ELECTRON-TRANSFER AND ENERGY-TRANSFER THROUGH BRIDGED SYSTEMS .3. TIGHT-BINDING LINKAGES WITH ZERO OR NONZERO ASYMPTOTIC BAND-GAP

The simplest analytical model for bridge-assisted electron or hole tra nsfer processes is that of McConnell. It gives an approximate solution for non-resonant transfer through a bridge described by a simple Huck el hamiltonian embodying no asymptotic band gap. We investigate the an alytical solution of this problem, recently obtained by Evenson and Ka rplus, for the weak donor/acceptor-to-bridge coupling limit. Exponenti al falloff of the coupling with increasing bridge length is predicted to occur in all regions of the parameter space, except that which is e xtremely close to or within the resonance region; this result is consi stent with McConnell's equation, but applies far more generally, and n ew and more accurate limits for the validity of the McConnell equation are derived. The analysis is extended to consider bridges described b y a Huckel hamiltonian containing two different intrabridge nearest-ne ighbour coupling parameters, as is appropriate for a sigma-bonded brid ge or a pi-bonded bridge with alternating single and double bonds. Suc h systems contain a finite asymptotic band gap between occupied and vi rtual bridge orbitals, and possibly also one or two non-bonding levels . Resonance conditions are obtained, as are analytical solutions for t he 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 difficul t to achieve inside a narrow band gap. Analogous equations to McConnel l's are derived and the effects of 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 eig enstate, a non-bonding level, at the centre of the band gap, and reson ance with this level must also be avoided in order to obtain regular e xponential falloff. However, rather than to avoid such situations, our ultimate goal is to design systems which are resonant or nearly reson ant in which the coupling decreases slowly or even oscillates with inc reasing bridge length. Odd-length bridges such as Brooker-dye ions, wi th their extra non-bonding level and necessarily smaller band gaps, ar e thus expected to conduct considerably better than systems studied so far which always tend to be even-length bridges.