Nonadiabatic Molecular Dynamics for Thousand Atom Systems: A

Article pubs.acs.org/JCTC

Nonadiabatic Molecular Dynamics for Thousand Atom Systems: A Tight-Binding Approach toward PYXAID Sougata Pal,† Dhara J. Trivedi,∥ Alexey V. Akimov,§ Bálint 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, Universität Bremen, Otto-Hahn-Alle 1, 28359 Bremen, Germany ∥

ABSTRACT: Excited state dynamics at the nanoscale requires treatment of systems involving hundreds and thousands of atoms. In the majority of cases, depending on the process under investigation, the electronic structure component of the calculation constitutes the computation bottleneck. We developed an efficient approach for simulating nonadiabatic molecular dynamics (NA-MD) of large systems in the framework of the self-consistent charge density functional tight binding (SCC-DFTB) method. SCC-DFTB is combined with the fewest switches surface hopping (FSSH) and decoherence induced surface hopping (DISH) techniques for NA-MD. The approach is implemented within the Python extension for the ab initio dynamics (PYXAID) simulation package, which is an open source NA-MD program designed to handle nanoscale materials. The accuracy of the developed approach is tested with ab initio DFT and experimental data, by considering intraband electron and hole relaxation, and nonradiative electron−hole recombination in a CdSe quantum dot and the (10,5) semiconducting carbon nanotube. The technique is capable of treating accurately and efficiently excitation dynamics in large, realistic nanoscale materials, employing modest computational resources.

1. INTRODUCTION Nanoscale materials, composed of hundreds and thousands of atoms, bridge the gap between molecules and bulk. They exhibit a variety of interesting and novel phenomena, which are being explored for technological and medical applications. For instance, nanoscale systems enable one to efficiently harvest alternative energy sources, using devices such as photovoltaic1−3 and photocatalytic3,4 cells. The applications stimulate extensive research on charge and energy transfer in hybrid nanoscale materials and interfaces,5 because these processes determine efficiencies of solar cells and other devices. Absorption of a photon creates an electronically excited state, which subsequently dissociates into a pair of spatially separated charges. Charge separation occurs at interfaces5−9 and competes with energy losses due to thermal relaxation of excited electrons and holes and electron−hole recombination. In order to understand and control the competition between the alternative dynamical pathways, with the goal of improving solar cell parameters, such as voltage, current, and photon-toelectron conversion yield, one needs to investigate the highly nonequilibrium nature of the excited state dynamics at the atomistic level of detail.10 Atomistic description is required in order to account for the intrinsically statistical origin of © 2016 American Chemical Society

nanoscale materials, involving relatively broad distributions of sizes, surfaces, and chiralities, nonstoichiometric and spatially varying compositions, etc. Perhaps even more importantly, nanoscale materials are synthesized by chemists and involve, intentionally or unintentionally, defects, ligands, dopants, and other atom-size features that often govern material’s properties and should be well understood. Numerous experimental11−17 and theoretical18−31 efforts are dedicated to studies of nonequilibrium, excited state dynamics on the nanoscale. Ultrafast, time-resolved spectroscopy32,33 is a particularly valuable experimental tool that helps characterizing the excited states dynamics in many materials. It allows one to obtain the time scale of each process and to determine the relationship between favorable carrier generation and transport and nonproductive energy losses. The analysis of these data leads to a thorough understanding of the origins of a particular device performance. Theoretical studies parallel the experimental characterization, since they interpret experimental results, establish mechanisms of the processes under investigation, and provide insights into many important details, Received: December 28, 2015 Published: March 8, 2016 1436

DOI: 10.1021/acs.jctc.5b01231 J. Chem. Theory Comput. 2016, 12, 1436−1448

Article

Journal of Chemical Theory and Computation

imation (CPA) for SH.97 DISH incorporates decoherence effects in a natural and efficient way. GFSH treats superexchange and many-particle Auger-type phenomena. CPA-SH greatly increases simulation efficiency. The utility of these and related methods has been demonstrated with different nonequilibrium phenomena occurring in a variety of nanoscale systems.101−105 Recently, the Prezhdo group released an open source package for simulation of NA dynamics on the nanoscale.103,104 Named PYXAID for PYthon eXtension for Ab Initio Dynamics, it interfaces the Kohn−Sham (KS)106 formulation of NA-MD70,107 with Quantum Espresso108 and Vienna Ab initio Simulation Package (VASP),109 which are used as DFT drivers for adiabatic electronic structure and MD calculations. PYXAID implements advanced techniques for integration of the time-dependent Schrodinger equation (TDSE), and a number of basic and more advanced NA-MD functionalities, including fewest-switches surface hopping (FSSH),67 DISH,96 multielectron adiabatic representation of the time-dependent KS (TD-KS) equations, and direct simulation of photoexcitation via explicit light-matter interaction. The CPA implemented in PYXAID achieves considerable computational savings. With the exception of processes involving hundreds of thousands of electronic states, such as multiple exciton generation,53,101,102,110,111 the computational bottleneck of PYXAID is determined by the scaling of the electronic structure computations. Realistic representation of many nanoscale systems requires hundreds and thousands of atoms, extending beyond the capabilities of the existing NA-MD codes. The self-consistent charge density functional tight binding method (SCCDFTB)112−118 is capable of increasing the system size by an order of magnitude. Parameterized against DFT, the SCCDFTB provides an accurate quantum-mechanical description of geometries, vibrational frequencies, reaction energies, and other properties115 of very large systems at a modest computational effort. In this article, we describe a technique for NA-MD simulation of many atom systems by combining SCCDFTB112−118 with PYXAID.103,104 The approach is illustrated with charge-phonon relaxation in a CdSe quantum dot119,120 (QD) and nonradiative electron−hole recombination a semiconducting carbon nanotube (CNT),121,122 including both inelastic and elastic electron−phonon scattering effects. The results obtained from NA-MD simulations based on the SCCDFTB formalism (DFTB-NA-MD) are then tested against that of ab initio DFT based NA-MD (DFT-NA-MD) and are compared with experimental data. The paper is organized as follows. Section 2 describes the computational method. Namely, subsection 2.1 presents the basic formalism of the electronic structure calculation by DFTB, while subsection 2.2 outlines the working principles of the NA-MD simulation as implemented in PYXAID. Section 3 reports the QD and CNT simulations, demonstrating and validating the approach. The paper concludes with a summary and outlook.

