Home
Search
Collections
Journals
About
Contact us
My IOPscience
Coupled quantum-classical method for long range charge transfer: relevance of the nuclear motion to the quantum electron dynamics
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 134206 (http://iopscience.iop.org/0953-8984/27/13/134206) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 202.28.191.34 This content was downloaded on 20/03/2015 at 11:45
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 134206 (7pp)
doi:10.1088/0953-8984/27/13/134206
Coupled quantum-classical method for long range charge transfer: relevance of the nuclear motion to the quantum electron dynamics Robson da Silva1 , Diego A Hoff2 and Luis G C Rego2 1 2
Department of Chemistry , Universidade Federal de Santa Catarina, SC 88040-900, Brazil Department of Physics Universidade Federal de Santa Catarina, SC 88040-900, Brazil
E-mail: [email protected] Received 2 July 2014, revised 14 August 2014 Accepted for publication 27 August 2014 Published 13 March 2015 Abstract
Charge and excitonic-energy transfer phenomena are fundamental for energy conversion in solar cells as well as artificial photosynthesis. Currently, much interest is being paid to light-harvesting and energy transduction processes in supramolecular structures, where nuclear dynamics has a major influence on electronic quantum dynamics. For this reason, the simulation of long range electron transfer in supramolecular structures, under environmental conditions described within an atomistic framework, has been a difficult problem to study. This work describes a coupled quantum mechanics/molecular mechanics method that aims at describing long range charge transfer processes in supramolecular systems, taking into account the atomistic details of large molecular structures, the underlying nuclear motion, and environmental effects. The method is applied to investigate the relevance of electron–nuclei interaction on the mechanisms for photo-induced electron–hole pair separation in dye-sensitized interfaces as well as electronic dynamics in molecular structures. Keywords: electron–nuclear, solar cells, quantum dynamics, molecular dynamics, supramolecular, electron transfer, energy transfer (Some figures may appear in colour only in the online journal)
by organic and inorganic materials such as dye-sensitized solar cells. There are two well developed theoretical approaches to describe the energy and charge separation processes [7, 8]. If the electronic coupling is weak and the environmental effects determine the driving, the Marcus theory and formalisms can both be safely applied to describe such processes. On the other hand, if donor and acceptor are strongly coupled and effects of the environment can be disregarded, the time-dependent Schrodinger equation and purely quantum mechanical propagators can be used to describe the charge transfer processes, in some instances even disregarding nuclear motion. A similar viewpoint can be extended to the Forster and Dexter energy transfer models in regard to the characteristics of the coupling between donor and acceptor sites [9]. The Forster resonance
1. Introduction
The quantum dynamics of electronic excitations is responsible for numerous energy and charge transfer phenomena in molecular and nanoscopic systems. In particular, these electronic processes are fundamental for the light-harvesting and energy transduction in natural photosynthesis [1, 2] and in excitonic photovoltaic devices [3–6] The operation of excitonic solar cells, which differ fundamentally from the usual bulk semiconductor p–n junction devices, rely on the processes of light-harvesting, excitonic energy transfer, charge separation, and charge transfer, which are carried out sequentially by underlying electron–nuclei molecular mechanisms [3, 6]. The excitonic solar cells comprehend the organic—polymeric and molecular included—and hybrid structures that are comprised 0953-8984/15/134206+07$33.00
1
© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 134206
R da Silva et al
energy transfer is adequate for describing incoherent energy transfer between small molecules in dilute solution whereas the Dexter method describes the energy transfer between donor and acceptor states in close contact. However, in molecular and supramolecular systems as well as small clusters, both driving mechanisms, the quantum electronic coupling and the thermodynamic fluctuations of the environment, play important roles and should be included in the theoretical description. Electron–nuclei interactions give rise to relaxation and dephasing, both of which destroy coherences between electronic states. Nuclear vibrations produced by thermal fluctuations break the symmetry of a system by distorting the ground state geometry of the molecular structure, causing localization of the electronic states [10, 11]. For heterogeneous electron transfer in dye-sensitized semiconductors [12, 13], it has been observed that nuclear dynamics improves the electronic transfer because it creates additional channels (pathways) for the wavepacket diffusion. In addition, it loosens symmetry constraints that can hinder electron transfer in the case of optimized ground state molecular geometries. For long range electron transfer through fluctuating bridges [14, 15], the transfer may be faster or slower depending on the interplay among donoracceptor (D–A) energy gap, D–A coupling strength, and period of bridge oscillations. Another situation in which nuclear dynamics influences the course of electronic dynamics is charge separation in molecules and small dye sensitizers [16, 17]. For some small molecules, the strong coupling between the photoexcited electron and hole pair can only be overcome to allow charge separation if the external driving force, due to fluctuations of the molecular orbitals, is taken into account. In this work we investigate the relevance of the electron– nuclear coupling in different electron transfer situations. In the case of interfacial electron transfer (IET) in dye-sensitized semiconductors, we show that the main features of the process can be described by assuming an incoherent electron– nuclei coupling, that is, by performing a sequential molecular dynamics/quantum dynamics approach. For intramolecular charge transfer and slower time scale phenomena, however, we show that the coherent coupling of electron and nuclei is more significant. The effect of the electron–nuclei coupling is quantified by correlation functions between electronic trajectories.
Figure 1. Schematic representation of the coupled QMMM method.
as the MM force field is updated by the time-dependent excitonic net charge produced by the evolving electron and hole wavepackets. The time-dependent tight-binding Hamiltonian is calculated upon the basis of time-evolving atomic orbitals that accompany the nuclei. The scheme in figure 1 illustrates this coupled quantum/classical method. The present approach differs from the usual QMMM formalisms because, in the regular methods, two spacial domains are defined [18]: a central quantum region, which generally describes a reaction center, and the classical surrounding, which represents the environment. The boundary between the two domains can be tricky to describe, particularly if it separates atoms on opposite sides of a chemical bond. Thus, the traditional method is, in principle, not well-suited for long range charge transfer since the boundary would have to evolve to accompany the traveling—or expanding—electronic excitation. In the present quantum/classical hybrid method, the dynamics of all the nuclei are described by a classical force field within the molecular mechanics formalism. On the other hand, the quantum wavepackets are described by adiabatic molecular orbitals delocalized over the entire system, including the atoms that comprise the eventual environmental degrees of freedom (solvent, for instance). Thus, quantum propagation can take the electronic excitations anywhere in the system, without the difficulties associated with QMMM domain boundaries. We describe the details of quantum and classical propagators in the remainder of this section. 2.1. Time propagation of electronic wavepackets
The time-propagation of electronic wavepackets is implemented by a dynamic tight-binding formalism. This is an essential feature of the method because, as it will be shown, the connection between successive molecular conformations is carried out through the diabatic atomic orbitals. The details of the tight-binding Hamiltonian were presented previously and can be found in previous works [19]. In principle any tightbinding implementation is viable as, for instance, the DFTTB (density functional theory—tight binding) method [20–22, 24, 25]. In this hybrid QMMM method, the electronic wavepacket is subject to the effects produced by the underlying nuclear dynamics. The nuclear coordinates {R(t)} yield positiondependent localized atomic orbitals (AO) |i(t) that are used to construct a time-dependent tight-binding Hamiltonian Hij (t), which can be diagonalized for each structural configuration {R(t)} to produce a time-dependent adiabatic basis of delocalized molecular orbitals (MO) |φ(t). Thus, for a given
2. Theory and methods
We use a coupled quantum/classical simulation procedure to study the quantum dynamics of charge excitations in time evolving molecular structures. The method describes the nuclear motion from a classical force field (FF) implemented in a molecular mechanics (MM) propagator. In addition to the internal nuclear degrees of freedom of the molecule, the influence of the environment can be taken into account through the nuclear motion of the neighboring molecules. Coupled with the MM propagator, electronic quantum dynamics describes the time evolution of excited states, represented by electron and hole wavepackets made up from the quantum states of a tight-binding Hamiltonian. The formalism connects the quantum and classical time propagation methods with each other 2
J. Phys.: Condens. Matter 27 (2015) 134206
R da Silva et al
instantaneous molecular conformation, we have {R(t)} ⇒ |i(t) ⇒ Hij (t) ⇒ |φ(t). Within the time-slice δt, between consecutive system configurations {Rn+1 } and {Rn }, the Hamiltonian Hˆ n is treated as time-independent. Projection operators change the representation of the wavepacket from one basis set to another, by making use of the generalized eigenvalue equation H C = SCEdiag , where Ediag is a diagonal matrix with the eigenvalues Eφ . Transformations between the localized {|i (n) } and the delocalized {|φ (n) } representations can be performed with the operators i (n) S −1(n) j (n) |φ (n) φ (n) Pˆ (n) = ij φ
Figure 2. Schematic representation of the QM propagator, which is applied within the coupled QMMM method illustrated in figure 1.
