Modeling Charge Transfer in Fullerene Collisions via Real-Time

May 18, 2012 - Cherry Emerson Center for Scientific Computing and Department of Chemistry, ... The consequences of real-time electron dynamics are ful...
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Modeling Charge Transfer in Fullerene Collisions via Real-Time Electron Dynamics Jacek Jakowski,*,† Stephan Irle,‡ Bobby G. Sumpter,¶ and Keiji Morokuma§,∥ †

National Institute for Computational Sciences, Oak Ridge, Tennessee 37831, United States Department of Chemistry, Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan ¶ Center for Nanophase Materials Sciences and Computer Science & Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States § Cherry Emerson Center for Scientific Computing and Department of Chemistry, Emory University, 1515 Dickey Drive, Atlanta, Georgia 30322, United States ∥ Fukui Institute for Fundamental Chemistry, Kyoto University, Sakyo, Kyoto 6006-8103, Japan ‡

S Supporting Information *

ABSTRACT: An approach for performing real-time dynamics of electron transfer in a prototype redox reaction that occurs in reactive collisions between neutral and ionic fullerenes is discussed. The quantum dynamical simulations show that the electron transfer occurs within 60 fs directly preceding the collision of the fullerenes, followed by structural changes and relaxation of electron charge. The consequences of real-time electron dynamics are fully elucidated for the far from equilibrium processes of collisions between neutral and multiply charged fullerenes.

SECTION: Energy Conversion and Storage; Energy and Charge Transport

U

fullerenes and carbon nanostructures have been reported (see, for example, refs 11 and 16−18 and references therein), whereas theoretical quantum dynamics studies mostly focused on electronic excitation in high-energy collisions with small ions3,7,17,19 or on the quantum dynamical treatment of vibrational wave functions.15 As noted by Campbell,16,20 the relatively large size of a fullerene, a highly delocalized electronic system, and the large number of vibrational degrees of freedom could significantly affect the charge-transfer process and presents a very challenging problem for theoretical treatment. A combined system of neutral and multiply charged fullerenes is not in its ground state, the collision-induced ET is a nonequilibrium process, and hence, it is very difficult to tackle using Born−Oppenheimer dynamics.21,22 In fact, electrontransfer processes are prototypical redox reactions and are intrinsically nonadiabatic in nature. Moreover, the selfinteraction error in density functional theory (DFT) methods typically causes overdelocalization of the electron density across several molecules, further diminishing chances for a correct simulation.23−26

nderstanding the motions of constituent atoms and electrons in molecules lies at the heart of chemical reactions. The idea behind performing molecular dynamics simulations is that an observer can “see” in detail where and how the reaction proceeds.1,2 In this regard, direct molecular dynamics with time-dependent quantum mechanical treatment of electronic structure is especially promising. Contrary to classical Born−Oppenheimer molecular dynamics or its extended Lagrangian approximations, the explicit time propagation of electronic structure is a promising approach for the simulation of (a) coupling between electrons and nuclei, (b) interaction with external time-dependent perturbations like laser fields, (c) excited-state dynamics, and (d) processes in which the electronic structure is far from equilibrium, such as a system consisting of a reductant and oxidant before reaction.3−8 The explicit treatment of electron transfer (ET) and nonadiabatic processes is fundamentally important to an enormous range of processes and is a key problem to tackle for the predictive theory and simulation challenge of the materials genome initiative (see, for example, ref 9). Aggregates of fullerenes and their derivatives are considered to be efficient semiconducting materials for organic solar cells10−12 and organic transistors.13−15 The ET between fullerenes is a key process of energy conversion of organic solar cells.15 A number of charge-transfer experiments involving © 2012 American Chemical Society

Received: April 10, 2012 Accepted: May 18, 2012 Published: May 18, 2012 1536

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Δt 2 [F ̅ , [F ̅ , P]] 2! Δt 3 Δt 4 +i [F ̅ , [F ̅ , [F ̅ , P]]] − 3! 4!

In this study, we investigate electron and energy transfer occurring far from equilibrium in collisions between neutral and multiply charged fullerenes by employing an explicit real-time time-dependent quantum mechanical approach for electronic structure. We focus on the coupling between electronic and nuclear degrees of freedom, and the effect of ET between the two fullerenes. We employ a direct dynamics method called Liouville−von Neumann molecular dynamics (LvNMD).2,27 In this method, all nuclei are treated classically, while for electrons, we solve the time-dependent Schrödinger equation. This is an initial value problem in which we simultaneously integrate differential equations for the evolution of electrons and for nuclei subject to the initial positions and velocities of nuclei and subject to the initial electronic structure. The electronic structure in our simulations is treated within spin-unpolarized density functional tight binding (DFTB).28,29 The electronic energy is given by E = Tr[HcoreP] +

