Ultrafast Evaluation of Two-Photon Absorption with Simplified Time-Dependent Density Functional Theory.

This work presents the theoretical background to evaluate two-photon absorption (2PA) cross-sections in the framework of simplified time-dependent density functional theory (sTD-DFT). Our new implementation allows the ultrafast evaluation of 2PA cross-sections for large molecules based on a regular DFT ground-state determinant as well as a variant employing our tight-binding sTD-DFT-xTX flavor for very large systems. The method is benchmarked against higher-level calculations for trans-stilbene and typical fluorescent protein chromophores. For eGFP, a quadrupolar chromophore and its branched version, the flavine mono-nucleotide, and the iLOV protein, we compare sTD-DFT 2PA spectra to experimental ones. This includes extension and testing of our all-atom quantum chemistry methodology for the evaluation of 2PA for a system of ∼2000 atoms, providing striking agreement with the experimental spectrum.

Self-assembling, structure and nonlinear optical properties of fluorescent organic nanoparticles in water.

Owing to their intense emission, low toxicity and solubility in aqueous medium, fluorescent organic nanoparticles (FONs) have emerged as promising alternatives to inorganic ones for the realization of exogenous probes for bioimaging applications. However, the intimate structure of FONs in solution, as well as the role played by intermolecular interactions on their optical properties, remains challenging to study. Following a recent Second-Harmonic Scattering (SHS) investigation led by two of us [Daniel et al., ACS Photonics, 2015, 2, 1209], we report herein a computational study of the structural organization and second-order nonlinear optical (NLO) properties of FONs based on dipolar chromophores incorporating a hydrophobic triphenylamine electron-donating unit and a slightly hydrophilic aldehyde electron-withdrawing unit at their extremities. Molecular dynamics simulations of the FON formation in water are associated with quantum chemical calculations, to provide insight into the molecular aggregation process, the molecular orientation of the dipolar dyes within the nanoparticles, and the dynamical behavior of their NLO properties. Moreover, the impact of intermolecular interactions on the NLO responses of the FONs is investigated by employing the tight-binding version of the recently developed simplified time-dependent density functional theory (sTD-DFT) approach, allowing the all-atom quantum mechanics treatment of nanoparticles.

All-Atom Quantum Mechanical Calculation of the Second-Harmonic Generation of Fluorescent Proteins.

Fluorescent proteins (FPs) are biotags of choice for second-harmonic imaging microscopy (SHIM). Because of their large size, computing their second-harmonic generation (SHG) response represents a great challenge for quantum chemistry. In this contribution, we propose a new all-atom quantum mechanics methodology to compute SHG of large systems. This is now possible because of two recent implementations: the tight-binding GFN2-xTB method to optimize geometries and a related version of the simplified time-dependent density functional theory (sTD-DFT-xTB) to evaluate quadratic response functions. In addition, a new dual-threshold configuration selection scheme is introduced to reduce the computational costs while retaining overall similar accuracy. This methodology was tested to evaluate the SHG of the proteins iLOV and bacteriorhodopsin (bR). In the case of bR, quantitative agreement with respect to experiment was reached for the out-of-resonance low-energy part of the ßHRS frequency dispersion. This work paves the way toward an accurate prediction of the SHG of large structures-a requirement for the design of new and improved SHIM biotags.

Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package.

This article summarizes technical advances contained in the fifth major release of the Q-Chem quantum chemistry program package, covering developments since 2015. A comprehensive library of exchange-correlation functionals, along with a suite of correlated many-body methods, continues to be a hallmark of the Q-Chem software. The many-body methods include novel variants of both coupled-cluster and configuration-interaction approaches along with methods based on the algebraic diagrammatic construction and variational reduced density-matrix methods. Methods highlighted in Q-Chem 5 include a suite of tools for modeling core-level spectroscopy, methods for describing metastable resonances, methods for computing vibronic spectra, the nuclear-electronic orbital method, and several different energy decomposition analysis techniques. High-performance capabilities including multithreaded parallelism and support for calculations on graphics processing units are described. Q-Chem boasts a community of well over 100 active academic developers, and the continuing evolution of the software is supported by an "open teamware" model and an increasingly modular design.

Perspective on Simplified Quantum Chemistry Methods for Excited States and Response Properties.

