Electron transfer and energy transfer through bridged systems III. Tight-binding linkages with zero or non-zero asymptotic band gap

Abstract

The simplest analytical model for bridge-assisted electron or hole transfer processes is that of McConnell. It gives an approximate solution for non-resonant transfer through a bridge described by a simple Hückel hamiltonian embodying no asymptotic band gap. We investigate the analytical solution of this problem, recently obtained by Evenson and Karplus, for the weak donor/acceptor-to-bridge coupling limit. Exponential fall of of the coupling with increasing bridge length is predicted to occur in all regions of the parameters space, except that which is extremely close to or within region; this result is consistent with McConnell's equation, but applies far more generally, and new and more accurate limits for the validity of the McConnell equation are derived. The analysis is extended to consider bridges described by a Hückel hamiltonian containing two different intrabridge nearest-neighbour coupling parameters, as is appropriate for a σ-bonded bridge or a π-bonded bridge with alternating single and double bonds. Such system contain a finite asymptotic band gap between occupied and virtual bridge orbitals, and possibly also one or two non-bonding levels. Resonance are obtained, as are analytical solutions for the coupling when the donor and acceptor levels lie either outside the occupied and virtual bands or within the band gap; exponential falloff is again predicted away from the resonances, but this may be difficult to achieve inside a narrow band gap. Analogous equations to McConnell's are derived and the effects for all approximations used, including that of an effective two-level hamiltonian, are considered numerically. Bridges with an odd number of functions are shown to contain one eigenstate, a non-bonding level, at the centre of the band gap, and the resonance with this level must also be avoided in order to obtain regular exponential falloff. However, rather than to avoid such situations, our ultimate goal is to design systems which are resonant or nearly resonant in which the coupling decreases slowly or even oscillates with increasing bridge length. Odd- length bridges such as Brooker-dye ions, with their extra non-bonding level and necessarily smaller band gaps, are thus expected to conduct considerably better than systems studied so far which always tend to be even- length bridges.