that the energy of the wavepacket has a vanishing imaginary component. It should be noted that the wavepacket coefficients Cφ (t) must be complex numbers to account for time-evolution. Thus, the combined AO/MO time-propagation method is implemented as follows: the initial wavepacket |Ψ (0) = T (0) A (0)|i is first projected by the operator Pˆ (0) onto the i i MO basis set, where time evolution takes place according to i ˆ T (5) |Ψ (δt) = exp − H0 δt Pˆ (0) |Ψ (0). h ¯
ij
i (n) C (n) φ (n) , =
(1)
iφ
whereas its transposition yields the inverse operation, T φ (n) φ (n) |i (n) S −1(n) i (n) Pˆ (n) = ij
φ
=
ij
φ (n) C T (n) j (n) ,
(2)
jφ
During the short propagation step described by equation (5) the Hamiltonian Hˆ (0) ≡ Hˆ 0 and the MO basis functions |φ (0) are assumed to be constant. At the end of the time-slice, the wavepacket |Ψ (δt) is projected back to the AO basis set by the operator Pˆ (0) , yielding |Ψ (δt) = i Ai (δt)|i (0) . From time-slice 0 to time-slice1, it is assumed that |i (0) = |i (1) , which yields |Ψ (δt) = i Ai (δt)|i (1) . Propagation through the next time slice is thus carried out as i T (6) |Ψ (2δt) = exp − Hˆ 1 δt Pˆ (1) |Ψ (δt). h ¯
with n designating the specific time-slice. The combined AO/MO method takes advantage of the simple form that the equations for time-propagation in the adiabatic MO basis set assume within the time slice. The time propagation of a wavepacket |Ψ (t) is given by the timedependent Schrodinger equation i¯h
d |Ψ (t) = Hˆ (t)|Ψ (t). dt
(3)
By projecting equation (3) onto the adiabatic MO basis, the following equation for the coefficients of the wavepacket is obtained: |Ψ (t) = φ Cφ (t)|φ(t) C˙ φ +
ϕ
i Cϕ φ|ϕ ˙ + Cφ Eφ = 0. h ¯
The procedure is repeated throughout the time-slices, as illustrated by the diagram in figure 2. The propagation of the bra state Ψ (t)| follows the same procedure. The consistency of the propagation results should be checked against time-slices of various sizes to attain good convergence for the quantum propagation. For the simulations presented herein, the time steps δt = 0.1 fs and δt = 0.01 fs were used for the short and long duration simulations, respectively. The time-dependent electronic occupation onto a determined fragment of the molecular structure is simply given by (7) Pfrag (t) = Re Ψ (t)| Pˆ frag |Ψ (t) ,
(4)
If the nuclear structure is treated as static within the time slice δt, we have φ|ϕ ˙ = 0 and the solution is readily obtained in the MO basis for the interval δt, Cφ (δt) = Cφ (0) exp(−iEφ δt/¯h). This approximation is correct in the limit δt → 0 and produces good results for δt ≈ 0.1– 0.5 fs if the nuclear trajectories are calculated by the a priori sequential MM–QM method. However, smaller time steps might be necessary if the nuclear trajectory is calculated on the fly with the quantum dynamics. Regardless, the proper choice of δt also depends on the total simulation time. To account for the nuclear motion, the transfer of the wavepacket coefficients between consecutive time slices is carried out in the diabatic AO basis of the tight-binding Hamiltonian. As a result of nuclear motion, the centers of the AO basis states change from one time slice to the next but if δt → 0, the translations are sufficiently small to justify the approximation |i (n) ≈ |i (n+1) . The procedure is robust. The stability of the electronic dynamics can be checked either by the convergence of the quantum trajectories or by verifying
where Pˆ frag can be obtained from equation (1). 3. Molecular mechanics under electronic excitations
Molecular mechanics formalism describes the nuclei by classical equations of motion, which are driven by parametrized classical FF [18]. These FF are comprised of a set of classical potentials build up to describe the internal bonding and intermolecular interactions of molecular systems. There are various 3
J. Phys.: Condens. Matter 27 (2015) 134206
R da Silva et al
FF implementations and, in fact, parametrization is not universal for a given system. Moreover, while some FF were created for the study of liquids, others were built to work with solids or even with a particular molecule. For the calculations presented herein, we used the OPLS [26, 27] and AMBER [28] force fields although, in principle, any force field can be used to perform the nuclear dynamics. The general features of the nuclear interactions are described by the OPLS force field equation, FF VOPLS = Kb (R − Ro )2 + Kθ (θ − θo )2 bonds
+
5
Cn (cosφ)n
torsions n=0
+
i,j =i
+
i,j =i
The equations of motion are solved numerically via the velocity Verlet method [18]. The intramolecular forces and short–range intermolecular interactions are calculated explicitly. However, the long range interactions (LJ and Coulomb potentials) require an efficient method to improve the implementation of molecular dynamics formalism. The Coulomb potential in particular has long range decay that cannot be treated via a simple truncation approach. To address this matter, we use a method developed by Wolf and collaborators [29–31], which converts the plain Coulomb potential into a damped, shifted force potential that exhibits a fast convergence. The method is easy to parallelize and is also applicable to the study of interfacial systems where the use of the Ewald sum can be problematic. Thus, the resulting FF is comprised of the FF that normally describes the molecular structure in the ground state and extra Coulombic energy terms that take into account the influence of the electron and hole excitations on nuclear dynamics. This approximation can, in principle, describe changes in bond length and bond angle; however, since it does not account for modifications in the bond order, the approximation would not, in principle, describe torsion effects adequately.
