Proton Quantization and Vibrational Relaxation in Nonadiabatic

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Proton Quantization and Vibrational Relaxation in Nonadiabatic Dynamics of Photoinduced Proton-Coupled Electron Transfer in a Solvated Phenol-Amine Complex Puja Goyal, Christine A. Schwerdtfeger, Alexander V. Soudackov, and Sharon Hammes-Schiffer J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.5b12015 • Publication Date (Web): 26 Jan 2016 Downloaded from http://pubs.acs.org on February 1, 2016

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The Journal of Physical Chemistry

Proton Quantization and Vibrational Relaxation in Nonadiabatic Dynamics of Photoinduced Proton-Coupled Electron Transfer in a Solvated Phenol-Amine Complex

Puja Goyal, Christine A. Schwerdtfeger, Alexander V. Soudackov, and Sharon Hammes-Schiffer* Department of Chemistry, 600 South Mathews Avenue, University of Illinois at UrbanaChampaign, Urbana, Illinois 61801

*corresponding author; e-mail: [email protected]; phone: 217-300-0335

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ACS Paragon Plus Environment


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Abstract Nonadiabatic dynamics simulations of photoinduced proton-coupled electron transfer (PCET) in a phenol-amine complex in solution were performed. The electronic potential energy surfaces were generated on-the-fly with a hybrid quantum mechanical/molecular mechanical approach that described the solute with a multiconfigurational method in a bath of explicit solvent molecules.

The transferring hydrogen nucleus was represented as a quantum mechanical

wavefunction calculated with grid-based methods, and surface hopping trajectories were propagated on the adiabatic electron-proton vibronic surfaces. Following photoexcitation to the excited S1 electronic state, the overall decay to the ground vibronic state was found to be comprised of relatively fast decay from a lower proton vibrational state of S1 to a highly excited proton vibrational state of the ground S0 electronic state, followed by vibrational relaxation within the S0 state. Proton transfer could occur either on the highly excited proton vibrational states of S0 due to small environmental fluctuations that shift the delocalized vibrational wavefunctions or on the low-energy proton vibrational states of S1 due to solvent reorganization that alters the asymmetry of the proton potential and reduces the proton transfer barrier. The isotope effect arising from replacing the transferring hydrogen with deuterium is predicted to be negligible because hydrogen and deuterium behave similarly in both types of proton transfer processes. Although an isotope effect could be observed for other systems, in general the absence of an isotope effect does not imply the absence of proton transfer in photoinduced PCET systems. This computational approach is applicable to a wide range of other photoinduced PCET processes.

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1. Introduction Proton-coupled electron transfer (PCET)1-4 is a vital component of a variety of energy conversion processes, including photosynthesis and respiration. Photoinduced PCET in photosynthesis leads to a series of steps that convert sunlight into chemical bond energy. Artificial solar energy conversion devices utilize the principles of natural photosynthesis to generate renewable fuel.5-8 Hence, an understanding of the fundamental physical principles that govern photoinduced PCET is important for the design of renewable and sustainable energy sources. The study of relatively simple model systems exhibiting photoinduced PCET provides a fundamental understanding of the various factors that dictate the nonequilibrium dynamics following photoexcitation.9-11 Transient absorption and coherent Raman spectroscopic experiments indicated the occurrence of photoinduced PCET in a hydrogen-bonded pnitrophenylphenol-t-butylamine complex in 1,2-dichloroethane solution (Figure 1).12 Previously the electronic states of this system were characterized with time-dependent density functional theory (TDDFT)13-14 and the complete active space self-consistent field and second-order perturbation theory (CASSCF and CASPT2) methods.15 In our studies, the solvent effects were included with either a dielectric continuum model13 or with explicit solvent molecules using a hybrid quantum mechanical/molecular mechanical (QM/MM) approach.15

Both excited

electronic states were identified as intramolecular charge transfer (ICT) states, indicating that photoexcitation is accompanied by a significant change in the dipole moment of the hydrogenbonded complex. Moreover, the first excited state was determined to favor proton transfer in solution on the basis of equilibrium free energy simulations and thus was identified as an electron-proton transfer (EPT) state.15

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In addition, we studied the nonadiabatic dynamics of this system by performing on-thefly surface hopping simulations using a semiempirical implementation of the floating occupation molecular orbital complete active space configuration interaction (FOMO-CASCI) method16-20 for the hydrogen-bonded complex surrounded by explicit solvent molecules within a QM/MM framework.15,21 These simulations demonstrated that this hydrogen-bonded complex exhibits proton transfer (PT) upon photoexcitation to the EPT state and provided insight into the relaxation pathways of the system. Subsequent analysis of these simulations highlighted the role of solvent dynamics in facilitating PT in the EPT state as well as relaxation from the EPT state to the ground state.22

In these previous nonadiabatic dynamics simulations,15 the transferring

proton was treated classically, and zero point energy was added to the solute by sampling the initial coordinates and velocities of the solute atoms from quantum harmonic oscillator distributions corresponding to the ground state normal modes of the solute. Explicit PT in the EPT state was observed in ~54% of the surface hopping trajectories. Most of these trajectories decayed to the ground electronic state at a configuration in which the proton was on the amine instead of on the phenol, and the proton transferred back to the phenol on the ground state.15,22 In the present paper, we investigate the impact of a quantum mechanical treatment of the transferring hydrogen nucleus on the nonadiabatic dynamics of this system by performing surface hopping simulations on electron-proton vibronic surfaces rather than on electronic surfaces.23-25 The electronic potential energy surfaces are still generated on-the-fly with the semiempirical implementation of FOMO-CASCI, but the transferring proton is represented by a one-dimensional quantum mechanical wavefunction. The dynamics of the system is monitored as it relaxes from an initial excited electron-proton vibronic state to the ground vibronic state. Comparison to available experimental data and our previous simulations with a classical proton

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provides insights into the significance of nuclear quantum effects, as well as the role of vibrational relaxation in this photoinduced process. These simulations also elucidate the nature of PT on the electron-proton vibronic surfaces and predict the isotope effect observed after replacing the transferring hydrogen with deuterium.26 An outline of the paper is as follows. Section 2 summarizes the theory underlying the calculation of electron-proton vibronic states for photoinduced PCET processes and describes the computational methodology used herein. Section 3 presents the results of the simulations, starting with the time evolution of the relaxation process on the electron-proton vibronic surfaces, followed by the analysis of PT and the hydrogen/deuterium isotope effect. Section 4 summarizes the new insights and provides concluding remarks.