We review recent developments in the framework of simplified quantum chemistry for excited state and optical response properties (sTD-DFT) and present future challenges for new method developments to improve accuracy and extend the range of application. In recent years, the scope of sTD-DFT was extended to molecular response calculations of the polarizability, optical rotation, first hyperpolarizability, two-photon absorption (2PA), and excited-state absorption for large systems with hundreds to thousands of atoms. The recently introduced spin-flip simplified time-dependent density functional theory (SF-sTD-DFT) variant enables an ultrafast treatment for diradicals and related strongly correlated systems. A few drawbacks were also identified, specifically for the computation of 2PA cross sections. We propose solutions to this problem and how to generally improve the accuracy of simplified schemes. New possible simplified schemes are also introduced for strongly correlated systems, e.g., with a second-order perturbative correlation correction. Interpretation tools that can extract chemical structure-property relationships from excited state or response calculations are also discussed. In particular, the recently introduced method-agnostic RespA approach based on natural response orbitals (NROs) as the key concept is employed.

A Unified Strategy for the Chemically Intuitive Interpretation of Molecular Optical Response Properties.

Interpreting response properties such as the polarizability, optical rotation (OR), or hyperpolarizabilities is a complex task for which a uniform strategy would be desirable. We propose a response analysis procedure called the RespA approach with two interrelated schemes to describe molecular optical response properties in terms of natural response orbitals (NROs) and chemical fragment response for convenient elucidation of structure-(optical)property relationships. These quantities can be easily extracted from the frequency-dependent perturbed one-electron transition/current density matrix obtained from any quantum mechanical response function calculation. NROs provide the most compact representation of the virtual excitations occurring in the (hyper)scattering process. It is decomposed in hole and electron NRO pairs providing a simple exciton picture. For a chemist, it is natural to interpret a property by decomposing it into functional groups or fragment contributions. In this spirit, the response is partitioned into on-site and between-site fragment responses, allowing a property mapping into real space. The new RespA procedure was implemented and tested at the simplified time-dependent density functional theory (sTD-DFT) level enabling calculations for large systems. The RespA strategy is a method-independent route for the understanding of a wide variety of response properties. We showcase how the chemically intuitive RespA approach extracts easily structure-property relationships for the particularly difficult case of OR. As examples, we demonstrate how to enhance the OR of [5]helicene and norbornenone, provide an analysis of the change of the OR observed for camphor and fenchone, and finally investigate the case of a (P,P)-bis-helicenic 2,2'-bipyridine chiroptical switch.

Simplified time-dependent density functional theory (sTD-DFT) for molecular optical rotation.

Theoretical methods able to screen large sets (e.g., conformers) of possibly large compounds are needed in many typical quantum chemistry applications. For this purpose, we here extend the well-established simplified time-dependent density functional theory (sTD-DFT) method for the calculation of optical rotation. This new scheme is benchmarked against 42 compounds of the OR45 set as well as thirteen helicene derivatives and one bio-molecular system. The sTD-DFT method yields optical rotations in good quantitative agreement with experiment for compounds with a valence-dominated response, e.g., conjugated π-systems, at a small fraction of the computational cost compared to TD-DFT (1-3 orders of magnitude speed-up). For smaller molecules with a Rydberg state dominated response, the agreement between TD-DFT and the simplified version using standard hybrid functionals is somewhat worse but still reasonable for typical applications. Our new implementation in the stda code enables computations for systems with up to 1000 atoms, e.g., for studying flexible bio-molecules.

Dynamic Structural Effects on the Second-Harmonic Generation of Tryptophane-Rich Peptides and Gramicidin A.

Peptide chains can model endogenous biotags for applications in second-harmonic imaging microscopy. Such structures are flexible which may strongly affect their structure-property relationship. Here, we explore quantum-mechanically the conformational space of a set of tryptophan-rich model peptides. This has become feasible because of the recently proposed meta-dynamics method based on efficient tight-binding (TB) calculations. The TB version of the simplified time-dependent density functional theory (sTD-DFT-xTB) method is used to evaluate the first hyperpolarizability (ß). These new tools enable us to calculate nonlinear optical properties for systems with several thousand atoms and/or to screen large structure ensembles. First, we show that the indole chromophore in tryptophan residues dominates the ß response of these systems. Their relative orientation mostly determines the global ß tensor and affects the static ß response. The results underline the importance of finding low-energy conformers for modeling ß of flexible molecules. Additionally, we compare calculated and extrapolated experimental static ß. The sTD-DFT-xTB method is capable of providing reliable second-harmonic generation values for tryptophan-rich systems at a fraction of the computational cost of the commonly used TD-DFT/TD-HF levels of theory.

