Inversion operator four-component

Characterization of Molecular Hyperpolarizabilities Using Third Harmonic Generation. Third harmonic generation (THG) is the generation of light at frequency 3co by the nonlinear interaction of a material and a fundamental laser field at frequency co. The process involves the third-order susceptibility x 3K-3 , , ) where —3 represents an output photon at 3 and the three s stand for the three input photons at . Since x(3) is a fourth (even) rank tensor property it can be nonzero for all material symmetry classes including isotropic media. This is easy to see since the components of x(3) transform like products of four spatial coordinates, e.g. x4 or x2y2. There are 21 components that are even under an inversion operation and thus can be nonzero in an isotropic medium. Since some of the terms are interrelated there are only four independent terms for the isotropic case. 

It should be noted that the formulas in eqs (10.84)-(10.93) have been derived by considering only the operators from the leading order in the inverse speed of light. One may obtain relativistic corrections by carrying out the expansion to higher orders, but this rapidly becomes quite involved, as many different operators and their combinations can make contributions to a given property. For systems where relativistic effects are important, a full four-component type calculation (Section 8.4) becomes attractive, at least conceptually, since it automatically includes all effects without the necessity of multiple perturbation operators. 

Of these classes of operations the first three are trivial to visualize, while the last two are represented schematically in Fig. 3.12. When inversion is added to these operations, the total number of images of the arbitrary point becomes 48, since inversion can be combined with any rotation to produce a new operation. It is easy to rationalize this result there are 48 different ways of rearranging the three cartesian components of the arbitrary point, including changes of sign the first coordinate can be any of six possibilities ( x, y, z), the second coordinate can be any of four possibilities, and the last coordinate any of two possibilities. 

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