Calculation of electronic coupling matrix elements for ground and excited state electron transfer reactions: Comparison of the generalized Mulliken–Hush and block diagonalization methods

Two independent methods are presented for the nonperturbative calculation of the electronic coupling matrix element ${\mathrm{(H}}_{\mathrm{ab}})$ for electron transfer reactions using ab initio electronic structure theory. The first is based on the generalized Mulliken–Hush (GMH) model, a multistate generalization of the Mulliken Hush formalism for the electronic coupling. The second is based on the block diagonalization (BD) approach of Cederbaum, Domcke, and co-workers. Detailed quantitative comparisons of the two methods are carried out based on results for (a) several states of the system ${\mathrm{Zn}}_{\mathrm{2}}{\mathrm{OH}}_{\mathrm{2}}^{\mathrm{+}}$ and (b) the low-lying states of the benzene–Cl atom complex and its contact ion pair. Generally good agreement between the two methods is obtained over a range of geometries. Either method can be applied at an arbitrary nuclear geometry and, as a result, may be used to test the validity of the Condon approximation. Examples of nonmonotonic behavior of the electronic coupling as a function of nuclear coordinates are observed for ${\mathrm{Zn}}_{\mathrm{2}}{\mathrm{OH}}_{\mathrm{2}}^{\mathrm{+}}.$ Both methods also yield a natural definition of the effective distance ${\mathrm{(r}}_{\mathrm{DA}})$ between donor $\mathrm{(D)}$ and acceptor $\mathrm{(A)}$ sites, in contrast to earlier approaches which required independent estimates of $r_{\mathrm{DA}},$ generally based on molecular structure data.