^{1,a)}and Joseph E. Subotnik

^{1,b)}

### Abstract

A new method for obtaining diabatic electronic states of a molecular system in a condensed environment is proposed and evaluated. This technique, which we denote as Edmiston-Ruedenberg (ER)-ɛ diabatization, forms diabatic states as a linear combination of adiabatic states by minimizing an approximation to the total coupling between states in a medium with temperature *T* and with a characteristic Pekar factor *C*. ER-ɛ diabatization represents an improvement upon previous localized diabatization methods for two reasons: first, it is sensitive to the energy separation between adiabatic states, thus accounting for fluctuations in energy and effectively preventing over-mixing. Second, it responds to the strength of system-solvent interactions via parameters for the dielectric constant and temperature of the medium, which is physically reasonable. Here, we apply the ER-ɛ technique to both intramolecular and intermolecular excitation energy transfer systems. We find that ER-ɛ diabatic states satisfy three important properties: (1) they have small derivative couplings everywhere; (2) they have small diabatic couplings at avoided crossings, and (3) they have negligible diabatic couplings everywhere else. As such, ER-ɛ states are good candidates for so-called “optimal diabatic states.”

The authors would like to thank David Reichman and Robin Hochstrasser for helpful discussions. This work was supported by the NSF CAREER grant (Grant No. CHE-1150851). J.E.S. also acknowledges an Alfred P. Sloan Research Fellowship.

I. INTRODUCTION

II. NOTATION

III. THEORY AND METHODOLOGY

IV. RESULTS

A. Multi-state mixing and total coupling minimization

B. Localization phase diagrams

V. CONCLUSIONS AND FUTURE DIRECTIONS

### Key Topics

- Solvents
- 21.0
- Electron transfer
- 8.0
- Cancer
- 5.0
- Dielectrics
- 4.0
- Potential energy surfaces
- 4.0

## Figures

DBA molecule in which the 4-benzaldehydeyl donor and the 2-naphthyl acceptor groups are joined at 1,4-equitorial positions on a cyclohexane bridge, henceforth known as C-1,4ee. Here, C-1,4ee is shown in the geometry optimized for the A*D configuration of the T_{1} excited state.

DBA molecule in which the 4-benzaldehydeyl donor and the 2-naphthyl acceptor groups are joined at 1,4-equitorial positions on a cyclohexane bridge, henceforth known as C-1,4ee. Here, C-1,4ee is shown in the geometry optimized for the A*D configuration of the T_{1} excited state.

PES of the seven lowest-energy triplet states of C-1,4ee in three bases: (a) the adiabatic basis, (b) the ER diabatic basis, and (c) the ER-ɛ diabatic basis. The reaction coordinate ζ denotes a linear interpolation between two geometries: ζ = 0 corresponds to the optimized A*D geometry, and ζ = 1 corresponds to the optimized AD* geometry. While the two lowest-energy ER diabatic states are clearly over-mixed with higher-energy states, this is not a problem for ER-ɛ diabatic states.

PES of the seven lowest-energy triplet states of C-1,4ee in three bases: (a) the adiabatic basis, (b) the ER diabatic basis, and (c) the ER-ɛ diabatic basis. The reaction coordinate ζ denotes a linear interpolation between two geometries: ζ = 0 corresponds to the optimized A*D geometry, and ζ = 1 corresponds to the optimized AD* geometry. While the two lowest-energy ER diabatic states are clearly over-mixed with higher-energy states, this is not a problem for ER-ɛ diabatic states.

Diabatic coupling between the two lowest-energy triplet states of C-1,4ee: the reaction coordinate ζ denotes a linear interpolation between two geometries; ζ = 0 corresponds to the optimized A*D geometry, and ζ = 1 corresponds to the optimized AD* geometry. The diabatic coupling calculated by ER localization is nearly constant over the reaction coordinate, while that produced by ER-ɛ peaks at ζ = 0.85, close to the avoided crossing at ζ = 0.887. Notice also that the two ER-ɛ couplings are very similar, despite having been generated from adiabatic bases of different size, indicating that ER-ɛ localization is not prone to over-mixing.

Diabatic coupling between the two lowest-energy triplet states of C-1,4ee: the reaction coordinate ζ denotes a linear interpolation between two geometries; ζ = 0 corresponds to the optimized A*D geometry, and ζ = 1 corresponds to the optimized AD* geometry. The diabatic coupling calculated by ER localization is nearly constant over the reaction coordinate, while that produced by ER-ɛ peaks at ζ = 0.85, close to the avoided crossing at ζ = 0.887. Notice also that the two ER-ɛ couplings are very similar, despite having been generated from adiabatic bases of different size, indicating that ER-ɛ localization is not prone to over-mixing.

