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Isotope Effects on the Nonequilibrium Dynamics of Ultrafast Photoinduced Proton-Coupled Electron Transfer Reactions in Solution Anirban Hazra, Alexander V. Soudackov, and Sharon Hammes-Schiffer* Department of Chemistry, Pennsylvania State University, University Park, Pennsylvania 16802, United States w This paper contains enhanced objects available on the Internet at http://pubs.acs.org/JPCL. n
ABSTRACT The hydrogen/deuterium isotope effects on the ultrafast dynamics of photoinduced proton-coupled electron transfer (PCET) are investigated with a recently developed nonadiabatic dynamics approach. An ensemble of surface hopping trajectories is propagated according to a Langevin equation on electronproton vibronic free energy surfaces that depend on a collective solvent coordinate. The calculations illustrate that ultrafast PCET reactions could exhibit a significant normal isotope effect, where PCET is faster for hydrogen than deuterium, but could also exhibit a negligible isotope effect or even a slight inverse isotope effect. The isotope effect is very small or absent when highly excited electron-proton vibronic states dictate the nonadiabatic dynamics and increases with greater participation of lower vibronic states. Thus, although the presence of a significant isotope effect strongly suggests that proton motion is coupled to electron transfer, the absence of an isotope effect does not exclude the possibility that proton transfer accompanies electron transfer in ultrafast photoinduced charge transfer processes. SECTION Energy Conversion and Storage
hotoinduced proton-coupled electron transfer (PCET) is a key process in photosynthesis, as well as in solar energy conversion systems such as dye-sensitized solar cells. Understanding the detailed mechanism of these processes is important for the further development of sustainable energy technology. Recently, the ultrafast dynamics of photoinduced PCET reactions has been investigated experimentally for a wide range of systems, including ruthenium bipyridyl complexes,1,2 double-stranded DNA,3,4 and methanol adsorbed on titanium dioxide surfaces.5,6 The coupling of proton motion to electron transfer for these systems has been probed by measurement of the isotope effect, which reveals the impact of replacing hydrogen with deuterium on the ultrafast dynamics. The presence of an isotope effect has often been used to distinguish PCET from pure electron transfer reactions. In this Letter, we perform nonadiabatic dynamics calculations to examine the hydrogen/deuterium isotope effects on the ultrafast dynamics of model photoinduced PCET systems. Our objective is to elucidate the physical basis for the experimentally observed isotope effects and to predict the dependence of the isotope effects on system properties. We also critically evaluate the premise that isotope effects can be used as a diagnostic for ultrafast PCET reactions. Our calculations are based on a recently developed theoretical formulation for modeling photoinduced nonequilibrium concerted PCET reactions in solution.7 In this formulation, the solvent is treated with dielectric continuum theory, and the solvent polarization
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is mapped onto a single collective solvent coordinate that evolves according to a Langevin equation.8-13 Nonadiabatic surface hopping trajectories are propagated on electron-proton vibronic free energy surfaces that depend on this collective solvent coordinate. Since electron-proton transfer can occur on a similar time scale as the solvent relaxation,7 the solvent dynamics can play an important role in the overall charge transfer process.14-20 Inclusion of the nonequilibrium solvent dynamics differentiates the present work from our previous isotope effect studies.21 In our theoretical formulation,7 the vibronic Hamiltonian representing the PCET system is expressed in the basis of two electronic states |Dæ and |Aæ corresponding to the electron localized on the electron donor and acceptor, respectively. This vibronic Hamiltonian is given by H vib ¼ Tp þ
2
1 2 6 U ðrp Þ þ 2 f0 x 4 VDA D
3
VDA 7 5 ð1Þ pffiffiffiffiffiffiffiffiffiffiffiffi 1 ~ U ðrp Þ þ f0 ðx - 2λ=f0 Þ2 - Δ 2 A
where Tp is the kinetic energy operator for the proton, rp is the proton coordinate, x is the collective solvent coordinate, λ is Received Date: November 11, 2010 Accepted Date: December 10, 2010 Published on Web Date: December 17, 2010
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the solvent reorganization energy, f0 = 4πε0ε¥/(ε0-ε¥) is the inverse Pekar factor, ε0 and ε¥ are the static and electronic ~ is the energy difference dielectric constants of the solvent, Δ between the solvated diabatic states, and VDA is the constant coupling between the two electronic states. In addition, UD(rp) and UA(rp) are two distinct harmonic proton potentials of the form U(rp) = mpω2p(rp - r0p)2/2 centered at r0p = rD p and r0p = rAp , respectively, with mass mp and frequency ωp. In general, the proton potentials could be anharmonic or even asymmetric double-well potentials. The shapes of the proton potentials will influence the detailed dynamics but are not expected to alter the qualitative trends described below. The adiabatic vibronic free energy surfaces Wk(x) are obtained by diagonalizing the Hamiltonian in eq 1 in an appropriate electron-proton vibronic basis, which is typically chosen to be products of the donor and acceptor electronic states |Dæ and |Aæ with proton vibrational basis functions χμ(rp) [i.e., |Dæχμ(rp) and |Aæχμ(rp)]. The dynamics of the collective solvent coordinate on a single adiabatic vibronic free energy surface Wk(x) is described by the Langevin equation7 dWk ðxÞ € ¼ - f0 τ~L x_ þ FðtÞ ð2Þ mx x dx
