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 $\chi^{(3)}(-(w_1+w_2+w_3);w_1,w_2,w_3)$ 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 $\chi^{(3)}(-w;0,0,w)$, DC-induced second harmonic generation $\chi^{(3)}(-2w;0,w,w)$, optical Kerr effect $\chi^{(3)}(-w;w,-w,w)$ and DC-electric-field-induced optical rectification $\chi^{(3)}(0;w,-w,0)$ are derived. By including or excluding ${\bf \nabla_k}$ 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 non-resonant regions. Hence it breaks the Kleinman 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 non-resonant region: $\chi^{(3)}(-3w;w,w,w)> \chi^{(3)}(-2w;0,w,w)> \chi^{(3)}(-w;w,-w,w)>\chi^{(3)}(-w; 0,0,w)>= \chi^{(3)}(0;w,-w,0)$, and in the resonant region: $\chi^{(3)}(-w;0,0,w)> \chi^{(3)}(-w;w,-w,w)> \chi^{(3)}(-2w;0,w,w)>\chi^{(3)}(0;w,-w,0)>\chi^{(3)}(-3w;w,w,w)$. (w=\omega) 
Shidong Jiang, Minzhong Xu