Magnitude of the approximate derivative coupling between the two lowest-energy triplet states of C-1,4ee: the reaction coordinate ζ denotes a linear interpolation between two geometries; ζ = 0 corresponds to the optimized A*D geometry, and ζ = 1 corresponds to the optimized AD* geometry. Derivative couplings in three bases are represented: the adiabatic basis, the ER diabatic basis, and the ER-ɛ diabatic basis. In the adiabatic basis, the derivative coupling becomes large close to the avoided crossing at ζ = 0.887, as expected, reaching values as large as 750 a . In the ER basis, the derivative couplings are consistently small, peaking at 0.05 . In the ER-ɛ basis, derivative couplings match adiabatic values for many nuclear geometries, and become as large as 0.17 . Close to the avoided crossing, however, this is not the case, and the enormous couplings present in the adiabatic basis are eliminated. The apparent “kink” in the ER-ɛ derivative coupling is in fact a sign change (see Fig. 5 ).

Magnitude of the approximate derivative coupling between the two lowest-energy triplet states of C-1,4ee: the reaction coordinate ζ denotes a linear interpolation between two geometries; ζ = 0 corresponds to the optimized A*D geometry, and ζ = 1 corresponds to the optimized AD* geometry. Derivative couplings in three bases are represented: the adiabatic basis, the ER diabatic basis, and the ER-ɛ diabatic basis. In the adiabatic basis, the derivative coupling becomes large close to the avoided crossing at ζ = 0.887, as expected, reaching values as large as 750 a . In the ER basis, the derivative couplings are consistently small, peaking at 0.05 . In the ER-ɛ basis, derivative couplings match adiabatic values for many nuclear geometries, and become as large as 0.17 . Close to the avoided crossing, however, this is not the case, and the enormous couplings present in the adiabatic basis are eliminated. The apparent “kink” in the ER-ɛ derivative coupling is in fact a sign change (see Fig. 5 ).

Approximate derivative coupling between the two lowest-energy triplet states of C-1,4ee: the reaction coordinate ζ denotes a linear interpolation between two geometries; ζ = 0 corresponds to the optimized A*D geometry, and ζ = 1 corresponds to the optimized AD* geometry. Derivative couplings in three bases are represented: the adiabatic basis, the ER diabatic basis, and the ER-ɛ diabatic basis. The derivative coupling values presented here are identical to those presented in Fig. 4 , with the exception that sign information has been included in this figure. The “kink” in the ER-ɛ derivative coupling near the avoided crossing in Fig. 4 is in fact a change of sign, as shown here.

Approximate derivative coupling between the two lowest-energy triplet states of C-1,4ee: the reaction coordinate ζ denotes a linear interpolation between two geometries; ζ = 0 corresponds to the optimized A*D geometry, and ζ = 1 corresponds to the optimized AD* geometry. Derivative couplings in three bases are represented: the adiabatic basis, the ER diabatic basis, and the ER-ɛ diabatic basis. The derivative coupling values presented here are identical to those presented in Fig. 4 , with the exception that sign information has been included in this figure. The “kink” in the ER-ɛ derivative coupling near the avoided crossing in Fig. 4 is in fact a change of sign, as shown here.

Localization phase diagram of S_{1} and S_{2} states of dibenzene: the degree of mixing between states S_{1} and S_{2} of the molecular system is presented via mixing angle θ as a function of Pekar factor *C* and temperature *T*. A mixing angle of θ = 0 represents total delocalization (unchanged from adiabatic states), and represents total localization. The model predicts complete localization of these electronic states at room temperature for a sufficiently polar (*C* ⪆ 0.6) solvent.

Localization phase diagram of S_{1} and S_{2} states of dibenzene: the degree of mixing between states S_{1} and S_{2} of the molecular system is presented via mixing angle θ as a function of Pekar factor *C* and temperature *T*. A mixing angle of θ = 0 represents total delocalization (unchanged from adiabatic states), and represents total localization. The model predicts complete localization of these electronic states at room temperature for a sufficiently polar (*C* ⪆ 0.6) solvent.

Attachment/detachment densities of the lowest-energy singlet excited state of the dibenzene system in in the ER-ɛ basis with three different sets of solvation parameters: (a) *C* = 0.4 and *T* = 200 K, (b) *C* = 0.8 and *T* = 200 K, and (c) *C* = 0.8 and *T* = 300 K. In each case, top figure is the attachment density, and the bottom figure is the detachment density. As solvent perturbations are applied, the delocalized excitation in the adiabatic basis begins to localize onto one ring. Increasing the polarity and temperature of the environment results in more rapid solvent fluctuations, which further trap electron density.

Attachment/detachment densities of the lowest-energy singlet excited state of the dibenzene system in in the ER-ɛ basis with three different sets of solvation parameters: (a) *C* = 0.4 and *T* = 200 K, (b) *C* = 0.8 and *T* = 200 K, and (c) *C* = 0.8 and *T* = 300 K. In each case, top figure is the attachment density, and the bottom figure is the detachment density. As solvent perturbations are applied, the delocalized excitation in the adiabatic basis begins to localize onto one ring. Increasing the polarity and temperature of the environment results in more rapid solvent fluctuations, which further trap electron density.

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