ultimately stimulating generalizations and the development of essential design principles. Arguably, nonadiabatic (NA) molecular dynamics (MD)34,35constitutes the most efficient means for modeling nonequilibrium excited state phenomena. Various theoretical groups perform NA-MD simulations of atomic, molecular, nanoscale, and bulk systems in order to capture photoinduced phenomena, such as electron−hole separation, energy exchange and recombination,36−46 charge and 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 the corresponding experimental investigations, partially due to computational limitations on system size and simulation time.58 More efficient computational NA-MD approaches are, therefore, required. NA-MD is often used in practice in its quantum-classical formulation. Within the quantum-classical framework, the motion of nuclei is treated classically, whereas the electronic evolution is propagated quantum mechanically. NA-MD generalizes classical ground-state MD to include NA transitions between quantum (electronic) states. NA-MD simulations involve two components: an algorithm for the electron−nuclear dynamics, including the possibility of NA transitions, and a method for electronic structure calculations, which provides nuclear forces and NA couplings.59,60 Typically, the electronic structure is computed “on the fly”61−66 and provides input to stochastic 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 configuration interaction,69 time-dependent density functional theory (TDDFT),70−74 and semiempirical methods.55,75−77 TD-DFT is particularly popular in this regard, since it provides a good compromise between accuracy and efficiency. DFT is transferrable and applicable to the majority systems. It can treat most of the relevant processes and can take into account the full complexity of a material, including composition, shape, defects, dopants, surface, ligands, and environment. Still, TD-DFT simulations are limited to about hundred atoms systems. Rooted in DFT and parametrized against it, the SCC-DFTB is capable 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 rely on computationally expensive electronic structure methods and, thus, are applicable to rather small systems. The codes typically implement the most basic NA-MD techniques, such as the Ehrenfest88−93 and standard TSH34,66 methods. These NA-MD approaches capture only some of the processes involved in operation of nanoscale devices. For instance, the Ehrenfest method mistreats electron-vibrational energy exchange and fails to equilibrate the system to Boltzmann distribution. The standard TSH cannot handle properly superexchange and many-particle transitions.94,95 Both the Ehrenfest and standard SH disregard decoherence effects, leading to discrepancies in computed and experimentally measured time scales for electronic dynamics by orders of magnitude. The Prezhdo group has developed a spectrum of advanced NA-MD techniques94−100 designed for studying the interplay between productive processes, such as charge and energy transfer, and energy losses to heat to due electron-vibrational relaxation and charge recombination. The focus is on nanoscale and condensed phase systems. Important practical advances include decoherence induced surface hopping (DISH),96 global flux surface hopping (GFSH),94 and classical path approx-

2. THEORETICAL METHOD 2.1. Electronic Structure Calculations and the DFTB+ Code. The SCC-DFTB method112−118 is implemented within the DFTB+ code.112,123It is used to calculate the electronic structure for the system of interest on-the-fly. The foremost idea of the SCC-DFTB method is to derive the total energy 1437


Article

Journal of Chemical Theory and Computation DFTB DFTB Etotal = Eelec + Erep

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

1 0 E = ∑ ni⟨ψi|Ĥ |ψi ⟩ + 2 i −

1 2

∫∫

where the second term is represented as

⎛ 1 δ 2Exc ⎞ ⎜ + | ρ ⎟δρδρ′ δρδρ′ 0 ⎠ ⎝ | r ⃗ − r ′⃗ |

Erep = −

ρ0 ρ0′

1 2

DFTB = Eelec

⎛

∑ ⎜⎜∑ aμaν⟨ϕμ|Ĥ αβ

⎝ μ,ν

⎞ 1 |ϕν⟩+ γαβ ΔqαΔq β ⎟⎟, 2 ⎠

0

∀ μ ∈ α, ν ∈ β

where aμ and aν are the combination coefficients used to expand the KS wave function. As the KS wave function has been written as a LCAO ansatz, thus, for a given set of atomic positions, a trial set of combination coefficients can be taken to construct a SCC-DFTB secular equation of the form

Here, Δqα and Δqβ denote atomic net Mulliken charges of atoms α and β. The function γαβ114 interpolates between a pure Coulomb interaction for large interatomic distance and an element-specific constant in the atomic limit, which is directly related to chemical hardness113 of the atomic species. The third term represents the Hartree Coulomb interaction. EXC, VXC, and VNN represent the exchange-correlation energy, the corresponding 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

∑ aμν(Hμν − εiSμν) = 0,

∀ μ ∈ α, ν ∈ β (8)

i

where aμν and εi represent the coefficient of the KS molecular orbital and its corresponding orbital energy, respectively. Hμν and Sμν are the Hamiltonian and overlap matrices given by 0 Hμν = Hμν +

( rr′ ). These are obtained variation-

ϕμ = ∑ζ ∑i (cζr l + i)e−ζrYlm

ally by solving the atomic KS equation with an additional harmonic confining potential, (r/r0)2 ⎡ ⎛ ⎞2 ⎤ ⎢ − 1 ∇2 + v (ρ) + ⎜ r ⎟ ⎥ϕα = ε αϕα eff μ μ ⎢⎣ 2 μ ⎝ r0 ⎠ ⎥⎦ μ

∫ Vxc[ρ0 ]ρ0 dr ⃗ + VNN

(7)

(2)

αβ

+ Exc [ρ0 ] −

The energy Erep in eq 6 can be approximated as a sum of αβ (R ). It can short-range two-body potentials i.e.,Erep = ∑αβ V rep be obtained by the difference of the total energy resulting from a self-consistent DFT calculation and the electronic part of the SCC-DFTB as a function of internuclear distance of chosen reference systems. The SCC-DFTB electronic energy is then given by

where Ĥ 0 in the first term of the above equation is the KS Hamiltonian at the reference density, ψi are the KS orbitals, and ni are the occupation numbers. The second term in eq 1 explicitly depends on the interactions between charge fluctuations with the Coulomb and exchange-correlation integral kernel. It can be decomposed into atom-centered monopole contributions113 of the form

∑ ΔqαΔqβγαβ

ρ′ρ

∫ ∫ |r −0 0r′|

(6)

⃗ ′⃗ + Exc[ρ0 ] − ∫ Vxc[ρ0 ]ρ0 dr ⃗ + VNN ∫ ∫ | r ⃗ − r ′⃗ | drdr (1)

1 2

(5)

1 Sμν ∑ Δq ω(γαω + γβω) 2 ω

0

0 Hμν = ⟨ϕμ|Ĥ |ϕν⟩

Sμν = ⟨ϕμ|ϕν⟩ (3)

∀ μ ∈ α, ν ∈ β

∀ μ ∈ α, ν ∈ β

(9) (10) (11)

Thus, eq 8 can be solved self-consistently with respect to the charge fluctuation Δq, in conjunction with eq 7 to find the optimal set of the linear combination coefficients.116,126 The computational efficiency of the SCC-DFTB approach is based on the fact that all necessary Hamiltonian and overlap matrix elements for each pair of atoms in a system versus distance only need to be calculated once, and stored in a separate file, known as the Slater-Koster table. 2.2. Nonadiabatic Molecular Dynamics Simulation and the PYXAID Code. “On-the-fly” NA-MD simulation developed in the present work requires calculation of adiabatic electronic state energies, NA couplings, dki, and forces acting on the nuclei. These are calculated in the framework of the SCCDFTB method.65,127,128 The time-dependent wave function of the system is represented in the basis of the KS orbitals as

