Electric hyperpolarizability tensor, quadratic

From, for instance, Bloembergen (1965) and Stevens et al. (1963), we see that the quadratic response function is the r , r , r "th component of the dynamic hyperpolarizability tensor, as it should be according to standard electric response theory. The residues of the quadratic response functions provide information about transition moments between two excited states, i.e. neither of the two states is the reference state 0>. More specifically we see from... 

The fundamental equation (1) describes the change in dipole moment between the ground state and an excited state jte expressed as a power series of the electric field E which occurs upon interaction of such a field, as in the electric component of electromagnetic radiation, with a single molecule. The coefficient a is the familiar linear polarizability, ft and y are the quadratic and cubic hyperpolarizabilities, respectively. The coefficients for these hyperpolarizabilities are tensor quantities and therefore highly symmetry dependent odd order coefficients are nonvanishing for all molecules but even order coefficients such as J3 (responsible for SHG) are zero for centrosymmetric molecules. Equation (2) is identical with (1) except that it describes a macroscopic polarization, such as that arising from an array of molecules in a crystal (10). 

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