2. Methods This section presents the computational methods developed and implemented for the onthe-fly QM/MM nonadiabatic dynamics simulations on electron-proton vibronic surfaces. The first subsection describes the method used to calculate the vibronic states, the forces on the classical nuclei, and the nonadiabatic couplings between all pairs of vibronic states. The second subsection provides the details of the generation of the electronic potential energy surfaces and the initial conditions chosen for the surface hopping trajectories. The last subsection describes the methodology used for the nonadiabatic dynamics surface hopping trajectories. All calculations were carried out using a modified version of MOPAC 2000 version 1.0.027 interfaced with AMBER 6.28

2.1. Theoretical framework for electron-proton vibronic states

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The quantum mechanical treatment of the phenolic proton in this study is based on the assumption that the proton transfer from phenol to ammonia occurs at essentially linear configurations of the O-H--N hydrogen-bonded interface, as observed in previous classical molecular dynamics simulations.15 Thus, the proton vibrational wavefunctions are calculated on a one-dimensional grid along the ON axis. To calculate these proton vibrational wavefunctions, the proton potential for each electronic state is constructed as a function of a one-dimensional proton coordinate rp defined relative to the midpoint between the proton donor and acceptor The relation between this one-dimensional proton coordinate rp and the three-

atoms.

dimensional position vectors of the proton donor and acceptor atoms, denoted by RD and RA , respectively, and the quantum mechanical proton, denoted by rH , is:

rH 

RD  RA R  rp DA R DA 2

,

(1)

where RDA  RD  RA . Within this framework, the vibronic Hamiltonian Hˆ tot is expressed as a sum of the electronic Hamiltonian Hˆ el and the proton kinetic energy:

2  2 tot el ˆ ˆ H (re , rp , R)  H (re , rp , R)  , 2mp rp2

(2)

where the electronic Hamiltonian is given by Ne 2 2 Hˆ el (re , rp , R)    i  V (re , rp , R) . i 1 2me

(3)

Here re denotes the electronic coordinates, R denotes the coordinates of all the classical nuclei,

mp is the mass of a proton, me is the mass of an electron, Ne is the number of electrons, and V is the total potential energy of the system.

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The electronic wavefunctions  i (re ; rp , R ) and energies i (rp , R) are obtained by the solution of the time-independent electronic Schrödinger equation given as Hˆ el (re , rp , R ) i (re ; rp , R )  i ( rp , R ) i (re ; rp , R ) .

(4)

Eq. (4) is solved for a series of grid points spanning the ON axis to obtain the proton potential

i (rp , R) for each electronic state i at a fixed classical nuclear configuration R. Subsequently, the proton vibrational wavefunctions  ( i ) ( rp ; R ) for each electronic state i are calculated by solving the following one-dimensional Schrödinger equation using the Fourier Grid Hamiltonian (FGH) method:23

 2  2   i (rp , R )   (i ) (rp ; R )   (i ) (R )  (i ) (rp ; R ) .  2  2mp rp 

(5)

The adiabatic electron-proton vibronic wavefunctions are calculated using a double adiabatic basis set defined as products of an electronic wavefunction and an associated proton vibrational wavefunction. The double adiabatic wavefunction i (re , rp ; R) associated with the ith electronic state and the μth associated proton vibrational state is defined as

i (re , rp ; R)  i (re ; rp , R)(i ) (rp ; R) .

(6)

The adiabatic electron-proton vibronic wavefunctions  k (re , rp ; R) are expanded in this basis,

 k (re , rp ; R)   cik  i (re ; rp , R )  (i ) (rp ; R) ,

(7)

i

and the coefficients are obtained by solving the following Schrödinger equation:24-25 Hˆ tot (re , rp , R ) k (re , rp ; R )  Ek (R ) k (re , rp ; R ) .

(8)

As established previously,25 the matrix elements of Hˆ tot in the basis of these double adiabatic states are given by 7



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(i ) H itot  , j   ij    

(ep)

 ( j ) 2  (i ) dij(ep)  mp rp

 p

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2  (i ) gij(ep) ( j ) 2mp

p

.

(9)

(ep)

Here dij and gij are the first-order and second-order nonadiabatic couplings, respectively, between electronic states i and j along the proton coordinate: d ij(ep) ( rp ; R )   i

g

(ep) ij

( rp ; R )   i

 j

(10)

rp

e

 2 j

(11)

.

rp2

e

The notations ... e and ... p denote integration over electronic coordinates re and the proton coordinate rp , respectively. Equation (9) can be rewritten as25

 (i )  2  (i ) (ep) ( j )    dij  ( j ) d (ep) ji rp rp 2mp  p  2  (i ) d ki(ep) d kj(ep) ( j ) p ,   2mp k

(i ) H itot  , j   ij    

   p

(12)

where the summation in the last term is over all electronic states k. The first-order nonadiabatic coupling dij(ep) is calculated as follows:

dij(ep)   i  i

 j rp  j xH

e

 j xH  i rp yH e

e

 j yH  i rp zH

zH . rp e

Here the partial derivatives of the components of rH  ( xH , yH , zH ) are given by

