A model for bond-breaking electron transfer at metal electrodes E. Santos a , M.T.M. Koper b , W. Schmickler c, * a Faculdad de Matematica, Astronomia y Fisica, Universidad Nacional de Cordoba, 5000 Cordoba, Argentina b Leiden Institute of Chemistry, Leiden University, P.O. Box 9502, 2300 RA Leiden, The Netherlands c Department of Theoretical Chemistry, University of Ulm, Abteilung Elektrochemie, D-89069 Ulm, Germany Received 23 October 2005; in final form 12 November 2005 Available online 27 December 2005 Abstract A model Hamiltonian is proposed for bond-breaking electrochemical electron transfer to an adsorbed molecule. The theory is applied to a homonuclear molecule, and self-consistent equations are derived for the occupation probabilities of the orbitals and the system energy. Model calculations result in an adiabatic potential energy surface with a minimum for the adsorbed state with the bond intact and a valley for the two dissociated anions. Ó 2005 Elsevier B.V. All rights reserved. 1. Introduction The theory of electrochemical electron transfer started with the work of Marcus [1] and Hush [2] on outer-sphere reactions; it is now about 50 years old and well established. However, most technologically important reactions do not have an outer-sphere character, but involve the breaking of bonds, and often the specific adsorption of the reactants at the electrode surface. Models for bond-breaking electron transfer, without specific adsorption, were proposed much later by Save ´ant [3,4] and German and Kuznetsov [5]. While these give important insights into the nature of bond-breaking, and often compare quite well with experiment, they are essentially extensions of the Marcus and Hush theories for outer-sphere reactions and treat the reaction as a purely classical, thermally activated process. A strictly quantum formulation was proposed by Koper and Voth [6,7], which made it possible to explore quantum effects, such as tunneling transitions [8] and Frank–Condon effects [9]. However, by ignoring spin this formalism is again limited to outer-sphere reactions. In addition, the potential energy curves for the breaking bond do not fol- low from the electronic Hamiltonian, but are introduced in an ad hoc manner by an operator which switches between the reduced and the oxidized states. Similarly, an ad hoc resonance splitting was also introduced in theories for dissociative adsorption on electrodes by Koper and Voth [10] and German et al. [11], to effect the transition between the bonded state and the separated adsorbed fragments. In this work, we want to take the theory of electrochem- ical reactions a significant step further by proposing a scheme that allows the treatment of adsorbed reactants, and which considers spin effects explicitly. The potential energy curves for the breaking bond will follow naturally from the Hamiltonian, without resource to ad hoc switch- ing terms between the various electronic states. We shall apply our model to the breaking of a bond in a diatomic molecule according to the scheme A 2 + 2e  ! 2A  , and develop a formalism to calculate the corresponding poten- tial energy surfaces. 2. The model Hamiltonian We formulate our model for the simple case of a dia- tomic molecule adsorbed on an electrode surface and con- sider one valence orbital explicitly; an extension to more 0009-2614/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.11.118 * Corresponding author. Fax: +49 731 502 5409. E-mail address: Wolfgang.Schmickler@Chemie.Uni-Ulm.de (W. Schmickler). www.elsevier.com/locate/cplett Chemical Physics Letters 419 (2006) 421–425