UDC 539.194

PHYSICS

A. A. OVCHINNIKOV, M. Ya. OVCHINNIKOVA

NONRADIATIVE TRANSITION WITH CHARGE TRANSFER IN A POLAR MEDIUM

(Presented by Academician V. N. Kondrat’ev, 18 X 1968)

Transitions with redistribution of charge in molecules or complexes placed in a polar liquid (water, alcohols, etc.) are characterized by strong interaction of the ionic states with the medium and by a large interaction region. Therefore the medium may be regarded as a continuous dielectric (\(^{1}\)). We shall assume that the average values of the field operators are described by the linear Maxwell equations, and that the medium is characterized by the dielectric permittivity \(\varepsilon(r,\omega)\). The aim of the present work is to find the transition probability \(W_{1,2}\) between two electronic states in such a system. For a solid, the theory of multiphonon transitions under strong coupling has been developed in (\(^{2,3}\)) and applied to a liquid in (\(^{4}\)). However, the derivation of the formula for the transition probability in these works makes essential use of a model of phonon oscillators and is unsuitable for a liquid, where the absorption of energy is connected mainly (\(^{5}\)) with reorientation of molecules (Debye absorption or resonance adsorption).

The Hamiltonian of the liquid–complex (or molecule) system in each electronic state \((i=1,2)\) in the Coulomb gauge is equal to

\[ H^{i}=H_{0}+\int \rho^{i}(r)\hat{\varphi}(r)\,dV+E_i; \tag{1} \]

\(H_0\) is the Hamiltonian of the medium without the complex; \(\hat{\varphi}(r)\) is the scalar potential of the field; \(\rho^{i}(r)\) is the charge density in each electronic state, and \(E_i\) are the electronic energy levels of the complex. Here the terms of the electron-vibrational interaction within the complexes are not written out, since their inclusion in the calculation of \(W_{1,2}\) can be done trivially within the framework of the known theory (\(^{2,3}\)) for oscillator Hamiltonians. For brevity the derivation is carried out for a homogeneous medium \(\varepsilon(r,\omega)=\varepsilon(\omega)\), although the final result (9) is written for arbitrary \(\varepsilon(r,\omega)\). The probability of the nonradiative transition \(W_{1,2}\), in first order of perturbation theory with respect to the interaction \(V_{1,2}\) between states 1, 2, is written (\(^{2,3}\)) in the form \((\hbar=1)\):

\[ W_{1,2}=|V_{1,2}|^{2}\int_{-\infty}^{\infty} G(t)\,dt;\qquad G(t)=\left\langle e^{iH_1t}e^{-iH_2t}\right\rangle_{H_1}, \tag{2} \]

where the brackets denote averaging over the equilibrium state of the system with Hamiltonian \(H_1\):
\[ \langle A\rangle_{H_1}=\mathrm{Sp}\{e^{-\beta(H_1-F_1)}A\},\quad \beta=1/kT. \]
In the Heisenberg representation with respect to \(H_1\), the function \(G(t)\) has the form:

\[ G(t)=e^{-i\Delta E t} \left\langle T\exp\left\{ -i\int_{0}^{t}dt\int dV\,\Delta\rho(r)\hat{\varphi}(r,t) \right\} \right\rangle_{H_1}, \tag{3} \]

where \(T\) is the chronological-ordering operator; \(\Delta\rho=\rho_2(r)-\rho_1(r)\), \(\Delta E=E_2-E_1\).