The introduction of the additional potential term124,125 into the equation forces the atomic wave functions to avoid areas far away from the nucleus resulting in compressed electron density rather than that of the free atom. The approach gives better electronic structure results for solids, molecules, and clusters.117 Once the basis set is defined, the matrix elements of the zeroth-order Hamiltonian can be calculated within the two center approximation ⎧ εμα if μ = v ⎪ ⎪ 0 Ĥμν = ⎨⟨ϕα| − 1 ∇2 + veff [ρ α + ρ β ]|ϕ β ⟩ if α ≠ β ν 0 0 ⎪ μ 2 ⎪ ⎩0 otherwise (4)

N

where veff is the effective KS potential (external, Hartree plus exchange-correlation) corresponding to the reference electron density ρ0. The latter is given as a superposition of the electron densities of the individual neutral atoms, α and β. The diagonal terms of the matrix elements are the DFT atomic eigenvalues. All off-diagonal terms are obtained via DFT calculations for an α−β dimer. With this approximation, the SCC-DFTB total energy can be expressed as follows

|ψ (r ; R(t ))⟩ =

∑ ck(t )|ϕk(r ; R(t ))⟩ k=0

(12)

where ck(t) is the time-dependent expansion coefficient, and | ϕk(r;R(t))⟩ is the adiabatic wave function representing the electronic state k, which is parametrically dependent on the classical nuclear trajectory R(t). N is the size of the electronic basis set used in temporal evolution. In the NA-MD simulation, 1438


Article


If the calculated dPki is negative, the hopping probability is set to zero. A hop from state k to state i can occur only when the electronic occupation of state k decreases and the occupation of state i increases, minimizing the number of hops. To conserve the total electron−nuclear energy after a hop, the original FSSH technique34 rescales the nuclear velocities along the direction of the NA coupling. If a NA transition to a higher energy electronic state is predicted by eq 16, while the kinetic energy available in the nuclear coordinates along the direction of the derivative coupling vector is insufficient to accommodate the increase in the electronic energy, the hop is rejected. Despite the great success of FSSH, absence of the decoherence effects and the problem due to frustrated hopping events, which prohibit modeling of superexchange and multiparticle phenomena, have hindered the range of FSSH applications. Advanced SH approaches, such as DISH96 and GFSH,94 have been developed and implemented in the PYXAID code by the Prezhdo group in order to address these issues. By combining PYXAID with DFTB, we have integrated the advanced techniques for simulating NA-MD within the tight binding approximation (DFTB-NA-MD).

nuclear degrees of freedom are treated classically, and nuclear trajectories, R(t), are obtained by solving the classical Newton’s equations of motion using quantum-mechanical forces. By using the ansatz, eq 12, time evolution of the coefficients ck(t) along a given classical trajectory can be obtained by solving the time-dependent Schrödinger equation dc (t ) iℏ k = dt

N

∑ (εkδki − iℏdki)ci(t )

(13)

i=0

where dki =

ϕk (r , R(t ))

∂ϕi(r , R(t )) ∂t

(14)

Here, εk is the energy of the adiabatic state k, and dki is the NA coupling between states k and i. The NA coupling arises because electronic wave functions depend parametrically on nuclear coordinates. It reflects the inelastic electron-vibrational interaction. The coupling is calculated numerically using the overlap of orbitals k and i at sequential time steps129 dki = −iℏ⟨ϕk (r , R(t ))|∇R |ϕi(r , R(t ))⟩ =−iℏ⟨ϕk (r , R(t )) ≈−

dR(t ) dt

3. RESULTS AND DISCUSSIONS To validate the accuracy of the developed DFTB-NA-MD approach, herein we compare the DFTB-NA-MD method with ab initio DFT-NA-MD and experiments for the relaxation dynamics of hot carriers in two low dimensional nanostructured materials. In particular, we study electron and hole relaxation in a zero-dimensional nanoscale cluster, known as QDs, and electron−hole recombination involving both inelastic and elastic electron−phonon scattering in a one-dimensional single-walled carbon nanotube (SWNT). Their potential in modern nanotechnology, in particular photovoltaic and photocat aly tic ap plicat ions, has received great at tention.5,26,119−122,132−135 The electronic structure of low dimensional systems is drastically different from the electronic structure of the corresponding bulk materials. Investigation of the dynamical processes in such systems provides solid validation of our method, which is designed for modeling of nanoscale objects. The considered dynamical processes carry both fundamental and practical importance. In this work, we have 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 Quantum Dot. CdSe gives rise to one of the most investigated types of semiconductor QDs. The hot carrier relaxation dynamics in these systems has been extensively studied, both experimentally120 and theoretically.119 With this motivation, we have chosen the Cd33Se33 QD as a model system to study the intraband electron and hole relaxation dynamics by using both DFTB-NA-MD and DFT-NA-MD simulation techniques. The Cd33Se33 QD is derived from wurtzite bulk structure and is a “magic” size cluster with diameter of 1.3 nm. It is one of the smallest stable CdSe QD with crystalline-like core.133,136,137 This makes the Cd33Se33 cluster an excellent model for quantum-mechanical studies of electronic properties of CdSe QDs. The ab initio DFT calculation has been carried out with a plane-wave basis, as incorporated in VASP.109 The valence electrons are described with the PBE exchange-correlation functional138 and a converged plane-wave basis. The core

∂ ϕ(r , R(t ))⟩ ∂t i

iℏ (⟨ϕk (r , R(t ))|ϕi(r , R(t + Δt ))⟩ 2Δt

−⟨ϕk (r , R(t + Δt ))|ϕi(r , R(t ))⟩)

(15)

where Δt is the time step used for the integration of the classical Newton’s equation of motion. The numerical solution of eq 13 yields the time-dependent amplitudes of the adiabatic states, ck(t). The amplitudes are utilized to calculate the hopping probabilities needed for switching the trajectory between the electronic states. Classical-mechanical prescription of nuclear dynamics in response to changes in electron density is known as the quantum backreaction problem.97 NA-MD provides a generalization of the original SH approach.130 Correlations between the nuclear motion and electronic states are built in using SH techniques. SH can be interpreted as a master equation, in which transition rates are nonperturbative and evolve in time. Tully’s FSSH algorithm34 (also called molecular dynamics with quantum transitions or MDQT) prescribes a probability for hopping between electronic states. FSSH minimizes the number of surface hops, while maintaining consistency with the Schrodinger equation.34 Moreover, it satisfies approximately the detailed balance between transitions upward and downward in energy,67 as required for proper description of electronvibrational energy exchange and relaxation to thermodynamic equilibrium.131 The probability of hopping from state k to i within the time step dt is given in FSSH by34 dPki =

bki dt cki

(16)

where cki = ck*(t )ci(t ) and bki = 2ℏ−1Im(aki⟨ϕk |H |ϕi⟩) − 2Re(cki dki) (17) 1439


Article


Figure 1. Projected density of states of the Cd33Se33 quantum dot calculated by (a) DFTB and (b) DFT. The optimized geometry is shown in the inset 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 middle of the HOMO−LUMO gap.