(13)

xH xA  xD  , rp R DA

where xA and xD are the x coordinates of the acceptor and donor atoms, respectively. The other partial derivatives in Eq. (13) are calculated analogously. 8


The matrix elements

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 (i ) dij(ep)

( j ) rp

in Eq. (12) are calculated numerically using Fourier grid methods, as p

described in the Supporting Information (SI). The Hellman-Feynman force along a classical coordinate R moving on a state defined by the adiabatic electron-proton vibronic wavefunction  k is given by

FkRλ    k

Hˆ tot k R

ep

  cik c kj  i  ( i ) i

j

Hˆ el  j ( j ) R

(14) ep

   cik c kj (1   ij )  ( i ) dijRλ ( j  i ) ( j )  i  j

p

  ij  (i )

i ( j )  R

 .  p

Here d ijRλ is the electronic nonadiabatic coupling between the pair of electronic states i and j along a classical coordinate R (i.e., the analog of Eq. (10) with rp replaced by R ). In contrast, the vibronic nonadiabatic coupling along a classical coordinate R between the pair of adiabatic vibronic states k and l can be calculated analytically as follows: k DklRλ 

Hˆ tot l R

ep

El  Ek

 cik clj  i  (i ) 

i

j

Hˆ el  j ( j ) R

ep

 Rλ k l (i ) ( j)  c c  i  j (1   ij )   d ij ( j  i )   i  j  El  Ek

9

(15)

El  Ek


p

  ij  (i )

i ( j )  R

   p .


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Note that i R and d ijRλ will include additional terms when R corresponds to a coordinate of the proton donor or acceptor atom due to contributions from derivatives with respect to the proton coordinate defined in Eq. (1).29 The expressions for these additional terms are provided in the SI. As will be discussed below, the full 3N-dimensional vibronic nonadiabatic coupling vector Dkl, where N is the number of classical nuclei, is calculated analytically for the adjustment of velocities following a nonadiabatic transition.30-31 Otherwise the scalar product of the vibronic

 , is calculated numerically using nonadiabatic coupling vector and the classical velocity, Dkl  R the norm-preserving interpolation (NPI) scheme.32

2.2. Electronic potential energy surfaces and initial conditions The ground and first two excited state electronic surfaces are generated on-the-fly using a QM/MM framework in which the hydrogen-bonded complex is treated quantum mechanically, while the surrounding sphere of explicit solvent molecules is treated molecular mechanically. The solute is described with a semiempirical implementation of FOMO-CASCI using the PM3 method reparameterized previously for this system.15 The explicit solvent is described with the general AMBER force field (GAFF).33-34 The interactions between the QM and MM regions are described through an electrostatic embedding approach. Soft restraints are applied to the outer shell of solvent to prevent evaporation. In this QM/MM treatment, the electronic wavefunctions

 i (re ; rp , R ) in Eq. (4) are polarized by the point charges in the MM region, and the interaction energy between the QM and MM regions, as well as the energy of the MM region, contribute to the associated energies i (rp , R) . Zero point energy is added to the solute by sampling the initial coordinates and velocities of the solute atoms from quantum harmonic oscillator distributions corresponding to the ground 10


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vibrational levels of the normal modes on the S0 state in the gas phase. Because of the quantum mechanical treatment of the phenolic proton, any contribution of this proton to the normal modes is eliminated by setting the Hessian matrix elements involving this proton to zero. Moreover, zero point energy is not added to the six normal modes with frequencies lower than 100 cm-1 to avoid initial solute geometries with very large deviations from the optimized geometry on the S0 state. A 5 ps QM/MM equilibration of the solvent is performed with the solute frozen at the geometry sampled from the quantum harmonic oscillator distributions. In this frozen solute geometry, the phenolic proton is placed on the ON axis at a distance of 1.0 Å from the O atom. After solvent equilibration, the system is instantaneously photoexcited to either the S1 or the S2 electronic state. The occupied vibronic state following photoexcitation is determined by the following procedure. According to the Franck-Condon principle, the proton vibrational wavefunction does not change during the electronic excitation. Thus, the proton vibrational wavefunction immediately following photoexcitation corresponds to the ground vibrational state of the ground electronic state: h  0 0(0)    i 0(0) ; i  1, 2 .

(16)

The photoexcited state can be expressed as a linear combination of the electron-proton vibronic states  k :

 i  0(0)    k  k  i  0(0) k

ep

  k

k j

k

  k k

The square of the expansion coefficient

 c 

c     k i

(i )

j

( j )  i  0(0)

j

c  

k i

0(0)

  (i )

ep

(17)

(0) 0 p

represents the initial occupation p

probability of the kth adiabatic vibronic state. The adiabatic vibronic state occupied immediately 11



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following photoexcitation is chosen based on these occupation probabilities, and surface hopping trajectories are propagated on the adiabatic vibronic states. Most trajectories start on the ground proton vibrational state of the S1 or S2 electronic states, as depicted in Figure 2 for the S1 state. The initial solvent velocities are sampled from a Maxwell-Boltzmann distribution at 296 K.

