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 initial and final states alone because of their nonlinear nature. Rather, they depend on the entire manifold of the excited states, which follows from the familiar sum-over-states 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 two-photon absorption (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-states analysis, by considering 2PA transitions in ethylene, trans-stilbene, and para-nitroaniline.