Quantum Dynamics Simulations of Excited State Energy Transfer in a

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Quantum Dynamics Simulations of Excited State Energy Transfer in a Zinc - Free-Base Porphyrin Dyad Judah Sterling High, Luis G.C. Rego, and Elena Jakubikova J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b05739 • Publication Date (Web): 23 Sep 2016 Downloaded from http://pubs.acs.org on September 24, 2016

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The Journal of Physical Chemistry

Quantum Dynamics Simulations of Excited State Energy Transfer in a Zinc - Free-Base Porphyrin Dyad Judah S. High,a Luis G.C. Rego,b Elena Jakubikovaa,* a

Department of Chemistry, North Carolina State University, Raleigh, North Carolina 27695,

United States b

Department of Physics, Federal University of Santa Catarina, Florianopolis, Brazil

ACS Paragon Plus Environment

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Abstract

Rational design of artificial light-harvesting molecular architectures entails building systems that absorb strongly in the visible and near-IR region of the electromagnetic spectrum and also funnel excited state energy to a single site. Ability to model nonadiabatic processes, such as excited state energy transfer (EET), that occur in such systems on picosecond time scale can aid in the development of novel artificial light harvesting arrays. A combination of density functional theory (DFT), time-dependent DFT, tight-binding molecular dynamics, and quantum dynamics is employed here to simulate EET in the ZnFbΦ dyad, a model artificial light-harvesting array that undergoes EET with an experimentally measured rate constant of (3.5 ps)-1 upon excitation at 550 nm in toluene [Yang et al., J. Phys. Chem. B 1998, 102, 9426-9436]. We find that in order to successfully simulate the EET process, it is important to (1) include coupling between nuclear and electronic degrees of freedom in the QD simulation, (2) account for Coulomb coupling between the electron and hole wavepackets, and (3) parameterize the extended Hückel model Hamiltonian employed in the QD simulations with respect to the DFT.


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1. Introduction Photosynthetic organisms employ light harvesting systems1-9 to transform light energy into chemical energy. Natural photosynthesis relies heavily on excitation energy transfer10-16 (EET) such that, upon photo-absorption, excited state energy is funneled toward a reaction center to ultimately be used in chemical reactions.17-21 In nature, bacteriochlorophylls (BChl) and chlorophylls (Chl) are arranged into large macrostructures, which undergo EET with near perfect quantum efficiency.22 Substantial effort has been dedicated to designing artificial systems that mimic the lightabsorption, energy transfer and charge separation capabilities of pigment assemblies in plants and bacteria.23-25 In particular, studies of porphyrin dyads connected through various linkers have shown that EET rates in such artificial systems depend on donor-acceptor distance as well as orientation,26-29 steric hindrance of the linker,30 linker-attachment site,31 array topology29,30,32 and the presence of electron donating or withdrawing groups.31,33 Yang and Kim have summarized important factors that affect EET rates in multiporphyrin arrays that include the extent of excitonic dipole-dipole interactions and extensive π-conjugation.34 Porphyrin dyads, such as the meso-meso phenylene-linked zinc-free-base porphyrin (ZnFbΦ, see Figure 1), represent simple model systems for studies of EET in multiporphyrin arrays. The ZnFbΦ dyad undergoes rapid EET from the zinc porphyrin (Zn) to the freebase porphyrin (Fb), (kEET = (3.5 ps)-1) upon photoexcitation at 550 nm.33 Experimental studies comparing absorption and redox properties of the ZnFbΦ dyad with those of the Zn and Fb monomers confirmed relatively weak electronic coupling between the monomers in the dyad.33 However, it was noted that the electronic coupling in ZnFbΦ is stronger than in porphyrin dyads with longer linkers and slower EET rates.33 Due to the abundance of experimental data, ZnFbΦ represents an ideal model system for the testing and development of new computational approaches to simulations of EET processes. Computational studies play an important role in elucidating the underlaying mechanism of charge and excited state energy transfer in molecular arrays as well as dye-nanoparticle assemblies. Approaches based on the reduced density matrix, such as the Redfield theory, are often used to model quantum dynamics in light-harvesting systems.35 For example, Redfield theory has been successfully employed in modeling energy transfer within the reaction center of photosystem II,36 and a modified Redfield theory was used to model energy transfer and trapping


