  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)
