Open-Ended Recursive Approach for the Calculation of Multiphoton Absorption Matrix Elements.

We present an implementation of single residues for response functions to arbitrary order using a recursive approach. Explicit expressions in terms of density-matrix-based response theory for the single residues of the linear, quadratic, cubic, and quartic response functions are also presented. These residues correspond to one-, two-, three- and four-photon transition matrix elements. The newly developed code is used to calculate the one-, two-, three- and four-photon absorption cross sections of para-nitroaniline and para-nitroaminostilbene, making this the first treatment of four-photon absorption in the framework of response theory. We find that the calculated multiphoton absorption cross sections are not very sensitive to the size of the basis set as long as a reasonably large basis set with diffuse functions is used. The choice of exchange-correlation functional, however, significantly affects the calculated cross sections of both charge-transfer transitions and other transitions, in particular, for the larger para-nitroaminostilbene molecule. We therefore recommend the use of a range-separated exchange-correlation functional in combination with the augmented correlation-consistent double-ζ basis set aug-cc-pVDZ for the calculation of multiphoton absorption properties.