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kinetics in Photosystem I.36 A number of other quantum dynamics methodologies, both fully quantum mechanical and mixed quantum-classical (MQC),37,38 have been developed and applied to simulate photoinduced dynamics in molecular and condensed matter systems. Fully quantum mechanical methodologies include the time dependent hartree (TDH) methods.37-44 Some of the predominant MQC methods inculde the Ehrenfest,45 fewest switches surface hopping,38,46-51 density matrix propagation,52,53 quantum Liouville46 and Bohmian mechanics methodologies.54-56

Figure 1. Molecular structure of meso-meso p-phenylene linked zinc-freebase porphyrin dyad (ZnFbΦ).

In this work, we investigate the EET in a model ZnFbΦ dyad (see Figure 1), utilizing a mixed time-dependent quantum-classical dynamics method that propagates wavepackets by solving the time-dependent Schrödinger equation using an extended Hückel (EH) Hamiltonian.57 This approach has been previously employed in the simulations of excited state charge separation in a carotenoid-porphyrin-fullerene system,58 dynamics of metal-to-ligand charge transfer excited states in solvated tris-bipyridine ruthenium(II) complexes,59 and interfacial electron transfer in sensitized TiO2 anatase.60-65 Compared to conventional (ground-state) molecular dynamics, nonadiabatic molecular dynamics (NA-MD) is a much more demanding computational method, owing to the fact that it is carried out in the excited state and also because the time steps of simulation must be considerably smaller than in common ground-state MD. The most common NA-MD methods separate the electron and nuclear dynamics propagation, by treating the nuclei by classical mechanics in mixed quantum–classical approaches. Some high level of ab initio NA-MD methods are Newton-X,66 Molpro,67 CPMD,68 Octopus69 and JADE.70 Essentially, the theory used to calculate the potential energy surfaces determines the computational cost of NA-MD methods. Therefore, the large number of degrees of freedom hinders the use of first principle methods, especially if there is the necessity to describe excited state dynamics. The demand for


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computationally feasible NA-MD methods promoted the use of semiempirical methods for mixed quantum–classical dynamics based on the extended Hückel method, such as the one used in this work. Another approach to accelerate the simulation consists of calculating the electron dynamics upon a previously calculated nuclear trajectory. This is known as sequential MM-QM approach58,65 or, equivalently, the classical path approximation (CPA),71 as implemented in the PYXAID package.72 It is based on the assumption that the excited potential energy surface does not change remarkably from ground to excited state, or when it changes more slowly than the process studied. Since the ZnFbΦ dyad studied in this work does not undergo large structural changes upon excitation (i.e., isomerization, dissociation, etc., see ref. 33), this first approximation to nuclear-electron coupling should provide a good qualitative description of the process investigated. The sequential MM-QD approach is faster from the computational viewpoint than the direct coupling between the electronic and nuclear degrees of freedom, and provides a semi-quantitative description of dynamic phenomena in the excited state. Thus it allows us to model the quantum dynamics of energy transfer in a molecular system comprised of almost 200 atoms within the atomistic framework on a picosecond time scale, using only moderate computing resources. In addition to the simulations of EET in ZnFbΦ dyad, a new approach for construction of initial wavepacket based on the results of time-dependent density functional theory calculations is also presented. This work represents the first time this particular QD approach has been used to simulate EET in a molecular system with a semi-quantitative accuracy and thus opens up its application to a wider class chemical systems. 2. Computational Methodology Electronic Structure Calculations Structure optimization of ZnFbΦ was performed with the B3LYP exchange-correlation functional.73-75 The 6-31G* basis set76,77 was employed for C, H, N and O atoms, while the LANL0878,79 effective core potential and associated basis set were used for zinc. A frequency calculation verified that the optimized structure lies at a minimum of its potential energy surface. The Gaussian 09 software package80 was employed for all DFT calculations.


