An Effective Hamiltonian Approach to Side Directed Tunneling
Written by Shira Weissman
Shira Weissman and Uri Peskin
The Schulich Faculty of Chemistry and The Lise Meitner Center for Computational Quantum Chemistry, Technion – Israel Institute of Technology, Haifa 32000, Israel
Understanding electronic dynamics in molecules, following perturbations such as charging, optical excitations etc, is of major importance. It is particularly interesting to predict the pathways for electron travel in complex molecular structures, following a particular initial local excitation. The interest in this topic had increased rapidly during the past years since understanding what affects the electron transfer can allow us to design new molecular structures that can be used as electric components, and to understanding existent biological components, based on electron transfer.
A convenient form of describing the molecular network is a tight binding model, where the molecule is divided into N discrete parts named "sites". Each site is characterized with an effective single electron potential (pseudo potential). The Hamiltonian of the system is an NxN matrix with the diagonal terms being the sites' potential energy and the off-diagonal being the coupling terms. One would be interested in predicting the electron dynamics for a given initial conditions, such as a full electron occupation of one of the sites which is termed "the donor". A special interest is in the case where the dynamics lead to a full occupation of one of the other site that would be termed "the acceptor".
Previous research showed that this kind of dynamics can be obsserved when the donor and the acceptor are close in energy and well separated from the other sites, termed "the bridge" (derived from the assumption that their energies are much higher than the donor/acceptor energies). One is interested in a reduced model for describing the direct interaction between the donor and acceptor, under the effective bridge influence. At the perturbative limit, where the coupling energy terms are much smaller then the bridge-donor/acceptor gap, such a reduced model was derived. Farther research showed however that this kind of dynamics exists beyond the perturbative regime as well.
The present research introduces a non-perturbative approach for calculating the reduced Donor-Acceptor model. We use Feshbach projection operators' formalism for dividing the Hamiltonian into two subspace, the donor and acceptor subspace and the bridge subspace. Using algebraic manipulations the bridge subspace appears in the donor/acceptor subspace as a self energy term. This model allows us to easily design a complex molecular network with an effective coupling between a donor and a particular acceptor. Additionally, the different pathways for electronic tunneling in the molecular network can be identified.