The electronic coupling was VDA =0.03 eV, and the energy ~ ~ bias was Δ=1.0 eV for models A and C and Δ=3.51 eV for model B. We propagated 1000 trajectories for each model system studied. Additional simulation details are provided in the Supporting Information. The photoexcitation was assumed to occur from the ground electronic state to the donor electronic state, followed by nonadiabatic dynamics on the adiabatic vibronic surfaces Wk(x). The initial solvent distribution corresponded to a classical Gaussian distribution associated with the acceptor diabatic electronic state. The initial populations of the donor vibronic states were determined by the Franck-Condon overlaps between the initial proton wavepacket in the ground electronic state and the proton vibrational wave functions corresponding to UD(rp). For models A and B, the initial proton wavepacket was the same as the ground vibrational state on the donor proton potential, so the system was photoexcited into the lowest-energy donor vibronic state. For model C, the initial proton wavepacket corresponded to the ground vibrational state of a proton potential centered midway between the donor and acceptor proton potentials, so the system was photoexcited into a coherent mixture of donor vibronic states. Although the MDQT trajectories evolve on the adiabatic surfaces Wk(x), the charge transfer dynamics are more easily interpreted in terms of transitions between the donor and acceptor vibronic states associated with the diabatic electronic states |Dæ and |Aæ. These diabatic electron-proton vibronic free energy surfaces can be obtained by diagonalization of the donor and acceptor blocks of the Hamiltonian in eq 1 in an electron-proton vibronic basis. The vibronic coupling between a pair of donor and acceptor vibronic states is the product of the constant electronic coupling VDA and the overlap integral between the corresponding donor and acceptor proton vibrational wave functions [i.e., the eigenfunctions of the harmonic potentials UD(rp) and UA(rp)]. Due to the relatively small vibronic coupling, the adiabatic and diabatic vibronic states are nearly identical everywhere except in the vicinity of an avoided crossing, and we can associate each adiabatic vibronic state with a diabatic vibronic state at virtually all values of the collective solvent coordinate. Figure 1 depicts snapshots of the ensemble of trajectories during the nonadiabatic dynamics on the adiabatic vibronic free energy surfaces. Movies depicting the full time-dependent dynamics are also provided. In this figure, the donor and acceptor vibronic surfaces appear as two sets of parabolas, where the donor surfaces are centered at x = 0 and the acceptor surfaces are centered at x=0.153 eV1/2. In models A and B, the system started with a nonequilibrium solvent distribution on the lowest-energy donor vibronic state. Model A exhibited an initial fast decay on the ground vibronic surface in ∼1 ps to the minimum of this surface, followed by slower nonadiabatic dynamics on the adiabatic vibronic surfaces, resulting in the majority of the population in the first-excited acceptor vibronic state after ∼40 ps. Model B exhibited a much faster relaxation process, with the majority of the population in the eighth and ninth acceptor vibronic states after less than 5 ps. In model C, the system started in a mixture of donor vibronic states and exhibited an initial fast decay on the
This equation was derived from a generalized Langevin equation with the Onodera model22 for dielectric relaxation of the solvent.7 Although the generalized Langevin equation with a more sophisticated solvent model could be utilized, this simple model is expected to capture the qualitative trends. In this case, the effective mass mx is calculated from the moment of inertia I of an individual solvent molecule, the temperature, and the dielectric constants of the solvent, and the parameter τ~L is calculated from the moment of inertia I, as well as the longitudinal relaxation time and dielectric constants of the solvent. The stochastic force F(t) is related to the friction f0~τL through the fluctuation-dissipation theorem. To simulate the nonadiabatic dynamics on multiple adiabatic vibronic surfaces following photoexcitation, we use Tully's fewest switches surface hopping method,23 also known as the molecular dynamics with quantum transitions (MDQT) method.24 In our implementation, each trajectory within an ensemble evolves according to eq 2 on a single adiabatic electron-proton vibronic free energy surface except for instantaneous transitions among the adiabatic vibronic states. In this model, proton vibrational relaxation cannot occur by a direct mechanism (i.e., within the donor or acceptor diabatic state) but can occur by an indirect mechanism via nonadiabatic transitions between the donor and acceptor states.21 Models that include the direct mechanism for proton vibrational relaxation, as well as more realistic proton potentials, will be the topic of future studies. We used this approach to study the isotope effect on the nonadiabatic dynamics for three different models denoted models A, B, and C. In all three models, the minima of the proton potentials for the donor and acceptor electronic states were at rpD = 0 and rpA = 0.5 Å. For hydrogen, mp = 1.0073 amu and ωp = 3000 cm-1, while for deuterium, mp = 2.0136 amu and ωp = 2121.8 cm-1. The solvent parameters were chosen to be those for water with reorganization energy λ = 0.65 eV.
