Spurious Oscillations Caused by Density Functional Approximations: Who is to Blame? Exchange or Correlation?

We analyze the varying susceptibilities of different density functional approximations (DFAs) to present spurious oscillations on the profiles of several vibrational properties. Among other problems, these spurious oscillations cause significant errors in harmonic and anharmonic IR and Raman frequencies and intensities. This work hinges on a judicious strategy to dissect the exchange and correlation components of DFAs and pinpoint the origins of these oscillations. We identify spurious oscillations in derivatives of all energy components with respect to nuclear displacements, including those energy terms that do not involve numerical integrations. These indirect spurious oscillations are attributed to suboptimal electron densities resulting from a self-consistent field procedure using a DFA that exhibits direct spurious oscillations. Direct oscillations stem from inaccurate numerical integration of the exchange and correlation energy density functionals. A thorough analysis of direct spurious oscillations reveals that only a handful of exchange and correlation components are insensitive to spurious oscillations, giving rise to three families of functionals, BH&H, LSDA, and BLYP. Among the functionals in these families, we encounter four widespread DFAs: BLYP, B3LYP, LC-BLYP, and CAM-B3LYP. Certain DFAs like PBE appear less sensitive to spurious oscillations due to compensatory cancellations between their energy components. Additionally, we found non-negligible but small oscillations in PBE and TPSS, which could be safely employed provided a sufficiently large integration grid is used in the calculations. These findings hint at the key components of current approximations to be improved and emphasize the necessity to develop accurate DFAs suitable for studying molecular spectroscopies.

One- and two-photon absorption spectra of organoboron complexes: vibronic and environmental effects.

We synthesized a series of four parent aza-ß-ketoiminate organoboron complexes and performed spectroscopic studies using both experimental and computational techniques. We studied how benzannulation influences the vibronic structure of the UV/Vis absorption bands with a focus on the bright lowest-energy π → π* electronic excitation. Theoretical simulations, accounting for inhomogeneous broadening effects using different embedding schemes, allowed gaining in-depth insights into the observed differences in band shapes induced by structural modifications. We observed huge variations in the distributions of vibronic transitions depending on the position of benzannulation. By and large, the harmonic approximation combined with the adiabatic hessian model delivers qualitatively correct band shapes for the one-photon absorption spectra, except in one case. We also assessed the importance of non-Condon effects (accounted for by the linear term in Herzberg-Teller expansion of the dipole moment) for S0 → S1 band shapes. It turned out that non-Condon contributions have no effect on the band shape in one-photon absorption spectra. In contrast, these effects significantly change the Franck-Condon band shapes of the two-photon absorption spectra. For one of the studied organoboron complexes we also performed a preliminary exploration of mechanical anharmonicity, resulting in an increase of the intensity of the 0-0 transition, which improves the agreement with the experimental data compared to the harmonic model.

Toward Accurate Two-Photon Absorption Spectrum Simulations: Exploring the Landscape beyond the Generalized Gradient Approximation.

In this Letter, we present a pioneering analysis of the density functional approximations (DFAs) beyond the generalized gradient approximation (GGA) for predicting two-photon absorption (2PA) strengths of a set of push-pull π-conjugated molecules. In more detail, we have employed a variety of meta-generalized gradient approximation (meta-GGA) functionals, including SCAN, MN15, and M06-2X, to assess their accuracy in describing the 2PA properties of a chosen set of 48 organic molecules. Analytic quadratic response theory is employed for these functionals, and their performance is compared against the previously studied DFAs and reference data obtained at the coupled-cluster CC2 level combined with the resolution-of-identity approximation (RI-CC2). A detailed analysis of the meta-GGA functional performance is provided, demonstrating that they improve upon their predecessors in capturing the key electronic features of the π-conjugated two-photon absorbers. In particular, the Minnesota functional MN15 shows very promising results as it delivers pleasingly accurate chemical rankings for two-photon transition strengths and excited-state dipole moments.