Are Fully Conjugated Expanded Indenofluorenes Analogues and Diindeno[n]thiophene Derivatives Diradicals? A Simplified (Spin-Flip) Time-Dependent Density Functional Theory [(SF-)sTD-DFT] Study.

Polycyclic hydrocarbons are often used to understand the electronic structure of nanographene systems. Among them, indeno[1,2b]fluorene and indeno[1,2c]fluorene isomers present a central p-quinodimethane unit leading to unique optical properties. In this work, we characterized the absorption spectra of indeno[1,2b]fluorene and [2,1-c]diindeno[n]thiophene derivatives with (spin-flip) simplified time-dependent density functional theory [(SF-)sTD-DFT] methods. Note that the SF-sTD-DFT level of theory allows a computationally efficient treatment for large diradicals. To interpret spectra, we implemented natural transition orbitals (NTOs) at both SF-sTD-DFT and sTD-DFT levels. This compact and method-independent representation of the electronic excitation provides a simple interpretation for the low-lying excited states of this set of molecules in terms of three different types of NTOs: "quinoid", "aromatic", and "π-bonded". When comparing with experiment, we found that only one molecule of this set is actually a high-spin triplet diradical. Others are almost closed-shell molecules with a very small contribution from a doubly excited configuration that only the spin-flip method could capture. The small amount of static correlation recovered by the spin-flip active space provides a linear relation between the first visible theoretical and experimental excitation energies among this set.

A Simplified Spin-Flip Time-Dependent Density Functional Theory Approach for the Electronic Excitation Spectra of Very Large Diradicals.

Experimentalists working with diradicals are often facing the question of what kind of species among singlet or triplet diradicals or closed-shell molecules are observed. To treat large diradicals with a high density of electronic states, we propose a simplified version of the spin-flip time-dependent density functional theory (SF-TD-DFT) method for a fast computation of their state energies and absorption spectra with an accuracy similar to the nonsimplified scheme. An ultrafast tight-binding variant called SF-sTD-DFT-xTB is also developed to treat even larger systems. For a benchmark set of nine diradicals, good agreement between simplified and conventional SF excitation energies for standard functionals is found. This shows that the proposed parameterization is robust for a wide range of Fock exchange mixing values. With the asymptotically correct response integrals used in SF-sTD-DFT and a correction factor of 2 for the transition moments, the SF-sTD-DFT/B5050LYP/cc-pVDZ method even outperforms the nonsimplified scheme at drastically reduced computational effort when comparing to the experimental absorption spectra for this set of diradicals. To showcase the actual performance of the method, absorption spectra of two µ-hydroxo-bridged dimers of corrole tape Ga(III) complex derivatives were computed and compared to the experiment, providing good qualitative agreement. Finally, a comparison with the high-spin triplet spectrum of a perylene bisimide biradical and the one determined at the SF-sTD-DFT level showed that at room temperature, mostly triplet diradicals are present in solution.

Nonlinear-response properties in a simplified time-dependent density functional theory (sTD-DFT) framework: Evaluation of excited-state absorption spectra.

The energy conversion efficiency of organic solar cells seems crucial for a clean future. The design of new light-harvesting devices needs an in-depth understanding of their optical properties, including the excited-state absorption (ESA). In biology, the optical characterization of photochemical/physical processes happening in photosynthetic pigments and proteins can be difficult to interpret due to their structural complexities. Experimentally, an ultrafast transient absorption experiment can probe the excited state interaction with light. Quantum chemistry could play an important role to model the transient absorption spectrum of excited states. However, systems that need to be investigated can be way too large for existent software implementations. In this contribution, we present the first sTDA/sTD-DFT (simplified time-dependent density functional theory with and without Tamm Dancoff approximation) implementation to evaluate the ESA of molecules. The ultrafast ESA evaluation presents a negligible extra cost with respect to sTDA/sTD-DFT original schemes for standard ground state absorption. The sTD-DFT method shows ability to assign ESA spectra to the correct excited state. We showed that in the literature, wrong assignments were proposed as for the L34/L44 mixture and N-methylfulleropyrrolidine. In addition, sTDA/sTD-DFT-xTB tight-binding variants are also available, allowing the evaluation of ESA for systems of a few thousands of atoms, e.g., the spectrum of the photoactive yellow protein composed of 1931 atoms.

