# Golden rule kinetics of transfer reactions in condensed phase: The microscopic model of electron transfer reactions in disordered solid matrices

## Abstract

The algorithm for a theoretical calculation of transfer reaction rates for light quantum particles (i.e., the electron and H-atom transfers) in non-polar solid matrices is formulated and justified. The mechanism postulated involves a local mode (an either intra- or inter-molecular one) serving as a mediator which accomplishes the energy exchange between the reacting high-frequency quantum mode and the phonon modes belonging to the environment. This approach uses as a background the Fermi golden rule beyond the usually applied spin-boson approximation. The dynamical treatment rests on the one-dimensional version of the standard quantum relaxation equation for the reduced density matrix, which describes the frequency fluctuation spectrum for the local mode under consideration. The temperature dependence of a reaction rate is controlled by the dimensionless parameter ξ{sub 0}=ℏω{sub 0}/k{sub B}T where ω{sub 0} is the frequency of the local mode and T is the temperature. The realization of the computational scheme is different for the high/intermediate (ξ{sub 0} < 1 − 3) and for low (ξ{sub 0}≫ 1) temperature ranges. For the first (quasi-classical) kinetic regime, the Redfield approximation to the solution of the relaxation equation proved to be sufficient and efficient in practical applications. The study of the essentially quantum-mechanical low-temperaturemore »

- Authors:

- Photochemistry Center, Russian Academy of Sciences, 7a, Novatorov ul., Moscow (Russian Federation)
- (Russian Federation)
- Karpov Institute of Physical Chemistry, 3-1/12, Building 6, Obuha pereulok, Moscow (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22220237

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Chemical Physics; Journal Volume: 139; Journal Issue: 23; Other Information: (c) 2013 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 37 INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL CHEMISTRY; DENSITY MATRIX; ELECTRON TRANSFER; QUANTUM MECHANICS; REACTION KINETICS; RELAXATION; SOLIDS; TEMPERATURE DEPENDENCE; TRANSFER REACTIONS

### Citation Formats

```
Basilevsky, M. V., Mitina, E. A., Odinokov, A. V., National Research Nuclear University “MEPhI,” 31, Kashirskoye shosse, Moscow, and Titov, S. V..
```*Golden rule kinetics of transfer reactions in condensed phase: The microscopic model of electron transfer reactions in disordered solid matrices*. United States: N. p., 2013.
Web. doi:10.1063/1.4838335.

```
Basilevsky, M. V., Mitina, E. A., Odinokov, A. V., National Research Nuclear University “MEPhI,” 31, Kashirskoye shosse, Moscow, & Titov, S. V..
```*Golden rule kinetics of transfer reactions in condensed phase: The microscopic model of electron transfer reactions in disordered solid matrices*. United States. doi:10.1063/1.4838335.

```
Basilevsky, M. V., Mitina, E. A., Odinokov, A. V., National Research Nuclear University “MEPhI,” 31, Kashirskoye shosse, Moscow, and Titov, S. V.. Sat .
"Golden rule kinetics of transfer reactions in condensed phase: The microscopic model of electron transfer reactions in disordered solid matrices". United States.
doi:10.1063/1.4838335.
```

```
@article{osti_22220237,
```

title = {Golden rule kinetics of transfer reactions in condensed phase: The microscopic model of electron transfer reactions in disordered solid matrices},

author = {Basilevsky, M. V. and Mitina, E. A. and Odinokov, A. V. and National Research Nuclear University “MEPhI,” 31, Kashirskoye shosse, Moscow and Titov, S. V.},

abstractNote = {The algorithm for a theoretical calculation of transfer reaction rates for light quantum particles (i.e., the electron and H-atom transfers) in non-polar solid matrices is formulated and justified. The mechanism postulated involves a local mode (an either intra- or inter-molecular one) serving as a mediator which accomplishes the energy exchange between the reacting high-frequency quantum mode and the phonon modes belonging to the environment. This approach uses as a background the Fermi golden rule beyond the usually applied spin-boson approximation. The dynamical treatment rests on the one-dimensional version of the standard quantum relaxation equation for the reduced density matrix, which describes the frequency fluctuation spectrum for the local mode under consideration. The temperature dependence of a reaction rate is controlled by the dimensionless parameter ξ{sub 0}=ℏω{sub 0}/k{sub B}T where ω{sub 0} is the frequency of the local mode and T is the temperature. The realization of the computational scheme is different for the high/intermediate (ξ{sub 0} < 1 − 3) and for low (ξ{sub 0}≫ 1) temperature ranges. For the first (quasi-classical) kinetic regime, the Redfield approximation to the solution of the relaxation equation proved to be sufficient and efficient in practical applications. The study of the essentially quantum-mechanical low-temperature kinetic regime in its asymptotic limit requires the implementation of the exact relaxation equation. The coherent mechanism providing a non-vanishing reaction rate has been revealed when T→ 0. An accurate computational methodology for the cross-over kinetic regime needs a further elaboration. The original model of the hopping mechanism for electronic conduction in photosensitive organic materials is considered, based on the above techniques. The electron transfer (ET) in active centers of such systems proceeds via local intra- and intermolecular modes. The active modes, as a rule, operate beyond the kinetic regimes, which are usually postulated in the existing theories of the ET. Our alternative dynamic ET model for local modes immersed in the continuum harmonic medium is formulated for both classical and quantum regimes, and accounts explicitly for the mode/medium interaction. The kinetics of the energy exchange between the local ET subsystem and the surrounding environment essentially determine the total ET rate. The efficient computer code for rate computations is elaborated on. The computations are available for a wide range of system parameters, such as the temperature, external field, local mode frequency, and characteristics of mode/medium interaction. The relation of the present approach to the Marcus ET theory and to the quantum-statistical reaction rate theory [V. G. Levich and R. R. Dogonadze, Dokl. Akad. Nauk SSSR, Ser. Fiz. Khim. 124, 213 (1959); J. Ulstrup, Charge Transfer in Condensed Media (Springer, Berlin, 1979); M. Bixon and J. Jortner, Adv. Chem. Phys. 106, 35 (1999)] underlying it is discussed and illustrated by the results of computations for practically important target systems.},

doi = {10.1063/1.4838335},

journal = {Journal of Chemical Physics},

number = 23,

volume = 139,

place = {United States},

year = {Sat Dec 21 00:00:00 EST 2013},

month = {Sat Dec 21 00:00:00 EST 2013}

}