BF2-Functionalized Benzothiazole Amyloid Markers: Effect of Donor Substituents on One- and Two-Photon Properties.

Investigation of amyloids with the aid of fluorescence microscopy provides crucial insights into the development of numerous diseases associated with the formation of aggregates. Here, we present a series of BF2-functionalized benzothiazoles with electron-donating methoxy group(s), which are tested as amyloid fluorescent markers. We evaluate how the position of donor functional group(s) influences optical properties (fluorescence lifetime (τ) and fluorescence quantum yield (FQY)) in a solution and upon binding to amyloids. We elucidate the importance of surrounding environmental factors (hydrogen-bonding network, polarity, and viscosity) on the observed changes in FQY and evaluate how the localization of a donor influences radiative and nonradiative decay pathways. We conclude that a donor attached to the benzothiazole ring contributes to the increment of radiative decay pathways upon binding to amyloids (kr), while the donor attached to the flexible part of a molecule (with rotational freedom) contributes to a decrease in nonradiative decay pathways (knr). We find that the donor-acceptor-donor architecture allows us to obtain 58 times higher FQY of the dye upon binding to bovine insulin amyloids. Finally, we measure two-photon absorption (2PA) cross sections (σ2) of the dyes and their change upon binding by the two-photon excited fluorescence (2PEF) technique. Measurements reveal that dyes that exhibit the increase/decrease of σ2 values when transferred from highly polar solvents to CHCl3 present a similar behavior upon amyloid binding. Our 2PA experimental values are supported by quantum mechanics/molecular mechanics (QM/MM) simulations. Despite this trend, the values of σ2 are not the same, which points out the importance of two-photon absorption measurements of amyloid-dye complexes in order to understand the performance of 2P probes upon binding.

Pitfall in simulations of vibronic TD-DFT spectra: diagnosis and assessment.

In this Communication, we study the effect of spurious oscillations in the profiles of energy derivatives with respect to nuclear coordinates calculated with density functional approximations (DFAs) for formaldehyde, pyridine, and furan in their ground and electronic excited states. These spurious oscillations, which can only be removed using extensive integration grids that increase enormously the CPU cost of DFA calculations, are significant in the case of third- and fourth-order energy derivatives of the ground and excited states computed by M06-2X and ωB97X functionals. The errors in question propagate to anharmonic vibronic spectra computed under the Franck-Condon approximation, i.e., positions and intensities of vibronic transitions are affected to a large extent (shifts as significant as hundreds of cm-1 were observed). On the other hand, the LC-BLYP and CAM-B3LYP functionals show a much less pronounced effect due to spurious oscillations. Based on the results presented herein, we recommend either LC-BLYP or CAM-B3LYP with integration grids (250, 974) (or larger) for numerically stable simulations of vibronic spectra including anharmonic effects.

Chromophore Planarity, -BH Bridge Effect, and Two-Photon Activity: Bi- and Ter-Phenyl Derivatives as a Case Study.

In this work, we have employed electronic structure theories to explore the effect of the planarity of the chromophore on the two-photon absorption properties of bi- and ter-phenyl systems. To that end, we have considered 11 bi- and 7 ter-phenyl-based chromophores presenting a donor-π-acceptor architecture. In some cases, the planarity has been enforced by bridging the rings at ortho-positions by -CH2 and/or -BH, -O, -S, and -NH moieties. The results presented herein demonstrate that in bi- and ter-phenyl systems, the planarity achieved via a -CH2 bridge increases the 2PA activity. However, the introduction of a bridge with the -BH moiety perturbs the electronic structure to a large extent, thus diminishing the two-photon transition strength to the lowest electronic excited state. As far as two-photon absorption activity is concerned, this work hints toward avoiding -BH bridge(s) to enforce planarity in bi- and ter-phenyl systems; however, one may use -CH2 bridge(s) to achieve the enhancement of the property in question. All of these conclusions have been supported by in-depth analyses based on generalized few-state models.

Photoswitchable Molecular Units with Tunable Nonlinear Optical Activity: A Theoretical Investigation.

