^{1}, Anirban Hazra

^{1}and Sharon Hammes-Schiffer

^{1,a)}

### Abstract

A theoretical approach for the multidimensional treatment of photoinduced proton-coupled electron transfer (PCET) processes in solution is presented. This methodology is based on the multistate continuum theory with an arbitrary number of diabatic electronic states representing the relevant charge distributions in a general PCET system. The active electrons and transferring proton(s) are treated quantum mechanically, and the electron-proton vibronic free energysurfaces are represented as functions of multiple scalar solvent coordinates corresponding to the single electron and proton transfer reactions involved in the PCET process. A dynamical formulation of the dielectric continuum theory is used to derive a set of coupled generalized Langevin equations of motion describing the time evolution of these collective solvent coordinates. The parameters in the Langevin equations depend on the solvent properties, such as the dielectric constants, relaxation time, and molecular moment of inertia, as well as the solute properties. The dynamics of selected intramolecular nuclear coordinates, such as the proton donor-acceptor distance or a torsional angle within the PCET complex, may also be included in this formulation. A surface hopping method in conjunction with the Langevin equations of motion is used to simulate the nonadiabaticdynamics on the multidimensional electron-proton vibronic free energysurfaces following photoexcitation. This theoretical treatment enables the description of both sequential and concerted mechanisms, as well as more complex processes involving a combination of these mechanisms. The application of this methodology to a series of model systems corresponding to collinear and orthogonal PCET illustrates fundamental aspects of these different mechanisms and elucidates the significance of proton vibrational relaxation and nonequilibrium solventdynamics.

This material is based upon work supported by the Air Force Office of Scientific Research under AFOSR Award No. FA9550-10-1-0081 and NSF Grant CHE-07-49646. We thank Ben Auer for helpful discussions and comments on the manuscript.

I. INTRODUCTION

II. THEORY

A. Multistate model of the PCET reaction complex

B. Interaction with continuum solvent environment: Vibronic free energysurfaces

C. Generalized Langevin equation for solvent coordinates

D. Mixed quantum-classical nonadiabaticdynamics on vibronic free energysurfaces

III. MODEL CALCULATIONS

A. Model systems

B. Initial conditions

C. Simulation details

IV. RESULTS

A. Collinear PCET

B. Orthogonal PCET (Model II)

V. CONCLUDING REMARKS

### Key Topics

- Solvents
- 129.0
- Protons
- 63.0
- Electron transfer
- 49.0
- Surface dynamics
- 47.0
- Surface states
- 35.0

## Figures

Model systems for collinear PCET (Model I) and orthogonal PCET (Model II). The green balls denote the electron donor and acceptor sites, the blue balls denote the proton donor and acceptor sites, and the gray ball denotes the transferring hydrogen. The cavities constructed from overlapping spheres around the sites are also shown. These cavities are used for the FRCM calculations of the solvent reorganization energy matrix elements. The arrows below depict the direction of PT and ET in these model systems.

Model systems for collinear PCET (Model I) and orthogonal PCET (Model II). The green balls denote the electron donor and acceptor sites, the blue balls denote the proton donor and acceptor sites, and the gray ball denotes the transferring hydrogen. The cavities constructed from overlapping spheres around the sites are also shown. These cavities are used for the FRCM calculations of the solvent reorganization energy matrix elements. The arrows below depict the direction of PT and ET in these model systems.

Initial conditions used for the simulations. (a) Diabatic harmonic proton potentials, , for the 1*a* and 1*b* diabatic electronic states and the initial proton wavepacket following photoexcitation (at *t* = 0) for the initial conditions A (red) and B (blue). The proton vibrational wavepacket prior to photoexcitation (green) corresponds to the ground proton vibrational state in the electronic ground state (not shown). (b) Initial distributions for the solvent coordinates *Z* _{p} and *Z* _{e} for the initial conditions A (red) and B (blue) for the symmetric Model II. The contour plot schematically depicts the four diabatic vibronic free energy surfaces with minima indicated as 1*a*, 1*b*, 2*a*, and 2*b*. For both initial conditions, the distributions are centered at *Z* _{e} = 0. The distributions are centered at the *Z* _{p} value corresponding to the 1*a*/2*a* minima for initial condition A and the 1*b*/2*b* minima for initial condition B.