angles
irrespective of the distance between them. For the induced dipole term, the 1–4 bond exclusion distance criterium is used. Nuclear dynamics is calculated with the usual ground state force field potential plus the excitation perturbation as,
∂ 2 Ri FF exc . (12) m i 2 = − ∇i V + V ∂t
4εij
σij Rij
12
−
σij Rij
6
qj qi 4πo Rij
(8)
where R is the atomic position and Rij = |Ri − Rj | is the distance between atoms i and j, which carry partial charges qi and qj in the ground state potential energy surface; Ro and θo are the equilibrium bond length and angle. Likewise, θ and φ are the angular and dihedral torsional variables; Kb , Kθ , and Cn are the intramolecular FF parameters and εij and σij are the Lennard–Jones parameters. To account for the electron–nuclei coupling effects, the method introduces extra Coulombic energy terms ind ˆ δqi δqj pj · R ij exc Vi = , (9) + 4πo j =i Rij Rij2
4. Relevance of electron–nuclei coupling: results and discussion 4.1. Interfacial electron transfer
which include the effect of the time-dependent electronic excitations on the ground state FF. In these extra terms, pind is the atomic induced-dipole moment [16] and δqj is the net excitonic charge on atom j , which is calculated as δqj (t) ∗ hl Ahl = Re i (t) Sij (t)Aj (t) e i ∗ el , (10) − Ael i (t) Sij (t)Aj (t)
For ultra-fast interfacial electron transfer events in some dye-sensitized structures, the influence of nuclear dynamics on electronic quantum dynamics can be disregarded if the electronic coupling between donor and acceptor states is very strong and the molecular structure is rigid. Electron– nuclear coupling may lead to coherent oscillations in electron transfer but this effect generally does not appreciably alter the characteristic time for electronic injection [13, 32, 33]. As a study case, we used the coupled QMMM method to investigate the effect of electron–nuclei coupling on the interfacial electron transfer produced by push–pull chromophores, as seen in figure 3. The model system is composed of a [TiO2 ]512 anatase cluster with a CT–CA3 push–pull dye molecule adsorbed on the (1 0 1) surface in a bidentate bridging mode; additional information about the system can be found in other sources [34]. For this system, molecular mechanics is applied to every atom of the dye molecule and to 30 atoms at the TiO2 cluster around the anchoring site. Three simulation conditions were adopted: the coupled QMMM method, described in the previous section; the sequential MM–QM method, which uses MM trajectories calculated a priori with only the ground state
for electron and hole wavepackets written as hl |Ψ el (t) = Ael Ahl i (t)|i(t) and |Ψ (t) = i (t)|i(t) i
i
(11)
in the atomic basis set, and Sij (t) = i(t)|j (t) is the timedependent overlap matrix. The V exc term differs from the ground state Coulomb term not just in the partial charges δqi . In the ground state FF, Coulomb coupling is only taken into account between atoms separated by at least four bonds in a given molecule and by the 1–4 pair interactions whereas, in the excited state, the Coulomb FF coupling term is calculated for each pair of atoms
3 The CT–CA chromophore:(E)-3-(5-((4-(9H-carbazol-9-yl)phenyl)ethynyl) thiophen-2-yl)-2-cyanoacrylic acid.
4
J. Phys.: Condens. Matter 27 (2015) 134206
R da Silva et al
Figure 3. Survival probability in the CT–CA dye during the interfacial electronic injection into TiO2 ; the system is presented in the inset. The different colored curves designate simulations performed with the QMMM method (black), the sequential MM–QM method (red), and when keeping the nuclear structure of the system rigid (green).
Figure 4. Time-dependent electronic occupation on a given ligand (BP1) produced by the time-evolving electronic wavepacket. The wavepacket quantum dynamics is calculated by the coupled QMMM (black) and the sequential MM–QM (red) methods. The inset illustrates the chemical structure of the [Ru(bpy)3 ]2+ complex.
We evaluated the influence of the electron–nuclei correlated coupling, as implemented in the present QMMM method, in comparison with the drive produced by the decoupled nuclear motion induced simply by thermal fluctuations, as obtained in the sequential MM–QM procedure. To perform this analysis, we used the time-dependent correlation function between electronic trajectories obtained from the coupled QMMM and the sequential MM–QM methods. The quantum dynamics simulations were carried out for the [Ru(bpy)3 ]2+ (bpy = 2,2’-bipyridine) complex, which is a prototypical chromophore that has been extensively investigated, both experimentally [39, 40] and theoretically [10, 11, 41]. Figure 4 shows the time-dependent electronic occupation on a given ligand (BP1) produced by the time-evolving electronic wavepacket; a similar curve (not shown in the figure) is also produced for the hole wavepacket. It is assumed that electron and hole excitations are produced in the same bpy ligand, as generated by a ligand–centered (LC) photoexcitation. Both occupation dynamics are very similar during the first 500 fs but start to diverge thereon. The ensuing decorrelation process is evaluated with
FF—in this case the nuclei are not coupled to the electronic excitations; and rigid conformation. Little influence of nuclear motion was observed due to the following reasons: (i) the electron transfer occurs with a characteristic time that is shorter than the reorganization energy of the molecular system; (ii) the electronic coupling between dye and TiO2 is strong, and the acceptor (TiO2 ) provides a large density of electronic states that are in resonance with the LUMO of the dye; and (iii) the push– pull nature of the dye gives rise to well-separated electron and hole photoexcited states. Overall, it can be observed that nuclear motion improves the electron injection by assisting the electron–hole separation; however, the sequential MM–QM approach tends to overestimate the effect of nuclear motion because it does not account for electron–nuclei coupling. 4.2. Relevance of the electron–nuclei coupling in intramolecular electron transfer
A different situation occurs for charge transfer within molecular or supramolecular systems, whose electronic structure is formed by discrete energy levels. In this case, nuclear dynamics is a major driving force as it brings the fragment molecular orbitals into resonance to create new channels for electronic transfer. Several approaches can be used to include the effect of nuclear dynamics in the description of electronic quantum dynamics. Although conceptually simple, the use of ground state molecular dynamics trajectories to modulate electronic energies or the electronic tight-binding Hamiltonian takes into account a great deal of the physics of the problem, including the localization of the electronic wavefunction, decoherence effects, and the thermal drive of the electronic charge transfer [11, 12, 14, 17, 35–38] Although useful for describing the basic effects produced by the interaction between electrons and nuclei, the use of ground state molecular trajectories does not account for electron–nuclei correlation effects. For instance, these effects have been observed in the coherent energy transfer dynamics in the Fenna–Mathews–Olson (FMO) complex for around 700 fs at 77 K [36]. The same effects have been observed in other photosynthetic complexes at higher temperatures and even in conjugate polymers and molecules [2].
Cf g (t) =
fα (t)gα (t) − fα (t)gα (t)
, (13) − gα2 (t)) · (fα (t)2 − gα (t)2 ) t where fα (t) = 1t 0 fα (t )dt , and fα (t) and gα (t) are the time-dependent electronic occupations—as produced by the coupled QMMM and sequential MM–QM methods, respectively, on the same ligand α. The results are shown in figure 5 for the electron and hole wavepackets. The curves are an average over nine correlation functions Cf g (t) with the standard deviation described by the vertical bars. The inset graphs show the early decorrelation dynamics. The pair-correlation Cf g (t) decays to zero within 3–4 ps for the quantum dynamics of the electron wavepacket, and a slightly slower for the hole wavepacket. The effect is caused by the electron–nuclei coupling included in the simulations by the V exc potential added to the ground state FF. The inset graphs show that the decay is not exponential at early times, which is in accordance with the results presented by figures 3 and 4 that evince a delay of a few hundred femtoseconds before the 5
(fα2 (t)
J. Phys.: Condens. Matter 27 (2015) 134206
R da Silva et al
and collaborators [21, 22] with the DFTB formalism and is also described in the overlooked, but very interesting, work of Field [23]. 5. Conclusions
We introduce a coupled quantum/classical simulation procedure to study the quantum dynamics of charge excitations in time evolving molecular structures. The method describes the nuclear motion by a classical FF implemented in a molecular mechanics propagator and the quantum dynamics of electronic excitations by means of a combined AO/MO quantum propagator. Despite approximations using a classical FF to describe the excited state molecular mechanics, the present method could be adopted as a platform for the study of other quantum mechanics procedures such as non-adiabatic surface hopping. In addition, the method could be adapted to other tight-binding formalisms. The calculations show that the use of a sequential MM– QM approach, in which ground state molecular dynamics trajectories are used to modulate quantum mechanical energies, provide good results for electron transfer events that endure up to several hundreds of femtoseconds. The coupled electron– nuclei dynamics and charge transfer are more sensitive within molecular or supramolecular systems, whose electronic structure is formed by discrete energy levels. In this case, nuclear dynamics is a major driving force as it brings the fragment molecular orbitals into resonance to create new channels for electronic transfer. The same conclusions apply to polymers, supramolecular structures, and small clusters, which are systems where electronic coupling and nuclear motion must be described at similar theoretical level.
