On the electronic structure analysis for one redox-active molecule S. Corni a, * , R. Di Felice a , E. Molinari a,b a INFM Centre on nanoStructures and bioSystems at Surfaces (S3), c/o Dipartimento di Fisica, Universit a di Modena e Reggio Emilia, v. Campi 213/A, I-41100 Modena, Italy b Dipartimento di Fisica, Universit a di Modena e Reggio Emilia, v. Campi 213/A, I-41100 Modena, Italy Received 3 March 2004 Abstract A theoretical method is proposed to circumvent the orbital relaxation problem (variations in the orbital composition when the electron number in the system changes) in the analysis of the calculated electronic structure of a redox-active molecule. The method is based on a convenient partition of the electron-transfer integral: one contribution depends on the coupling of the molecule with the external redox partner; the other is an intrinsic feature of the molecule under study and takes into account the orbital relaxation upon reduction/oxidation. The method is applied to the electronic structure analysis of the active site of the electron-transfer protein azurin. Ó 2004 Elsevier B.V. All rights reserved. 1. Introduction The rate of an electron transfer (ET) reaction de- pends, following Marcus [1], on three main ingredients [2]: the reorganisation energy (the free energy variation that accompanies the rearrangement of the nuclear de- grees of freedom from the initial state-reactants to the final state-products once the ET has taken place); the nuclear frequency factor (the frequency of passage across the transition state barrier); and the electronic coupling matrix element between the electronic states of the reactants and the products. In the present work we shall focus on the latter quantity, which is also called transfer integral [3]. An intuitive framework for a qualitative under- standing of the electronic coupling is the single particle picture [3]. In such a picture each electron has a well- defined state, represented in the spin-coordinates space by a spin–orbital. These states are the same in both the reactants (the complex formed by the donor and the acceptor) and the products (the same complex where an electron moved from the donor to the acceptor) of the ET process. The only change is the variation in the oc- cupation numbers of two single-particle states: one, mainly localised on the donor, was occupied and be- comes virtual, the other, mainly localised on the accep- tor, was virtual and becomes occupied. All the other (‘core’) states maintain their shape and occupation numbers unaltered. This simple picture can be straightforwardly applied if the spin–orbitals are calculated by using empirical electronic structure techniques such as the Extended H€ uckel method [4]. However, if more accurate tech- niques are used, such as Hartree–Fock (HF) or Kohn– Sham density functional theory (DFT), one has to face the problem that, when the occupation numbers of the donor and the acceptor orbitals are changed, their shape (and the shape of all the other orbitals in the donor– acceptor complex) varies (spin–orbital relaxation), be- cause the mean-field in the system changes. Even if one is interested only in simple qualitative considerations on the strength of the electronic coupling (i.e., the magni- tude of the transfer integral), these spin–orbital rota- tions may create problems, since it is no longer possible to clearly assign the occupation number variation to just two orbitals. To overcome this problem, Newton [5] exploited the corresponding orbital method [6] to define * Corresponding author. Fax: +39-059-374-794. E-mail address: corni.stefano@unimore.it (S. Corni). 0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.06.025 Chemical Physics Letters 393 (2004) 118–123 www.elsevier.com/locate/cplett