Initial conditions used for the simulations. (a) Diabatic harmonic proton potentials, , for the 1*a* and 1*b* diabatic electronic states and the initial proton wavepacket following photoexcitation (at *t* = 0) for the initial conditions A (red) and B (blue). The proton vibrational wavepacket prior to photoexcitation (green) corresponds to the ground proton vibrational state in the electronic ground state (not shown). (b) Initial distributions for the solvent coordinates *Z* _{p} and *Z* _{e} for the initial conditions A (red) and B (blue) for the symmetric Model II. The contour plot schematically depicts the four diabatic vibronic free energy surfaces with minima indicated as 1*a*, 1*b*, 2*a*, and 2*b*. For both initial conditions, the distributions are centered at *Z* _{e} = 0. The distributions are centered at the *Z* _{p} value corresponding to the 1*a*/2*a* minima for initial condition A and the 1*b*/2*b* minima for initial condition B.

Results for Model IA, the symmetric collinear PCET model system with initial condition A. (a) The time-dependent populations of the lowest two adiabatic vibronic states, , and the diabatic electronic states, (*i* = 1*a*, 1*b*, 2*a*, 2*b*), where the notation for these populations is slightly altered for convenience. (b) The time-dependent marginal distributions, *P* ^{(solv)}(*Y* _{p}) and *P* ^{(solv)}(*Y* _{e}), for the collective solvent coordinates *Y* _{p} and *Y* _{e}, corresponding predominantly to PT and ET, respectively. The darker blue corresponds to a greater value of the marginal distribution function. The values of the solvent coordinates corresponding to the 1*a* and 2*b* minima on the diabatic vibronic surfaces are indicated by dashed lines. (c) Snapshots of the distributions, , of the transformed solvent coordinates *Z* _{p} and *Z* _{e} in the lowest two adiabatic vibronic states along the trajectories. In the first snapshot, the minima of the ground adiabatic vibronic surface are labeled according to the dominant diabatic electronic state. See supplementary material for a movie of the time-dependent solvent distribution (Ref. 55).

Results for Model IA, the symmetric collinear PCET model system with initial condition A. (a) The time-dependent populations of the lowest two adiabatic vibronic states, , and the diabatic electronic states, (*i* = 1*a*, 1*b*, 2*a*, 2*b*), where the notation for these populations is slightly altered for convenience. (b) The time-dependent marginal distributions, *P* ^{(solv)}(*Y* _{p}) and *P* ^{(solv)}(*Y* _{e}), for the collective solvent coordinates *Y* _{p} and *Y* _{e}, corresponding predominantly to PT and ET, respectively. The darker blue corresponds to a greater value of the marginal distribution function. The values of the solvent coordinates corresponding to the 1*a* and 2*b* minima on the diabatic vibronic surfaces are indicated by dashed lines. (c) Snapshots of the distributions, , of the transformed solvent coordinates *Z* _{p} and *Z* _{e} in the lowest two adiabatic vibronic states along the trajectories. In the first snapshot, the minima of the ground adiabatic vibronic surface are labeled according to the dominant diabatic electronic state. See supplementary material for a movie of the time-dependent solvent distribution (Ref. 55).

Results for Model IB, the symmetric collinear PCET model system with initial condition B. (a) The time-dependent populations of the lowest five adiabatic vibronic states, , and the diabatic electronic states, (*i* = 1*a*, 1*b*, 2*a*, 2*b*). (b) The time-dependent marginal distributions, *P* ^{(solv)}(*Y* _{p}) and *P* ^{(solv)}(*Y* _{e}), for the collective solvent coordinates *Y* _{p} and *Y* _{e}, corresponding predominantly to PT and ET, respectively. The darker blue corresponds to a greater value of the marginal distribution function. The values of the solvent coordinates corresponding to the 1*a* and 2*b* minima on the diabatic vibronic surfaces are indicated by dashed lines. (c) Snapshots of the distributions, , of the transformed solvent coordinates *Z* _{p} and *Z* _{e} in the lowest five adiabatic vibronic states along the trajectories. In the first snapshot, the minimum of the ground adiabatic vibronic surface is labeled according to the dominant diabatic electronic state. See supplementary material for a movie of the time-dependent solvent distribution (Ref. 55).