Figure 5. Decay of the cross-correlation between the time-dependent wavepacket dynamics, as produced by the QMMM and MM–QM methods, calculated by equation (13). The curves were obtained after an average of nine correlation functions Cf g (t) with the standard deviation described by the vertical bars. The inset graphs show the early decorrelation dynamics.
electronic excitations have an effect on the nuclear dynamics and vice versa. Finally, we point out that the [Ru(bpy)3 ]2+ complex has a rigid molecular structure that reduces the electron–nuclear coupling. The effect is stronger—faster paircorrelation decay—in flexible molecules and molecular chains.
Acknowledgments 4.3. Discussion
The authors are indebted to the Brazilian National Counsel of Technological and Scientific Development (CNPq) and the Coordenac¸a˜ o de Aperfeic¸oamento de Pessoal de N´ıvel Superior (CAPES) for funding the project.
The range of simulation time in this coupled quantum/classical hybrid method is constrained by its quantum mechanics aspect, depending on the size and sensitivity of the system. For instance, charge transfer simulations enduring tens of picoseconds (hundreds of femtoseconds) can be performed for systems comprised of a coupled of hundred (couple of thousand) atoms can be performed in a single cluster node with 16 cores. Thus, the method is capable of describing processes within the picosecond regime, which is the time scale for critical charge and energy transfer events in molecular systems as observed in artificial and natural light-harvesting structures. The photons also play an important role in the photoinduced charge separation process, particularly in DSSC and light-harvesting structures. The present method does not describe the photoexcitation process but assumes vertical instantaneous photoexcitation within the chromophore, as widely assumed in the literature. To describe the photoabsorption process more rigorously, the light-matter coupling term µˆ · E(t) can be included in the Hamiltonian, where µˆ is the transition dipole matrix and E(t) is the radiation electric field. The simulation thus starts from the ground state. This approach has been successfully implemented by Sanchez
References [1] McConnell I, Li G and Brudvig G 2010 Chem. Biol. Rev. 17 434–47 [2] Scholes G D, Fleming G R, Olaya-Castro and van Grondelle R 2011 Nat. Chem. 3 763–74 [3] Gregg B A 2003 J. Phys. Chem. B 107 4688–98 [4] Gust D, Moore T A and Moore A L 2001 Acc. Chem. Res. 34 40–8 [5] Gust D, Moore T A and Moore A L 2009 Acc. Chem. Res. 42 1890–8 [6] Clarke T M and Durrant J 2010 Chem. Rev. 110 6736–67 [7] May V and K¨uhn O 2011 Charge and Energy Transfer Dynamics in Molecular Systems (Berlin: Wiley) [8] Nitzan A 2006 Chemical Dynamics in Condensed Phases: Relaxation, Transfer, and Reactions in Condensed Molecular Systems (Oxford: Oxford University Press) [9] Beljonne D, Curutchet C, Scholes G D and Silbey R J 2009 J. Phys. Chem. B 113 6583–99 [10] Moret M, Tavernelli I, Chergui M and Rothlisberger U 2010 Chem. Eur. J. 16 5889–94 6
J. Phys.: Condens. Matter 27 (2015) 134206
R da Silva et al
[26] Jorgensen W L, Maxwell D S and Tirado-Rives J 1996 J. Am. Chem. Soc. 118 11225–36 [27] Kaminski G A, Friesner R A, Tirado-Rives J and Jorgensen W L 2001 J. Phys. Chem. B 105 6474–87 [28] Cornell W D, Cieplak P, Bayly C I, Gould I R, Merz K M Jr, Ferguson D M, Spellmeyer D C, Fox T, Caldwell J W and Kollman P A 1995 J. Am. Chem. Soc. 117 5179–97 [29] Fennell C J and Gezelter J D 2006 J. Chem. Phys. 124 234104 [30] Wolf D, Keblinski P, Phillpot S R and Eggebrecht J 1999 J. Chem. Phys. 110 8254–82 [31] Zahn D, Schilling B and Kast S M 2002 J. Phys. Chem. B 106 10725–32 [32] Gundlach L and Willig F 2012 Chem. Phys. Chem. 13 2877–81 [33] Gundlach L, Burfeindt B, Mahrt J and Willig F 2012 Chem. Phys. Lett. 545 35–9 [34] Bairu S G et al 2013 J. Phys. Chem. C 117 4824–35 [35] Olbrich C, Str¨umpfer J, Schulten K and Kleinekath¨ofer U 2011 J. Chem. Phys. Lett. 2 1771–6 [36] Olbrich C, Jansen T L C, Liebers J, Aghtar M, Str¨umpfer J, Schulten K, Knoester J and Kleinekath¨ofer U 2011 J. Chem. Phys. B 115 8609–21 [37] Madrid A B, Hyeon-Deuk K, Habenicht B F and Prezhdo O V 2009 ACS Nano 9 2487–94 [38] Akimov A V and Prezhdo O V 2013 J. Phys. Chem. Lett. 4 3857–64 [39] Damrauer N H, Cerullo G, Yeh A, Boussie T R, Shank C V and McCusker J K 1997 Science 275 54–7 [40] Wallin S, Davidsson J, Modin J and Hammarstr¨om L 2005 J. Chem. Phys. A 109 4697–704 [41] Tavernelli I, Curchod B F E and Rothlisberger U 2011 Chem. Phys. 391 101–9
[11] Hoff D A, da Silva R and Rego L G C 2011 J. Phys. Chem. C 115 15617–26 [12] Abuabara S G, Rego L G C and Batista V S 2005 J. Am. Chem. Soc. 127 18234–42 [13] Prezhdo O, Duncam W and Prezhdo V 2009 Prog. Surf. Sci. 84 30–68 [14] Skourtis S S 2003 Chem. Phys. Lett. 372 224–31 [15] Skourtis S S, Waldeck D and Beratan D N 2010 Ann. Rev. Phys. Chem. 61 461–85 [16] Rego L G C, Hames B C, Mazon K T, Joswig J-O 2014 J. Phys. Chem. C 118 126–34 [17] Hoff D A, da Silva R and Rego L G C 2012 J. Phys. Chem. C 116 21169–78 [18] Cramer C J 2002 Essentials of Computational Chemistry: Theory and Methods (West Sussex: Wiley) [19] Hoff D A and Rego L G C 2013 Specialist Periodical Reports—Chemical Modelling: Modelling Electron Quantum Dynamics in Large Molecular Systems vol 10, ed M Springboard and J-O Joswig (London: RSC Publishing) pp 102–26 [20] Elstner M, Porezag D, Jungnickel G, Elsner J, Haugk M, Frauenheim T, Suhai S and Seifert G 1998 Phys. Rev. B 58 7260–68 [21] Negre C F A, Fuertes V C, Oviedo M B, Oliva F Y and Sanchez C G 2012 J. Phys. Chem. C 116 14748–53 [22] Oviedo M B, Zarate X, Negre C F A, Schott E, Perez R A and Sanchez C G 2012 J. Phys. Chem. Lett. 3 2548–55 [23] Field M J 1992 Chem. Phys. Lett. 195 388–92 [24] Seifert G, Porezag D and Frauenheim T 1996 Int. J. Quantum Chem. 58 185–92 [25] Porezag D, Frauenheim T, K¨ohler T, Seifert G and Kaschner R 1995 Phys. Rev. B 51 12947–57
7
Search
Collections
Journals
About
Contact us
My IOPscience
Coupled quantum-classical method for long range charge transfer: relevance of the nuclear motion to the quantum electron dynamics
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 134206 (http://iopscience.iop.org/0953-8984/27/13/134206) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 202.28.191.34 This content was downloaded on 20/03/2015 at 11:45
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 134206 (7pp)
doi:10.1088/0953-8984/27/13/134206
Coupled quantum-classical method for long range charge transfer: relevance of the nuclear motion to the quantum electron dynamics Robson da Silva1 , Diego A Hoff2 and Luis G C Rego2 1 2
Department of Chemistry , Universidade Federal de Santa Catarina, SC 88040-900, Brazil Department of Physics Universidade Federal de Santa Catarina, SC 88040-900, Brazil
E-mail: [email protected] Received 2 July 2014, revised 14 August 2014 Accepted for publication 27 August 2014 Published 13 March 2015 Abstract
Charge and excitonic-energy transfer phenomena are fundamental for energy conversion in solar cells as well as artificial photosynthesis. Currently, much interest is being paid to light-harvesting and energy transduction processes in supramolecular structures, where nuclear dynamics has a major influence on electronic quantum dynamics. For this reason, the simulation of long range electron transfer in supramolecular structures, under environmental conditions described within an atomistic framework, has been a difficult problem to study. This work describes a coupled quantum mechanics/molecular mechanics method that aims at describing long range charge transfer processes in supramolecular systems, taking into account the atomistic details of large molecular structures, the underlying nuclear motion, and environmental effects. The method is applied to investigate the relevance of electron–nuclei interaction on the mechanisms for photo-induced electron–hole pair separation in dye-sensitized interfaces as well as electronic dynamics in molecular structures. Keywords: electron–nuclear, solar cells, quantum dynamics, molecular dynamics, supramolecular, electron transfer, energy transfer (Some figures may appear in colour only in the online journal)
