Quantitative Prediction of Two-Photon Absorption Cross Section Based on Linear Spectroscopic Properties † Aleksander Rebane, ‡,|,⊥ Nikolay S. Makarov, | Mikhail Drobizhev, | Brenda Spangler,* ,# E. Scott Tarter, # Benjamin D. Reeves, # Charles W. Spangler, §,∇,O Fanqing Meng, O and Zhiyong Suo | Department of Physics, Montana State UniVersity, Bozeman, Montana 59717, National Institute of Chemical Physics and Biophysics, Tallinn, EE 12618, Estonia, Sensopath Technologies, Inc., Bozeman, Montana 59715, Department of Chemistry and Biochemistry, Montana State UniVersity, Bozeman, Montana 59717, and MPA Technologies, Inc., Bozeman, Montana 59715 ReceiVed: January 6, 2008; ReVised Manuscript ReceiVed: February 28, 2008 We study two-photon absorption (2PA) spectra in a broad class of organic dye molecules, such as substituted diphenylaminostilbenes, push-pull porphyrins, and carbazol-substituted stilbenes. We show, for the first time, that the 2PA cross section in the lowest-energy dipole-allowed transition may be predicted with better than 50% absolute accuracy based solely on the molecular parameters obtained from linear spectroscopic measurements. 1. Introduction Designing molecules for two-photon fluorescence micros- copy, 1 and other emerging applications of two-photon absorption (2PA) including 3D optical memory, 2–5 nanofabrication, 6 optical power limiting, 7 and photodynamic therapy, 8,9 requires knowl- edge of underlying structure-property relationships for 2PA cross sections (σ 2 ) in a broad range of different chromophores. Usually, such information is obtained from direct quantitative measurement of the nonlinear absorption. This task is often tedious and requires special wavelength-tunable short-pulsed lasers. Alternatively, 2PA spectra and σ 2 values may be forecast on the basis of theoretical quantum-chemical calculations. 10–13 In this case, one computes molecular transition and permanent dipole moments with the associated transition frequencies and then uses these values in perturbation expansion formulas (or equivalent relations) to obtain the nonlinear-optical properties. Although quantum-chemical methods have an advantage that they can account for many excited states, they are also limited by finite computational accuracy. If the number of atoms involved in the calculation becomes large (e.g., over 20), then the quantitative prediction of the 2PA cross sections becomes less accurate. Here, we are proposing an alternative approach, where the 2PA cross sections are found based on the quantities determined solely from linear (one-photon) measurements. In this way, one may be able to avoid the limitations of quantum-chemical calculations as well as circumvent complicated nonlinear-optical experiments. Previously, two, three, and four energy level models have been successful in establishing correspondence between theo- retical and experimental 2PA cross section values. 14–20 It was shown that some molecules, especially those lacking center of symmetry, may be qualitatively described by the two-level model. 16 At the same time, in symmetrical molecules and in the molecules with small permanent dipole moments, the 2PA cross sections were better approximated with three- or four- level models, which does require accounting for high-energy excited states. 16–20 Scaling of the maximum 2PA cross section with an effective number of electrons contributing to the nonlinear response has been discussed in ref 21. Here we focus on the lowest-energy dipole-allowed transition in a selection of organic fluorophores with different degrees of dipolar character, including substituted diphenylaminostilbenes, push-pull porphyrins, and carbazol-substituted stilbenes. If we may assume that the 2PA is described by the two-level model, then there are only two key parameters that need to be evaluated: the permanent dipole moment difference between the ground and the excited state and the transition dipole moment from the ground state to the excited state. We determine both dipole moment values by using standard spectroscopic techniques: The transition dipole moment is directly related to the molar extinction, while the permanent dipole moment difference may be found from solvent-specific Stokes shifts. 22,23 We evaluate the corresponding σ 2 from the perturbation expansion formula, taking into account absorption line shape and local field correction. The same 2PA cross sections are also measured directly by the femtosecond fluorescence-excitation method, showing a remarkably good coincidence with the predicted values. 2. Theoretical Background The interaction of a two-level molecule with applied external electric field, E f (t), is described by the density matrix equation of motion: 24,25 † Part of the “Larry Dalton Festschrift”. * Corresponding author. Phone: (406) 587-6338; fax: 406-585-8390; e-mail: brenda.spangler@sensopath.com. ‡ Phone: (406)-994-7831; fax: (406)-994-4452; e-mail: rebane@ physics.montana.edu. § Phone: 406-585-8192; fax: 406-585-8390; e-mail: spangler.charles@ gmail.com. | Department of Physics, Montana State University. ⊥ National Institute of Chemical Physics and Biophysics. # Sensopath Technologies, Inc. ∇ Department of Chemistry and Biochemistry, Montana State University. O MPA Technologies, Inc. J. Phys. Chem. C 2008, 112, 7997–8004 7997 10.1021/jp800104q CCC: $40.75  2008 American Chemical Society Published on Web 05/22/2008