Semiclassical Theory of Multistage Nonequilibrium Electron Transfer in Macromolecular Compounds in Polar Media with Several Relaxation Timescales

Abstract

Many specific features of ultrafast electron transfer (ET) reactions in macromolecular compounds can be attributed to nonequilibrium configurations of intramolecular vibrational degrees of freedom and the environment. In photoinduced ET, nonequilibrium nuclear configurations are often produced at the stage of optical excitation, but they can also be the result of electron tunneling itself, i.e., fast redistribution of charges within the macromolecule. A consistent theoretical description of ultrafast ET requires an explicit consideration of the nuclear subsystem, including its evolution between electron jumps. In this paper, the effect of the multi-timescale nuclear reorganization on ET transitions in macromolecular compounds is studied, and a general theory of ultrafast ET in non-Debye polar environments with a multi-component relaxation function is developed. Particular attention is paid to designing the multidimensional space of nonequilibrium nuclear configurations, as well as constructing the diabatic free energy surfaces for the ET states. The reorganization energies of individual ET transitions, the equilibrium energies of ET states, and the relaxation properties of the environment are used as input data for the theory. The effect of the system-environment interaction on the ET kinetics is discussed, and mechanisms for enhancing the efficiency of charge separation in macromolecular compounds are analyzed.

Keywords: electron transfer; macromolecular compounds; nonequilibrium processes; photochemistry; ultrafast reactions.

Figure 1

Schematic presentation of subspaces $\mathbf{Q}$ and $\mathbf{R}$, as well as the composite space $\mathbf{q}=\mathbf{Q}\times \mathbf{R}$ of nuclear configurations of the environment.

Figure 2

(a) Three-center molecular compound of the type DA${}_1$A${}_2$, where D is the electron-donating, and A${}_1$/A${}_2$ are the electron-accepting units. The spatial structure of the compound is shown schematically including the effective radii of the units and the center-to-center distances. (b) Photochemical processes in this compound. Abbreviations CS, CR, CSh, and IC denote charge separation, charge recombination, charge shift, and internal conversion, respectively.

Figure 3

(a) Free energy surfaces of the ET states $|{\phi }_1\rangle ={\mathrm{D}}^{*}{\mathrm{A}}_1{\mathrm{A}}_2$, $|{\phi }_2\rangle ={\mathrm{D}}^{+}{\mathrm{A}}_1^{-}{\mathrm{A}}_2$ and $|{\phi }_3\rangle ={\mathrm{D}}^{+}{\mathrm{A}}_1{\mathrm{A}}_2^{-}$ as functions of the polarization coordinates. These curves are constructed by projection of the 4-dimensional $G^{\left(n\right)}\left(\mathbf{q}\right)$ surfaces into the 2-dimensional $\mathbf{Q}$ subspace. The values of the model parameters are indicated in the text. (b) FESs minima on the $\left(Q_1,Q_2\right)$ plane. The displacement vectors ${\mathbf{D}}^{\left(n{n}^{\prime }\right)}$ and the $\theta$ angle are also shown.

Figure 4

(a) Free energy surfaces of the ET states as functions of the relaxation coordinates $R_1$ and $R_2$. The surfaces are calculated by orthogonal projection from the $\mathbf{q}$ space onto the. The $x_1$ and $x_2$ parameters correspond to acetonitrile (given in the text), other parameters are the same as in Figure 3. (b) Arrangement of the FESs minima on the $\left(R_1,R_2\right)$ plane: the ${\check{\mathbf{R}}}^{\left(n\right)}$ points are located on the straight line given by the equation $R_2/R_1=\sqrt{x_2/x_1}$.