The first-, second-, and third-order molecular nonlinear optical properties, including two-photon absorption of a series of derivatives, involving two dithienylethene (DTE) groups connected by several molecular linkers (bis(ethylene-1,2-dithiolato)Ni- (NiBDT), naphthalene, quasilinear oligothiophene chains), are investigated by employing density functional theory (DFT). These properties can be efficiently controlled by DTE switches, in connection with light of appropriate frequency. NiBDT, as a linker, is associated with a greater contrast, in comparison to naphthalene, between the first and second hyperpolarizabilities of the "open-open" and the "closed-closed" isomers. This is explained by invoking the low-lying excited states of NiBDT. It is shown that the second hyperpolarizability can be used as an index, which follows the structural changes induced by photochromism. Assuming a Förster type transfer mechanism, the intramolecular excited-state energy transfer (EET) mechanism is studied. Two important parameters related to this are computed: the electronic coupling (VDA) between the donor and acceptor fragments as well as the overlap between the absorption and emission spectra of the donor and acceptor groups. NiBDT as a linker is associated with a low electronic coupling, VDA, value. We found that VDA is affected by molecular geometry. Our results predict that the linker strongly influences the communication between the open-closed DTE groups. The sensitivity of the molecular nonlinear optical properties could assist with identification of molecular isomers.

Decoding the infrared spectra changes upon formation of molecular complexes: the case of halogen bonding in pyridine⋯perfluorohaloarene complexes.

A recently developed computational scheme is employed to interpret changes in the infrared spectra of halogen-bonded systems in terms of intermolecular interaction energy components (electrostatic, exchange, induction, dispersion) taking pyridine⋯perfluorohaloarene complexes as examples. For all complexes, we find a strong linear correlation between the different terms of the interaction-induced changes of the IR band associated with an intermolecular halogen bond stretching mode and the corresponding terms of the interaction energy, which implies that the interaction components play similar roles in both properties. This is not true for other vibrational modes localized in one of the monomers studied here, for which the corresponding interaction-induced changes in IR bands may present a completely different decomposition than the interaction energy.

A new computational tool for interpreting the infrared spectra of molecular complexes.

The popularity of infrared (IR) spectroscopy is due to its high interpretive power. This study presents a new computational tool for analyzing the IR spectra of molecular complexes in terms of intermolecular interaction energy components. In particular, the proposed scheme enables associating the changes in the IR spectra occurring upon complex formation with individual types of intermolecular interactions (electrostatic, exchange, induction, and dispersion), thus providing a completely new insight into the relations between the spectral features and the nature of interactions in molecular complexes. To demonstrate its interpretive power, we analyze, for selected vibrational modes, which interaction types rule the IR intensity changes upon the formation of two different types of complexes, namely π⋯π stacked (benzene⋯1,3,5-trifluorobenzene) and hydrogen-bonded (HCN⋯HNC) systems. The exemplary applications of the new scheme to these two molecular complexes revealed that the interplay of interaction energy components governing their stability might be very different from that behind the IR intensity changes. For example, in the case of the dispersion-bound π⋯π-type complex, dispersion contributions to the interaction induced IR intensity of the selected modes are notably smaller than their first-order (electrostatic and exchange) counterparts.

Computational Investigations into Two-Photon Fibril Imaging Using the DANIR-2c Probe.

The design of novel fibril imaging molecules for medical diagnosis requires the simultaneous optimization of fibril-specific optical properties and binding specificity toward amyloid fibrils. Because of the possibility to monitor internal organs and deep tissues, the two-photon probes that can absorb in the infrared (IR) and near-IR (NIR) region with a significant two-photon absorption cross section are of immense interest. To contribute to this exploration of chemical compounds suitable for two-photon fibril imaging, we have computationally studied the one- and two-photon properties of a donor-acceptor-substituted DANIR-2c probe, which was used for in vivo detection of ß-amyloid deposits using fluorescence spectroscopy. In particular, a multiscale computational approach was employed involving molecular docking, molecular dynamics, hybrid QM/MM molecular dynamics, and coupled-cluster/MM to study the binding of the studied probe to amyloid fibril and its one- and two-photon absorption properties in the fibrillar environment. Multiple binding sites are available for this probe in amyloid fibril, and the one corresponding to the largest binding affinity exhibits also the largest and experimentally meaningful two-photon absorption cross section, thus demonstrating the potential of the studied probe in two-photon microscopy.