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All UV-vis spectra were calculated using time-dependent DFT (TD-DFT),81-83 as implemented in Gaussian 09,84 at the B3LYP/LANL08(Zn),6-31G*(C,H,N) level of theory. Natural transition orbitals (NTOs) were employed to characterize all excitations (see Figures S2-S11 in the SI).85 Molecular Dynamics Simulations Using the optimized ZnFbΦ as the initial structure, ten independent MD simulations of ZnFbΦ were performed using the tight-binding density functional theory (DFTB) methodology employing the DFTB+ software package.50,86-92 The MD simulations consisted of a 20 ps equilibration period (NVT ensemble) at 298 K, followed by a production simulation of 10 ps in the NVE ensemble. Nuclear configurations used in quantum dynamics were those obtained in the NVE ensemble. A 5-chain Nosé-Hoover93-97 thermostat with a coupling frequency of 3600 cm-1 was employed during the 20 ps equilibration period. All simulations were run with a 0.1 fs time step. Due to computational limitations, no solvent model was used during the MD simulations. Extended Hückel Parametrization The EH Hamiltonian was reparametrized with respect to the optimized ZnFbΦ Kohn-Sham Molecular Orbital (KS MO) shapes, energetic ordering, and energy gaps. Reparametrization involved first tuning atomic orbital (AO) ionization potentials (IP). The Wolfsberg-Helmholtz98101

parameter, κ, was set to 2.05 (κdefault = 1.75) (see Table S1 in the SI) to assist in reproducing

the KS HOMO-LUMO gap as well as various MO energy gaps from the HOMO-3 to LUMO+3. All other IPs were left default as prescribed by Alvarez.102 The manual EH parameters were than optimized further using the genetic algorithm employed in DynEMol.57 The optimized EH parameters were utilized during all QD simulations. Excited State Energy Transfer Simulations The quantum dynamics methodology developed by Rego and Batista57-59,65,103,104 was utilized to model EET in ZnFbΦ. In this model, initial electron and hole wavepackets, Φ El (0) and

Φ Hl (0) , are constructed as linear combinations of the EH MOs of the donor zinc trimesitylporphyrin subunit. The EH molecular orbitals (MOs) of Zn selected to construct the wavepackets correspond to Kohn-Sham (KS) MOs describing Q-band excitations as calculated by the TD-DFT.82-84,105-108 The selected excitations correspond with the excitation (550 nm) employed in experimental investigation of the EET rate in ZnFbΦ.33,109,110


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The donor KS MOs and weights used in the construction of Φ El (0) and Φ Hl (0) were calculated from the transition density eigenvectors of the Casida equations81,111 for selected excitations of the Zn subunit. The EH MOs that best represent the KS MOs were determined before construction of the initial linear combinations of EH MOs. The mathematical details for wavepacket construction are given in the Supporting Information. Ten quantum dynamics simulations of EET were done through time propagation of electron and hole wavepackets, Φ El (t) and Φ Hl (t) , by solving the time-dependent Schrödinger equation with the EH98,112-122 Hamiltonian.59 The wavepackets were propagated for 10 ps, with a 0.1 fs time step, on pre-computed trajectories obtained from molecular dynamics (MD) simulations. Electrostatic Coulomb coupling between Φ El (t) and Φ Hl (t) was calculated at each step of the QD simulation. At each step in the time-slice propagation procedure, Φ El (t) and Φ Hl (t) were projected onto the atomic basis of Zn, p-phenylene, and Fb to obtain survival probabilities P(t), i.e., partial occupations of Φ El (t) and Φ Hl (t) on the Zn, p-phenylene, or Fb subunits.59 Since the wavepacket at t = 0 is constructed to be fully localized on the Zn-porphyrin donor, P(t=0) is always one for the Zn-subunit and zero for the acceptor (Fb) or bridge (pphenylene). To describe EET in ZnFbΦ, P(t) for the electron and hole wavepackets on the Zn subunit must decrease to zero, while simultaneously increasing to unity for the Fb subunit. 3. Results and Discussion 3.1. Ground state structure The optimized structure of ZnFbΦ is such that the Zn and Fb porphyrin subunits are coplanar while the p-phenylene bridge plane lies at an angle of about 60° from the planes of each tetrapyrrole. The incorporation of phenylene linkers in multiporphyrin arrays is thought to preserve the electronic characteristics of the porphyrin subunits.33 This is supported by the shapes and energies of the frontier molecular orbitals of the optimized ZnFbΦ system (see Figure 2). The eight frontier Kohn-Sham (KS) MOs of ZnFbΦ spanning from the HOMO-3 to LUMO+3 are localized on either the Zn or Fb subunits. These localized MOs resemble the Zn and Fb subunit frontier MOs as shown in Figure 2. Furthermore, there is little splitting of the frontier KS


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MO energy levels in going from a situation in which the chromophores are isolated to the situation in which they are covalently linked via the p-phenylene bridge (see Figure 2).

