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 chi(3)[-(omega1 + omega2 + omega3);omega1,omega2,omega3] 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 chi(3)(-omega;0,0,omega), dc-induced second-harmonic generation chi(3)(-2omega;0,omega,omega), optical Kerr effect chi(3)(-omega;omega,-omega,omega), and dc electric-field-induced optical rectification chi(3)(0;omega,-omega,0) are derived. By including or excluding nabla(k) terms in the calculations, comparisons show that the intraband contributions dominate the hyperpolarizabilities if they are included. nabla(k) term or intraband transition leads to the break of the overall permutation symmetry in chi(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 chi(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: chi(nonres)(3)(-3omega;omega,omega,omega)>chi(nonres)(3)(-2omega;0,omega,omega)>chi(nonres)(3)(-omega;omega,-omega,omega)>chi(nonres)(3)(-omega;0,0,omega) > or = chi(nonres)(3)(0;omega,-omega,0), and in the resonant region: chi(res)(3)(-omega;0,0,omega)>chi(res)(3)(-omega;omega,-omega,omega)>chi(res)(3)(-2omega;0,omega,omega)>chi(res)(3)(0;omega,-omega,0)>chi(res)(3)(-3omega;omega,omega,omega).