On the quasi-classical limit of the quadratic susceptibility

For autonomous Hamiltonian systems, the quasi-classical limit ($\hbar\to0$) of the quadratic susceptibility to an external harmonic field is considered. To calculate this limit, the coordinate matrix elements and the quantum transition frequencies are expanded in powers of $\hbar$ up to terms of order $\hbar^2$ based on symmetry relations and sum rules. The quasi-classical limit of the quadratic susceptibility is calculated in terms of classical parameters and can be used to determine the response functions of chaotic systems.