Electron transfer to heteronuclear diatomic molecules E. Santos a,b , S. Bartenschlager a , W. Schmickler a,⇑ a Institute of Theoretical Chemistry, Ulm University, D-89069 Ulm, Germany b Facultad de Matemáticas, Astronomı ´ a y Fı ´ sica, IFEG-CONICET, Universidad Nacional de Córdoba, Argentina article info Article history: Received 8 July 2010 Received in revised form 18 October 2010 Accepted 16 February 2011 Available online 21 March 2011 Keywords: Electron transfer Metal d-band Density of states abstract A previously proposed theory for electron exchange between a metal electrode and a diatomic molecule is generalized to the case where the two atoms are not equivalent. The Green’s function for the molecule is derived and used to calculate the densities of states of the atoms and of the molecule. A few examples demonstrate the effect of the interaction with a wide metal sp-band and with a narrow d-band. It is sug- gested that the formalism can be linked to density-functional theory in order to perform quantitative cal- culations for particular reactions. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction Understanding electrocatalytic reactions is one of the big chal- lenges for theoretical electrochemistry, and much research is being devoted to this task. At present, two different approaches can be discerned, which do not necessarily exclude each other. One ap- proach follows and extends the theory of electron transfer to cata- lytic processes, the other approach is computational and relies on density-functional theory (DFT) to describe electrochemical reac- tions. The theoretical line started with the pioneering works of Marcus [1] and Hush [2], and was then taken up by the Soviet school [3], to which Alexander Kuznetsov made so many important contributions [4]. In recent years, it has focused on bond-breaking or forming reactions. Early work by Saveant [5] was an extension of Marcus theory, while later theories, to which Alexander Kuznetsov again contributed much, were based on model Hamiltonians [6–11]. By their very nature, DFT calculations are less general and always performed for specific systems. This is not the place to give a survey of the extensive work that has been performed, but good examples and further references can be found in two recent volumes dedicated to electrocatalysis [12,13]. Recent work from our group has started from theory, which we later combined with DFT in order to investigate the hydrogen reac- tion. Thus, together with M. Koper, we have proposed a model Hamiltonian for bond-breaking reactions [14,15], which served as a basis to explain the general principles of electrocatalysis [16]. In order to apply this work to hydrogen evolution, we used exten- sive DFT calculations to obtain the system parameter, and to correct for the exchange and correlation terms that are missing in the theory [17,18]. In this work we return to pure theory and generalize the Hamiltonian for bond-breaking. In our previous work [14,15] we had only considered the symmetric case, in which the molecule consisted of two identical atoms A. Here we present the formalism for the general case of a heteronuclear molecule A–B, in which the atoms A and B can either be chemically different species, or the same atoms but in different positions. A good example for the latter process is the Heyrowsky reaction, in which initially one hydrogen atom is adsorbed on the surface, the other approaches in the form of proton, and the two combine to form a hydrogen molecule. 2. The model Hamiltonian The model that we investigate is based on the Hamiltonian pro- posed by Santos, Koper, and Schmickler (SKS) [14,15], which had previously been solved only for the case of two identical atoms. In order to keep this article self-contained, we present the Hamil- tonian, before we proceed to calculate the Green’s functions for the general case. Thus, we consider two atoms A and B which interact with each other, with a metal electrode, and with the solvent. It is convenient to separate the Hamiltonian into several parts. We start with the terms for the two atoms A and B; on each atom we con- sider a valence orbital labeled a and b, respectively, each of which can take up two electrons with opposite spin. In the tight-binding (Hückel) approximation the corresponding terms are: H ab ¼ X r  a n a;r þ  b n b;r þ U a n a;r n a;r þ U b n b;r n b;r  þ bc þ ar c br þ bc þ br c ar  ð1Þ 1572-6657/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2011.02.014 ⇑ Corresponding author. E-mail address: wolfgang.schmickler@uni-ulm.de (W. Schmickler). Journal of Electroanalytical Chemistry 660 (2011) 314–319 Contents lists available at ScienceDirect Journal of Electroanalytical Chemistry journal homepage: www.elsevier.com/locate/jelechem