Figure 2. Frontier molecular orbitals of the Zn and Fb porphyrin subunits as well as the ZnFbΦ dyad calculated at the B3LYP level of theory.

Previous experimental work showed that the one-electron redox potentials of the ZnFbΦ system remain unchanged in comparison to the one-electron redox potentials of isolated Zn and Fb chromophores.33 Taking Koopman’s theorem as a first approximation to ground state redox potentials,123,124 the ground state electronic structure of ZnFbΦ obtained here is consistent with the experimental observations. This is reflected in the small changes to the HOMO and LUMO energies going from the isolated Zn and Fb subunits to ZnFbΦ. Since EET simulations are done at the extended Hückel level of theory, the EH Hamiltonian was reparametrized to reproduce the ZnFbΦ frontier KS MO shapes and energies. A comparison of the DFT and reparametrized EH MOs is shown in Figure 3. The optimized EH parameters used in all EET simulations are shown in the Table S1 of the SI. Note that the EH MOs are in very close agreement with the DFT KS MOs. As both the EH MO energies and shapes are used during the wavepacket propagation procedure in DynEMol,57-59,65,103,104 it is critical that the EH Hamiltonian reproduces these properties with respect to results obtained at higher level of theory, DFT in this case.


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Figure 3. A comparison of the MOs of ZnFbΦ obtained at the B3LYP level (left) and those calculated using optimal parameters at the EH level of theory (right).

3.2. Absorption properties and initial state construction Previous experimental investigation of ZnFbΦ revealed that its UV-vis absorption spectrum can be approximately described as the superposition of the Zn and Fb subunit’s absorption spectra, suggesting that the excitation at 550 nm is a Q-band excitation localized on Zn subunit.33 The excitation at 550 nm was also utilized as the initial excitation for experimental EET studies.33 The calculated electronic absorption spectra of ZnFbΦ and the constituent monomers shown in Figure 4 reproduce these experimental findings. The calculated absorption spectrum of ZnFbΦ displays two near-degenerate Zn-localized excitations at 542.25 nm and 540.35 nm, which were confirmed to be nearly indistinguishable from the Q-band excitations of the isolated Zn subunit at 531.60 nm and 531.49 nm by the NTO analysis (see Figure 5). The initial electron and hole wavepackets for the EET simulations were therefore constructed based on the results of TD-DFT calculations on the Zn subunit of the ZnFbΦ dyad (see Appendix).


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Figure 4. B3LYP calculated electronic absorption spectrum of the Zn and Fb subunits as well as the ZnFbΦ dyad. The subunits exhibit the Q and Soret band and the ZnFbΦ spectrum resembles the superposition of the subunit spectra.

Figure 5. Comparison of NTOs for the second and third lowest energy calculated excitations of ZnFbΦ and the near-degenerate Q-band excitations of the isolated Zn subunit.

Since EET in ZnFbΦ occurs in the pico-second regime,33 electronic coupling to nuclear degrees of freedom is likely to play an important role in the EET process. Ten MD simulations of ZnFbΦ equilibrated at 298 K were carried out to obtain nuclear trajectories over which the


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electron and hole wavepackets propagate in time. The electronic absorption spectra for each of the initial structures of Zn subunit, taken from the 10 MD simulations (MDSIM1-10), are shown in Figure 6.

Figure 6. B3LYP calculated UV-vis spectra of Zn subunit structure after 20 ps of equilibration at 298 K of ZnFbΦ via MD. All spectra exhibit Soret- and Q-band excitations.

