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We present a quaternion inspired formalism specifically developed to evaluatethe intensity of the electrical current that traverses a single moleculeconnected to two semi-infinite electrodes as the applied external bias isvaried. The self-adjustment of the molecular levels is fully described at adensity functional ab initio quantum chemical level. Use of a quaternionapproach allows for an integrated treatment of both coherent (ballistic) andnon-coherent (co-tunneling) contributions to the effective charge transport,where the latter involve the existence of transient charged states of thecorresponding molecular species. An expression for the net current iscalculated by using second-order perturbation theory to take into account allpossible transitions between states localized at the two different electrodesthat involve intermediary levels in the so-called "extended molecule" complexthat comprises the system of interest attached to two small metallic clusters.We show that by a judicious choice of the relevant molecular parameters, theformalism can be extended to describe the electronic transport both inconjugated as in saturated molecules, where localized orbitals are more likelyto be found. In this manner, the method can be applied to the full range ofcoupling regimes, not only to the weak or strong cases, but also inintermediate situations, where ballistic and co-tunneling processes maycoexist.

A Quaternion Based Quantum Chemical ab initio Treatment of Coherent and Non-Coherent Electron Transport in Molecules