Simulations of Two-Photon Absorption Spectra of Fluorescent Dyes: The Impact of Non-Condon Effects

Computer simulations play a pivotal role in interpreting experimental two-photon absorption (2PA) spectra. One of the key aspects of the simulation of these spectra is to take into account the vibrational fine structure of the bands in electronic spectra. This is typically done by employing Franck-Condon (FC) term and low-order terms in the Herzberg-Teller (HT) expansion. In this work, we present a systematic study of first-order HT effects on the vibronic structure of π → π* electronic bands in 2PA spectra of 13 common fluorophores. We begin by evaluating the performance of several density functional approximations (DFAs) against the second-order coupled cluster singles and doubles model (CC2) for reproducing two-photon transition moments and their first- and second-order derivatives with respect to normal modes of vibration on a set of six donor-acceptor molecules. Our findings reveal that most DFAs produce inaccurate values for these derivatives, with the exception of the LC-BLYP functionals with range-separation parameters of 0.33 and 0.47. Although these functionals underestimate the HT contribution to the 2PA total intensities of the π → π* electronic bands, they offer a reasonable qualitative reproduction of the HT vibrational fine structure of the reference spectra. We further explore HT effects on fluorescent chromophores, finding that HT contributions are secondary to FC effects, leading to small shifts of the wavelengths peaks, and minimal changes in the intensities. Additionally, the adiabatic Hessian, vertical Hessian, and vertical gradient vibronic models are assessed. The general agreement among these models confirms that the harmonic approximation is suitable for studying the selected fluorophores.