Nonlinear-response properties in a simplified time-dependent density functional theory (sTD-DFT) framework: Evaluation of the first hyperpolarizability.

Recent developments in nonlinear imaging microscopy show the need to implement new theoretical tools, which are able to characterize nonlinear optical properties in an efficient way. For second-harmonic imaging microscopy (SHIM), quantum chemistry could play an important role to design new exogenous dyes with enhanced first hyperpolarizabilities or to characterize the response origin in large endogenous biological systems. Such methods should be able to screen a large number of compounds while reproducing their trends and to treat large systems in reasonable computation times. To fulfill these requirements, we present a new simplified time-dependent density functional theory (sTD-DFT) implementation to evaluate the first hyperpolarizability where the Coulomb and exchange integrals are approximated by short-range damped Coulomb interactions of transition density monopoles. For an ultra-fast computation of the first hyperpolarizability, a tight-binding version (sTD-DFT-xTB) is also proposed. In our implementation, a sTD-DFT calculation is more than 600 time faster with respect to a regular TD-DFT treatment, while the xTB version speeds up the entire calculation further by at least two orders of magnitude. We challenge our implementation on three test cases: typical push-pull π-conjugated compounds, fluorescent proteins, and a collagen model, which were selected to model requirements for SHIM applications.

ONIOM Investigation of the Second-Order Nonlinear Optical Responses of Fluorescent Proteins.

The first hyperpolarizability (ß) of six fluorescent proteins (FPs), namely, enhanced green fluorescent protein, enhanced yellow fluorescent protein, SHardonnay, ZsYellow, DsRed, and mCherry, has been calculated to unravel the structure-property relationships on their second-order nonlinear optical properties, owing to their potential for multidimensional biomedical imaging. The ONIOM scheme has been employed and several of its refinements have been addressed to incorporate efficiently the effects of the microenvironment on the nonlinear optical responses of the FP chromophore that is embedded in a protective ß-barrel protein cage. In the ONIOM scheme, the system is decomposed into several layers (here two) treated at different levels of approximation (method1/method2), from the most elaborated method (method1) for its core (called the high layer) to the most approximate one (method2) for the outer surrounding (called the low layer). We observe that a small high layer can already account for the variations of ß as a function of the nature of the FP, provided the low layer is treated at an ab initio level to describe properly the effects of key H-bonds. Then, for semiquantitative reproduction of the experimental values obtained from hyper-Rayleigh scattering experiments, it is necessary to incorporate electron correlation as described at the second-order Møller-Plesset perturbation theory (MP2) level as well as implicit solvent effects accounted for using the polarizable continuum model (PCM). This led us to define the MP2/6-31+G(d):HF/6-31+G(d)/IEFPCM scheme as an efficient ONIOM approach and the MP2/6-31+G(d):HF/6-31G(d)/IEFPCM as a better compromise between accuracy and computational needs. Using these methods, we demonstrate that many parameters play a role on the ß response of FPs, including the length of the π-conjugated segment, the variation of the bond length alternation, and the presence of π-stacking interactions. Then, noticing the small diversity of the FP chromophores, these results highlight the key role of the ß-barrel and surrounding residues on ß, not only because they can locally break the noncentrosymmetry vital to a ß response but also because it can impose geometrical constraints on the chromophore.

Two-photon absorption spectroscopy of stilbene and phenanthrene: Excited-state analysis and comparison with ethylene and toluene.

