Numerical Liouville approach: A calculation method for nonlinear optical susceptibilities of N-state systems

We develop an alternative calculation method for the nth-order nonlinear optical susceptibilities ${\mathrm{\chi }}_{\mathrm{g}}^{\mathrm{}\left(\mathrm{n}\right)\mathrm{}}$(ω), for N-state quantum systems interacting with intense polychromatic fields, by means of the Fourier transformation of numerically exact solutions of the Liouville equation. This is a nonperturbative method that can provide both real and imaginary nonlinear optical spectra, valid for arbitrary laser intensities, frequencies, and relaxation. As applications of the method, we examine two types of three-, four-, and sixteen-state models that mimic the electronic excited states of trans octatetraene obtained from a full configuration-interaction calculation using the Pariser-Parr-Pople Hamiltonian. We also analyze the characteristics of the spectra in the off-resonance region for these models based on virtual excitation processes derived from the perturbative approach, and show that the second ${\mathrm{A}}_{\mathrm{g}}^1$ excited state above the first ionic ${\mathrm{B}}_{\mathrm{u}}^1$ state is essential for describing a qualitative ${\mathrm{\chi }}_{\mathrm{g}}^{\mathrm{}\left(3\right)\mathrm{}}$(ω) value rather than the first ${\mathrm{A}}_{\mathrm{g}}^1$ excited state below the first ${\mathrm{B}}_{\mathrm{u}}^1$ state. Furthermore, the characteristics of variations in the real part of ${\mathrm{\chi }}_{\mathrm{g}}^{\mathrm{}\left(3\right)\mathrm{}}$(ω) with the transition properties (transition energies and moments) of each excited state are elucidated both for a model only including the off-resonance process and a model including a two-photon resonance process.