1 Tr[G(P) ·P] + 2

P(t ) = P(0) − iΔt[F ̅ , P] −

[F ̅ , [F ̅ , [F ̅ , [F ̅ , P]]]] + ...

where F̅ = 1/2[F(t) + F(t + Δt)] is the time-averaged Fock matrix. The details of implementation and evaluation of forces can be found elsewhere.2 Any physically meaningful density matrix can be used as an initial starting density in LvNMD. Practical choices may include ground- or excited-state densities (either idempotent or fractional, finite electronic temperature densities),30 constrained DFT densities,31 or as a direct product of electron densities of isolated molecules. Here, we borrow from the perturbational theory of intermolecular interaction and construct the initial total density matrix as a direct product of isolated molecular densities.32 The total density matrix of a combined C60···C2+ 60 is constructed as

Atoms

∑ A>B

AB Erep

(1)

in which P is an electronic density matrix representing valence electrons and Hcore describes the interaction of valence electrons with core ions and also the interaction of valence electrons with themselves included in the Hartree and correlation−exchange potentials at the reference density. Erep is a sum of pairwise potentials between atomic centers A and B and describes the interaction of reference atomic densities, core electrons, and nuclei.7 The matrix G(P) describes the interaction between valence electrons, and it depends on the density matrix elements and on chemical hardness-based parameters γAB that depend on the interatomic distance and assumes the value of 1/RAB in the limit RAB → ∞. In LvNMD, the nuclei are moving classically, while we propagate electronic density matrices quantum mechanically using the von Neumann’s equation iℏ

P(t ) = [F(t ), P(t )] dt

P(0) =

PA 0 0 PB

(5)

where PA and PB are densities for, respectively, a neutral C60 and ionic C2+ 60 fullerene. Both matrices, PA and PB, are obtained in two separate BO calculations for isolated fullerene subunits. In fact, the large initial separation between fullerene subunits ensures that there is no covalent interfullerene interaction (overlap between A and B is zero) and that the interaction can be treated as a perturbation.32 The current approach allows efficient construction of the initial nonequilibrium electronic state of a large system consisting of several molecules (such as reductant and oxidant in the solvent before a redox reaction). This approach is especially useful in bulk phase simulations, where preparation of the system of many molecules in a desired electronic state might be otherwise difficult to achieve. Finally, it is worthwhile to mention that LvNMD is an attractive approach for massively parallel computer architectures due to its heavy dependence on linear algebra matrix operations.33,34 To simplify the analysis, the initial fullerenes were set to be vibrationally cold (no ZPE), and the initial electronic structure of each C60 subunit was in its ground electronic state (electronic temperature Tel = 0 K). Nevertheless, transfer of collision energy into heating of the carbon cage and excitation of the electronic structure are allowed during the quantum dynamics, and we observe such transfers resulting from the collisions. In all of our simulations, the initial separation between fullerenes was 20 Å, we used a time step equal to 0.02 fs, and the dynamics simulations were performed in the microcanonical ensemble for 3 ps with different collision energies ranging from 1.4 eV up to 143 eV with respect to the center of mass of the total system. Overall, we ran 19 central collision trajectories for C60···C60 and 19 trajectories for C60···C2+ 60 collisions. The conservation of the total (kinetic + potential) energy was within microhartrees. The initial electronic structure for the LvNMD simulations for the collisions was prepared as a direct product of the groundstate noninteracting densities of neutral and dication molecules. Figure 1 presents the time dependence of the kinetic and potential energy for C60···C60 collisions in comparison with the corresponding ground-state energies. Similarly to our previous

(2)

Here, the density matrix, P(t), depends on time, and F(t) is a time-dependent Hamiltonian matrix for electrons such that

F(t ) = Hcore + G(P)

(4)

(3)

The last two expressions can be used to propagate the electronic structure within Hartree−Fock (HF), DFT, and tight binding methods. The simultaneous dynamics of electrons and nuclei is achieved as follows. At the start of the trajectories (t = 0), initial positions of nuclei, velocities, and the corresponding density matrix need to be generated. The procedure used to initialize electronic structure is presented below. After the initialization, at every time step of the trajectory, we first move all nuclei using the velocity Verlet algorithm and then propagate the electronic structure by solving eq 2. To solve eq 2, one needs to know the time dependence of F, but F depends on P via the electron−electron repulsion. We solve this problem iteratively by repeating the density matrix propagation and refining F until the final F is known. The sequence of density propagation followed by Fock update converges quickly (within a few iterations), and it greatly improves the long time stability of the dynamics. The solution to eq 2 is given by 1537

