Visualizing the contributions of virtual states to two-photon absorption cross-sections by natural transition orbitals of response transition density matrices
Observables such as two-photon absorption cross-sections cannot be computed from the wave functions of the initial and final states alone because of their non-linear nature. Rather, they depend on the entire manifold of the excited states, which follows from the familiar sum-over-state expressions of second- and higher-order properties. Consequently, the interpretation of the computed nonlinear optical properties in terms of molecular orbitals is not straightforward and usually relies on approximate few-states models. Here, we show that the 2PA transitions can be visualized using response one-particle transition density matrices, which are defined as transition density matrices between the zero-order and first-order perturbed states. We also extend the concept of natural transition orbitals to 2PA transitions. We illustrate the utility of this new tool, which provides a rigorous black-box alternative to traditional qualitative few-state analysis, by considering 2PA transitions in ethylene, trans-stilbene, and para-nitroaniline.