Cost-Effective Simulations of Vibrationally-Resolved Absorption Spectra of Fluorophores with Machine-Learning-Based Inhomogeneous Broadening.

The results of electronic and vibrational structure simulations are an invaluable support for interpreting experimental absorption/emission spectra, which stimulates the development of reliable and cost-effective computational protocols. In this work, we contribute to these efforts and propose an efficient first-principle protocol for simulating vibrationally-resolved absorption spectra, including nonempirical estimations of the inhomogeneous broadening. To this end, we analyze three key aspects: (i) a metric-based selection of density functional approximation (DFA) so to benefit from the computational efficiency of time-dependent density function theory (TD-DFT) while safeguarding the accuracy of the vibrationally-resolved spectra, (ii) an assessment of two vibrational structure schemes (vertical gradient and adiabatic Hessian) to compute the Franck-Condon factors, and (iii) the use of machine learning to speed up nonempirical estimations of the inhomogeneous broadening. In more detail, we predict the absorption band shapes for a set of 20 medium-sized fluorescent dyes, focusing on the bright ππ★ S0 → S1 transition and using experimental results as references. We demonstrate that, for the studied 20-dye set which includes structures with large structural variability, the preselection of DFAs based on an easily accessible metric ensures accurate band shapes with respect to the reference approach and that range-separated functionals show the best performance when combined with the vertical gradient model. As far as band widths are concerned, we propose a new machine-learning-based approach for determining the inhomogeneous broadening induced by the solvent microenvironment. This approach is shown to be very robust offering inhomogeneous broadenings with errors as small as 2 cm-1 with respect to genuine electronic-structure calculations, with a total CPU time reduced by 98%.

Are Accelerated and Enhanced Wave Function Methods Accurate to Compute Static Linear and Nonlinear Optical Properties?

Key components of organic-based electro-optic devices are challenging to design or optimize because they exhibit nonlinear optical responses, which are difficult to model or rationalize. Computational chemistry furnishes the tools to investigate extensive collections of molecules in the quest for target compounds. Among the electronic structure methods that provide static nonlinear optical properties (SNLOPs), density functional approximations (DFAs) are often preferred because of their low cost/accuracy ratio. However, the accuracy of the SNLOPs critically depends on the amount of exact exchange and electron correlation included in the DFA, precluding the reliable calculation of many molecular systems. In this scenario, wave function methods such as MP2, CCSD, and CCSD(T) constitute a reliable alternative to compute SNLOPs. Unfortunately, the computational cost of these methods significantly restricts the size of molecules to study, a limitation that hampers the identification of molecules with significant nonlinear optical responses. This paper analyzes various flavors and alternatives to MP2, CCSD, and CCSD(T) methods that either drastically reduce the computational cost or improve their performance but were scarcely and unsystematically employed to compute SNLOPs. In particular, we have tested RI-MP2, RIJK-MP2, RIJCOSX-MP2 (with GridX2 and GridX4 setups), LMP2, SCS-MP2, SOS-MP2, DLPNO-MP2, LNO-CCSD, LNO-CCSD(T), DLPNO-CCSD, DLPNO-CCSD(T0), and DLPNO-CCSD(T1). Our results indicate that all these methods can be safely employed to calculate the dipole moment and the polarizability with average relative errors below 5% with respect to CCSD(T). On the other hand, the calculation of higher-order properties represents a challenge for LNO and DLPNO methods, which present severe numerical instabilities in computing the single-point field-dependent energies. RI-MP2, RIJK-MP2, or RIJCOSX-MP2 are cost-effective methods to compute first and second hyperpolarizabilities with a marginal average error with respect to canonical MP2 (up to 5% for ß and up to 11% for γ). More accurate hyperpolarizabilities can be obtained with DLPNO-CCSD(T1); however, this method cannot be employed to obtain reliable second hyperpolarizabilities. These results open the way to obtain accurate nonlinear optical properties at a computational cost that can compete with current DFAs.