2.3. MDQT simulations on electron-proton vibronic surfaces This section presents the computational details for the MDQT simulations on electronproton vibronic surfaces. The electron-proton vibronic wavefunctions and energies, as well as the corresponding nuclear forces and nonadiabatic couplings, are calculated on-the-fly during the MDQT trajectories. At each configuration of the classical nuclei, R, QM/MM calculations are performed with the transferring proton placed sequentially at 24 grid points along the ON axis. The origin rp  0 is chosen to be the midpoint between O and N, and the grid spacing is

( RON  1.1Å) / 24 , where RON is the distance between O and N. These calculations yield the energies i (rp , R) for each electronic state i by solving Eq. (4) for each value of the proton coordinate rp on the grid. In addition, the energy gradient vector for each electronic state, as well as the electronic nonadiabatic coupling vector between each pair of electronic states, is obtained at each grid point. Subsequently, cubic spline interpolation is performed to obtain the values of these quantities on a finer grid comprised of 64 points. The energy values on the fine grid are used to construct the proton potential energy curve for each electronic state. Subsequently, the proton vibrational wavefunctions  ( i ) ( rp ; R ) corresponding to the proton potential are calculated for each electronic state on the fine grid by solving Eq. (5) using the FGH method.23 For each of the three electronic states S0, S1 and S2, the

12


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20 or 28 lowest energy proton vibrational states are calculated for hydrogen or deuterium, respectively. The double adiabatic states i (re , rp ; R) are constructed as products of each electronic state  i (re ; rp , R ) with the associated proton vibrational wavefunctions  ( i ) ( rp ; R ) , as given in Eq. (6). These double adiabatic states form the basis set for construction of the full vibronic Hamiltonian matrix with matrix elements given by Eq. (9).24-25 These matrix elements depend on the first-order and second-order nonadiabatic couplings along the one-dimensional proton coordinate and are calculated using Eq. (12) and Eq. (13). Diagonalization of this Hamiltonian matrix to solve Eq. (8) yields the adiabatic vibronic surfaces Ek (R) on which the surface hopping trajectories are propagated.31 In the MDQT trajectories, restraints on the O-N-HN angle (HN denotes any of the three hydrogen atoms bonded to the N atom of ammonia) and the ON distance are applied to prevent self-consistent field (SCF) errors due to unconventional geometries in the electronic structure calculations for certain positions of the transferring proton and to prevent dissociation of the hydrogen-bonded complex. The restraining potentials are defined as follows. For the ON distance, V ( RON )  0 

, RON  4 Å

1 eq 2 kON ( RON  RON ) , RON  4 Å , 2

(18)

and for the O-N-HN angle,

V ( O-N-H N )  0 

, cos( O-N-H N )  0

1 eq kO-N-H N [cos( O-N-H N )  cos( O-N-H )]2 , cos( O-N-H N )  0. N 2

(19)

eq , k O-N-H N and O-N-HN are assigned values of 800 kcal mol-1 Å-2, 4 Å, The parameters k ON , RON

eq

2000 kcal mol-1 and 90°, respectively. These values allow sampling of ON distances up to ~4.2 13



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Å and O-N-HN angles larger than ~82°. These restraints are consistent with the treatment of the quantum mechanical proton in one dimension along the ON axis, while also enabling sampling of the conformational space in which PT from O to N is possible. However, these restraints prevent the breaking of the hydrogen bond between the phenol and the amine and therefore could lead to slightly faster computed relaxation rates than those observed experimentally. In our implementation of the MDQT approach, an ensemble of trajectories is propagated on the adiabatic vibronic surfaces Ek (R) with instantaneous nonadiabatic transitions incorporated according to Tully’s fewest switches surface hopping algorithm.30-31 The timedependent electron-proton vibronic wavefunction is expanded in the basis of adiabatic vibronic states:

 (re , rp , R , t )   Ck (t )  k (re , rp ; R ) .

(20)

k

The quantum amplitudes Ck are determined by integration of the time-dependent Schrödinger equation (TDSE) in parallel with integration of the classical equations of motion:  . i C k   Cl  E k  kl  i D kl  R

(21)

l

The fewest switches surface hopping algorithm is designed to ensure that the population on a 2

given adiabatic vibronic state k at time t is equal to the quantum probability C k , neglecting the deviations due to classically forbidden transitions. After a nonadiabatic transition, the classical velocities are adjusted along the nonadiabatic coupling vector to conserve energy. In the event of a forbidden transition, the velocities remain unchanged. The classical coordinates are propagated using the velocity Verlet algorithm and a time step of 0.1 fs. The forces along the classical coordinates are calculated with Eq. (14), which depends on the energies of the electronic states, the gradients of the electronic states with respect 14


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to the classical coordinates, and the electronic nonadiabatic couplings between all pairs of electronic states along the classical coordinates. The matrix elements between the proton vibrational wavefunctions are calculated numerically using the proton vibrational wavefunctions on the 64-point grid. The integration of the TDSE in Eq. (21) is performed using a smaller time step of 0.0002 fs and a fourth-order Runge-Kutta scheme. The TDSE integration requires knowledge of the

 , for all pairs of scalar product of the vibronic nonadiabatic coupling and velocity vectors, Dkl  R adiabatic vibronic states k and l. These scalar quantities are obtained using the NPI scheme, which allows the use of a computationally affordable classical time step of 0.1 fs and avoids the expensive analytical calculation of the vibronic nonadiabatic coupling vector, with elements given in Eq. (15), when a nonadiabatic transition does not occur.32 The vibronic nonadiabatic coupling from the NPI scheme at the midpoint of a classical time step is kept constant over the classical time step. The TDSE integration also requires the energies of the adiabatic vibronic states, which are linearly interpolated over the classical time step. After a nonadiabatic transition, the nonadiabatic coupling vector is calculated analytically for the adjustment of the classical velocities required to maintain energy conservation.31 The vibronic nonadiabatic coupling vector is calculated with Eq. (15), which depends on the energies of the electronic states, the gradients of the electronic states along the classical coordinates, and the electronic nonadiabatic couplings between all pairs of electronic states along the classical coordinates. The classical and quantum time steps are reduced by a factor of 400 for 20 steps after a nonadiabatic transition to ensure stable propagation of the trajectory. We propagated ~80 trajectories after photoexcitation to either the S1 or the S2 state. A trajectory is assumed to be complete after it remains on the ground vibronic state for 100 fs

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without any nonadiabatic transitions. Decoherence effects35-38 are not expected to play a significant role for these ultrafast decay processes.