Results for Model IB, the symmetric collinear PCET model system with initial condition B. (a) The time-dependent populations of the lowest five adiabatic vibronic states, , and the diabatic electronic states, (*i* = 1*a*, 1*b*, 2*a*, 2*b*). (b) The time-dependent marginal distributions, *P* ^{(solv)}(*Y* _{p}) and *P* ^{(solv)}(*Y* _{e}), for the collective solvent coordinates *Y* _{p} and *Y* _{e}, corresponding predominantly to PT and ET, respectively. The darker blue corresponds to a greater value of the marginal distribution function. The values of the solvent coordinates corresponding to the 1*a* and 2*b* minima on the diabatic vibronic surfaces are indicated by dashed lines. (c) Snapshots of the distributions, , of the transformed solvent coordinates *Z* _{p} and *Z* _{e} in the lowest five adiabatic vibronic states along the trajectories. In the first snapshot, the minimum of the ground adiabatic vibronic surface is labeled according to the dominant diabatic electronic state. See supplementary material for a movie of the time-dependent solvent distribution (Ref. 55).

Results for Model I^{′}A, the biased collinear PCET model system with initial condition A. (a) The time-dependent populations of the lowest five adiabatic vibronic states, , and the diabatic electronic states, (*i* = 1*a*, 1*b*, 2*a*, 2*b*). (b) The time-dependent marginal distributions, *P* ^{(solv)}(*Y* _{p}) and *P* ^{(solv)}(*Y* _{e}), for the collective solvent coordinates *Y* _{p} and *Y* _{e}, corresponding predominantly to PT and ET, respectively. The darker blue corresponds to a greater value of the marginal distribution function. The values of the solvent coordinates corresponding to the 1*a*, 2*a,* and 2*b* minima on the diabatic vibronic surfaces are indicated by dashed lines. (c) Snapshots of the distributions, , of the transformed solvent coordinates *Z* _{p} and *Z* _{e} in the lowest five adiabatic vibronic states along the trajectories. In the first snapshot, the minimum of the ground adiabatic vibronic surface is labeled according to the dominant diabatic electronic state. See supplementary material for a movie of the time-dependent solvent distribution (Ref. 55).

Results for Model I^{′}A, the biased collinear PCET model system with initial condition A. (a) The time-dependent populations of the lowest five adiabatic vibronic states, , and the diabatic electronic states, (*i* = 1*a*, 1*b*, 2*a*, 2*b*). (b) The time-dependent marginal distributions, *P* ^{(solv)}(*Y* _{p}) and *P* ^{(solv)}(*Y* _{e}), for the collective solvent coordinates *Y* _{p} and *Y* _{e}, corresponding predominantly to PT and ET, respectively. The darker blue corresponds to a greater value of the marginal distribution function. The values of the solvent coordinates corresponding to the 1*a*, 2*a,* and 2*b* minima on the diabatic vibronic surfaces are indicated by dashed lines. (c) Snapshots of the distributions, , of the transformed solvent coordinates *Z* _{p} and *Z* _{e} in the lowest five adiabatic vibronic states along the trajectories. In the first snapshot, the minimum of the ground adiabatic vibronic surface is labeled according to the dominant diabatic electronic state. See supplementary material for a movie of the time-dependent solvent distribution (Ref. 55).

Results for Model IIA, the symmetric orthogonal PCET model system with initial condition A. (a) The time-dependent populations of the lowest two adiabatic vibronic states, , and the diabatic electronic states, (*i* = 1*a*, 1*b*, 2*a*, 2*b*). (b) The time-dependent marginal distributions, *P* ^{(solv)}(*Y* _{p}) and *P* ^{(solv)}(*Y* _{e}), for the collective solvent coordinates *Y* _{p} and *Y* _{e}, corresponding predominantly to PT and ET, respectively. The darker blue corresponds to a greater value of the marginal distribution function. The values of the solvent coordinates corresponding to the 1*a*, 1*b*, 2*a*, and 2*b* minima on the diabatic vibronic surfaces are indicated by dashed lines. (c) Snapshots of the distributions, , of the transformed solvent coordinates *Z* _{p} and *Z* _{e} in the lowest two adiabatic vibronic states along the trajectories. In the first snapshot, the minima of the ground adiabatic vibronic surface are labeled according to the dominant diabatic electronic state. See supplementary material for a movie of the time-dependent solvent distribution (Ref. 55).