Cd-localized atomic orbitals. The projected DOS (PDOS) of both DFT and DFTB calculations shows that there is a notable contribution of the Se atom electron to the total electron density of the valence band edge, while the p orbitals of Cd atoms contribute to the total electron density of the conduction band edge of the CdSe QD. The DOS plot also shows that for both cases the state structure in the valence band (VB) is less pronounced as compared to the conduction band (CB), since the states are closer in energy in the VB. This fact agrees with the effective-mass theory, which uses a higher effective mass for holes than for electrons. Our results show that the geometric relaxation and the electronic structure obtained from DFTB agree with those of the DFT calculation, which is computationally much more demanding. Next, we compare the average absolute values of the NA couplings between representative states computed by DFT and DFTB, Table 1.We considered first four energy states in both

electrons are accounted for with the projector-augmented-wave (PAW) pseudopotentials.139 A periodically replicated cubic cell with at least 8 Å of vacuum between the QD replicas is used as a molecular model of the nonperiodic cluster. DFTB calculations have been performed by using the SCCDFTB 112−118 scheme as implemented in the DFTB+ code112,123 using the recently developed parameters, which have been tested extensively for a variety of Cd-chalcogenide systems.140 Similar simulation strategies were adopted for both DFT and DFTB. Starting with the initially optimized ground state cluster, at 0 K, we heated the system to 300 K with repeated velocity rescaling. Microcanonical trajectories were generated for 3 ps using the Verlet algorithm141 with 1 fs time step and Hellman-Feynman forces. At each snapshot, the energies of the KS molecular orbitals and the corresponding state-to-state NA couplings were calculated at both the DFTB and DFT levels. This time dependent information was stored and used in the NA-MD simulations, performed with CPAFSSH. In order to make a general analysis of the DFTB and DFT calculations, we first compare the geometric parameters of the QD obtained from both methods. Surface reconstruction provides a typical test, since it determines the interionic interactions and the force tolerance on individual ions. The DOS plot obtained by DFTB of this bare QD is shown in Figure 1 (a) with the optimized structure given in the inset. It is noteworthy that, for the stoichiometric cluster such Cd33Se33, the surface relaxation affects all surface atoms similarly, and thus all surface Cd−Se bonds are approximately equal. The surface reconstruction predicted by the DFTB and DFT is similar, even though it is known that DFTB overbinds.142 The DFTB DOS is consistent with DOS from the DFT calculations as shown in Figure 1 (b). The calculated zero temperature band gap obtained with DFTB is slightly larger than the one computed at the DFT level: ca. 1.84 eV and ca. 1.6 eV, respectively. A common feature of all semilocal exchangecorrelation functionals, such as PBE used in the present work, ab initio DFT tends to underestimate the band gap.143,144 On the contrary, the DFTB-derived band structure is not affected in the same way as in LDA or GGA-PBE calculations. This is a consequence of the small basis set employed. As a result, the band gap obtained in DFTB is much closer to the experimentally determined value.145 The DOS shows that the HOMO is dominated by the Se atoms, while the isolated LUMO state is mostly composed of

Table 1. Average Absolute Values of Nonadiabatic Coupling between Representative Pairs of States in the Cd33Se33 Quantum Dot Computed by DFTB and VASP ⟨|NAC|⟩, meV

DFTB

DFT

(HOMO)-(HOMO−1) (HOMO)-(HOMO−2) (HOMO)-(HOMO−3) (HOMO−1)-(HOMO−2) (HOMO−1)-(HOMO−3) (HOMO−2)-(HOMO−3) (LUMO+1)-(LUMO) (LUMO+2)-(LUMO) (LUMO+3)-(LUMO) (LUMO+2)-(LUMO+1) (LUMO+3)-(LUMO+1) (LUMO+3)-(LUMO+2)

8.86 1.97 1.13 9.85 3.01 10.61 5.72 1.70 1.29 10.41 3.87 13.61

5.57 2.25 1.56 8.55 2.91 9.40 3.42 1.71 1.36 7.43 4.00 8.79

VB (HOMO to HOMO−3) and CB (LUMO to LUMO+3). The NA couplings calculated by the two methods range from 1.13 to 13.61 meV. In both VB and CB, the NA coupling is higher between two neighboring energy levels than between more distant levels. Also, the coupling generally increases with an increase in the energy level. The NA couplings obtained by DFTB and DFT follow similar trends and are consistent with each other. The agreement between the DOS and NA 1440


Article


dynamics of electrons and holes, we compute populationweighted energy of the evolving electron and hole (Figure 4). The zero of energy is chosen to be at the average energy of LUMO+1 for electron and HOMO for hole. Black and red curves are obtained from DFTB-NA-MD and DFT-NA-MD simulations, respectively. The decay curves predicted by both simulation techniques are composed of two parts. The shorttime Gaussian decay gradually switches to exponential dynamics. The observed nonexponential relaxation component is in agreement with the strongly non-Lorentzian line shapes observed experimentally.146 The exponential component becomes important at lower energies for the electrons and at higher energies for the holes. The results indicate that DFTBNA-MD and DFT-NA-MD predict comparable electron and hole transfer dynamics and the energy decay times. We characterize the phonon modes participating in the intraband relaxation of electron and hole by computing Fourier transforms (FT) of the energy gaps between the initial and final states of the relaxation processes. Parts (a) and (c) of Figure 5 refer to electron, while parts (b) and (d) represent hole. These influence spectra are computed using both DFT (panels a, b) and DFTB (panels c, d). Two phonon frequencies are mainly associated with electron relaxation, one below 100 cm−1 and another around 200 cm−1. The peak below 100 cm−1 is less intense in DFT than DFTB. The difference can be attributed to a difference in the extent of surface relaxation between the two methods. Both DFTB and DFT predict that the hole couples to the phonons with a wide range of frequencies spreading over 400 cm−1. The corresponding spectra are broader than those for the electron relaxation. The overall appearance of the FT spectra for both electron and hole relaxation is qualitatively consistent between DFTB and DFT, although notable quantitative differences are seen in this case. We conclude that the newly developed DFTB-NA-MD formalism performs well in characterizing the charge-phonon interaction and energy exchange during intraband relaxation of electrons and holes in the CdSe QD. The results are in agreement with DFT-NA-MD simulations. The band gap in particular is more accurate in DFTB and DFT. At the same time, the DFTB-NA-MD scheme is computationally much more efficient that the ab initio method. 3.2. Electron−Hole Recombination in a Carbon Nanotube. As a further validation and robustness test of the DFTB-NA-MD scheme, we applied it to study the electron− hole recombination across the fundamental bank gap in a semiconducting SWNT. We compared the simulation results with the experimental time scales for both inelastic and elastic electron−phonon scattering in this case. The carrier relaxation dynamics in SWNT is well investigated both theoretically121,147,148 and experimentally.122,132,135,149−151 Reproducing the exciton lifetime of a semiconducting SWNT is a good benchmark for the new implementation of the NA-MD methodologies. Exciton life times are notably shorter in SWNTs compared to other semiconductor materials. Experiments produce electron−hole recombination times on the order of 100 ps.122,132 Motivated by these experiments, we compute 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 is found to be semiconducting with the experimental band gap of 0.86 eV.152 Figure 6 depicts the DFTB calculated DOS of the (10,5) SWNT at 0 K along with its optimized geometry. The