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superposition of adiabatic states. An important feature of nonadiabatic molecular dynamics is that the electronic state can be described as a coherent superposition on many electronic states. For the incident energy larger than 70 eV, in addition to collision-induced electronic excitation and vibrational heating, we also observe electronic stopping leading to postcollision transfer of energy from the nuclei to electronic excitation. That is, the immediate effects of inelastic collisions are two competing effects, (a) transfer of incident kinetic energy into carbon cage vibrations and (b) transfer of incident energy into electronic excitation. The majority of the kinetic energy is transferred into vibrational energy of the carbon cage, resulting in a rapid heating of the carbon network. The hot carbon nuclei then move in a “cold” field of electrons, while slowing themselves down. The term used for this phenomenon in the literature is “electron stopping”, and it is suggested that its effects lead to equilibration of energy between electrons and nuclei that could be included into the classical MD through a friction or drag force proportional to the velocity of the nuclei, F⃗ = −β v.⃗ 38−40 We find that electronic friction leading to slowing of rapidly vibrating nuclei becomes dominant for incident energies larger than 70 eV. None of these effects was captured in our previous BOMD simulations.21 Figures 2−4 present results for the collisions of C60 and C2+ 60 . For all collision simulations, we observe an ET from the neutral Figure 1. Time dependence of the potential, Epot(LvN), and kinetic energy, Ekin, of the C60···C60 system from the microcanonical LvNMD simulations. The conserved quantity during dynamics is the sum of Epot(LvN) and Ekin, representing the total energy. BO corresponds to the ground-state SCC-DFTB energy (Tel = 0 K) obtained for selected structures from the LvNMD trajectory. The difference between BO and Epot(LvN) corresponds to the electronic excitation (bottom panel). Collisions with the incident energy not exceeding 30 eV do not lead to noticeable effects on the electronic structure, and the dynamics proceeds on its ground state.

BOMD and Mermin studies,21 we observe that for low incident energies (140

elastic inelastic inelastic inelastic inelastic inelastic

ground state ground state excited state friction friction friction

none none none reorganization fusion fragmentation

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provided through Teragrid/Xsede and resources of the Oak Ridge Leadership Computing facility and the National Center for Computational Sciences. B.G.S. acknowledges support from the Center for Nanophase Materials Sciences sponsored at Oak Ridge National Laboratory by the Scientific User Facilities Division, U.S. Department of Energy. S.I. also acknowledges support from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan under the Strategic Programs for Innovative Research (SPIRE) and the Computational Materials Science Initiative (CMSI). K.M. acknowledges support from AFOSR (Grant No. FA9550-10-1-0304) at Emory University.

considered elastic, with the postcollision trajectory proceeding on the ground electronic state. The collisions with incident energy from 10 to 30 eV are inelastic. We observe the transfer of collision energy to the vibrational degrees of freedom, resulting in heating of the system but without electronic excitations. The collisions with incident energies from 30 to 70 eV are inelastic collisions with associated transfer of the collision kinetic energy into heat and possible electronic excitations. For collision energies larger than 70 eV but smaller than 100 eV, we observe that the impact energy is large enough to result in bond breaking and reorganization of the carbon network as well as energy transfer between electrons and nuclei in an electronic stopping mechanism. For collision energies between 100 and 140 eV, we observe the highest probability of fusion between fullerenes. For collision energies larger than 140 eV, we observe increased fragmentation of the carbon network. 2+ The quantum dynamical simulations of C 60 and C60 collisions show that the ET occurs within 60 fs directly preceding the collision of fullerenes, followed by structural changes of colliding fullerenes and relaxation of electron charge (see Table 1). Overall, the results show that dynamics can be divided into four well-separated phases, (1) nonoverlap dynamics of reactants weakly interacting through long-range forces, (2) a short and fast phase in which both species are in close contact and the electron charge is equilibrated, (3) covalent dynamics with strong contact in which most structural changes can occur and the collision energy is transferred into vibrational degrees of freedom and subsequent heating the molecules, and (4) postcollision dynamics in which the species interact through long-range interactions and without overlap. We note that the results presented here were obtained for central collisions between fullerenes. In practice, many collisions between fullerenes will be off-center with a finite impact parameter. The effect of the finite impact parameter on the outcome of fullerene collisions will be a subject of detailed, future studies. In summary, we have demonstrated an explicit timedependent treatment of nonequilibrium ET which is fundamentally important to an enormous range of processes such as redox reactions. We expect that the role of explicit timedependent quantum mechanical approaches will increase as its applicability to a wide range of problems becomes recognized and more efficient implementations become available.

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ASSOCIATED CONTENT

S Supporting Information *

Video illustration showing a fragment of the LvNMD trajectory with electron transfer for the collision of C60 and C2+ 60 with incident energy at 1.4 eV. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +1 (865) 574-6093. Fax: +1 (865) 576-4368. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The present research is supported by the National Science Foundation (Grant No. ARRA-NSF-EPS-0919436). We gratefully acknowledge the allocation time on kraken at NICS 1541

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