All absorption spectra of Zn subunits shown in Figure 6 exhibit a Soret- and a Q-band with excited state energies and intensities that are within 0.18 eV of the excitation energies and intensities of the optimized Zn subunit. The Q-band excitations found near 550 nm for each Zn subunit involve two contributions from one-electron transitions between the HOMO, HOMO-1 and the LUMO, LUMO+1 of each Zn structure. The contributions from the one-electron transitions to the Q-band excitations are used directly in the wavepacket construction procedure summarized in the Appendix. Ultimately, weighting coefficients are calculated for EH MOs, which are used in forming the electron and hole wavepackets as a linear combination of EH MOs. Care is taken during the wavepacket construction procedure to identify the EH MOs that match the KS MOs contributing to the excitation, which were calculated for the thermalized Zn subunit structures. Figure 7 illustrates the result of the wavepacket construction procedure by showing the Zn EH MOs contributing to the electron and hole wavepacket along with their weighting coefficients. The electron and hole wavepacket densities resulting from the linear combination of EH MOs are also shown in Figure 7.


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Figure 7. Initial EH electron and hole wavepacket densities for a representative QD simulation. Wavepackets were constructed as a linear combination of the EH MOs that best resemble the geometry optimized KS MOs involved with a given excitation as prescribed by TD-DFT. The coefficients are calculated using the procedure outlined in the Appendix.

3.3. Excited state energy transfer simulations Wavepacket dynamics was employed to perform EET simulations in ZnFbΦ dyad, with the initial hole and electron excitations set up based on TD-DFT calculations on the Zn subunit. The survival probabilities, P(t), for the electron and hole wavepackets on the Zn, p-phenylene and Fb subunits of the ZnFbΦ dyad are shown in Figure 8 for a series of QD simulations utilizing computational models of increasing complexity. Case A corresponds to the simplest surmise, in which QD simulations were performed upon the static optimized ZnFbΦ structure, with Coulomb coupling between the electron and hole wavepackets neglected. Both electron and hole wavepackets remain mainly localized on the Zn subunit throughout the simulation. Neither the addition of nuclear motion (model B) nor the addition of Coulomb coupling (model C) into the model alone results in the EET in ZnFbΦ dyad on 1 ps time scale. The complete EET transfer is only observed when both nuclear motion and Coulomb coupling are included in the model (model D). Reparametrization of EH Hamiltonian is equally important for the success of the EET simulations. Utilization of the default Alvarez parameters in the QD simulations leads to incorrect description of the electron and hole wavepacket dynamics set, even if nuclear motion and Coulomb coupling are included in the model (model E). It is clear that in order to describe EET in ZnFbΦ system, one must include the following in the model: (1) nuclear motion, (2) electron-hole Coulomb coupling, and (3) EH Hamiltonian parametrized with respect to the DFT results.


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Figure 8. Survival probabilities for the Zn, p-phenylene and Fb subunits during QDSIM1 in which combinations of having nuclear motion, Coulombic wavepacket coupling and optimal EH parameters were included.

Figure 9 displays survival probabilities of electron and hole wavepackets on Zn subunit from ten independent QD simulations of EET in ZnFbΦ. All simulations utilize parametrized extended Hückel Hamiltonian and include nuclear motion as well as Coulomb coupling between electron and hole wavepackets. The survival probability for both electron and hole wavepackets on Zn subunit becomes negligible within 1.5 ps upon initial excitation of Zn subunit, suggesting a complete excited state energy transfer. Complete EET can be confirmed by inspecting survival probabilities on Fb and p-phenylene subunits, shown in Figures S15 and S16 in the SI. To further evaluate the importance of nuclear motion for simulating EET in molecular systems, ten additional QD simulations were performed on ten different static structures. The static structures were chosen to be the same as the initial structures of the ten MD production runs. The QD simulations utilized optimized EH parameters and included electron-hole Coulomb coupling in the model. Interestingly, two of the ten QD simulations displayed complete EET transfer in the time frame of 1.5 ps, while the wavepacket remained localized on the Zn subunit in the rest of the simulations (see Figure S17 in the SI). This suggests that certain conformations of ZnFbΦ dyad will facilitate the EET even at the frozen geometry. The structures resulting in EET are only accessible through the exploration of phase space and it may prove difficult to consistently


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predict which static structures will result in EET a priori. A reliable simulation of the EET in this system thus requires the inclusion of nuclear motion into the model.

Figure 9. Survival probability for the electron and hole on the Zn subunit of ZnFbΦ for QDSIM1-10. All simulations show EET by means of the survival probabilities decaying to zero.