couplings leads to agreement in the electron and hole relaxation dynamics. To study the intraband relaxation dynamics of excited electron and hole in the Cd33Se33 QD we prepare initial excitations by creating a hole approximately 1 eV below the HOMO level and an electron approximately 1 eV above the LUMO level. The dynamics of the electronic excitation is restricted to the dense mainfold of unoccupied states above the LUMO, excluding the latter. The LUMO is energetically wellseparated from the rest of the CB, giving rise to the so-called phonon bottleneck to the electron relaxation.119 Simulation of such slow relaxation processes requires incorporation of decoherence effects, as exemplified in the following section. The quasi-continuous DOS in the VB suggests ultrafast relaxation 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)

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

and (b), respectively. The figures show a three-dimensional plot of the state populations as a function of energy and time. In both instances, the initial photoexcitation peak at around 0.8 eV vanishes and reappears in the final states. Although the carriers visit multiple intermediate states during relaxation, none of them play any special role. Analogous results computed for hole relaxation are summarized in Figure 3. Figures 2 and 3 indicate that the dynamics of electron and hole obtained with DFTBNA-MD and DFT-NA-MD are very similar, validating the former method. To obtain further insight into the relaxation

Figure 3. Intraband relaxation of hole energy at 300 K computed with FSSH using (a) DFTB and (b) ab initio DFT. 1441


Article


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.

simulations are performed in a periodically replicated cell with 30 Å of vacuum between SWNT replicas in the X- and Ydirections. The tube is periodic in the Z-direction with the optimized cell length of 11.33 Å. The optimized tube diameter is 10.30 Å, and the total number of carbon atoms in the simulation cell is 140. Ab initio DFT-NA-MD simulations are rather challenging in this case. The DFTB calculations were performed using the DFTB+ code112,123 with a well tested parameter set developed for carbon based systems.114 The kpoint sampling was done with the (1 × 1 × 256) Monkhorst− Pack k-point grid. The calculated DOS agrees well with the previously reported theoretical results.153 The calculated energy gap 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 between nearly degenerate states. Further, quantum transitions across large gaps are slower than loss of quantum coherence, requiring an explicit treatment of decoherence effects in the quantum-

Figure 6. Density of states of the (10,5) single wall carbon nanotube at 0 K calculated by DFTB. The optimized geometry of the unit cell is shown in the inset. The zero of energy is set equal to the middle of the band gap.

1442


Article


transitions. In the latter case, within the framework of the optical response theory, elastic electron−phonon scattering is known as pure-dephasing.155 In particular, its inverse determines the line width of single particle luminescence. The pure-dephasing time is inversely related to the magnitude of the energy gap fluctuation: the smaller the fluctuation of the gap, the larger the dephasing time.156 The pure-dephasing time is also related to the memory of the energy gap fluctuation. Using the optical response theory, one computes the gap autocorrelation function (ACF)

classical NA-MD simulation. Hence, we used the DISH method.96 Figure 7 shows the decay of population of the lowest energy excitation in the (10,5) SWNT. The relaxation follows

C(t ) = ⟨ΔE(t )ΔE(0)⟩T

(18)

where the bracket indicates canonical averaging. The ACF normalized by its initial value ⟨ΔE(t )EΔ(0)⟩T

Cnorm(t ) =

⟨ΔE2(0)⟩T

(19)

is shown in Figure 9 (a). The decay is very slow and indicates long memory in the gap fluctuation. The extremely long decay Figure 7. Electron−hole recombination dynamics in the (10, 5) carbon nanotube at 300 K, computed using DFTB electronic structure and DISH nonadiabatic molecular dynamics.

exponential decay, P(t) = exp(−t/τ1). Using the short-time, first-order Taylor expansion, we obtained the electron−hole recombination time of 206 ps, which is comparable with the experimental result of Huang and Krauss.122 To elucidate the atomic motions that are responsible for the energy loss, we have identified the phonon modes that couple with the electronic subsystem. Figure 8 presents FT of the

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

VBM-CBM energy gap in the (10,5) SWNT. The FT spectrum indicates that the electronic subsystem couples to the G-type longitudinal optical (LO) phonon. The computed frequency is around 1800 cm−1 and is overestimated compared to the experimental value of 1600 cm−1. Malola et al.154 showed that gamma-point calculation by the DFTB method overestimates the frequency of the high-energy G-mode in SWNTs. The overestimation occurs due to the incomplete atomic basis set used 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 electronic transition of from CBM to VBM is accommodated via inelastic process leading to energy exchange between the electronic and vibrational subsystems. Elastic electron−phonon interaction destroys coherence between the CBM and VBM states. The coherence is formed during both nonradiative and radiative

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

of the ACF arises because the electronic subsystem couples only to the G-mode (Figure 8), and because this phonon is very harmonic.111 The pure-dephasing function is computed using the secondorder cumulant expansion of the optical response function as155 Dcumu(t ) = exp( −g (t ))

(20)

where g(t) is g (t ) = 1443

∫0

t

dτ1

∫0

τ1

dτ2C(τ2)

