullo, G. Intersubband exciton relaxation dynamics insingle-walled carbon nanotubes. Phys. Rev. Lett. 2005, 94 (20), 207401.(151) Song, D.; Wang, F.; Dukovic, G.; Zheng, M.; Semke, E.; Brus,L. E.; Heinz, T. F. Direct measurement of the lifetime of optical

Journal of Chemical Theory and Computation Article

DOI: 10.1021/acs.jctc.5b01231J. Chem. Theory Comput. 2016, 12, 1436−1448

1447

phonons in single-walled carbon nanotubes. Phys. Rev. Lett. 2008, 100(22), 225503.(152) Wong, H.-S. P.; Akinwande, D. Carbon nanotube and graphenedevice physics; Cambridge University Press: New York, 2011.(153) Reich, S.; Thomsen, C.; Ordejon, P. Electronic band structureof isolated and bundled carbon nanotubes. Phys. Rev. B: Condens.Matter Mater. Phys. 2002, 65 (15), 155411.(154) Malola, S.; Hakkinen, H.; Koskinen, P. Raman spectra ofsingle-walled carbon nanotubes with vacancies. Phys. Rev. B: Condens.Matter Mater. Phys. 2008, 77 (15), 155412.(155) Mukamel, S. Principles of nonlinear optical spectroscopy; OxfordUniversity Press: New York, 1999.(156) Akimov, A. V.; Prezhdo, O. V. Persistent electronic coherencedespite rapid loss of electron−nuclear correlation. J. Phys. Chem. Lett.2013, 4 (22), 3857−3864.(157) Hartschuh, A.; Pedrosa, H. N.; Peterson, J.; Huang, L.; Anger,P.; Qian, H.; Meixner, A. J.; Steiner, M.; Novotny, L.; Krauss, T. D.Single carbon nanotube optical spectroscopy. ChemPhysChem 2005, 6(4), 577−582.

Journal of Chemical Theory and Computation Article

DOI: 10.1021/acs.jctc.5b01231J. Chem. Theory Comput. 2016, 12, 1436−1448

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