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Characterizing theoretically nonlinear optical properties of compounds useful for non- linear imaging microscopy such as multicolor two-photon imaging microscopy (M2PIM)1 or second harmonic imaging microscopy (SHIM)2 is a real challenge for quantum chem- istry where new methods need to be implemented and tested. These two imaging technics are based on nonlinear optical phenomena: two-photon absorption (2PA) (and emission) and second-harmonic generation (SHG)3,4. SHIM of biological tissues requires endogenous and/or exogenous dyes with large first hyperpolarizability (β). Endogenous dyes encompass all sorts of structural proteins where their molecular arrangements can be probed by SHIM. Among endogenous dyes, collagen is probably the most important one for the diagnosis of tissue alteration due to diseases.5 For exemple, in case of fibrosis, the three-dimensional dis- tribution of collagen is modified and fibrillar collagen accumulates in the tissue. In the case of exogenous dyes, molecular properties need to be carefully adjusted to provide efficient dyes with limited phototoxicity.6 The desired properties are (i) a large first hyperpolarizability at the wavelength of illumination to maximize the signal, (ii) charge transfer bands enhanced at approximately the wavelength (one-photon enhancement) and half the wavelength (two- photon enhancement) of the biological tissue transparency window (700-900 nm), to achieve resonance enhancement, (iii) a high affinity for the biological membranes, and (iv) the ability to aggregate in a non-centrosymmetric way since centrosymmetric structures have no SHG responses. Amphiphilic dyes are well suited to satisfy conditions (iii) and (iv). Biological systems contain membranes and interfacial structures, which are asymmetric environments and are therefore favorable to organize the chromophores, leading to constructive interfer- ences and enhanced SHG responses. Unfortunately, condition (ii) is counterproductive with respect to the design of dyes exhibiting limited photodamage. Most of SHG exogenous probes currently used for SHIM can lead to photodamage7,8 where their SHG signals are always accompanied by two-photon fluorescence9,10 indicating that the excited states are populated via a multi-photon process. This demonstrates the need for designing new dyes with large SHG responses since they could be detected with smaller laser intensities as well as low or non-existent 2PA activity in the experimental range of frequencies.

Designing new SHIM dyes or rationalizing the response origin in endogenous biological structures could really benefit of a fast and systematic quantum chemical evaluation of β.


But predicting the β value of molecules and matter is, however, a challenging task, owing to many subtle issues that need to be considered. The description of the first hyperpolarizability requires accurate quantum chemistry methods accounting for i) electron correlation effects, ii) frequency dispersion, and iii) environment effects.11,12 Concerning electron-correlation effects, two major types of methods are usually employed to evaluate β: wave function methods and density functional theory (DFT) approaches. Due to their computational re- quirements, wave function methods can only be applied to small (or medium-sized) systems. They can be used as references in order to assess the reliability of DFT used with differ- ent approximate exchange-correlation (XC) functionals.13–16 Among wave function meth- ods, de Wergifosse and Champagne13 showed that the MP2 method predicts β of push-pull π-conjugated systems with an accuracy similar to coupled cluster singles and doubles cal- culations with a perturbative estimate of triples [CCSD(T)]. MP2 can treat medium-sized system but usually only the static value for β is obtained using a finite-field method.17 In that case, approximate schemes have been developed where the static β value is calculated at a correlated level, while the frequency dispersion is estimated at a lower level, typically using the time-dependent Hartree-Fock (TDHF) scheme. Carefully used, TD-DFT (time- dependent density functional theory) can provide an efficient way to determine nonlinear properties for medium to large systems but sometimes for none negligible computational costs. This implies a wise selection of the XC functional and the basis set. In the case of push-pull π-conjugated compounds, de Wergifosse and Champagne13 showed that among XC correlation functionals, LC-BLYP seems the most reliable to characterize trends for a set of molecules in a systematic manner. Solvent effects are commonly accounted for by using continuum models. However, the reliability of implicit solvent models was recently challenged by comparison against an explicit scheme where the solvent molecules are rep- resented by point charges18–20, and where the first solvatation shell is considered explicitly in the QM calculation as well19. It was shown that β calculated with the implicit solvation model are usually larger than those obtained with the multiscale approach.18 The effects of the environment can also be treated explicitly using the ONIOM method,21 where the sys- tem is divided into successive layers: the core layer, which should be treated at the highest level of approximation (quantum level with electron correlation), and the outermost layer that can be treated at a low level (semi-empirical Hamiltonian or Hartree-Fock level). In studies on fluorescent proteins22,23, de Wergifosse et al. showed that to achieve quantitative


agreement with experiment, electron correlation is crucial to get accurate molecular first hyperpolarizabilities as well as to describe the near environment involving H-bonds and van der Waals interactions.

The aim of this article is to provide the theoretical foundations for a new simplified for- mulation of TD-DFT to evaluate first hyperpolarizabilities, providing a fast and efficient method able to screen large numbers of compounds and to treat very large systems. This method shall be a real asset for new developments in SHIM. The density-matrix based linear and nonlinear-response TD-DFT formalism24–30 is modified in the same sTD-DFT (simpli- fied TD-DFT) framework as proposed by Grimme and coworkers31–33 for the determination of electronic excitation spectra of very large molecules. This formalism is also extended to a tight-binding based sTD-DFT approximation (sTD-DFT-xTB), following the method proposed by Grimme and Bannwarth33. After a recapitulation of the density-matrix based linear and nonlinear-response TD-DFT formalism and its sTD-DFT counterpart in section II, three challenging cases are tested: push-pull π-conjugated compounds13,34, fluorescent proteins22,23,35, and a collagen model36, comprising more than 1035 atoms in the largest system considered. Section III gives details about the implementation, systems and com- putations. Section IV discuss the results. Conclusions and outlooks are outlined in section V.

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