(21) DOI: 10.1021/acs.jctc.5b01231 J. Chem. Theory Comput. 2016, 12, 1436−1448

Article


(7) Grätzel, M. Solar energy conversion by dye-sensitized photovoltaic cells. Inorg. Chem. 2005, 44 (20), 6841−6851. (8) Liu, F.; Meyer, G. J. A nuclear isotope effect for interfacial electron transfer: Excited-state electron injection from Ru ammine compounds to nanocrystalline TiO2. J. Am. Chem. Soc. 2005, 127 (3), 824−825. (9) McConnell, I.; Li, G.; Brudvig, G. W. Energy conversion in natural and artificial photosynthesis. Chem. Biol. 2010, 17 (5), 434− 447. (10) Wang, L.; Long, R.; Trivedi, D.; Prezhdo, O. V. Time-Domain Ab Initio Modeling of Charge and Exciton Dynamics in Nanomaterials. In Green Processes for Nanotechnology; Springer International Publishing: 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.; Grätzel, M.; Ho-Baillie, A. Utilization of Direct and Diffuse Sunlight in a DyeSensitized Solar Cell - Silicon Photovoltaic Hybrid Concentrator System. J. Phys. Chem. Lett. 2011, 2 (6), 581−585. (12) Barea, E. M.; Shalom, M.; Giménez, S.; Hod, I.; Mora-Seró, I.; Zaban, A.; Bisquert, J. Design of injection and recombination in quantum 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 fission for solar energy conversion: one-versus two-electron transfer from the quantum mechanical superposition. J. Am. Chem. Soc. 2012, 134 (44), 18295−18302. (14) Johnson, J. C.; Nozik, A. J.; Michl, J. High triplet yield from singlet 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. Hybrid Si 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 fission in 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 solar cells with efficient light management and self-cleaning. Nano Lett. 2010, 10 (6), 1979−1984. (18) Akimov, A. V.; Prezhdo, O. V. Nonadiabatic dynamics of charge transfer 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 electronic couplings in organic semiconductors. J. Chem. Theory Comput. 2011, 7 (3), 594−601. (20) Fischer, S. A.; Duncan, W. R.; Prezhdo, O. V. Ab initio nonadiabatic 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 and optical absorption of small semiconducting clusters: Dopants, defects and 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 initio study. J. Phys. Chem. Lett. 2010, 1 (1), 232−237. (23) Hyeon-Deuk, K.; Prezhdo, O. V. Time-domain ab initio study of auger and phonon-assisted auger processes in a semiconductor quantum dot. Nano Lett. 2011, 11 (4), 1845−1850. (24) Isborn, C. M.; Kilina, S. V.; Li, X.; Prezhdo, O. V. Generation of multiple excitons in PbSe and CdSe quantum dots by direct photoexcitation: first-principles calculations on small PbSe and CdSe clusters. J. Phys. Chem. C 2008, 112 (47), 18291−18294. (25) Kilin, D. S.; Pereversev, Y. V.; Prezhdo, O. V. Electron-nuclear correlations 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 initio time-domain study of phonon-assisted relaxation of charge carriers in a PbSe 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

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

4. CONCLUSIONS Concluding, we have developed and implemented a novel methodology for studying NA charge and energy transfer dynamics in large systems. The tight-binding model provides the electronic structure at a very good compromise between accuracy and computational demand. Electronic states, and thereafter NA couplings between pairs of these states, calculated within the tight-binding scheme have been utilized for studying excited state dynamics explicitly in the timedomain. We have tested the method by considering excitation dynamics in zero- and one-dimensional materials: a CdSe QD and a SWNT. The relaxation dynamics of the hot charge carriers, the time scale of the electron−hole recombination, and the elastic electron−phonon scattering time match well with the time-domain ab initio DFT calculations and the experimental results. The test studies justify the accuracy and efficiency of the approach, which is expected to perform well for other large systems and to require modest computer resources.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors are grateful to Parmeet Ninjar for comments on the manuscript. A.V.A. acknowledges financial support from the University at Buffalo, The State University of New York startup package. 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.; Grätzel, M. Molecular photovoltaics. Acc. Chem. Res. 2000, 33 (5), 269−277. (2) Anderson, N. A.; Lian, T. Ultrafast electron transfer at the molecule-semiconductor nanoparticle interface. Annu. Rev. Phys. Chem. 2005, 56, 491−519. (3) Zhao, W.; Ma, W.; Chen, C.; Zhao, J.; Shuai, Z. Efficient degradation of toxic organic pollutants with Ni2O3/TiO2‑x Bx under visible 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 bipyridyl CO2 reduction catalyst to rutile TiO2. J. Am. Chem. Soc. 2011, 133 (18), 6922−6925. (5) Akimov, A. V.; Neukirch, A. J.; Prezhdo, O. V. Theoretical insights into photoinduced charge transfer and catalysis at oxide interfaces. Chem. Rev. 2013, 113 (6), 4496−4565. (6) Ardo, S.; Meyer, G. J. Photodriven heterogeneous charge transfer with transition-metal compounds anchored to TiO2 semiconductor surfaces. Chem. Soc. Rev. 2009, 38 (1), 115−164. 1444


Article

Journal of Chemical Theory and Computation 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 dyesensitized solar cells: Can a suitable sensitizer be found? J. Am. Chem. Soc. 2006, 128 (51), 16546−16553. (29) Sánchez-de-Armas, R. o.; Oviedo López, J.; San-Miguel, M. A.; Sanz, J. F.; Ordejón, P.; Pruneda, M. Real-time TD-DFT simulations in dye sensitized solar cells: the electronic absorption spectrum of Alizarin 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 anchors for robust TiO2 sensitization. J. Am. Chem. Soc. 2011, 133 (23), 9014− 9022. (31) Yost, S. R.; Wang, L.-P.; Van Voorhis, T. Molecular insight into the energy levels at the organic donor/acceptor interface: A quantum mechanics/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 electroninjection processes in a photoexcited dye-sensitized nanocrystalline zinc 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. Ultrafast plasmon-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 thermal fluctuations 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 ab initio study of charge relaxation and recombination in dye-sensitized TiO2. J. Am. Chem. Soc. 2007, 129 (27), 8528−8543. (38) Duncan, W. R.; Prezhdo, O. V. Theoretical studies of photoinduced 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 Surfaces Sensitized with Ru (II)- Polypyridine Complexes†. J. Phys. Chem. A 2009, 113 (45), 12532−12540. (40) Kilin, D. S.; Micha, D. A. Surface photovoltage at nanostructures on 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 Doped Si 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: a step toward the understanding of dye-sensitized solar cell performances. J. Am. Chem. Soc. 2009, 131 (40), 14290−14298. (43) Labat, F. d. r.; Le Bahers, T.; Ciofini, I.; Adamo, C. Firstprinciples modeling of dye-sensitized solar cells: challenges and perspectives. Acc. Chem. Res. 2012, 45 (8), 1268−1277. (44) Maggio, E.; Martsinovich, N.; Troisi, A. Theoretical study of charge recombination at the TiO2-electrolyte interface in dye sensitised solar cells. J. Chem. Phys. 2012, 137 (22), 22A508. (45) Maggio, E.; Martsinovich, N.; Troisi, A. Evaluating charge recombination rate in dye-sensitized solar cells from electronic structure calculations. J. Phys. Chem. C 2012, 116 (14), 7638−7649. (46) Rego, L. G.; Batista, V. S. Quantum dynamics simulations of interfacial 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 Energy Transfer between Building Blocks in a Phenylene Ethynylene Dendrimer†. 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: The Crucial Role of High-Frequency CC 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 of photoinduced dynamics in conjugated molecules. J. Phys. Chem. B 2011, 115 (18), 5402−5414. (50) Wu, C.; Malinin, S. V.; Tretiak, S.; Chernyak, V. Y. Multiscale Modeling of Electronic Excitations in Branched Conjugated Molecules Using 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 Investigation of Triplet Yield in the Regime of Localized Excitation and Fast Coherent 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 generation and recombination dynamics in small Si and CdSe quantum dots: An ab initio time-domain study. ACS Nano 2012, 6 (2), 1239−1250. (54) Isborn, C. M.; Li, X. Modeling the doubly excited state with time-dependent Hartree−Fock and density functional theories. J. Chem. Phys. 2008, 129 (20), 204107. (55) Fabiano, E.; Keal, T.; Thiel, W. Implementation of surface hopping 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 cisdienes. Faraday Discuss. 2012, 157, 193−212. (57) Tao, H.; Allison, T.; Wright, T.; Stooke, A.; Khurmi, C.; Van Tilborg, J.; Liu, Y.; Falcone, R.; Belkacem, A.; Martinez, T. Ultrafast internal 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 in Chemistry: A Bird’s Eye View of a Vibrant Field. Chem. Rev. 2015, 115 (12), 5797. (59) Barbatti, M. Nonadiabatic dynamics with trajectory surface hopping 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. Entropy 2014, 16 (1), 62−85. (62) Kapral, R. Progress in the theory of mixed quantum-classical dynamics. 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 and advanced methods; Cambridge University Press: New York, 2009. (65) Mitric, R.; Werner, U.; Wohlgemuth, M.; Seifert, G.; BonačićKoutecký, V. Nonadiabatic Dynamics within Time-Dependent Density Functional 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-classical equilibrium. J. Chem. Phys. 2005, 122 (9), 094102. (68) Hartmann, M.; Pittner, J.; Bonačić-Koutecký, V. Ab initio nonadiabatic dynamics involving conical intersection combined with Wigner distribution approach to ultrafast spectroscopy illustrated on Na3F2 cluster. J. Chem. Phys. 2001, 114 (5), 2123−2136. (69) Mitrić, R.; Bonačić-Koutecký, V.; Pittner, J.; Lischka, H. Ab initio nonadiabatic dynamics study of ultrafast radiationless decay over conical intersections illustrated on the Na3F cluster. J. Chem. Phys. 2006, 125 (2), 024303. 1445