Results for Model IIA, the symmetric orthogonal PCET model system with initial condition A. (a) The time-dependent populations of the lowest two adiabatic vibronic states, , and the diabatic electronic states, (*i* = 1*a*, 1*b*, 2*a*, 2*b*). (b) The time-dependent marginal distributions, *P* ^{(solv)}(*Y* _{p}) and *P* ^{(solv)}(*Y* _{e}), for the collective solvent coordinates *Y* _{p} and *Y* _{e}, corresponding predominantly to PT and ET, respectively. The darker blue corresponds to a greater value of the marginal distribution function. The values of the solvent coordinates corresponding to the 1*a*, 1*b*, 2*a*, and 2*b* minima on the diabatic vibronic surfaces are indicated by dashed lines. (c) Snapshots of the distributions, , of the transformed solvent coordinates *Z* _{p} and *Z* _{e} in the lowest two adiabatic vibronic states along the trajectories. In the first snapshot, the minima of the ground adiabatic vibronic surface are labeled according to the dominant diabatic electronic state. See supplementary material for a movie of the time-dependent solvent distribution (Ref. 55).

Results for Model IIB, the symmetric orthogonal PCET model system with initial condition B. (a) The time-dependent populations of the lowest five adiabatic vibronic states, , and the diabatic electronic states, (*i* = 1*a*, 1*b*, 2*a*, 2*b*). (b) The time-dependent marginal distributions, *P* ^{(solv)}(*Y* _{p}) and *P* ^{(solv)}(*Y* _{e}), for the collective solvent coordinates *Y* _{p} and *Y* _{e}, corresponding predominantly to PT and ET, respectively. The darker blue corresponds to a greater value of the marginal distribution function. The values of the solvent coordinates corresponding to the 1*a*, 1*b*, 2*a*, and 2*b* minima on the diabatic vibronic surface are indicated by dashed lines. (c) Snapshots of the distributions, , of the transformed solvent coordinates *Z* _{p} and *Z* _{e} in the lowest five adiabatic vibronic states along the trajectories. In the first snapshot, the minima of the diabatic vibronic surfaces are labeled according to the dominant diabatic electronic state. See supplementary material for a movie of the time-dependent solvent distribution (Ref. 55).

Results for Model IIB, the symmetric orthogonal PCET model system with initial condition B. (a) The time-dependent populations of the lowest five adiabatic vibronic states, , and the diabatic electronic states, (*i* = 1*a*, 1*b*, 2*a*, 2*b*). (b) The time-dependent marginal distributions, *P* ^{(solv)}(*Y* _{p}) and *P* ^{(solv)}(*Y* _{e}), for the collective solvent coordinates *Y* _{p} and *Y* _{e}, corresponding predominantly to PT and ET, respectively. The darker blue corresponds to a greater value of the marginal distribution function. The values of the solvent coordinates corresponding to the 1*a*, 1*b*, 2*a*, and 2*b* minima on the diabatic vibronic surface are indicated by dashed lines. (c) Snapshots of the distributions, , of the transformed solvent coordinates *Z* _{p} and *Z* _{e} in the lowest five adiabatic vibronic states along the trajectories. In the first snapshot, the minima of the diabatic vibronic surfaces are labeled according to the dominant diabatic electronic state. See supplementary material for a movie of the time-dependent solvent distribution (Ref. 55).

## Tables

Radii and charges for the model PCET systems. ET and PT donor sites are denoted D_{e} and D_{p}, respectively; ET and PT acceptor sites are denoted A_{e} and A_{p}, respectively; and the transferring proton is denoted H. All of the quantities in this table are used to calculate the solvent reorganization energy matrix elements, and the charges are used to calculate the electrostatic contributions to the gas phase potentials of the reaction complex.

Radii and charges for the model PCET systems. ET and PT donor sites are denoted D_{e} and D_{p}, respectively; ET and PT acceptor sites are denoted A_{e} and A_{p}, respectively; and the transferring proton is denoted H. All of the quantities in this table are used to calculate the solvent reorganization energy matrix elements, and the charges are used to calculate the electrostatic contributions to the gas phase potentials of the reaction complex.

Solvent reorganization energies, λ_{PT} and λ_{ET}, for PT and ET reactions, respectively, and cross-reorganization energy *γ* for the model PCET systems studied in this paper. Quantities given in units of kcal/mol.

Solvent reorganization energies, λ_{PT} and λ_{ET}, for PT and ET reactions, respectively, and cross-reorganization energy *γ* for the model PCET systems studied in this paper. Quantities given in units of kcal/mol.

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