Analytical expressions for the vibrational static second hyperpolarizability of polyacetylene

Analytical results for the vibrational static second hyperpolarizability of polyacetylene are derived. The vibrational second hyperpolarizability of polyacetylene is calculated through the lattice relaxation expression in the solid-state limit. The exact results indicate that the vibrational contribution to the second hyperpolarizabilities of polyacetylene is comparable to the pure electronic contribution. Additionally, the vibrational and electronic second hyperpolarizabilities of polyacetylene under the parabolic approximation for the energy band are also obtained, and they are in excellent qualitative agreement with the exact results. Therefore, the vibrational contribution to the static second hyperpolarizability of polyacetylene is as important as the electronic contribution.