3. Results and discussions 3.1. Time evolution of the electronic and vibronic state populations Analysis of the time evolution of the adiabatic vibronic state populations is challenging because these states change character often during each trajectory. However, the ground adiabatic vibronic state always maintains its character of being the ground proton vibrational state of the S0 electronic state. Figure 3 depicts the time evolution of the population of the ground adiabatic vibronic state after photoexcitation to the S1 or S2 electronic state. The population of this state increases on a timescale of ~0.6 ps, which is similar to the timescale of ~0.9 ps observed for the increase of the ground electronic state population in our previous work that treated the transferring proton classically.15

This similarity indicates that the quantum

mechanical treatment of the proton does not significantly affect the timescale of overall relaxation to the ground state following photoexcitation. To obtain additional insight into the relaxation pathways, we identified the dominant double adiabatic vibronic state for the occupied adiabatic vibronic state at each time step along all of the trajectories. The time evolution of the double adiabatic vibronic state populations is provided in Figure S1 of the SI. The population of each electronic state S0, S1, and S2 was determined by summing over the populations of the double adiabatic vibronic states associated with each of these electronic states at each time step (i.e., summing over all of the proton vibrational states for each electronic state). The time evolution of the populations of the electronic states is depicted in Figure 4. After initial photoexcitation to the S1 state, the

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population of the S1 state decays to the S0 state on a timescale of ~0.2 ps, which is faster than the calculated timescale of ~0.9 ps observed when the transferring proton is treated classically with zero point energy included in the initial conditions.15 This faster decay is observed when the proton is treated quantum mechanically because the higher-energy excited proton vibrational states of the S0 electronic state are very close in energy to the lower-energy proton vibrational states of the S1 state, thereby providing many more pathways for relaxation to the S0 state. This physical phenomenon is depicted in Figures 5 and 6. Although the surface hopping trajectories are propagated on the adiabatic vibronic states, analysis of the double adiabatic vibronic states provides useful insights.

In Figure 5, the energies of the double adiabatic

vibronic states are depicted as a function of time for a representative trajectory. The solid lines are associated with the S0 state, and the dashed lines are associated with the S1 state. The intersections between the ground proton vibrational state of the S1 electronic state and the 14th and 15th excited proton vibrational states of the S0 electronic state are identified with vertical dotted lines. These intersections between double adiabatic vibronic states correspond to avoided crossings between adiabatic vibronic states, thereby allowing the decay from the S1 to the S0 state. The ground proton vibrational wavefunction in the S1 state and the closely lying excited proton vibrational wavefunctions in the S0 state are depicted in Figure 6. The green arrow indicates the decay from the ground proton vibrational state of S1 to an excited proton vibrational state of S0, followed by proton vibrational relaxation within the S0 electronic state on a slower timescale. After photoexcitation to the S2 state, the population of the S2 state decays within ~100 fs (Figure 4b), similar to the observation in our previous simulations treating the transferring proton classically.15 However, again due to the accessibility of excited proton vibrational states of the S0

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state, the S0 state population increases faster when the transferring proton is treated quantum mechanically. Moreover, the increase of the S0 state population is found to be faster after photoexcitation to the S2 state than after photoexcitation to the S1 state. Specifically, the S0 state population increases on a timescale of ~206 fs following photoexcitation to the S1 state and ~114 fs following photoexcitation to the S2 state (Figure 4). One explanation for this observation is that the higher-energy proton vibrational states of S0 are also very close in energy to the lowerenergy proton vibrational states of S2, thereby allowing direct transitions from the S2 state to the S0 state. These additional decay pathways to the S0 state lead to a faster increase in the population of the S0 state. Another possible contributing factor is that the velocities on the S1 state may be larger for trajectories initially photoexcited to the S2 state compared to those initially photoexcited to the S1 state. Experimentally, a transient absorption band at 395 nm assigned to the S1 state was observed to decay on a timescale of ~4.5 ps.12 In our previous classical proton simulations,15 both the decay of the S1 state and the rise of the S0 state occurred on a timescale of ~0.9 ps and were interpreted as corresponding to the decay of the transient absorption band at 395 nm. In the quantum proton simulations, however, the S1 state decays on a faster timescale of ~0.2 ps. In the experimental study, the pump pulse used for the S0 to S1 excitation had a wavelength of 388 nm. According to our analysis, the system could exhibit significant absorption at 395 nm because of excitation from excited vibronic states associated with the S0 electronic state to vibronic states associated with the S1 electronic state (or to higher electronic states). Thus, the transient absorption band at 395 nm may persist as long as the ground vibronic state is not occupied, given that the experimental transient absorption excludes equilibrium absorption from the ground vibronic state. Within this interpretation, the experimental timescale of ~4.5 ps for the decay of

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the transient absorption band at 395 nm corresponds to the computed timescale of ~0.6 ps for the rise of the ground vibronic state in the quantum proton simulations.

3.2. PT on the vibronic states and the role of solvent dynamics In the classical proton simulations, PT was defined to occur when the NH distance became less than 1.15 Å.15 In the quantum proton simulations, based on the ON distances sampled on the S0 and S1 states, we define PT to occur when the expectation value of the proton coordinate, rp , is greater than 0.40 Å on the S0 state or greater than 0.25 Å on the S1 state (see Figure S2 and caption for determination of these criteria). For the trajectories initially photoexcited to the S1 state, ~33% and ~28% exhibit PT on the S0 and S1 states, respectively. For the trajectories initially photoexcited to the S2 state, ~70% and ~27% exhibit PT on the S0 and S1 states, respectively. More trajectories exhibit PT on the S0 state following photoexcitation to the S2 state because higher-energy proton vibrational states of S0 are more populated in this case. In our previous classical proton simulations, forward PT from O to N was observed to occur only on the S1 state, and solvent reorganization was found to play an important role in facilitating PT.22 Figures S3-S5 in the SI illustrate the changes in solvent electrostatic potential at the amine N atom, the PT reaction energy, and the PT barrier for the dominant double adiabatic states. These figures indicate that PT on the ground proton vibrational state of the S1 electronic state (blue proton vibrational wavefunction in Figure 6) is accompanied by significant solvent reorganization, as well as a decrease in both the PT reaction energy and the PT barrier. To a lesser extent, PT on the first excited proton vibrational state of the S1 electronic state also exhibits similar behavior. In contrast, PT on the vibronic states associated with S0 does not involve any significant solvent reorganization or change in PT reaction energy or barrier. 19