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w Snapshots of the ensemble of trajectories during the nonadiabatic dynamics on the adiabatic vibronic free energy surfaces, which Figure 1. n depend on the collective solvent coordinate x, for (a) model A, (b) model B, and (c) model C. This figure corresponds to hydrogen transfer, and the analogous figure for deuterium transfer is given in the Supporting Information. The full movies for both hydrogen (Movies 1, 2, and 3) and deuterium (Movies 4, 5, and 6) transfer are also provided.
donor vibronic surfaces, followed by slower nonadiabatic dynamics with the majority of the population ending up in the first through third acceptor vibronic states. Figure 2 depicts the population decay of the donor state for hydrogen and deuterium for models A, B, and C. For all three models, the qualitative behavior of the nonadiabatic dynamics was similar for hydrogen and deuterium, but the isotope effect on the donor state population decay varied among the models. For model A, the donor state population decayed approximately three times slower for deuterium than for hydrogen. In contrast, for models B and C, the donor state population decay time scales were very similar for hydrogen and deuterium, with deuterium decaying slightly faster than hydrogen for model B and the reverse for model C. The physical basis for these isotope effects can be understood by the analysis of the hydrogen and deuterium vibronic couplings that impact the probability of nonadiabatic transitions. As depicted in Figure 3, initially both the hydrogen and deuterium vibronic couplings increase with increasing quantum number of the acceptor vibronic state because the overlap integral is dominated by the tails of the vibrational wave
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functions, and the excited state vibrational wave functions become progressively more delocalized. In this regime, the vibronic coupling is significantly greater for hydrogen than for deuterium because the hydrogen wave functions are more delocalized. For highly excited acceptor vibronic states, the overlap integral starts to decrease because of cancellation effects due to oscillations of the vibrational wave functions. In the relatively flat turnover regime, the vibronic couplings are similar for hydrogen and deuterium. Note that the energy splittings between vibronic states are smaller for deuterium than for hydrogen, so the excited acceptor vibronic states that are populated after photoexcitation and participate in the nonadiabatic dynamics will be shifted to higher quantum numbers for deuterium compared to hydrogen. This trend is illustrated in Figure 3 by the identification of the acceptor vibronic states that participate in the nonadiabatic dynamics for models A and B. Figure 4 depicts the time-dependent populations of the individual acceptor vibronic states for models A and B. For model A, the first and second excited acceptor vibronic states participate in the dynamics for hydrogen, and the ground
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Figure 4. (a) Hydrogen (left) and deuterim (right) diabatic vibronic state populations as functions of time for (a) model A and (b) model B. The lines are labeled D and A for donor and acceptor vibronic states, respectively, with 0 denoting the ground state.
in the dynamics for deuterium. As shown in Figure 3, the vibronic couplings between the ground donor vibronic state and the participating acceptor vibronic states are very similar for hydrogen and deuterium in this regime, leading to similar donor state population decay time scales for hydrogen and deuterium in this model. The slight inverse isotope effect, where the donor state population decay is slightly faster for deuterium than for hydrogen, arises from subtle differences in the detailed dynamics. For model C, the initial donor state population decay is similar for hydrogen and deuterium, but, subsequently, the population decay is slightly faster for hydrogen. This behavior can be explained using the concepts understood from models A and B. For the first picosecond, the population is distributed among higher vibronic states, where the vibronic couplings are similar for hydrogen and deuterium, so no isotope effect is observed. After the initial relaxation, the population is distributed among the lower vibronic states, where the vibronic couplings are significantly larger for hydrogen than for deuterium, so the population decay is faster for hydrogen. This model provides an explanation for the experimental observation of the onset of an isotope effect after an initial fast decay in the ultrafast dynamics of double-stranded DNA3,4 and methanol adsorbed on titanium dioxide surfaces.5,6 The calculations in this Letter illustrate that ultrafast PCET reactions could exhibit a significant normal hydrogen/deuterium isotope effect, where the donor state population decays faster for hydrogen, but could also exhibit a negligible isotope effect or even a slight inverse isotope effect. The isotope effect is predicted to be very small or absent when highly excited electron-proton vibronic states dictate the nonadiabatic dynamics and is predicted to increase with greater participation of the lower vibronic states. The participation of the lower vibronic states can be enhanced by decreasing the driving force of the reaction and minimizing the difference between the proton potential energy curves in the ground and donor electronic states. In some cases, an isotope effect may be
Figure 2. Population of the donor state as a function of time for hydrogen (red) and deuterium (blue) transfer for (a) model A, (b) model B, and (c) model C.