Article

Journal of Chemical Theory and Computation (70) Craig, C. F.; Duncan, W. R.; Prezhdo, O. V. Trajectory surface hopping in the time-dependent Kohn-Sham approach for electronnuclear dynamics. Phys. Rev. Lett. 2005, 95 (16), 163001. (71) Mitrić, R.; Werner, U.; Bonačić-Koutecký, V. Nonadiabatic dynamics and simulation of time resolved photoelectron spectra within time-dependent density functional theory: Ultrafast photoswitching in benzylideneaniline. J. Chem. Phys. 2008, 129 (16), 164118. (72) Tapavicza, E.; Tavernelli, I.; Rothlisberger, U. Trajectory surface hopping within linear response time-dependent density-functional theory. 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.; Mitrić, R.; Suzuki, T.; Bonačić-Koutecký, V. Nonadiabatic dynamics within the time dependent density functional theory: Ultrafast photodynamics in pyrazine. Chem. Phys. 2008, 349 (1), 319−324. (75) Fabiano, E.; Thiel, W. Nonradiative deexcitation dynamics of 9H-adenine: An OM2 surface hopping study. J. Phys. Chem. A 2008, 112 (30), 6859−6863. (76) Granucci, G.; Persico, M.; Toniolo, A. Direct semiclassical simulation of photochemical processes with semiempirical wave functions. J. Chem. Phys. 2001, 114 (24), 10608−10615. (77) Lan, Z.; Fabiano, E.; Thiel, W. Photoinduced nonadiabatic dynamics of pyrimidine nucleobases: on-the-fly surface-hopping study with semiempirical methods. J. Phys. Chem. B 2009, 113 (11), 3548− 3555. (78) CPMD; IBM Corp.: Armonk, NY, 1990−2008; MPI für Festkö rperforschung 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 in massively 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-Maksić, M.; Lischka, H. The on-the-fly surfacehopping program system Newton-X: Application to ab initio simulation 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: a package 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 for the 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 molecular dynamics. 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 quantum chemistry package. Chem. Phys. 2008, 347 (1), 3−16. (85) Marques, M. A.; Castro, A.; Bertsch, G. F.; Rubio, A. Octopus: a first-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.; Schütz, 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.; Köppl, 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.; Pflüger, 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; Cardiff University, UK, 2012. See http://www.molpro.net/ (accessed Mar 1, 2016).

(88) Ehrenfest, P. Adiabatische Transformationen in der Quantentheorie und ihre Behandlung durch Niels Bohr. Naturwissenschaften 1923, 11 (27), 543−550. (89) Ehrenfest, P. Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik. Eur. Phys. J. A 1927, 45 (7−8), 455−457. (90) Fischer, S. A.; Chapman, C. T.; Li, X. Surface hopping with Ehrenfest excited potential. J. Chem. Phys. 2011, 135 (14), 144102. (91) Käb, G. Mean field Ehrenfest quantum/classical simulation of vibrational 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 initio Ehrenfest dynamics. J. Chem. Phys. 2005, 123 (8), 084106. (93) Subotnik, J. E. Augmented Ehrenfest dynamics yields a rate for surface hopping. J. Chem. Phys. 2010, 132 (13), 134112. (94) Wang, L.; Trivedi, D.; Prezhdo, O. V. Global flux surface hopping approach for mixed quantum-classical dynamics. J. Chem. Theory Comput. 2014, 10 (9), 3598−3605. (95) Akimov, A. V.; Prezhdo, O. V. Second-quantized surface hopping. Phys. Rev. Lett. 2014, 113 (15), 153003. (96) Jaeger, H. M.; Fischer, S.; Prezhdo, O. V. Decoherence-induced surface hopping. J. Chem. Phys. 2012, 137 (22), 22A545. (97) Prezhdo, O. V.; Duncan, W. R.; Prezhdo, V. V. Photoinduced electron dynamics at the chromophore−semiconductor interface: A time-domain ab initio perspective. Prog. Surf. Sci. 2009, 84 (1), 30−68. (98) Wang, L.; Prezhdo, O. V. A simple solution to the trivial crossing problem in surface hopping. J. Phys. Chem. Lett. 2014, 5 (4), 713−719. (99) Akimov, A. V.; Prezhdo, O. V. Formulation of quantized Hamiltonian dynamics in terms of natural variables. J. Chem. Phys. 2012, 137 (22), 224115. (100) Akimov, A. V.; Long, R.; Prezhdo, O. V. Coherence penalty functional: A simple method for adding decoherence in Ehrenfest dynamics. J. Chem. Phys. 2014, 140 (19), 194107. (101) Hyeon-Deuk, K.; Prezhdo, O. V. Photoexcited electron and hole dynamics in semiconductor quantum dots: phonon-induced relaxation, 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-domain ab 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 nonadiabatic 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 the PYXAID program: integration schemes, decoherence effects, multiexcitonic 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 Initio Nonadiabatic Molecular Dynamics of the Ultrafast Electron Injection across the Alizarin-TiO2 Interface. J. Am. Chem. Soc. 2005, 127 (21), 7941−7951. (106) Kohn, W.; Sham, L. J. Self-consistent equations including exchange 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 surface hopping. 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 project for quantum simulations of materials. J. Phys.: Condens. Matter 2009, 21 (39), 395502. (109) Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio 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. Exciton multiplication from first principles. Acc. Chem. Res. 2013, 46 (6), 1280−1289. 1446