Difluoroborate-based bichromophores: Symmetry relaxation and two-photon absorption.

Given potential applications of multiphoton absorbers, in the present work we have studied the symmetry-relaxation effects in one- and two-photon absorption spectra in two bichromophore systems based on difluoroborate core linked by biphenylene or bianthracene moieties. We have employed a palette of experimental methods (synthesis, one- and two-photon spectroscopy, X-ray crystallography) and state-of-the-art computational methods to shed light on how symmetry relaxation, a result of twisting of building blocks, affects one- and two-photon absorption of the two studied fluorescent dyes. Electronic-structure calculations revealed that the planarity of central biphenyl moiety, as well as deviations from planarity up to 30-40 deg., ensure maximum values of two-photon transition strengths. Perpendicular arrangement of phenylene units in biphenylene moiety leads to 20% drop in the two-photon transition strengths. More detailed studies demonstrated that equilibrium structures of both compounds in chloroform solution show very different values of two-photon absorption cross sections at absorption band maxima, i.e. 224 GM for and 134 GM for biphenyle and bianthracene linkers, respectively. The latter value is in good agreement with experimental value obtained using Z-scan method. The difference in two-photon absorption cross section between both compounds can be rationalized based on equilibrium geometry differences, i.e. interplanar angle is 35 deg and 91 deg in the case of biphenylene and bianthracene moiety, respectively. It is thus not beneficial to introduce conformationally locked central linker based on bianthracene moiety.

Hyperpolarizabilities of Push-Pull Chromophores in Solution: Interplay between Electronic and Vibrational Contributions.

Contemporary design of new organic non-linear optical (NLO) materials relies to a large extent on the understanding of molecular and electronic structure-property relationships revealed during the years by available computational approaches. The progress in theory-hand-in-hand with experiment-has enabled us to identify and analyze various physical aspects affecting the NLO responses, such as the environmental effects, molecular vibrations, frequency dispersion, and system dynamics. Although it is nowadays possible to reliably address these effects separately, the studies analyzing their mutual interplay are still very limited. Here, we employ density functional theory (DFT) methods in combination with an implicit solvent model to examine the solvent effects on the electronic and harmonic as well as anharmonic vibrational contributions to the static first hyperpolarizability of a series of push-pull α,ω-diphenylpolyene oligomers, which were experimentally shown to exhibit notable second-order NLO responses. We demonstrate that the magnitudes of both vibrational and electronic contributions being comparable in the gas phase significantly increase in solvents, and the enhancement can be, in some cases, as large as three- or even four-fold. The electrical and mechanical anharmonic contributions are not negligible but cancel each other out to a large extent. The computed dynamic solute NLO properties of the studied systems are shown to be in a fair agreement with those derived from experimentally measured electric-field-induced second-harmonic generation (EFISHG) signals. Our results substantiate the necessity to consider concomitantly both solvation and vibrational effects in modeling static NLO properties of solvated systems.

Unveiling Halogen-Bonding Interactions between a Pyridine-Functionalized Fluoroborate Dye and Perfluorohaloarenes with Fluorescence Spectroscopy.

We have studied the halogen-bonding interactions of a pyridine-functionalized fluoroborate dye with perfluorohaloarenes (C6F6, C6F5Cl, C6F5Br, and C6F5I) in the two-component-only liquid phase using fluorescence spectroscopy. Based on the results of spectroscopic measurements and electronic-structure calculations, we have confirmed the stability only for the complex between C6F5I and the emissive dye, and it has been demonstrated that halogen-bonding interactions are accompanied by significant Stokes shifts for the ππ* band. We also provide experimental evidence that for this complex, the emission is quenched due to a simultaneous decrease of radiative and increase of nonradiative decay rate constants upon halogen-bonding interactions.