Two-photon absorption (2PA) spectra of several prototypical molecules (ethylene, toluene, trans- and cis-stilbene, and phenanthrene) are computed using the equation-of-motion coupled-cluster method with single and double substitutions. The states giving rise to the largest 2PA cross sections are analyzed in terms of their orbital character and symmetry-based selection rules. The brightest 2PA transitions correspond to Rydberg-like states from fully symmetric irreducible representations. Symmetry selection rules dictate that totally symmetric transitions typically have the largest 2PA cross sections for an orientationally averaged sample when there is no resonance enhancement via one-photon accessible intermediate states. Transition dipole arguments suggest that the strongest transitions also involve the most delocalized orbitals, including Rydberg states, for which the relative transition intensities can be rationalized in terms of atomic selection rules. Analysis of the 2PA transitions provides a foundation for predicting relative 2PA cross sections of conjugated molecules based on simple symmetry and molecular orbital arguments.

Two-photon absorption spectroscopy of trans-stilbene, cis-stilbene, and phenanthrene: Theory and experiment.

Two-photon absorption (2PA) spectroscopy provides complementary, and sometimes more detailed, information about the electronic structure of a molecule relative to one-photon absorption (1PA) spectroscopy. However, our understanding of the 2PA processes is rather limited due to technical difficulties in measuring experimental 2PA spectra and theoretical challenges in computing higher-order molecular properties. This paper examines the 2PA spectroscopy of trans-stilbene, cis-stilbene, and phenanthrene by a combined experimental and theoretical approach. The broadband 2PA spectra of all three compounds are measured under identical conditions in order to facilitate a direct comparison of the absolute 2PA cross sections in the range 3.5-6.0 eV. For comparison, the theoretical 2PA cross sections are computed using the equation-of-motion coupled-cluster method with single and double substitutions. Simulated 2PA spectra based on the calculations reproduce the main features of the experimental spectra in solution, although the quantitative comparison is complicated by a number of uncertainties, including limitations of the theoretical model, vibronic structure, broadening of the experimental spectra, and solvent effects. The systematic comparison of experimental and theoretical spectra for this series of structurally similar compounds provides valuable insight into the nature of 2PA transitions in conjugated molecules. Notably, the orbital character and symmetry-based selection rules provide a foundation for interpreting the features of the experimental 2PA spectra in unprecedented detail.

Approximate Spin-Projected Density-Based Romberg Differentiation Procedure to Evaluate the Second-Hyperpolarizability of p-Quinodimethane and Twisted Ethylene and Their Diradical Character Dependence.

The evaluation of the static second hyperpolarizability (γ) of diradical species is a challenging task due to the use of spin-unrestricted methods, which may suffer from spin contamination. Here, we present the methodological aspect of a density-based differentiation procedure to evaluate static polarizability and hyperpolarizabilities. The finite-field calculations are done on the spin-projected electron density to remove the spin contamination, and the automatized Romberg's differentiation procedure is used to improve the numerical accuracy in the finite-field method. This implementation is tested in the present report for the challenging case of the evaluation of the second hyperpolarizability of the singlet ground state of p-quinodimethane (PQM) for the equilibrium geometry as well as for a stretched geometry where the diradical character of PQM is increased, and for twisted ethylene models where the diradical character changes with the dihedral angle. The application of the approximate spin-projected (ASP) scheme leads to a major improvement of the density functional theory calculations. In particular, for PQM models, BHandHLYP functional reproduces the UCCSD(T) values when the diradical character is below 0.5. The visualization of the γ-densities shows that (i) when increasing the diradical character, the amount of γ-density increases on the -CH2(•) extremities, and (ii) the ASP scheme decreases the amount of "p-like" γ-density for diradical character below 0.4, and increases it for larger diradical character. For twisted ethylene model, we show that the UCCSD(T) reference values can be reproduced by the ASP-UB3LYP method for y < 0.4 and by the ASP-UBHandHLYP method for y > 0.6. To best reproduce the UCCSD(T) reference calculations, the amount of exact exchange in hybrid functionals needs to be tuned along the range of diradical characters.

Challenging compounds for calculating molecular second hyperpolarizabilities: the triplet state of the trimethylenemethane diradical and two derivatives.

