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

We develop an alternative calculation method for the nth-order nonlinear optical susceptibilities χg⁽ⁿ⁾({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 Ag¹ excited state above the first ionic Bᵤ¹ state is essential for describing a qualitative χg⁽³⁾({ω}) value rather than the first Ag¹ excited state below the first Bᵤ¹ state. Furthermore, the characteristics of variations in the real part of χg⁽³⁾({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.