Figure 3. Square of the vibronic coupling between the ground donor vibronic state and the νth acceptor vibronic state as a function of the quantum number ν of the acceptor vibronic state for hydrogen (red) and deuterium (blue) transfer. The vibronic couplings that impact the nonadiabatic dynamics, as determined from analysis of Figure 4, are indicated with filled squares for model A and with filled triangles for model B.
through second excited acceptor vibronic states participate in the dynamics for deuterium. As shown in Figure 3, the squared vibronic couplings between the ground donor vibronic state and the participating acceptor vibronic states are significantly larger for hydrogen than for deuterium by ∼10 - 104 in this regime, leading to the faster donor state population decay for hydrogen. In contrast, for model B, the 8th through 11th excited acceptor vibronic states participate in the dynamics for hydrogen, while the 11th through 15th excited acceptor vibronic states participate
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observable only after an initial fast decay involving vibrational relaxation to the lower vibronic states. Thus, although the presence of a significant isotope effect strongly suggests that proton motion is coupled to electron transfer, the absence of an isotope effect does not exclude the possibility that proton transfer accompanies electron transfer in ultrafast photoinduced charge transfer processes.
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SUPPORTING INFORMATION AVAILABLE Simulation
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details and analogue of Figure 1 for deuterium transfer. This material is available free of charge via the Internet at http:// pubs.acs.org.
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AUTHOR INFORMATION (18)
Corresponding Author: *To whom correspondence should be addressed. E-mail: shs@ chem.psu.edu.
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ACKNOWLEDGMENT We gratefully acknowledge funding from
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AFOSR Grant FA9550-10-1-0081 and NSF Grant CHE-07-49646.
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REFERENCES (1)
Concepcion, J. J.; Brennaman, M. K.; Deyton, J. R.; Lebedeva, N. V.; Forbes, M. D. E.; Papanikolas, J. M.; Meyer, T. J. ExcitedState Quenching by Proton-Coupled Electron Transfer. J. Am. Chem. Soc. 2007, 129, 6968–6969. (2) Gagliardi, C. J.; Westlake, B. C.; Papanikolas, J. M.; Kent, C. A.; Paul, J. J.; Meyer, T. J. Integrating Proton Coupled Electron Transfer (PCET) and Excited States. Coord. Chem. Rev. 2010, 254, 2459–2471. (3) Crespo-Hernandez, C. E.; Cohen, B.; Kohler, B. Base Stacking Controls Excited-State Dynamics in AT DNA. Nature 2005, 436, 1141–1144. (4) de La Harpe, K.; Crespo-Hernandez, C. E.; Kohler, B. Deuterium Isotope Effect on Excited-State Dynamics in an Alternating GC Oligonucleotide. J. Am. Chem. Soc. 2009, 131, 17557– 17559. (5) Li, B.; Zhao, J.; Onda, K.; Jordan, K. D.; Yang, J.; Petek, H. Ultrafast Proton-Coupled Electron Transfer. Science 2006, 311, 1436–1440. (6) Zhao, J.; Li, B.; Onda, K.; Feng, M.; Petek, H. Solvated Electrons on Metal Oxide Surfaces. Chem. Rev. 2006, 106, 4402–4427. (7) Hazra, A.; Soudackov, A. V.; Hammes-Schiffer, S. Role of Solvent Dynamics in Ultrafast Photoinduced Proton-Coupled Electron Transfer Reactions in Solution. J. Phys. Chem. B 2010, 114, 12319–12332. (8) Zusman, L. D. Outer-Sphere Electron Transfer in Polar Solvents. Chem. Phys. 1980, 49, 295–304. (9) Calef, D. F.; Wolynes, P. G. Classical Solvent Dynamics and Electron Transfer: 1. Continuum Theory. J. Phys. Chem. 1983, 87, 3387–3400. (10) Hynes, J. T. Outer-Sphere Electron-Transfer Reactions and Frequency-Dependent Friction. J. Phys. Chem. 1986, 90, 3701–3706. (11) Sparpaglione, M.; Mukamel, S. Dielectric Friction and the Transition from Adiabatic to Nonadiabatic Electron Transfer I. Solvation Dynamics in Liouville Space. J. Chem. Phys. 1988, 88, 3263–3280. (12) Basilevsky, M. V.; Chudinov, G. E.; Newton, M. D. The Multiconfigurational Adiabatic Electron Transfer Theory and Its