To calculate (3) we use the long-wavelength approximation developed in (\(^{6}\)). In this case the interaction of the particles of the medium with the field is divided into a short-range part and an interaction with the long-wavelength-

by photons. After expanding the exponentials (3) in a series, finding the \(n\)-th term of the series reduces to calculating the average:

\[ \int_0^t dt_1 \ldots \int_0^t dt_n \int \ldots \int dV_1 \ldots dV_n \Delta \rho(r_1) \ldots \Delta \rho(r_n)\, \frac{1}{n!}\, \langle T\varphi(r_1 t_1)\ldots \varphi(r_n,t_n)\rangle_{H_1} \tag{4} \]

(4) is represented by the sum of the contributions of all diagrams with \(n\) free photon ends. Following \({}^{(6)}\), we shall take into account only those diagrams in which there is no integration over the momenta of long-wavelength photons. In the language of diagrams, the quantities \(\langle \varphi(r)\rangle_{H_1}\) and \(D_{00}(r_1t_1,r_2t_2)=-i\langle T\delta\varphi(r_1,t_1)\times \delta\varphi(r_1,t_2)\rangle_{H_1}\), \(\delta\varphi=\varphi-\langle\varphi\rangle_{H_1}\), are equal to:

\[ \langle \hat{\varphi}\rangle_{H_1} = ---\bigcirc + ---\bigcirc---\bigcirc +\cdots, \qquad D_{\bullet\bullet} = ---\bigcirc--- + ---\bigcirc---\bigcirc--- +\cdots \tag{5} \]

Summing all diagrams of the indicated type, we obtain for (4) an expression of the form

\[ \frac{1}{n!}\sum_{m=0}^{[n/2]} f^{\,n-2m}g^m C_n^{2m}(2m-1)!!, \tag{6} \]

where

\[ g(t)=-i\int_0^t dt_1\int_0^t dt_2 \iint dV_1dV_2\,\Delta\rho(r_1)\Delta\rho(r_2)D_{00}(r_1,r_2,t_1-t_2). \tag{7} \]

\[ f(t)=-it\int \langle\varphi(r)\rangle_{H_1}\Delta\rho(r)dV = -\frac{it}{4\pi}\int \frac{\left|D^{\mathrm{II}}(r)-D^{\mathrm{I}}(r)\right|D^{\mathrm{I}}(r)} {\varepsilon(0)}\,dV. \]

Here \(D^i(r)\) is the induction vector in the \(i\)-th state. Summing all terms (6), we obtain

\[ G(t)=e^{f(t)}e^{g(t)}e^{-i\Delta Et}. \tag{8} \]

The function \(D_{00}(r_1,r_2,\omega)\) is related by a known relation to the retarded Green’s function \(D_{00}^R(r_1,r_2,\omega)\), which in turn is expressed through the dielectric permittivity \(\varepsilon(r,\omega)\) of the medium \({}^{(6)}\).

Taking all this into account and carrying out simple transformations, we obtain:

\[ G(t)=\exp\left\{ -i\widetilde{\Delta E}t + \int \frac{|\Delta D(r)|^2}{8\pi^2} \left[ \int_{-\infty}^{\infty} \frac{d\omega}{\omega^3} \frac{\varepsilon''(r,\omega)}{|\varepsilon(r,\omega)|^2} \times \frac{\operatorname{ch}(\hbar\omega/2kT-i\omega t)-\operatorname{ch}\hbar\omega/2kT} {\operatorname{sh}\hbar\omega/2kT} \right]dV \right\}, \tag{9} \]

where \(\Delta D(r)=D^{\mathrm{II}}(r)-D^{\mathrm{I}}(r)\) and

\[ \widetilde{\Delta E} = \Delta E + \int \left\{ \frac{|D^{\mathrm{I}}(r)|^2}{8\pi} - \frac{|D^{\mathrm{II}}(r)|^2}{8\pi} \right\} \left(1-\frac{1}{\varepsilon(r,0)}\right)dV - \]

is the transition energy with allowance for the change in the electromagnetic energy of the dielectric during the transition. Formulas (2), (9) solve the problem posed.

The authors express their gratitude for useful discussions of the work to N. D. Sokolov, I. V. Aleksandrov, V. G. Levich, R. R. Dogonadze, and A. Kuznetsov.

Institute of Chemical Physics
Academy of Sciences of the USSR

Received
27 IX 1968

CITED LITERATURE

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