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Further analysis of the proton potentials provides additional insight into the nature of PT on the S0 and S1 states. Figure 7 depicts the type of PT that can occur on the S0 state. Following photoexcitation to the ground proton vibrational state of the S1 electronic state, the system decays to the 12th proton vibrational state of the S0 electronic state, where rp ≈ 0. Subsequently, small solute and solvent fluctuations can cause the delocalized proton vibrational wavefunction to become more biased toward the acceptor N atom, with rp ≈ 0.40, thus apparently leading to PT. Figure 8 depicts the type of PT that can occur on the S1 state. After photoexcitation to the ground proton vibrational state of the S1 electronic state, the minimum of the proton potential is near the O atom (i.e., the donor well is lower than the acceptor well), and the proton vibrational wavefunction is localized near the O atom. Subsequently, solvent reorganization alters the proton potential until the minimum is near the N atom (i.e., the acceptor well is lower), so the proton vibrational wavefunction becomes localized near the N atom. When the system is in the first excited proton vibrational state of the S1 electronic state, PT can also occur due to changes in the shape of the proton potential (i.e., when both the ground and first excited proton vibrational wavefunctions are localized on the acceptor side or when the ground proton vibrational wavefunction is localized on the donor side but the first excited proton vibrational wavefunction is localized on the acceptor side). Overall, solvent dynamics plays an important role in PT on the S1 state, as observed in our previous classical proton simulations.22 As determined in this previous study, the solvent dynamics that results in the reorganization required to facilitate PT on the S1 state occurs in ~240 fs. As depicted in Figure 8, PT does not occur instantaneously upon photoexcitation to the S1 state but rather occurs only after solvent reorganization leads to a proton potential with the acceptor well almost isoenergetic with or lower than the donor well and a reduced proton transfer 20


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barrier. In other words, the solvent reorganization alters the asymmetry of the proton potential. The type of PT that occurs on the S0 state is less well-defined in that it occurs in the highly excited proton vibrational states and does not require significant solvent reorganization but rather arises from small environmental fluctuations that shift the delocalized proton vibrational wavefunction.

3.3. Hydrogen/deuterium isotope effect on decay rates We repeated all of the simulations replacing the transferring hydrogen with deuterium, leading to smaller splittings between the vibronic energy levels. Figure 9 indicates that the time evolution of the population of the ground adiabatic vibronic state is very similar for the hydrogen and deuterium isotopes. The negligible isotope effect implies that the transferring proton is not tunneling from one localized state to another. Instead, as discussed in Section 3.2, PT occurs either on the highly excited proton vibrational states of S0 due to small environmental fluctuations or on the ground or first excited proton vibrational states of S1 due to significant solvent reorganization that alters the asymmetry of the proton potential. In both types of PT, hydrogen and deuterium are expected to behave similarly, resulting in the absence of a significant isotope effect. As emphasized in our previous study of photoinduced PCET in a model system,26 the absence of an isotope effect does not imply the absence of PT. We analyzed PT for deuterium using the same criteria used for the analysis of PT for hydrogen. For trajectories initially photoexcited to the S1 state, ~59% and ~16% exhibit PT on the S0 and S1 states, respectively, for the deuterium isotope. For trajectories photoexcited to the S2 state, ~75% and ~30% exhibit PT on the S0 and S1 states, respectively. Thus, more PT is observed on the S0 state and less PT is observed on the S1 state (especially after photoexcitation

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to the S1 state) for deuterium than for hydrogen. These differences are due mainly to the faster decay from the excited electronic states to the S0 electronic state for deuterium than for hydrogen. Figure 10 depicts the time evolution of the populations of the electronic states following photoexcitation. A comparison of Figures 4 and 10 illustrates that the decay from the S1 state to the S0 state is faster for deuterium than for hydrogen, although the overall rise of the ground vibronic state population occurs on a similar timescale for the two isotopes. The faster decay to the excited proton vibrational states of the S0 state for deuterium is most likely due to the smaller splittings between the vibrational energy levels, resulting in larger nonadiabatic couplings between the adiabatic vibronic states. However, the subsequent vibrational relaxation within the S0 state down to the ground vibronic state occurs on a similar or slightly longer timescale for deuterium because of a somewhat larger number of vibrational states involved in the relaxation process. Following photoexcitation to the S2 electronic state, the population decay from the S2 state to the S1 state is faster for deuterium than for hydrogen, presumably due to the smaller energy level splittings that increase the nonadiabatic couplings between adiabatic vibronic states for deuterium. Moreover, the percentage of trajectories exhibiting PT on the S1 state after photoexcitation to the S2 state is slightly higher for deuterium. The physical basis for these observations is that the decay of the S2 state leads to greater populations of the higher and more delocalized proton vibrational states of S1 because of the smaller splittings between vibrational energy levels. Analogous to the manner in which PT occurs on the highly excited proton vibrational states of S0, occupation of the highly excited proton vibrational states of S1 can also result in PT. Although experimentally measured hydrogen/deuterium isotope effects are not

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available for this system, our nonadiabatic dynamics calculations predict that a significant isotope effect will not be observed in the overall decay to the ground vibronic state.

