A Bit on Models

Science is largely about the world around us, about reality insofar as we can grasp it. But since the days of Euclid, and particularly since Lucretius, scientists have constructed models - that is, scientists have made simulacra, either conceptual or physical, in an 

Several categories of models appear as the basis for the study of molecular electronics in general, and molecular transport junctions in particular. These are the geometrical (or molecular), Hamiltonian, and transport analysis models. 

The molecular models are in a sense a subset of the geometrical ones - we assume that we know which molecules are present and we assume that we know their geometries (indeed sometimes we assume more than that, such as the usual assumption that thiol end groups lose their protons when forming their asymmetric bond with gold). In this we also necessarily assume that there are no other species, either on the electrode surface or in the surrounding media, that influence the current flow through the system. 

there are model Hamiltonians. Effectively a model Hamiltonian includes only some effects, in order to focus on those effects. It is generally simpler than the true full Coulomb Hamiltonian, but is made that way to focus on a particular aspect, be it magnetization, Coulomb interaction, diffusion, phase transitions, etc. A good example is the set of model Hamiltonians used to describe the IETS experiment and (more generally) vibronic and vibrational effects in transport junctions. Special models are also used to deal with chirality in molecular transport junctions [42, 43], as well as optical excitation, Raman excitation [44], spin dynamics, and other aspects that go well beyond the simple transport phenomena associated with these systems. 

The Hamiltonian models are broadly variable. Even for an isolated molecule, it is necessary to make models for the Hamiltonian - the Hamiltonian is the operator whose solutions give both the static energy and the dynamical behavior of quantum mechanical systems. In the simplest form of quantum mechanics, the Hamiltonian is the sum of kinetic and potential energies, and, in the Cartesian coordinates that are used, the Hamiltonian form is written as 

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