by organic and inorganic materials such as dye-sensitized solar cells. There are two well developed theoretical approaches to describe the energy and charge separation processes [7, 8]. If the electronic coupling is weak and the environmental effects determine the driving, the Marcus theory and formalisms can both be safely applied to describe such processes. On the other hand, if donor and acceptor are strongly coupled and effects of the environment can be disregarded, the time-dependent Schrodinger equation and purely quantum mechanical propagators can be used to describe the charge transfer processes, in some instances even disregarding nuclear motion. A similar viewpoint can be extended to the Forster and Dexter energy transfer models in regard to the characteristics of the coupling between donor and acceptor sites [9]. The Forster resonance
1. Introduction
The quantum dynamics of electronic excitations is responsible for numerous energy and charge transfer phenomena in molecular and nanoscopic systems. In particular, these electronic processes are fundamental for the light-harvesting and energy transduction in natural photosynthesis [1, 2] and in excitonic photovoltaic devices [3–6] The operation of excitonic solar cells, which differ fundamentally from the usual bulk semiconductor p–n junction devices, rely on the processes of light-harvesting, excitonic energy transfer, charge separation, and charge transfer, which are carried out sequentially by underlying electron–nuclei molecular mechanisms [3, 6]. The excitonic solar cells comprehend the organic—polymeric and molecular included—and hybrid structures that are comprised 0953-8984/15/134206+07$33.00
1
© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 134206
R da Silva et al
energy transfer is adequate for describing incoherent energy transfer between small molecules in dilute solution whereas the Dexter method describes the energy transfer between donor and acceptor states in close contact. However, in molecular and supramolecular systems as well as small clusters, both driving mechanisms, the quantum electronic coupling and the thermodynamic fluctuations of the environment, play important roles and should be included in the theoretical description. Electron–nuclei interactions give rise to relaxation and dephasing, both of which destroy coherences between electronic states. Nuclear vibrations produced by thermal fluctuations break the symmetry of a system by distorting the ground state geometry of the molecular structure, causing localization of the electronic states [10, 11]. For heterogeneous electron transfer in dye-sensitized semiconductors [12, 13], it has been observed that nuclear dynamics improves the electronic transfer because it creates additional channels (pathways) for the wavepacket diffusion. In addition, it loosens symmetry constraints that can hinder electron transfer in the case of optimized ground state molecular geometries. For long range electron transfer through fluctuating bridges [14, 15], the transfer may be faster or slower depending on the interplay among donoracceptor (D–A) energy gap, D–A coupling strength, and period of bridge oscillations. Another situation in which nuclear dynamics influences the course of electronic dynamics is charge separation in molecules and small dye sensitizers [16, 17]. For some small molecules, the strong coupling between the photoexcited electron and hole pair can only be overcome to allow charge separation if the external driving force, due to fluctuations of the molecular orbitals, is taken into account. In this work we investigate the relevance of the electron– nuclear coupling in different electron transfer situations. In the case of interfacial electron transfer (IET) in dye-sensitized semiconductors, we show that the main features of the process can be described by assuming an incoherent electron– nuclei coupling, that is, by performing a sequential molecular dynamics/quantum dynamics approach. For intramolecular charge transfer and slower time scale phenomena, however, we show that the coherent coupling of electron and nuclei is more significant. The effect of the electron–nuclei coupling is quantified by correlation functions between electronic trajectories.
Figure 1. Schematic representation of the coupled QMMM method.
as the MM force field is updated by the time-dependent excitonic net charge produced by the evolving electron and hole wavepackets. The time-dependent tight-binding Hamiltonian is calculated upon the basis of time-evolving atomic orbitals that accompany the nuclei. The scheme in figure 1 illustrates this coupled quantum/classical method. The present approach differs from the usual QMMM formalisms because, in the regular methods, two spacial domains are defined [18]: a central quantum region, which generally describes a reaction center, and the classical surrounding, which represents the environment. The boundary between the two domains can be tricky to describe, particularly if it separates atoms on opposite sides of a chemical bond. Thus, the traditional method is, in principle, not well-suited for long range charge transfer since the boundary would have to evolve to accompany the traveling—or expanding—electronic excitation. In the present quantum/classical hybrid method, the dynamics of all the nuclei are described by a classical force field within the molecular mechanics formalism. On the other hand, the quantum wavepackets are described by adiabatic molecular orbitals delocalized over the entire system, including the atoms that comprise the eventual environmental degrees of freedom (solvent, for instance). Thus, quantum propagation can take the electronic excitations anywhere in the system, without the difficulties associated with QMMM domain boundaries. We describe the details of quantum and classical propagators in the remainder of this section. 2.1. Time propagation of electronic wavepackets
The time-propagation of electronic wavepackets is implemented by a dynamic tight-binding formalism. This is an essential feature of the method because, as it will be shown, the connection between successive molecular conformations is carried out through the diabatic atomic orbitals. The details of the tight-binding Hamiltonian were presented previously and can be found in previous works [19]. In principle any tightbinding implementation is viable as, for instance, the DFTTB (density functional theory—tight binding) method [20–22, 24, 25]. In this hybrid QMMM method, the electronic wavepacket is subject to the effects produced by the underlying nuclear dynamics. The nuclear coordinates {R(t)} yield positiondependent localized atomic orbitals (AO) |i(t) that are used to construct a time-dependent tight-binding Hamiltonian Hij (t), which can be diagonalized for each structural configuration {R(t)} to produce a time-dependent adiabatic basis of delocalized molecular orbitals (MO) |φ(t). Thus, for a given
2. Theory and methods
We use a coupled quantum/classical simulation procedure to study the quantum dynamics of charge excitations in time evolving molecular structures. The method describes the nuclear motion from a classical force field (FF) implemented in a molecular mechanics (MM) propagator. In addition to the internal nuclear degrees of freedom of the molecule, the influence of the environment can be taken into account through the nuclear motion of the neighboring molecules. Coupled with the MM propagator, electronic quantum dynamics describes the time evolution of excited states, represented by electron and hole wavepackets made up from the quantum states of a tight-binding Hamiltonian. The formalism connects the quantum and classical time propagation methods with each other 2
J. Phys.: Condens. Matter 27 (2015) 134206
R da Silva et al
instantaneous molecular conformation, we have {R(t)} ⇒ |i(t) ⇒ Hij (t) ⇒ |φ(t). Within the time-slice δt, between consecutive system configurations {Rn+1 } and {Rn }, the Hamiltonian Hˆ n is treated as time-independent. Projection operators change the representation of the wavepacket from one basis set to another, by making use of the generalized eigenvalue equation H C = SCEdiag , where Ediag is a diagonal matrix with the eigenvalues Eφ . Transformations between the localized {|i (n) } and the delocalized {|φ (n) } representations can be performed with the operators i (n) S −1(n) j (n) |φ (n) φ (n) Pˆ (n) = ij φ
Figure 2. Schematic representation of the QM propagator, which is applied within the coupled QMMM method illustrated in figure 1.