4. Conclusions We have developed a general approach for simulating the nonadiabatic dynamics of photoinduced PCET processes in solution. In this approach, the electronic potential energy surfaces are generated on-the-fly using a QM/MM method that describes the solute with a semiempirical implementation of the FOMO-CASCI method and the explicit solvent molecules with a molecular mechanical potential. To include the most significant nuclear quantum effects, the transferring hydrogen nucleus is represented as a one-dimensional quantum mechanical wavefunction that is calculated along the proton donor-acceptor axis in the environment of the other nuclei. The extension to a three-dimensional grid is conceptually straightforward39-40 although computationally much more expensive. The adiabatic electron-proton vibronic states are linear combinations of double adiabatic vibronic states, which are products of electronic states and associated proton vibrational states.

Following photoexcitation to an excited

electronic state, an ensemble of nonadiabatic surface hopping trajectories is propagated on the adiabatic electron-proton vibronic surfaces to simulate relaxation down to the ground vibronic state. The application of this nonadiabatic dynamics approach to a phenol-amine complex in solution provides physical insights into the relaxation process following photoexcitation. After photoexcitation to the S1 electronic state, the system decays to the S0 electronic state on the relatively fast timescale of ~0.2 ps because the higher-energy excited proton vibrational states of the S0 electronic state are close in energy to the lower-energy proton vibrational states of the S1

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state. The decay from a low-energy proton vibrational state of S1 to a highly excited proton vibrational state of S0 is followed by proton vibrational relaxation within the S0 electronic state. The overall decay to the ground vibronic state occurs on the longer timescale of ~0.6 ps, which includes both the decay from the S1 to the S0 electronic state, as well as vibrational relaxation within the S0 state. Photoexcitation to the S2 electronic state is followed by decay to S1 or to an excited proton vibrational state of S0 within ~100 fs. According to this analysis, the experimentally measured timescale of 4.5 ps for the decay of the transient absorption band at 395 nm, which was originally interpreted as the decay from the S1 to the S0 electronic state, corresponds to the computed timescale of ~0.6 ps for the decay to the ground vibronic state and therefore includes vibrational relaxation within the S0 electronic state. The computed timescale is faster than the experimentally measured timescale. A possible explanation for this discrepancy is that the simulations impose restraints on the phenol-amine complex to prevent dissociation and to maintain a strong hydrogen-bonding interaction with the proton remaining on the proton donor-acceptor axis.

In addition, limitations of the

semiempirical FOMO-CASCI method and the conformational sampling of initial conditions, as well as a relatively small number of surface hopping trajectories, reduce the quantitative accuracy of the calculations, although the qualitative insights are still expected to be meaningful. The simulations also address the key question as to whether PT occurs in this system, as suggested by the experimental data12 and previous nonadiabatic dynamics simulations that treated the transferring proton classically.15 When the proton is treated quantum mechanically, PT is not as easily identifiable but can be defined in terms of the expectation value of the proton coordinate. Two different types of PT were observed in the simulations following photoexcitation to the ground proton vibrational state of the S1 electronic state. At the instant of

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photoexcitation, the S1 electronic state is similar to the S0 electronic state in that the donor well of the proton potential is lower than the acceptor well. If the system decays to a highly excited proton vibrational state of the S0 electronic state, PT can occur on this vibronic state due to small environmental fluctuations that shift the delocalized proton vibrational wavefunction. Alternatively, PT can occur on the ground proton vibrational state of the S1 electronic state after solvent reorganization leads to a proton potential with the acceptor well almost isoenergetic with or lower than the donor well and a reduced proton transfer barrier.

Previous simulations

indicated that the solvent reorganization required to facilitate PT on the S1 state occurs in ~200300 fs. Solvent reorganization on a timescale of ~120 fs was also found to play a significant role for PT in experimental studies of photoinduced PCET in a nanocage.41-42 These simulations predict that the deuterium isotope effect on the decay time will be negligible. The physical explanation for the lack of an isotope effect is that PT occurs either on the highly excited proton vibrational states of S0 due to small environmental fluctuations or on the ground or first excited proton vibrational states of S1 due to solvent reorganization that alters the asymmetry of the proton potential. Neither of these processes corresponds to tunneling of the proton from one localized state to another, and hydrogen and deuterium are expected to behave similarly in both processes. Thus, although the initial relaxation from S1 to S0 is slightly faster for deuterium because of the smaller splittings between the vibrational energy levels, the overall decay to the ground vibronic state occurs on similar timescales for hydrogen and deuterium. However, an isotope effect may be observed for other photoinduced PCET systems, depending on the relative timescales of the solvent dynamics and vibrational relaxation, as well as the specific characteristics of the potential energy surface. In general, the absence of an isotope effect does not imply the absence of PT for photoinduced PCET processes.26 This computational

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approach is applicable to a wide range of other photoinduced PCET processes and can be utilized to further explore these fundamental issues. Notes The authors declare no competing financial interest. Acknowledgments This material is based upon work supported by the Air Force Office of Scientific Research under AFOSR Award No. FA9550-14-1-0295. This research is part of the Blue Waters sustainedpetascale computing project, which is supported by the National Science Foundation (Award Nos. OCI-0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications. We are grateful to Todd Martinez and James Stewart for providing a modified MOPAC code that includes the semiempirical FOMO-CASCI QM/MM method. We also thank Benjamin G. Levine for helpful discussions regarding the norm-preserving interpolation method for calculating nonadiabatic couplings.

Supporting Information. Additional details of the methodology for calculation of adiabatic vibronic states; time evolution of the population of double adiabatic vibronic states; choice of proton coordinate value for defining proton transfer on the S0 and S1 electronic states; analysis of proton transfer on the double adiabatic vibronic states. This material is available free of charge via the Internet at http://pubs.acs.org.

