Origin density dipole polarizability

The microscopic origin of the nonlinear response is the distortion induced in the molecular charge distribution due to the electrical field. The presence of a microscopic dipole produces a macroscopic polarization in the unit volume P = N r) where N is the number density of polarizable units and (er) the expectation value of the dipole moment induced in each unit. In order to evaluate (sr) we will use the density matrix formalism, because it is the easiest way to relate microscopic properties to macroscopic ones and to cope with macroscopic coherence effects. In the absence of fields, the medium is supposed to be described by an unperturbed Hamiltonian Hq and to be at equilibrium. When the fields are applied, the field-matter interaction contributes a time-dependent term V(t) =-E(t)P(t) to the global energy. The evolution of the system under this perturbation can be described through the equation of motion of the density operator ... 

The first-order correction to the charge density is independent of the choice of origin for the dipole moment operator, as the extra contribution to Do arising from a shift 5R in origin is — e5R<0 /c>, a term which vanishes because of the orthogonality of the zero-order states. The expression for the electric polarizability density (r) using the above expression for the perturbed density is... 

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