Article

Journal of Chemical Theory and Computation (111) Madrid, A. B.; Hyeon-Deuk, K.; Habenicht, B. F.; Prezhdo, O. V. Phonon-induced dephasing of excitons in semiconductor quantum dots: Multiple exciton generation, fission, and luminescence. ACS Nano 2009, 3 (9), 2487−2494. (112) Aradi, B.; Hourahine, B.; Frauenheim, T. DFTB+, a sparse matrix-based implementation of the DFTB method. J. Phys. Chem. A 2007, 111 (26), 5678−5684. (113) Elstner, M. SCC-DFTB: What is the proper degree of selfconsistency? 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 densityfunctional tight-binding method for simulations of complex materials properties. Phys. Rev. B: Condens. Matter Mater. Phys. 1998, 58 (11), 7260. (115) Krüger, T.; Elstner, M.; Schiffels, P.; Frauenheim, T. Validation of the density-functional based tight-binding approximation method for 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-dependent density-functional response theory. Phys. Rev. B: Condens. Matter Mater. Phys. 2001, 63 (8), 085108. (117) Porezag, D.; Frauenheim, T.; Köhler, T.; Seifert, G.; Kaschner, R. Construction of tight-binding-like potentials on the basis of densityfunctional theory: Application to carbon. Phys. Rev. B: Condens. Matter Mater. Phys. 1995, 51 (19), 12947. (118) Seifert, G. Tight-binding density functional theory: an approximate Kohn-Sham DFT scheme. J. Phys. Chem. A 2007, 111 (26), 5609−5613. (119) Trivedi, D. J.; Wang, L.; Prezhdo, O. V. Auger-Mediated Electron Relaxation Is Robust to Deep Hole Traps: Time-Domain Ab Initio 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 photoemission study of competing Auger and surface-mediated relaxation of hot electrons in CdSe quantum dot solids. Nano Lett. 2013, 13 (4), 1655− 1661. (121) Habenicht, B. F.; Craig, C. F.; Prezhdo, O. V. Time-domain ab initio simulation of electron and hole relaxation dynamics in a singlewall semiconducting carbon nanotube. Phys. Rev. Lett. 2006, 96 (18), 187401. (122) Huang, L.; Krauss, T. D. Quantized bimolecular Auger recombination of excitons in single-walled carbon nanotubes. Phys. Rev. Lett. 2006, 96 (5), 057407. (123) Frauenheim, T.; Seifert, G.; Elstner, M.; Niehaus, T.; Köhler, C.; Amkreutz, M.; Sternberg, M.; Hajnal, Z.; Di Carlo, A.; Suhai, S. Atomistic simulations of complex materials: ground-state and excitedstate properties. J. Phys.: Condens. Matter 2002, 14 (11), 3015. (124) Eschrig, H.; Bergert, I. An optimized LCAO version for band structure calculations application to copper. Phys. Status Solidi B 1978, 90 (2), 621−628. (125) Eschrig, H. Optimized LCAO method and the electronic structure of extended systems, 1st ed.; Akademie Verlag and Springer: Berlin, Germany, 1989; 221 pages. (126) Oliveira, A. F.; Seifert, G.; Heine, T.; Duarte, H. A. Densityfunctional 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 Charge Transport 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 based treatment 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 hopping approach to nonadiabatic molecular collisions: the reaction of H+ with D2. J. Chem. Phys. 1971, 55 (2), 562−572. (131) Prezhdo, O. V. Mean field approximation for the stochastic Schrödinger equation. J. Chem. Phys. 1999, 111 (18), 8366−8377. (132) Carlson, L. J.; Krauss, T. D. Photophysics of individual singlewalled carbon nanotubes. Acc. Chem. Res. 2008, 41 (2), 235−243. (133) Kilina, S.; Ivanov, S.; Tretiak, S. Effect of surface ligands on optical 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 phonon bottleneck in PbSe and CdSe quantum dots: Time-domain density functional 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 in single-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 mass spectrometry. Nat. Mater. 2004, 3 (2), 99−102. (137) Puzder, A.; Williamson, A.; Gygi, F.; Galli, G. Self-healing of CdSe nanocrystals: first-principles calculations. Phys. Rev. Lett. 2004, 92 (21), 217401. (138) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77 (18), 3865. (139) Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B: Condens. Matter Mater. Phys. 1999, 59 (3), 1758. (140) Sarkar, S.; Pal, S.; Sarkar, P.; Rosa, A.; Frauenheim, T. SelfConsistent-Charge Density-Functional Tight-Binding Parameters for Cd−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.; Schlüter, M.; Sham, L. Self-energy operators and exchange-correlation potentials in semiconductors. Phys. Rev. B: Condens. Matter Mater. Phys. 1988, 37 (17), 10159. (144) Gygi, F.; Baldereschi, A. Quasiparticle energies in semiconductors: 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-binding description of zinc-containing compounds. J. Chem. Theory Comput. 2009, 5 (3), 605−614. (146) Califano, M.; Zunger, A.; Franceschetti, A. Efficient inverse Auger 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 in semiconducting carbon nanotubes. Nano Lett. 2007, 7 (11), 3260− 3265. (148) Habenicht, B. F.; Prezhdo, O. V. Nonradiative quenching of fluorescence in a semiconducting carbon nanotube: A time-domain ab initio 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 in single-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 1447


Article

Journal of Chemical Theory and Computation phonons in single-walled carbon nanotubes. Phys. Rev. Lett. 2008, 100 (22), 225503. (152) Wong, H.-S. P.; Akinwande, D. Carbon nanotube and graphene device physics; Cambridge University Press: New York, 2011. (153) Reich, S.; Thomsen, C.; Ordejón, P. Electronic band structure of isolated and bundled carbon nanotubes. Phys. Rev. B: Condens. Matter Mater. Phys. 2002, 65 (15), 155411. (154) Malola, S.; Häkkinen, H.; Koskinen, P. Raman spectra of single-walled carbon nanotubes with vacancies. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 77 (15), 155412. (155) Mukamel, S. Principles of nonlinear optical spectroscopy; Oxford University Press: New York, 1999. (156) Akimov, A. V.; Prezhdo, O. V. Persistent electronic coherence despite 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.

1448


Nonadiabatic Molecular Dynamics for Thousand Atom Systems: A

Recommend Documents