that the energy of the wavepacket has a vanishing imaginary component. It should be noted that the wavepacket coefficients Cφ (t) must be complex numbers to account for time-evolution. Thus, the combined AO/MO time-propagation method is implemented as follows: the initial wavepacket |Ψ (0) = T (0) A (0)|i is first projected by the operator Pˆ (0) onto the i i MO basis set, where time evolution takes place according to i ˆ T (5) |Ψ (δt) = exp − H0 δt Pˆ (0) |Ψ (0). h ¯
ij
i (n) C (n) φ (n) , =
(1)
iφ
whereas its transposition yields the inverse operation, T φ (n) φ (n) |i (n) S −1(n) i (n) Pˆ (n) = ij
φ
=
ij
φ (n) C T (n) j (n) ,
(2)
jφ
During the short propagation step described by equation (5) the Hamiltonian Hˆ (0) ≡ Hˆ 0 and the MO basis functions |φ (0) are assumed to be constant. At the end of the time-slice, the wavepacket |Ψ (δt) is projected back to the AO basis set by the operator Pˆ (0) , yielding |Ψ (δt) = i Ai (δt)|i (0) . From time-slice 0 to time-slice1, it is assumed that |i (0) = |i (1) , which yields |Ψ (δt) = i Ai (δt)|i (1) . Propagation through the next time slice is thus carried out as i T (6) |Ψ (2δt) = exp − Hˆ 1 δt Pˆ (1) |Ψ (δt). h ¯
with n designating the specific time-slice. The combined AO/MO method takes advantage of the simple form that the equations for time-propagation in the adiabatic MO basis set assume within the time slice. The time propagation of a wavepacket |Ψ (t) is given by the timedependent Schrodinger equation i¯h
d |Ψ (t) = Hˆ (t)|Ψ (t). dt
(3)
By projecting equation (3) onto the adiabatic MO basis, the following equation for the coefficients of the wavepacket is obtained: |Ψ (t) = φ Cφ (t)|φ(t) C˙ φ +
ϕ
i Cϕ φ|ϕ ˙ + Cφ Eφ = 0. h ¯
The procedure is repeated throughout the time-slices, as illustrated by the diagram in figure 2. The propagation of the bra state Ψ (t)| follows the same procedure. The consistency of the propagation results should be checked against time-slices of various sizes to attain good convergence for the quantum propagation. For the simulations presented herein, the time steps δt = 0.1 fs and δt = 0.01 fs were used for the short and long duration simulations, respectively. The time-dependent electronic occupation onto a determined fragment of the molecular structure is simply given by (7) Pfrag (t) = Re Ψ (t)| Pˆ frag |Ψ (t) ,
(4)
If the nuclear structure is treated as static within the time slice δt, we have φ|ϕ ˙ = 0 and the solution is readily obtained in the MO basis for the interval δt, Cφ (δt) = Cφ (0) exp(−iEφ δt/¯h). This approximation is correct in the limit δt → 0 and produces good results for δt ≈ 0.1– 0.5 fs if the nuclear trajectories are calculated by the a priori sequential MM–QM method. However, smaller time steps might be necessary if the nuclear trajectory is calculated on the fly with the quantum dynamics. Regardless, the proper choice of δt also depends on the total simulation time. To account for the nuclear motion, the transfer of the wavepacket coefficients between consecutive time slices is carried out in the diabatic AO basis of the tight-binding Hamiltonian. As a result of nuclear motion, the centers of the AO basis states change from one time slice to the next but if δt → 0, the translations are sufficiently small to justify the approximation |i (n) ≈ |i (n+1) . The procedure is robust. The stability of the electronic dynamics can be checked either by the convergence of the quantum trajectories or by verifying
where Pˆ frag can be obtained from equation (1). 3. Molecular mechanics under electronic excitations
Molecular mechanics formalism describes the nuclei by classical equations of motion, which are driven by parametrized classical FF [18]. These FF are comprised of a set of classical potentials build up to describe the internal bonding and intermolecular interactions of molecular systems. There are various 3
J. Phys.: Condens. Matter 27 (2015) 134206
R da Silva et al
FF implementations and, in fact, parametrization is not universal for a given system. Moreover, while some FF were created for the study of liquids, others were built to work with solids or even with a particular molecule. For the calculations presented herein, we used the OPLS [26, 27] and AMBER [28] force fields although, in principle, any force field can be used to perform the nuclear dynamics. The general features of the nuclear interactions are described by the OPLS force field equation, FF VOPLS = Kb (R − Ro )2 + Kθ (θ − θo )2 bonds
+
5
Cn (cosφ)n
torsions n=0
+
i,j =i
+
i,j =i
The equations of motion are solved numerically via the velocity Verlet method [18]. The intramolecular forces and short–range intermolecular interactions are calculated explicitly. However, the long range interactions (LJ and Coulomb potentials) require an efficient method to improve the implementation of molecular dynamics formalism. The Coulomb potential in particular has long range decay that cannot be treated via a simple truncation approach. To address this matter, we use a method developed by Wolf and collaborators [29–31], which converts the plain Coulomb potential into a damped, shifted force potential that exhibits a fast convergence. The method is easy to parallelize and is also applicable to the study of interfacial systems where the use of the Ewald sum can be problematic. Thus, the resulting FF is comprised of the FF that normally describes the molecular structure in the ground state and extra Coulombic energy terms that take into account the influence of the electron and hole excitations on nuclear dynamics. This approximation can, in principle, describe changes in bond length and bond angle; however, since it does not account for modifications in the bond order, the approximation would not, in principle, describe torsion effects adequately.