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Invariance under Transformations of Charge Density Basis Functions. Chem. Phys. 1994, 179, 263–278. Malhado, J. P.; Spezia, R.; Hynes, J. T. Dynamical Friction Effects on the Photoisomerization of a Model Protonated Schiff Base in Solution. J. Phys. Chem. A [Online early access]. DOI: 10.1021/jp106096m. Rips, I.; Jortner, J. Dynamic Solvent Effects on Outer-Sphere Electron Transfer. J. Chem. Phys. 1987, 87, 2090–2104. Sumi, S.; Marcus, R. A. Dynamical Effects in Electron Transfer Reactions. J. Chem. Phys. 1986, 84, 4894–4914. Borgis, D.; Hynes, J. T. Dynamical Theory of Proton Tunneling Transfer Rates in Solution: General Formulation. Chem. Phys. 1993, 170, 315–346. Bixon, M.; Jortner, J. Solvent Relaxation Dynamics and Electron Transfer. Chem. Phys. 1993, 176, 467–481. Prezhdo, O. V.; Rossky, P. J. Solvent Mode Participation in the Nonradiative Relaxation of the Hydrated Electron. J. Phys. Chem. 1996, 100, 17094–17102. Barzykin, A. V.; Frantsuzov, P. A.; Seki, K.; Tachiya, M. Solvent Effects in Nonadiabatic Electron Transfer Reactions: Theoretical Aspects. Adv. Chem. Phys. 2002, 123, 511–616. Grimminger, J.; Schmickler, W. Dynamics of Combined Electron- and Proton Transfer at Metal Electrodes. Chem. Phys. 2007, 334, 8–17. Venkataraman, C.; Soudackov, A. V.; Hammes-Schiffer, S. Photoinduced Homogeneous Proton-Coupled Electron Transfer: Model Study of Isotope Effects on Reaction Dynamics. J. Chem. Phys. 2009, 131, 154502. Onodera, Y. Breakdown of Debye Model for Dielectric Relaxation in High Frequencies. J. Phys. Soc. Jpn. 1993, 62, 4104–4107. Tully, J. C. Molecular Dynamics with Electronic Transitions. J. Chem. Phys. 1990, 93, 1061–1071. Hammes-Schiffer, S.; Tully, J. C. Proton Transfer in Solution: Molecular Dynamics with Quantum Transitions. J. Chem. Phys. 1994, 101, 4657–67.
DOI: 10.1021/jz101532g |J. Phys. Chem. Lett. 2011, 2, 36–40
Isotope Effects on the Nonequilibrium Dynamics of Ultrafast Photoinduced Proton-Coupled Electron Transfer Reactions in Solution Anirban Hazra, Alexander V. Soudackov, and Sharon Hammes-Schiffer* Department of Chemistry, Pennsylvania State University, University Park, Pennsylvania 16802, United States w This paper contains enhanced objects available on the Internet at http://pubs.acs.org/JPCL. n
ABSTRACT The hydrogen/deuterium isotope effects on the ultrafast dynamics of photoinduced proton-coupled electron transfer (PCET) are investigated with a recently developed nonadiabatic dynamics approach. An ensemble of surface hopping trajectories is propagated according to a Langevin equation on electronproton vibronic free energy surfaces that depend on a collective solvent coordinate. The calculations illustrate that ultrafast PCET reactions could exhibit a significant normal isotope effect, where PCET is faster for hydrogen than deuterium, but could also exhibit a negligible isotope effect or even a slight inverse isotope effect. The isotope effect is very small or absent when highly excited electron-proton vibronic states dictate the nonadiabatic dynamics and increases with greater participation of lower vibronic states. Thus, although the presence of a significant isotope effect strongly suggests that proton motion is coupled to electron transfer, the absence of an isotope effect does not exclude the possibility that proton transfer accompanies electron transfer in ultrafast photoinduced charge transfer processes. SECTION Energy Conversion and Storage
hotoinduced proton-coupled electron transfer (PCET) is a key process in photosynthesis, as well as in solar energy conversion systems such as dye-sensitized solar cells. Understanding the detailed mechanism of these processes is important for the further development of sustainable energy technology. Recently, the ultrafast dynamics of photoinduced PCET reactions has been investigated experimentally for a wide range of systems, including ruthenium bipyridyl complexes,1,2 double-stranded DNA,3,4 and methanol adsorbed on titanium dioxide surfaces.5,6 The coupling of proton motion to electron transfer for these systems has been probed by measurement of the isotope effect, which reveals the impact of replacing hydrogen with deuterium on the ultrafast dynamics. The presence of an isotope effect has often been used to distinguish PCET from pure electron transfer reactions. In this Letter, we perform nonadiabatic dynamics calculations to examine the hydrogen/deuterium isotope effects on the ultrafast dynamics of model photoinduced PCET systems. Our objective is to elucidate the physical basis for the experimentally observed isotope effects and to predict the dependence of the isotope effects on system properties. We also critically evaluate the premise that isotope effects can be used as a diagnostic for ultrafast PCET reactions. Our calculations are based on a recently developed theoretical formulation for modeling photoinduced nonequilibrium concerted PCET reactions in solution.7 In this formulation, the solvent is treated with dielectric continuum theory, and the solvent polarization
