Band-resolved analysis of nonlinear optical properties of crystalline and molecular materials

Through three case studies, plane wave pseudopotential density functional theory calculations are performed to investigate the mechanism of second harmonic generation (SHG) of the nonlinear optical materials. We want to know how SHG coefficients are affected by: (1) anion substitution in $\mathrm{AgGa}{({\mathrm{S}}_{x}{\mathrm{Se}}_{1\ensuremath{-}x})}_{2}$; (2) the isomeric effect in push-pull benzenes; and (3) the length of $\ensuremath{\pi}$-conjugate chain in push-pull polyenes. A sum-over-states type formalism is used for the evaluation of static SHG coefficients. Orbital contribution to the $\ensuremath{\beta}$ or ${\ensuremath{\chi}}^{(2)}$ can be decomposed on a (energy) level-by-level basis by partially summing only two out of all three band indices. Through such process that we proposed, the dominant orbitals that give major contribution to a SHG process can then be identified and analyzed. These so-called ``band-resolved'' plots of SHG coefficient are sensitive to the variation of structure and configuration of the materials, and can thus be used to study the detailed mechanism of SHG processes in crystals and molecules.