1232 ISSN 1023-1935, Russian Journal of Electrochemistry, 2017, Vol. 53, No. 10, pp. 1232–1238. © Pleiades Publishing, Ltd., 2017. Original Russian Text © A.S. Berezin, R.R. Nazmutdinov, 2017, published in Elektrokhimiya, 2017, Vol. 53, No. 10, pp. 1390–1397. Monte Carlo Simulations of Heterogeneous Electron Transfer: New Challenges 1, 2 A. S. Berezin and R. R. Nazmutdinov* Kazan National Research Technological University, Kazan, 420015 Russia *e-mail: nazmutdi@mail.ru Received February 14, 2017; in final form, April 4, 2017 Abstract—We report results of MC simulations of electron transfer across a metal electrode/electrolyte solu- tion interface. The model presumes the Landau–Zener theory and a random walk on a two-dimensional lat- tice formed by crossing parabolic reaction free energy surfaces along the solvent coordinate. Emphasis is put on investigating the activationless discharge regime; the bridge-assisted electron transfer is also partially addressed. We have calculated effective electronic transmission coefficient as a function of the electrode over- potential and temperature in a wide range of orbital overlap. The dependence of the transmission coefficient on the electronic density of states is analyzed as well. Keywords: electron transfer, Monte Carlo simulations, electronic transmission coefficient, activationless dis- charge, bridge-assisted electron transfer DOI: 10.1134/S1023193517100032 1. INTRODUCTION The elementary act of electron transfer (ET) across a solid/electrolyte solution interface is broadly more complicated as compared with homogeneous redox processes. The first reason is the adsorption of reac- tant (product) for inner-sphere reactions, while the second complication results from a manifold of elec- tronic energy levels which might contribute to the ele- mentary act. The latter presumes a family of crossing free energy surfaces along the reaction coordinate which should be addressed when considering a hetero- geneous charge transfer in terms of the quantum mechanical theories (see, for example, Refs. [1–4]). This issue is important for calculations of electronic transmission coefficient and can be treated analyti- cally in two kinetic regimes: diabatic (i.e. weak elec- tronic coupling) and adiabatic (strong coupling) lim- its. However, in the important intermediate region only numerical approaches seem to be efficient. Monte Carlo (MC) simulations which mimic a ran- dom walk through nodes of a 2D-lattice formed by crossing reaction free energy surfaces (RFES) along the solvent coordinate were performed first in work [5]; the crossing plots of RFES were considered in a linear approximation. Later this model was extended to parabolic RFES in Ref. [6]; the electrode overpo- tential effect was also partially addressed. The aim of the present work is to elucidate three issues of heterogeneous ET using the MC technique employed in Ref. [5]. First of all we investigate not only normal but also activationless region when calcu- lating an effective electronic transmission coefficient ( ). It has been shown in Refs. [7, 8] on the basis of quantum mechanical theory that for example, the reduction of the [Fe(CN) 6 ] 3– and peroxodisulphate anions at a mercury electrode (thoroughly investigated experimentally by the Frumkin’s school) proceeds in a near-activationsless region. Secondly, we consider in a more detail the dependence of on the electronic density of states (DOS) in an arbitrary region of orbital overlap. Finally the dependences of on the DOS and temperature for a bridge assisted ET are calculated as well. These problems were not discussed earlier in works [5, 6]. 2. MODEL AND COMPUTATIONAL DETAILS Let us consider for the sake of simplicity an outer- sphere one electron reduction proceeding at a metal electrode/electrolyte solution interface without bond break and with a small intramolecular reorganization. It is also assumed that the influence of the reactant— electrode orbital overlap on the activation barrier is small and can be neglected. Then the reaction energy 1 This article is a contribution of the authors to the special journal issue dedicated to the centenary of the birth of outstanding elec- trochemist, corresponding member of the Academy of Sciences of the USSR, Veniamin Grigor’evich Levich (1917–1987). 2 The article was translated by the authors. κ κ κ