Examination of the electron and hole wavepacket densities for each EET simulation, as shown in Figure 10 for one of the QD simulations, reveals how electronic excitation energy is transferred from the Zn to the Fb. It is evident from Figure 10 that by 1.0 fs, wavepacket density has begun to delocalize onto the p-phenylene bridge and by 10 fs, wavepacket density has delocalized across the entire dyad. By 1.0 ps, EET has been achieved and the wavepacket density is completely localized on the Fb subunit.


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Figure 10. Shown here are snapshots of the electron and hole wavepacket density of QDSIM1 at various times. Note that by 1.0 ps the wavepacket densities are fully localized on the Fb subunit.

Since the transient absorption experiments used to monitor EET rates in ZnFbΦ measure the average response of a collection of ZnFbΦ molecules to light at 550 nm, the survival probabilities from all 10 QD simulations were averaged (see Figure 11), with the goal of obtaining an average EET rate to be compared to that determined from experiment ((3.5 ps)-1).33 Survival probabilities for the electron and hole wavepackets on the Zn- and Fb- porphyrin were fit to bi-exponential decay functions to obtain rate constants and characteristic times for EET in ZnFbΦ (see Figure 11). No fitting scheme was applied to the survival probability of the pphenylene bridge as it stays close to zero throughout the simulation. The calculated average characteristic time for the EET from the Zn onto the Fb subunit is 0.45 ps. This characteristic time is one order of magnitude faster than the experimentally determined characteristic time of 3.5 ps, suggesting that while our model correctly describes the qualitative behavior of this system, it does not possess quantitative accuracy. We attribute this lack of quantitative accuracy to the neglect of the solvent effects in our model. Solvation dynamics have been previously


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shown to play an important role in obtaining the correct description of excited state dynamics in molecular systems.59

Figure 11. Average survival probabilities of the electron and hole wavepackets averaged over QDSIM110. The average survival probabilities were fitted to bi-exponentials with characteristic times τ. No fit was applied to the p-phenylene electron and hole P(t).

4. Conclusions We have presented a successful simulation of non-adiabatic excited state energy transfer in a prototypical meso-meso phenylene linked zinc-free-base porphyrin dyad, ZnFbΦ, utilizing a mixed time-dependent quantum mechanics/molecular mechanics computational method.57,59,103A procedure was also developed for constructing the initial wavepackets as a linear combination of extended Hückel MOs whose weighting coefficients were determined through excitation analysis of the donor Zn subunit via TD-DFT. This was necessary, since porphyrin excitations cannot be approximated as single one-electron MO-to-MO transitions. The inclusion of nuclear motion, Coulomb coupling between electron and hole wavepackets, and a reparametrized Extended Hückel Hamiltonian was critical for achieving Zn to Fb excitation energy migration in the quantum dynamics simulations. Our simulations yielded EET with a time constant of 0.45 ps, which is about an order of magnitude faster than the EET time


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constant of 3.5 ps determined from transient absorption measurements.33 We attribute this discrepancy to the neglect of solvation dynamics in our computational model. Overall, we have shown that a mixed time-dependent quantum mechanics/molecular dynamics computational approach can be used to simulate EET in rigid molecular systems with a semi-quantitative accuracy. Associated Content Supporting Information Description of a procedure for construction of initial wavepackets based on TD-DFT calculations. Optimal extended Hückel parameters used in QD simulations; frontier B3LYP MOs of initial ZnFbΦ structures used in QD simulations; NTOs of Q band excitations in ZnFbΦ and Zn donor at all initial QD structures; initial electron and hole wavepacket densities for QDSIM110; QD simulation P(t) plots demonstrating importance of factors resulting in EET; P(t) associated with the p-phenylene bridge and Fb-porphyrin acceptor for QDSIM1-10; and P(t) for ten static QD simulations. Author Information Corresponding Author *E-mail: [email protected], Tel. 919-515-1808 Notes The authors declare no competing financial interest. Acknowledgments This work was supported by the Department of Chemistry at North Carolina State University (NCSU) and the High Performance Computing (HPC) services at NCSU.


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TOC Figure


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Quantum Dynamics Simulations of Excited State Energy Transfer in a

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