The second hyperpolarizability γ of trimethylenemethane (TMM) and two 1,3-dipole derivatives (NXA and OXA) in their triplet ground state has been evaluated at the UCCSD(T) level with the d-aug-cc-pVDZ extended basis set, highlighting that γ decreases from TMM to NXA and OXA, following the opposite order of their permanent dipole moments. These results are then used to benchmark a broad range of levels of approximation. So, the UMP2, UMP4, and UCCSD methods can be used to characterize γ of TMM and NXA but not of OXA. In that case, the large field-induced charge transfer contribution is difficult to handle using the MPn methods and only the UCCSD method provides values close to the UCCSD(T) reference. Turning to the performance of DFT with typical exchange-correlation functionals, the UM06-2X functional, which contains 54% of HF exchange, performs very well with a maximum of 4.5% of difference with respect to the reference values. On the other hand, employing less HF exchange leads to an overestimation of the responses whereas range-separated hybrids generally underestimate the second hyperpolarizabilities. Finally, the use of spin-projected methods for these 1,3-dipole triplet molecules has a little impact since the spin contamination is almost negligible.

Frequency dispersion of the first hyperpolarizabilities of reference molecules for nonlinear optics.

The frequency dispersion of the hyper-Rayleigh scattering first hyperpolarizabilities (ßHRS) of five reference molecules for nonlinear optics, namely, carbon tetrachloride, chloroform, dichloromethane, acetonitrile, and trichloroacetonitrile, is described using the coupled-cluster singles and doubles quadratic response function (CCSD-QRF) as well as approximate schemes. Comparisons to approximate schemes in which the frequency dispersion is evaluated as either a multiplicative or an additive correction to the static hyperpolarizability yield the following observations: (i) errors of the order of 10% or less are usually encountered when using the multiplicative scheme for photon energies far from the lowest dipole-allowed excitation energies, (ii) spurious cases cannot be excluded as evidenced by carbon tetrachloride where the multiplicative scheme predicts a decrease of ßHRS in contradiction to the increase obtained using the CCSD-QRF method, and (iii) the additive scheme is at best as reliable as the multiplicative approximation. The two-state approximation presents the advantage of correcting the wrong behavior of the additive and multiplicative schemes for carbon tetrachloride, but it is not an improved solution for the other compounds, while the question of selecting the appropriate dominant excited state remains unanswered. Finally, a new ß(xyz) value of 18.9 a.u. is proposed for carbon tetrachloride in gas phase at λ = 1064 nm, to be compared with the measured 16.9 ± 1.4 a.u. value due to Shelton.

Challenging compounds for calculating hyperpolarizabilities: p-quinodimethane derivatives.

The hyperpolarizabilities of three p-quinodimethane derivatives with low diradical character have been evaluated. As electron correlation effects rule the electric field response properties, wave function and density functional theory-based methods have been compared to benchmark values calculated with the coupled cluster method including single and double excitations as well as perturbative estimate of the triples [CCSD(T)]. The basis set effects have been further assessed. This study shows that the determination of the second hyperpolarizability with the CCSD method provides results in closest agreement with the CCSD(T) reference values. The use of MP2 level of theory performs well for the closed-shell compound but not for open-shell ones. Spin-projection UMP3 and UMP4 methods reproduce well UCCSD(T) values for the p-quinodimethane but not for the charged compound. Without spin projection correction, density functional theory with a large range of exchange-correlation functionals does not perform well for these systems. Similar effects have been observed for the polarizability and first hyperpolarizability, although these effects are smaller.

Improving the second-order nonlinear optical response of fluorescent proteins: the symmetry argument.

We have successfully designed and expressed a new fluorescent protein with improved second-order nonlinear optical properties. It is the first time that a fluorescent protein has been rationally altered for this particular characteristic. On the basis of the specific noncentrosymmetry requirements for second-order nonlinear optical effects, we had hypothesized that the surprisingly low first hyperpolarizability (ß) of the enhanced yellow fluorescent protein (eYFP) could be explained by centrosymmetric stacking of the chromophoric Tyr66 and the neighboring Tyr203 residue. The inversion center was removed by mutating Tyr203 into Phe203, with minor changes in the linear optical properties and even an improved fluorescence quantum yield. Structure determination by X-ray crystallography as well as linear optical characterization corroborate a correct folding and maturation. Measurement of ß by means of hyper-Rayleigh scattering (HRS) as well as their analysis using quantum chemistry calculations validate our hypothesis. This observation can eventually lead to improved red fluorescent proteins for even better performance. On the basis of the specific function (second-harmonic generation), the color of its emission, and in analogy with the "fruit" names, we propose SHardonnay as the name for this Tyr203Phe mutant of eYFP.