P
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is mapped onto a single collective solvent coordinate that evolves according to a Langevin equation.8-13 Nonadiabatic surface hopping trajectories are propagated on electron-proton vibronic free energy surfaces that depend on this collective solvent coordinate. Since electron-proton transfer can occur on a similar time scale as the solvent relaxation,7 the solvent dynamics can play an important role in the overall charge transfer process.14-20 Inclusion of the nonequilibrium solvent dynamics differentiates the present work from our previous isotope effect studies.21 In our theoretical formulation,7 the vibronic Hamiltonian representing the PCET system is expressed in the basis of two electronic states |Dæ and |Aæ corresponding to the electron localized on the electron donor and acceptor, respectively. This vibronic Hamiltonian is given by H vib ¼ Tp þ
2
1 2 6 U ðrp Þ þ 2 f0 x 4 VDA D
3
VDA 7 5 ð1Þ pffiffiffiffiffiffiffiffiffiffiffiffi 1 ~ U ðrp Þ þ f0 ðx - 2λ=f0 Þ2 - Δ 2 A
where Tp is the kinetic energy operator for the proton, rp is the proton coordinate, x is the collective solvent coordinate, λ is Received Date: November 11, 2010 Accepted Date: December 10, 2010 Published on Web Date: December 17, 2010
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the solvent reorganization energy, f0 = 4πε0ε¥/(ε0-ε¥) is the inverse Pekar factor, ε0 and ε¥ are the static and electronic ~ is the energy difference dielectric constants of the solvent, Δ between the solvated diabatic states, and VDA is the constant coupling between the two electronic states. In addition, UD(rp) and UA(rp) are two distinct harmonic proton potentials of the form U(rp) = mpω2p(rp - r0p)2/2 centered at r0p = rD p and r0p = rAp , respectively, with mass mp and frequency ωp. In general, the proton potentials could be anharmonic or even asymmetric double-well potentials. The shapes of the proton potentials will influence the detailed dynamics but are not expected to alter the qualitative trends described below. The adiabatic vibronic free energy surfaces Wk(x) are obtained by diagonalizing the Hamiltonian in eq 1 in an appropriate electron-proton vibronic basis, which is typically chosen to be products of the donor and acceptor electronic states |Dæ and |Aæ with proton vibrational basis functions χμ(rp) [i.e., |Dæχμ(rp) and |Aæχμ(rp)]. The dynamics of the collective solvent coordinate on a single adiabatic vibronic free energy surface Wk(x) is described by the Langevin equation7 dWk ðxÞ € ¼ - f0 τ~L x_ þ FðtÞ ð2Þ mx x dx
The electronic coupling was VDA =0.03 eV, and the energy ~ ~ bias was Δ=1.0 eV for models A and C and Δ=3.51 eV for model B. We propagated 1000 trajectories for each model system studied. Additional simulation details are provided in the Supporting Information. The photoexcitation was assumed to occur from the ground electronic state to the donor electronic state, followed by nonadiabatic dynamics on the adiabatic vibronic surfaces Wk(x). The initial solvent distribution corresponded to a classical Gaussian distribution associated with the acceptor diabatic electronic state. The initial populations of the donor vibronic states were determined by the Franck-Condon overlaps between the initial proton wavepacket in the ground electronic state and the proton vibrational wave functions corresponding to UD(rp). For models A and B, the initial proton wavepacket was the same as the ground vibrational state on the donor proton potential, so the system was photoexcited into the lowest-energy donor vibronic state. For model C, the initial proton wavepacket corresponded to the ground vibrational state of a proton potential centered midway between the donor and acceptor proton potentials, so the system was photoexcited into a coherent mixture of donor vibronic states. Although the MDQT trajectories evolve on the adiabatic surfaces Wk(x), the charge transfer dynamics are more easily interpreted in terms of transitions between the donor and acceptor vibronic states associated with the diabatic electronic states |Dæ and |Aæ. These diabatic electron-proton vibronic free