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Interfacial Electrochemistry$
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Wolfgang Schmickler

Print publication date: 1996

Print ISBN-13: 9780195089325

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780195089325.001.0001

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PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 27 July 2021

Quantum theory of electron-transfer reactions

Quantum theory of electron-transfer reactions

Chapter:
(p.259) 19 Quantum theory of electron-transfer reactions
Source:
Interfacial Electrochemistry
Author(s):

Wolfgang Schmickler

Publisher:
Oxford University Press
DOI:10.1093/oso/9780195089325.003.0026

The theory of electron-transfer reactions presented in Chapter 6 was mainly based on classical statistical mechanics. While this treatment is reasonable for the reorganization of the outer sphere, the inner-sphere modes must strictly be treated by quantum mechanics. It is well known from infrared spectroscopy that molecular vibrational modes possess a discrete energy spectrum, and that at room temperature the spacing of these levels is usually larger than the thermal energy kT. Therefore we will reconsider electron-transfer reactions from a quantum-mechanical viewpoint that was first advanced by Levich and Dogonadze. In this course we will rederive several of, the results of Chapter 6, show under which conditions they are valid, and obtain generalizations that account for the quantum nature of the inner-sphere modes. By necessity this chapter contains more mathematics than the others, but the calculations are not particularly difficult. Readers who are not interested in the mathematical details can turn to the summary presented in Section 6. To be specific we consider electron transfer from a reactant in a solution, such as [Fe(H2O)6]2+, to an acceptor, which may be a metal or semiconductor electrode, or another molecule. To obtain wavefunctions for the reactant in its reduced and oxidized state, we rely on the Born-Oppenheimer approximation, which is commonly used for the calculation of molecular properties. This approximation is based on the fact that the masses of the nuclei in a molecule are much larger than the electronic mass. Hence the motion of the nuclei is slow, while the electrons are fast and follow the nuclei almost instantaneously. The mathematical consequences will be described in the following. Let us denote by R the coordinates of all the nuclei involved, those of the central ion, its ligarids, and the surrounding solvation sphere, and by r the coordinates of all electrons.

Keywords:   Born-Oppenheimer approximation, Coulomb repulsion, Fermi's golden rule, Marcus theory, Schrödinger equation, adiabatic, effective mass, harmonic oscillator, inner-sphere mode, nuclear wavefunctions

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