^{1,a)}, Peter J. Rossky

^{2,b)}and László Turi

^{3,c)}

### Abstract

A new, alternative form of the golden rule formula defining the nonadiabatic transition rate between two quantum states in condensed phase is presented. The formula involves the quantum time correlation function of the energy gap, of the nonadiabatic coupling, and their cross terms. Those quantities can be inferred from their classical counterparts, determined via molecular dynamics simulations. The formalism is applied to the problem of the nonadiabatic relaxation of an equilibrated -electron in water and methanol. We find that, in both solvents, the relaxation is induced by the coupling to the vibrational modes and the quantum effects modify the rate by a factor of 2–10 depending on the quantization procedure applied. The resulting -state lifetime for a hypothetical equilibrium excited state appears extremely short, in the sub- regime. Although this result is in contrast with all previous theoretical predictions, we also illustrate that the lifetimes computed here are very sensitive to the simulated electronic quantum gap and to the strongly correlated nonadiabatic coupling.

This work was supported by research grants to one of the authors (L.T.) from the Eötvös Fellowship, the Bolyai Research Fellowship, and the National Research Fund of Hungary (OTKA, T049715). One of the authors (P.J.R.) acknowledges the support of the U. S. National Science Foundation (CHE-0134775) and the R. A. Welch Foundation (F-0019).

I. INTRODUCTION

II. FERMI GOLDEN RULE EXPRESSION FOR NONADIABATIC ELECTRONIC TRANSITIONS

III. APPLICATION TO A SOLVATED ELECTRON IN WATER AND METHANOL

A. Motivations

B. Simulation results

C. Classical rates

D. Quantized correlation functions and quantum transition rates: Time-domain formulation

E. Quantized correlation functions and quantum transition rates: Decoherence

F. Quantized correlation functions and quantum transition rates: Frequency dependent expression of the rate

G. Discussion

IV. CONCLUSION

### Key Topics

- Correlation functions
- 36.0
- Excited states
- 31.0
- Band gap
- 28.0
- Solvents
- 27.0
- Non adiabatic reactions
- 25.0

## Figures

slices of the trajectories obtained for water (top) and for methanol (bottom). In each frame, the top curve reports the time-dependent energy gap and, for clarity, the bottom one gives the absolute value of the coupling, with a minus sign to avoid overlaps.

slices of the trajectories obtained for water (top) and for methanol (bottom). In each frame, the top curve reports the time-dependent energy gap and, for clarity, the bottom one gives the absolute value of the coupling, with a minus sign to avoid overlaps.

Normalized energy gap correlation function for an equilibrated solvated -electron in water (top) and in methanol (bottom).

Normalized energy gap correlation function for an equilibrated solvated -electron in water (top) and in methanol (bottom).

Spectral density (in arbitrary units) of the nonadiabatic coupling in water (top) and in methanol (bottom).

Spectral density (in arbitrary units) of the nonadiabatic coupling in water (top) and in methanol (bottom).

Classical reactive flux correlation function for water (top) and methanol (bottom). The circles indicate the direct numerical integration of Eqs. (12) and (17), whereas the solid line involves the Gaussian approximation for the dephasing function with neglect of the cross-correlation terms, as in Eq. (24).

Classical reactive flux correlation function for water (top) and methanol (bottom). The circles indicate the direct numerical integration of Eqs. (12) and (17), whereas the solid line involves the Gaussian approximation for the dephasing function with neglect of the cross-correlation terms, as in Eq. (24).

Quantum reactive flux for water (top) and methanol (bottom). The solid line is for the direct integration of Eq. (22), including all terms, and the dashed line involves a Gaussian approximation for the dephasing function and the neglect of the cross-correlation terms, Eq. (25). The dot-dashed line recalls the classical results of Fig. 4.

Quantum reactive flux for water (top) and methanol (bottom). The solid line is for the direct integration of Eq. (22), including all terms, and the dashed line involves a Gaussian approximation for the dephasing function and the neglect of the cross-correlation terms, Eq. (25). The dot-dashed line recalls the classical results of Fig. 4.

Frequency dependent rate for water (top) and methanol (bottom). The solid curve indicates the quantized result using the harmonic quantization scheme, and the dashed-dotted curves denote the standard quantization procedure [Eqs. (20)–(23)]. The Gaussian-type dashed curve on top is the “window” function of Eq. (34) (renormalized to fit in the figure).

Frequency dependent rate for water (top) and methanol (bottom). The solid curve indicates the quantized result using the harmonic quantization scheme, and the dashed-dotted curves denote the standard quantization procedure [Eqs. (20)–(23)]. The Gaussian-type dashed curve on top is the “window” function of Eq. (34) (renormalized to fit in the figure).

The probability distribution of the energy gap, , and its approximation by a Gaussian distribution (dashed, upper frame). The average value of sampled at each particular energy value (lower frame).

The probability distribution of the energy gap, , and its approximation by a Gaussian distribution (dashed, upper frame). The average value of sampled at each particular energy value (lower frame).

The lifetime of the excited state electron for alternative approximations as a function of the mean energy gap (see text). Classical case (solid), standard (dashed), and harmonic quantization schemes (dotted). The harmonic quantization with the very high frequency coupling contributions removed is also shown (dash-dot).

The lifetime of the excited state electron for alternative approximations as a function of the mean energy gap (see text). Classical case (solid), standard (dashed), and harmonic quantization schemes (dotted). The harmonic quantization with the very high frequency coupling contributions removed is also shown (dash-dot).

## Tables

Averaged classical quantities collected along the MD runs and the corresponding quantized quantities (denoted by the subscript) computed using the harmonic quantization scheme. All quantities are in .

Averaged classical quantities collected along the MD runs and the corresponding quantized quantities (denoted by the subscript) computed using the harmonic quantization scheme. All quantities are in .

Classical and quantized nonadiabatic transition times, and dephasing times for electronic relaxation of an equilibrated excited state solvated electron in water and methanol (see text). The quantized rates are computed using either the harmonic or the standard quantization scheme. The classical dephasing times and the quantum decoherence times using the harmonic quantization are also shown. All quantities are in fs.

Classical and quantized nonadiabatic transition times, and dephasing times for electronic relaxation of an equilibrated excited state solvated electron in water and methanol (see text). The quantized rates are computed using either the harmonic or the standard quantization scheme. The classical dephasing times and the quantum decoherence times using the harmonic quantization are also shown. All quantities are in fs.

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