energy surfaces can be obtained by diagonalization of the donor and acceptor blocks of the Hamiltonian in eq 1 in an electron-proton vibronic basis. The vibronic coupling between a pair of donor and acceptor vibronic states is the product of the constant electronic coupling VDA and the overlap integral between the corresponding donor and acceptor proton vibrational wave functions [i.e., the eigenfunctions of the harmonic potentials UD(rp) and UA(rp)]. Due to the relatively small vibronic coupling, the adiabatic and diabatic vibronic states are nearly identical everywhere except in the vicinity of an avoided crossing, and we can associate each adiabatic vibronic state with a diabatic vibronic state at virtually all values of the collective solvent coordinate. Figure 1 depicts snapshots of the ensemble of trajectories during the nonadiabatic dynamics on the adiabatic vibronic free energy surfaces. Movies depicting the full time-dependent dynamics are also provided. In this figure, the donor and acceptor vibronic surfaces appear as two sets of parabolas, where the donor surfaces are centered at x = 0 and the acceptor surfaces are centered at x=0.153 eV1/2. In models A and B, the system started with a nonequilibrium solvent distribution on the lowest-energy donor vibronic state. Model A exhibited an initial fast decay on the ground vibronic surface in ∼1 ps to the minimum of this surface, followed by slower nonadiabatic dynamics on the adiabatic vibronic surfaces, resulting in the majority of the population in the first-excited acceptor vibronic state after ∼40 ps. Model B exhibited a much faster relaxation process, with the majority of the population in the eighth and ninth acceptor vibronic states after less than 5 ps. In model C, the system started in a mixture of donor vibronic states and exhibited an initial fast decay on the
This equation was derived from a generalized Langevin equation with the Onodera model22 for dielectric relaxation of the solvent.7 Although the generalized Langevin equation with a more sophisticated solvent model could be utilized, this simple model is expected to capture the qualitative trends. In this case, the effective mass mx is calculated from the moment of inertia I of an individual solvent molecule, the temperature, and the dielectric constants of the solvent, and the parameter τ~L is calculated from the moment of inertia I, as well as the longitudinal relaxation time and dielectric constants of the solvent. The stochastic force F(t) is related to the friction f0~τL through the fluctuation-dissipation theorem. To simulate the nonadiabatic dynamics on multiple adiabatic vibronic surfaces following photoexcitation, we use Tully's fewest switches surface hopping method,23 also known as the molecular dynamics with quantum transitions (MDQT) method.24 In our implementation, each trajectory within an ensemble evolves according to eq 2 on a single adiabatic electron-proton vibronic free energy surface except for instantaneous transitions among the adiabatic vibronic states. In this model, proton vibrational relaxation cannot occur by a direct mechanism (i.e., within the donor or acceptor diabatic state) but can occur by an indirect mechanism via nonadiabatic transitions between the donor and acceptor states.21 Models that include the direct mechanism for proton vibrational relaxation, as well as more realistic proton potentials, will be the topic of future studies. We used this approach to study the isotope effect on the nonadiabatic dynamics for three different models denoted models A, B, and C. In all three models, the minima of the proton potentials for the donor and acceptor electronic states were at rpD = 0 and rpA = 0.5 Å. For hydrogen, mp = 1.0073 amu and ωp = 3000 cm-1, while for deuterium, mp = 2.0136 amu and ωp = 2121.8 cm-1. The solvent parameters were chosen to be those for water with reorganization energy λ = 0.65 eV.
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w Snapshots of the ensemble of trajectories during the nonadiabatic dynamics on the adiabatic vibronic free energy surfaces, which Figure 1. n depend on the collective solvent coordinate x, for (a) model A, (b) model B, and (c) model C. This figure corresponds to hydrogen transfer, and the analogous figure for deuterium transfer is given in the Supporting Information. The full movies for both hydrogen (Movies 1, 2, and 3) and deuterium (Movies 4, 5, and 6) transfer are also provided.