angles
irrespective of the distance between them. For the induced dipole term, the 1–4 bond exclusion distance criterium is used. Nuclear dynamics is calculated with the usual ground state force field potential plus the excitation perturbation as,
∂ 2 Ri FF exc . (12) m i 2 = − ∇i V + V ∂t
4εij
σij Rij
12
−
σij Rij
6
qj qi 4πo Rij
(8)
where R is the atomic position and Rij = |Ri − Rj | is the distance between atoms i and j, which carry partial charges qi and qj in the ground state potential energy surface; Ro and θo are the equilibrium bond length and angle. Likewise, θ and φ are the angular and dihedral torsional variables; Kb , Kθ , and Cn are the intramolecular FF parameters and εij and σij are the Lennard–Jones parameters. To account for the electron–nuclei coupling effects, the method introduces extra Coulombic energy terms ind ˆ δqi δqj pj · R ij exc Vi = , (9) + 4πo j =i Rij Rij2
4. Relevance of electron–nuclei coupling: results and discussion 4.1. Interfacial electron transfer
which include the effect of the time-dependent electronic excitations on the ground state FF. In these extra terms, pind is the atomic induced-dipole moment [16] and δqj is the net excitonic charge on atom j , which is calculated as δqj (t) ∗ hl Ahl = Re i (t) Sij (t)Aj (t) e i ∗ el , (10) − Ael i (t) Sij (t)Aj (t)
For ultra-fast interfacial electron transfer events in some dye-sensitized structures, the influence of nuclear dynamics on electronic quantum dynamics can be disregarded if the electronic coupling between donor and acceptor states is very strong and the molecular structure is rigid. Electron– nuclear coupling may lead to coherent oscillations in electron transfer but this effect generally does not appreciably alter the characteristic time for electronic injection [13, 32, 33]. As a study case, we used the coupled QMMM method to investigate the effect of electron–nuclei coupling on the interfacial electron transfer produced by push–pull chromophores, as seen in figure 3. The model system is composed of a [TiO2 ]512 anatase cluster with a CT–CA3 push–pull dye molecule adsorbed on the (1 0 1) surface in a bidentate bridging mode; additional information about the system can be found in other sources [34]. For this system, molecular mechanics is applied to every atom of the dye molecule and to 30 atoms at the TiO2 cluster around the anchoring site. Three simulation conditions were adopted: the coupled QMMM method, described in the previous section; the sequential MM–QM method, which uses MM trajectories calculated a priori with only the ground state
for electron and hole wavepackets written as hl |Ψ el (t) = Ael Ahl i (t)|i(t) and |Ψ (t) = i (t)|i(t) i
i
(11)
in the atomic basis set, and Sij (t) = i(t)|j (t) is the timedependent overlap matrix. The V exc term differs from the ground state Coulomb term not just in the partial charges δqi . In the ground state FF, Coulomb coupling is only taken into account between atoms separated by at least four bonds in a given molecule and by the 1–4 pair interactions whereas, in the excited state, the Coulomb FF coupling term is calculated for each pair of atoms
3 The CT–CA chromophore:(E)-3-(5-((4-(9H-carbazol-9-yl)phenyl)ethynyl) thiophen-2-yl)-2-cyanoacrylic acid.
4
J. Phys.: Condens. Matter 27 (2015) 134206
R da Silva et al
Figure 3. Survival probability in the CT–CA dye during the interfacial electronic injection into TiO2 ; the system is presented in the inset. The different colored curves designate simulations performed with the QMMM method (black), the sequential MM–QM method (red), and when keeping the nuclear structure of the system rigid (green).
Figure 4. Time-dependent electronic occupation on a given ligand (BP1) produced by the time-evolving electronic wavepacket. The wavepacket quantum dynamics is calculated by the coupled QMMM (black) and the sequential MM–QM (red) methods. The inset illustrates the chemical structure of the [Ru(bpy)3 ]2+ complex.
We evaluated the influence of the electron–nuclei correlated coupling, as implemented in the present QMMM method, in comparison with the drive produced by the decoupled nuclear motion induced simply by thermal fluctuations, as obtained in the sequential MM–QM procedure. To perform this analysis, we used the time-dependent correlation function between electronic trajectories obtained from the coupled QMMM and the sequential MM–QM methods. The quantum dynamics simulations were carried out for the [Ru(bpy)3 ]2+ (bpy = 2,2’-bipyridine) complex, which is a prototypical chromophore that has been extensively investigated, both experimentally [39, 40] and theoretically [10, 11, 41]. Figure 4 shows the time-dependent electronic occupation on a given ligand (BP1) produced by the time-evolving electronic wavepacket; a similar curve (not shown in the figure) is also produced for the hole wavepacket. It is assumed that electron and hole excitations are produced in the same bpy ligand, as generated by a ligand–centered (LC) photoexcitation. Both occupation dynamics are very similar during the first 500 fs but start to diverge thereon. The ensuing decorrelation process is evaluated with
FF—in this case the nuclei are not coupled to the electronic excitations; and rigid conformation. Little influence of nuclear motion was observed due to the following reasons: (i) the electron transfer occurs with a characteristic time that is shorter than the reorganization energy of the molecular system; (ii) the electronic coupling between dye and TiO2 is strong, and the acceptor (TiO2 ) provides a large density of electronic states that are in resonance with the LUMO of the dye; and (iii) the push– pull nature of the dye gives rise to well-separated electron and hole photoexcited states. Overall, it can be observed that nuclear motion improves the electron injection by assisting the electron–hole separation; however, the sequential MM–QM approach tends to overestimate the effect of nuclear motion because it does not account for electron–nuclei coupling. 4.2. Relevance of the electron–nuclei coupling in intramolecular electron transfer
A different situation occurs for charge transfer within molecular or supramolecular systems, whose electronic structure is formed by discrete energy levels. In this case, nuclear dynamics is a major driving force as it brings the fragment molecular orbitals into resonance to create new channels for electronic transfer. Several approaches can be used to include the effect of nuclear dynamics in the description of electronic quantum dynamics. Although conceptually simple, the use of ground state molecular dynamics trajectories to modulate electronic energies or the electronic tight-binding Hamiltonian takes into account a great deal of the physics of the problem, including the localization of the electronic wavefunction, decoherence effects, and the thermal drive of the electronic charge transfer [11, 12, 14, 17, 35–38] Although useful for describing the basic effects produced by the interaction between electrons and nuclei, the use of ground state molecular trajectories does not account for electron–nuclei correlation effects. For instance, these effects have been observed in the coherent energy transfer dynamics in the Fenna–Mathews–Olson (FMO) complex for around 700 fs at 77 K [36]. The same effects have been observed in other photosynthetic complexes at higher temperatures and even in conjugate polymers and molecules [2].
Cf g (t) =
fα (t)gα (t) − fα (t)gα (t)
, (13) − gα2 (t)) · (fα (t)2 − gα (t)2 ) t where fα (t) = 1t 0 fα (t )dt , and fα (t) and gα (t) are the time-dependent electronic occupations—as produced by the coupled QMMM and sequential MM–QM methods, respectively, on the same ligand α. The results are shown in figure 5 for the electron and hole wavepackets. The curves are an average over nine correlation functions Cf g (t) with the standard deviation described by the vertical bars. The inset graphs show the early decorrelation dynamics. The pair-correlation Cf g (t) decays to zero within 3–4 ps for the quantum dynamics of the electron wavepacket, and a slightly slower for the hole wavepacket. The effect is caused by the electron–nuclei coupling included in the simulations by the V exc potential added to the ground state FF. The inset graphs show that the decay is not exponential at early times, which is in accordance with the results presented by figures 3 and 4 that evince a delay of a few hundred femtoseconds before the 5
(fα2 (t)
J. Phys.: Condens. Matter 27 (2015) 134206
R da Silva et al
and collaborators [21, 22] with the DFTB formalism and is also described in the overlooked, but very interesting, work of Field [23]. 5. Conclusions
We introduce a coupled quantum/classical simulation procedure to study the quantum dynamics of charge excitations in time evolving molecular structures. The method describes the nuclear motion by a classical FF implemented in a molecular mechanics propagator and the quantum dynamics of electronic excitations by means of a combined AO/MO quantum propagator. Despite approximations using a classical FF to describe the excited state molecular mechanics, the present method could be adopted as a platform for the study of other quantum mechanics procedures such as non-adiabatic surface hopping. In addition, the method could be adapted to other tight-binding formalisms. The calculations show that the use of a sequential MM– QM approach, in which ground state molecular dynamics trajectories are used to modulate quantum mechanical energies, provide good results for electron transfer events that endure up to several hundreds of femtoseconds. The coupled electron– nuclei dynamics and charge transfer are more sensitive within molecular or supramolecular systems, whose electronic structure is formed by discrete energy levels. In this case, nuclear dynamics is a major driving force as it brings the fragment molecular orbitals into resonance to create new channels for electronic transfer. The same conclusions apply to polymers, supramolecular structures, and small clusters, which are systems where electronic coupling and nuclear motion must be described at similar theoretical level.