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34. van der Spoel, D. http://virtualchemistry.org/molecules/107-06-2/index.php (accessed 01/31/2014). 35. Webster, F.; Wang, E. T.; Rossky, P. J.; Friesner, R. A., Stationary Phase Surface Hopping for Nonadiabatic Dynamics: Two-State Systems. J. Chem. Phys. 1994, 100, 4835-4847. 36. Bittner, E. R.; Rossky, P. J., Quantum Decoherence in Mixed Quantum-Classical Systems: Nonadiabatic Processes. J. Chem. Phys. 1995, 103, 8130-8143. 37. Landry, B. R.; Subotnik, J. E., How to Recover Marcus Theory with Fewest Switches Surface Hopping: Add Just a Touch of Decoherence. J. Chem. Phys. 2012, 137, 22A513. 38. Subotnik, J. E.; Ouyang, W.; Landry, B. R., Can We Derive Tully's Surface-Hopping Algorithm from the Semiclassical Quantum Liouville Equation? Almost, but only with Decoherence. J. Chem. Phys. 2013, 139, 214107. 39. Billeter, S. R.; Webb, S. P.; Agarwal, P. K.; Iordanov, T.; Hammes-Schiffer, S., Hydride Transfer in Liver Alcohol Dehydrogenase: Quantum Dynamics, Kinetic Isotope Effects, and Role of Enzyme Motion. J. Am. Chem. Soc. 2001, 123, 11262-11272. 40. Agarwal, P. K.; Billeter, S. R.; Hammes-Schiffer, S., Nuclear Quantum Effects and Enzyme Dynamics in Dihydrofolate Reductase Catalysis. J. Phys. Chem. B 2002, 106, 32833293. 41. Gera, R.; Das, A.; Jha, A.; Dasgupta, J., Light-Induced Proton-Coupled Electron Transfer Inside a Nanocage. J. Am. Chem. Soc. 2014, 136, 15909-15912. 42. Das, A.; Jha, A.; Gera, R.; Dasgupta, J., Photoinduced Charge Transfer State Probes the Dynamic Water Interaction with Metal-Organic Nanocages. J. Phys. Chem. C 2015, 119, 2123421242.

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Figure 1. The experimentally studied hydrogen-bonded complex between p-nitrophenylphenol and t-butylamine.12 The computational studies are carried out on a reduced system in which tbutylamine has been replaced with ammonia. The full and reduced systems were shown to exhibit very similar ground and excited state properties in our previous study.15

Figure 2. Depiction of the proton potential energy curves and ground state proton vibrational wavefunctions for the S0 and S1 electronic states at the instant of photoexcitation. Photoexcitation from the ground proton vibrational state of the S0 electronic state to the S1 electronic state leads mainly to population of the ground proton vibrational state of S1. At the instant of photoexcitation, the donor well is lower in energy than the acceptor well in both proton potentials, although the asymmetry is greater for the S0 state. Subsequent solvent relaxation often alters the proton potential in the S1 electronic state so that the acceptor well becomes lower in energy than the donor well.

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Figure 3. Time evolution of the population of the ground adiabatic electron-proton vibronic state after photoexcitation to the S1 and S2 electronic states. The population rises with a timescale of ~0.6 ps in both cases.

Figure 4. Time evolution of the populations of the electronic states S0 (green), S1 (blue), and S2 (red) after initial photoexcitation to the (a) S1 and (b) S2 electronic states.

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Figure 5. Energies of double adiabatic electron-proton vibronic states for the initial 7 fs of a representative trajectory following photoexcitation to the ground proton vibrational state of S1 (double adiabatic state 21, dashed black curve). The high-energy proton vibrational states of the S0 electronic state (11-20, solid curves) are close in energy to the low-energy proton vibrational states of the S1 electronic state (21-25, dashed curves). The relative energies of the double adiabatic vibronic states change during the trajectory, resulting in numerous crossings between these double adiabatic states. The crossings between double adiabatic vibronic states 15 and 21 and between states 14 and 21 are denoted by small black boxes and vertical black dotted lines. Note that the surface hopping trajectories are propagated on the adiabatic vibronic states, although the double adiabatic vibronic states are depicted here.

Figure 6. Illustration of the population decay from the ground proton vibrational state of the S1 electronic state to the high-energy proton vibrational states of the S0 electronic state. This process is associated with the shorter lifetime of the S1 electronic state in the quantum proton simulations than in the classical proton simulations.15

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Figure 7. Illustration of PT on the 12th proton vibrational state of the S0 electronic state for a representative trajectory. The proton potentials and associated proton vibrational wavefunctions are depicted at the instant this vibrational state is first occupied in the trajectory (top panel, rp  0 ) and at the instant of the first PT on this state (bottom panel, rp  0.40 ). In this case, PT occurs with negligible accompanying solvent reorganization or change in PT barrier. See Figure S6 in the SI for a more detailed analysis of this type of PT process.

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Figure 8. Illustration of PT on the ground proton vibrational state of the S1 electronic state for a representative trajectory. The proton potentials and associated proton vibrational wavefunctions are depicted at the instant this vibrational state is first occupied in the trajectory (top panel, rp  0.46 ) and at the instant of the first PT on this state (bottom panel, rp  0.25 ). In this case, PT is accompanied by significant solvent reorganization and changes in the PT reaction energy and barrier. Specifically, the acceptor well of the proton potential becomes lower in energy than the donor well, and the PT barrier is significantly reduced. See Figure S7 in the SI for a more detailed analysis of this type of PT process.

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Figure 9. Time evolution of the populations of the ground adiabatic electron-proton vibronic state after initial photoexcitation to the (a) S1 and (b) S2 electronic states. Replacing the transferring hydrogen (red curves) with deuterium (blue curves) leads to only negligible differences in the timescale on which the population of the ground vibronic state increases.

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Figure 10. For the system with deuterium substitution of the transferring hydrogen, time evolution of the populations of the electronic states S0 (green), S1 (blue), and S2 (red) after initial photoexcitation to the (a) S1 and (b) S2 electronic states.

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Proton Quantization and Vibrational Relaxation in Nonadiabatic

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