donor vibronic surfaces, followed by slower nonadiabatic dynamics with the majority of the population ending up in the first through third acceptor vibronic states. Figure 2 depicts the population decay of the donor state for hydrogen and deuterium for models A, B, and C. For all three models, the qualitative behavior of the nonadiabatic dynamics was similar for hydrogen and deuterium, but the isotope effect on the donor state population decay varied among the models. For model A, the donor state population decayed approximately three times slower for deuterium than for hydrogen. In contrast, for models B and C, the donor state population decay time scales were very similar for hydrogen and deuterium, with deuterium decaying slightly faster than hydrogen for model B and the reverse for model C. The physical basis for these isotope effects can be understood by the analysis of the hydrogen and deuterium vibronic couplings that impact the probability of nonadiabatic transitions. As depicted in Figure 3, initially both the hydrogen and deuterium vibronic couplings increase with increasing quantum number of the acceptor vibronic state because the overlap integral is dominated by the tails of the vibrational wave
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functions, and the excited state vibrational wave functions become progressively more delocalized. In this regime, the vibronic coupling is significantly greater for hydrogen than for deuterium because the hydrogen wave functions are more delocalized. For highly excited acceptor vibronic states, the overlap integral starts to decrease because of cancellation effects due to oscillations of the vibrational wave functions. In the relatively flat turnover regime, the vibronic couplings are similar for hydrogen and deuterium. Note that the energy splittings between vibronic states are smaller for deuterium than for hydrogen, so the excited acceptor vibronic states that are populated after photoexcitation and participate in the nonadiabatic dynamics will be shifted to higher quantum numbers for deuterium compared to hydrogen. This trend is illustrated in Figure 3 by the identification of the acceptor vibronic states that participate in the nonadiabatic dynamics for models A and B. Figure 4 depicts the time-dependent populations of the individual acceptor vibronic states for models A and B. For model A, the first and second excited acceptor vibronic states participate in the dynamics for hydrogen, and the ground
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Figure 4. (a) Hydrogen (left) and deuterim (right) diabatic vibronic state populations as functions of time for (a) model A and (b) model B. The lines are labeled D and A for donor and acceptor vibronic states, respectively, with 0 denoting the ground state.
in the dynamics for deuterium. As shown in Figure 3, the vibronic couplings between the ground donor vibronic state and the participating acceptor vibronic states are very similar for hydrogen and deuterium in this regime, leading to similar donor state population decay time scales for hydrogen and deuterium in this model. The slight inverse isotope effect, where the donor state population decay is slightly faster for deuterium than for hydrogen, arises from subtle differences in the detailed dynamics. For model C, the initial donor state population decay is similar for hydrogen and deuterium, but, subsequently, the population decay is slightly faster for hydrogen. This behavior can be explained using the concepts understood from models A and B. For the first picosecond, the population is distributed among higher vibronic states, where the vibronic couplings are similar for hydrogen and deuterium, so no isotope effect is observed. After the initial relaxation, the population is distributed among the lower vibronic states, where the vibronic couplings are significantly larger for hydrogen than for deuterium, so the population decay is faster for hydrogen. This model provides an explanation for the experimental observation of the onset of an isotope effect after an initial fast decay in the ultrafast dynamics of double-stranded DNA3,4 and methanol adsorbed on titanium dioxide surfaces.5,6 The calculations in this Letter illustrate that ultrafast PCET reactions could exhibit a significant normal hydrogen/deuterium isotope effect, where the donor state population decays faster for hydrogen, but could also exhibit a negligible isotope effect or even a slight inverse isotope effect. The isotope effect is predicted to be very small or absent when highly excited electron-proton vibronic states dictate the nonadiabatic dynamics and is predicted to increase with greater participation of the lower vibronic states. The participation of the lower vibronic states can be enhanced by decreasing the driving force of the reaction and minimizing the difference between the proton potential energy curves in the ground and donor electronic states. In some cases, an isotope effect may be
Figure 2. Population of the donor state as a function of time for hydrogen (red) and deuterium (blue) transfer for (a) model A, (b) model B, and (c) model C.
Figure 3. Square of the vibronic coupling between the ground donor vibronic state and the νth acceptor vibronic state as a function of the quantum number ν of the acceptor vibronic state for hydrogen (red) and deuterium (blue) transfer. The vibronic couplings that impact the nonadiabatic dynamics, as determined from analysis of Figure 4, are indicated with filled squares for model A and with filled triangles for model B.
through second excited acceptor vibronic states participate in the dynamics for deuterium. As shown in Figure 3, the squared vibronic couplings between the ground donor vibronic state and the participating acceptor vibronic states are significantly larger for hydrogen than for deuterium by ∼10 - 104 in this regime, leading to the faster donor state population decay for hydrogen. In contrast, for model B, the 8th through 11th excited acceptor vibronic states participate in the dynamics for hydrogen, while the 11th through 15th excited acceptor vibronic states participate
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observable only after an initial fast decay involving vibrational relaxation to the lower vibronic states. Thus, although the presence of a significant isotope effect strongly suggests that proton motion is coupled to electron transfer, the absence of an isotope effect does not exclude the possibility that proton transfer accompanies electron transfer in ultrafast photoinduced charge transfer processes.
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SUPPORTING INFORMATION AVAILABLE Simulation
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details and analogue of Figure 1 for deuterium transfer. This material is available free of charge via the Internet at http:// pubs.acs.org.
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AUTHOR INFORMATION (18)
Corresponding Author: *To whom correspondence should be addressed. E-mail: shs@ chem.psu.edu.
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ACKNOWLEDGMENT We gratefully acknowledge funding from
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AFOSR Grant FA9550-10-1-0081 and NSF Grant CHE-07-49646.
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REFERENCES (1)
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