Electron transfer reactions: generalized spin-boson approach
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Electron transfer reactions: generalized spin-boson approach M. Merkli · G. P. Berman · R. Sayre
Received: 4 September 2012 / Accepted: 28 November 2012 / Published online: 13 December 2012 © Springer Science+Business Media New York 2012
Abstract We introduce a mathematically rigorous analysis of a generalized spinboson system for the treatment of a donor–acceptor (reactant-product) quantum system coupled to a thermal quantum noise. The donor/acceptor probability dynamics describes transport reactions in chemical processes in presence of a noisy environment – such as the electron transfer in a photosynthetic reaction center. Besides being rigorous, our analysis has the advantages over previous ones that (1) we include a general, non energy-conserving system-environment interaction, and that (2) we allow for the donor or acceptor to consist of multiple energy levels lying closely together. We establish explicit expressions for the rates and the efficiency (final donor–acceptor population difference) of the reaction. In particular, we show that the rate increases for a multi-level acceptor, but the efficiency does not. Keywords Donor-acceptor quantum system · Reactant-product quantum system · Thermal quantum noise · Electron transfer · Photosynthetic reaction center · Degenerate donor/acceptor · Mathematically rigorous spin-boson model · Quantum resonances · Relaxation time
M. Merkli (B) Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7, Canada e-mail: [email protected] http://www.math.mun.ca/~merkli G.P. Berman Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA e-mail: [email protected] R. Sayre Los Alamos National Laboratory and New Mexico Consortium, 202B Research Center, Los Alamos, NM 87544, USA e-mail: [email protected]
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J Math Chem (2013) 51:890–913
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1 Introduction 1.1 Transfer reactions and spin-boson model An important problem in chemistry and biology is to find electron transfer rates and transfer efficiencies in chemical reactions. A prominent example is the electron transfer in proteins carrying out photosynthesis [2–5]. The simplest reactions are described by two states, a reactant (electron donor) and a product (electron acceptor). Before the reaction, the system is localized mainly in the reactant state, and after mainly in the product state. The passage from reactant to product is induced by two effects: a direct tunneling (hopping) and an indirect transition. The former originates from electron tunneling between reactant and product moieties, while the latter is due to the presence of thermal noise created by the many protein atoms and molecules in which the electron donor and acceptor are embedded. Denoting the reactant and product states by |R and |P, respectively, a “Marcus model” Hamiltonian for the electron exchange has been used in [6,7], HMarcus = |RE R R| + |PE P P| + |RV P| + |PV R|, where E R and E P are the reactant and product energies, and V is the direct tunneling constant. Both E
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