Hyperpolarizabilities for the one-dimensional infinite single-electron periodic systems. I. Analytical solutions under dipole-dipole correlations

The analytical solutions for the general-four-wave-mixing hyperpolarizabilities χ(3) [- (ω1 + ω2 + ω3); ω1, ω2, ω3] on infinite chains under both Su-Shrieffer-Heeger and Takayama-Lin-Liu-Maki models of trans-polyacetylene are obtained through the scheme of dipole-dipole correlation. Analytical expressions of dc Kerr effect χ(3) (-ω;0,0,ω), dc-induced second-harmonic generation χ(3) (-2ω;0,ω,ω), optical Kerr effect χ(3) (-ω;ω,- ω,ω), and dc electric-field-induced optical rectification χ(3) (0;ω,-ω,0) are derived. By including or excluding ∇k terms in the calculations, comparisons show that the intraband contributions dominate the hyperpolarizabilities if they are included. ∇k term or intraband transition leads to the break of the overall permutation symmetry in χ(3) even for the low-frequency and nonresonant regions. Hence it breaks the Kleinman [Phys. Rev. 126, 1977 (1962)] symmetry that is directly based on the overall permutation symmetry. Our calculations provide a clear understanding of the Kleinman symmetry breaks that are widely observed in many experiments. We also suggest a feasible experiment on χ(3) to test the validity of overall permutation symmetry and our theoretical prediction. Finally, our calculations show the following trends for the various third-order nonlinear optical processes in the low-frequency and nonresonant region: χnonres (3) (-3ω;ω,ω,ω) > χnonres (3) (-2ω;0,ω, ω) > χnonres (3) (-ω;ω,-ω,ω) > χnonres (3) (-ω;0,0,ω) ≥ χnonres (3) (0;ω,-ω, 0), and in the resonant region: χres (3) (-ω;0,0,ω) > χres (3) (-ω;ω,-ω,ω) > χres (3) (-2ω;0,ω,ω) > χres (3) (0;ω,-ω,0) > χres (3) (-3ω;ω,ω,ω).