Figure 5. Decay of the cross-correlation between the time-dependent wavepacket dynamics, as produced by the QMMM and MM–QM methods, calculated by equation (13). The curves were obtained after an average of nine correlation functions Cf g (t) with the standard deviation described by the vertical bars. The inset graphs show the early decorrelation dynamics.
electronic excitations have an effect on the nuclear dynamics and vice versa. Finally, we point out that the [Ru(bpy)3 ]2+ complex has a rigid molecular structure that reduces the electron–nuclear coupling. The effect is stronger—faster paircorrelation decay—in flexible molecules and molecular chains.
Acknowledgments 4.3. Discussion
The authors are indebted to the Brazilian National Counsel of Technological and Scientific Development (CNPq) and the Coordenac¸a˜ o de Aperfeic¸oamento de Pessoal de N´ıvel Superior (CAPES) for funding the project.
The range of simulation time in this coupled quantum/classical hybrid method is constrained by its quantum mechanics aspect, depending on the size and sensitivity of the system. For instance, charge transfer simulations enduring tens of picoseconds (hundreds of femtoseconds) can be performed for systems comprised of a coupled of hundred (couple of thousand) atoms can be performed in a single cluster node with 16 cores. Thus, the method is capable of describing processes within the picosecond regime, which is the time scale for critical charge and energy transfer events in molecular systems as observed in artificial and natural light-harvesting structures. The photons also play an important role in the photoinduced charge separation process, particularly in DSSC and light-harvesting structures. The present method does not describe the photoexcitation process but assumes vertical instantaneous photoexcitation within the chromophore, as widely assumed in the literature. To describe the photoabsorption process more rigorously, the light-matter coupling term µˆ · E(t) can be included in the Hamiltonian, where µˆ is the transition dipole matrix and E(t) is the radiation electric field. The simulation thus starts from the ground state. This approach has been successfully implemented by Sanchez
References [1] McConnell I, Li G and Brudvig G 2010 Chem. Biol. Rev. 17 434–47 [2] Scholes G D, Fleming G R, Olaya-Castro and van Grondelle R 2011 Nat. Chem. 3 763–74 [3] Gregg B A 2003 J. Phys. Chem. B 107 4688–98 [4] Gust D, Moore T A and Moore A L 2001 Acc. Chem. Res. 34 40–8 [5] Gust D, Moore T A and Moore A L 2009 Acc. Chem. Res. 42 1890–8 [6] Clarke T M and Durrant J 2010 Chem. Rev. 110 6736–67 [7] May V and K¨uhn O 2011 Charge and Energy Transfer Dynamics in Molecular Systems (Berlin: Wiley) [8] Nitzan A 2006 Chemical Dynamics in Condensed Phases: Relaxation, Transfer, and Reactions in Condensed Molecular Systems (Oxford: Oxford University Press) [9] Beljonne D, Curutchet C, Scholes G D and Silbey R J 2009 J. Phys. Chem. B 113 6583–99 [10] Moret M, Tavernelli I, Chergui M and Rothlisberger U 2010 Chem. Eur. J. 16 5889–94 6
J. Phys.: Condens. Matter 27 (2015) 134206
R da Silva et al
[26] Jorgensen W L, Maxwell D S and Tirado-Rives J 1996 J. Am. Chem. Soc. 118 11225–36 [27] Kaminski G A, Friesner R A, Tirado-Rives J and Jorgensen W L 2001 J. Phys. Chem. B 105 6474–87 [28] Cornell W D, Cieplak P, Bayly C I, Gould I R, Merz K M Jr, Ferguson D M, Spellmeyer D C, Fox T, Caldwell J W and Kollman P A 1995 J. Am. Chem. Soc. 117 5179–97 [29] Fennell C J and Gezelter J D 2006 J. Chem. Phys. 124 234104 [30] Wolf D, Keblinski P, Phillpot S R and Eggebrecht J 1999 J. Chem. Phys. 110 8254–82 [31] Zahn D, Schilling B and Kast S M 2002 J. Phys. Chem. B 106 10725–32 [32] Gundlach L and Willig F 2012 Chem. Phys. Chem. 13 2877–81 [33] Gundlach L, Burfeindt B, Mahrt J and Willig F 2012 Chem. Phys. Lett. 545 35–9 [34] Bairu S G et al 2013 J. Phys. Chem. C 117 4824–35 [35] Olbrich C, Str¨umpfer J, Schulten K and Kleinekath¨ofer U 2011 J. Chem. Phys. Lett. 2 1771–6 [36] Olbrich C, Jansen T L C, Liebers J, Aghtar M, Str¨umpfer J, Schulten K, Knoester J and Kleinekath¨ofer U 2011 J. Chem. Phys. B 115 8609–21 [37] Madrid A B, Hyeon-Deuk K, Habenicht B F and Prezhdo O V 2009 ACS Nano 9 2487–94 [38] Akimov A V and Prezhdo O V 2013 J. Phys. Chem. Lett. 4 3857–64 [39] Damrauer N H, Cerullo G, Yeh A, Boussie T R, Shank C V and McCusker J K 1997 Science 275 54–7 [40] Wallin S, Davidsson J, Modin J and Hammarstr¨om L 2005 J. Chem. Phys. A 109 4697–704 [41] Tavernelli I, Curchod B F E and Rothlisberger U 2011 Chem. Phys. 391 101–9
[11] Hoff D A, da Silva R and Rego L G C 2011 J. Phys. Chem. C 115 15617–26 [12] Abuabara S G, Rego L G C and Batista V S 2005 J. Am. Chem. Soc. 127 18234–42 [13] Prezhdo O, Duncam W and Prezhdo V 2009 Prog. Surf. Sci. 84 30–68 [14] Skourtis S S 2003 Chem. Phys. Lett. 372 224–31 [15] Skourtis S S, Waldeck D and Beratan D N 2010 Ann. Rev. Phys. Chem. 61 461–85 [16] Rego L G C, Hames B C, Mazon K T, Joswig J-O 2014 J. Phys. Chem. C 118 126–34 [17] Hoff D A, da Silva R and Rego L G C 2012 J. Phys. Chem. C 116 21169–78 [18] Cramer C J 2002 Essentials of Computational Chemistry: Theory and Methods (West Sussex: Wiley) [19] Hoff D A and Rego L G C 2013 Specialist Periodical Reports—Chemical Modelling: Modelling Electron Quantum Dynamics in Large Molecular Systems vol 10, ed M Springboard and J-O Joswig (London: RSC Publishing) pp 102–26 [20] Elstner M, Porezag D, Jungnickel G, Elsner J, Haugk M, Frauenheim T, Suhai S and Seifert G 1998 Phys. Rev. B 58 7260–68 [21] Negre C F A, Fuertes V C, Oviedo M B, Oliva F Y and Sanchez C G 2012 J. Phys. Chem. C 116 14748–53 [22] Oviedo M B, Zarate X, Negre C F A, Schott E, Perez R A and Sanchez C G 2012 J. Phys. Chem. Lett. 3 2548–55 [23] Field M J 1992 Chem. Phys. Lett. 195 388–92 [24] Seifert G, Porezag D and Frauenheim T 1996 Int. J. Quantum Chem. 58 185–92 [25] Porezag D, Frauenheim T, K¨ohler T, Seifert G and Kaschner R 1995 Phys. Rev. B 51 12947–57
7