How Reliable Are Modern Density Functional Approximations to Simulate Vibrational Spectroscopies?

We show that properties of molecules with low-frequency modes calculated with density functional approximations (DFAs) suffer from spurious oscillations along the nuclear displacement coordinate due to numerical integration errors. Occasionally, the problem can be alleviated using extensive integration grids that compromise the favorable cost-accuracy ratio of DFAs. Since spurious oscillations are difficult to predict or identify, DFAs are exposed to severe performance errors in IR and Raman intensities and frequencies or vibrational contributions to any molecular property. Using Fourier spectral analysis and digital signal processing techniques, we identify and quantify the error due to these oscillations for 45 widely used DFAs. LC-BLYP and BH&H are revealed as the only functionals showing robustness against the spurious oscillations of various energy, dipole moment, and polarizability derivatives with respect to a nuclear displacement coordinate. Given the ubiquitous nature of molecules with low-frequency modes, we warrant caution in using modern DFAs to simulate vibrational spectroscopies.

First-order hyperpolarizabilities of propellanes: elucidating structure-property relationships.

Following recent experimental work demonstrating strong nonlinear optical properties, namely second harmonic generation of light, in crystals composed of 16,20-dinitro-(3,4,8,9)-dibenzo-2,7-dioxa-5,10-diaza[4.4.4]propellane molecules [A. Miniewicz, S. Bartkiewicz, E. Wojaczynska, T. Galica, R. Zalesny and R. Jakubas, J. Mater. Chem. C, 2019, 7, 1255-1262] in this paper we aim to investigate "structure-property" relationships for a series of 16 propellanes presenting a wide palette of substituents with varying electron-accepting/donating capabilities. To that end, we use electronic- and vibrational-structure theories and a recently developed generalized few-state model combined with a range-separated CAM-B3LYP functional to analyze electronic and vibrational contributions to the first hyperpolarizability for the whole series of molecules. The variations in computed properties are large among the studied set of substituents and can reach an order of magnitude. It has been demonstrated that the maximum values of frequency-independent first hyperpolarizability are expected for strong electron-accepting NO2 substituents, but only at the preferred position with respect to the electronegative oxygen atom in the 1,4-oxazine moiety. This holds for electronic as well as vibrational counterparts.

Choosing Bad versus Worse: Predictions of Two-Photon-Absorption Strengths Based on Popular Density Functional Approximations.

We present a benchmark study of density functional approximation (DFA) performances in predicting the two-photon-absorption strengths in π-conjugated molecules containing electron-donating/-accepting moieties. A set of 48 organic molecules is chosen for this purpose, for which the two-photon-absorption (2PA) parameters are evaluated using different DFAs, including BLYP, PBE, B3LYP, PBE0, CAM-B3LYP, LC-BLYP, and optimally tuned LC-BLYP. Minnesota functionals and ωB97X-D are also used, applying the two-state approximation, for a subset of molecules. The efficient resolution-of-identity implementation of the coupled-cluster CC2 model (RI-CC2) is used as a reference for the assessment of the DFAs. Two-state models within the framework of both DFAs and RI-CC2 are used to gain a deeper insight into the performance of different DFAs. Our results give a clear picture of the performance of the density functionals in describing the two-photon activity in dipolar π-conjugated systems. The results show that global hybrids are best suited to reproduce the absolute values of 2PA strengths of donor-acceptor molecules. The range-separated functionals CAM-B3LYP and optimally tuned LC-BLYP, however, show the highest linear correlations with the reference RI-CC2 results. Hence, we recommend the latter DFAs for structure-property studies across large series of dipolar compounds.

Much of a Muchness: On the Origins of Two- and Three-Photon Absorption Activity of Dipolar Y-Shaped Chromophores.

The molecular origin of two- (2PA) and three-photon absorption (3PA) activity in three experimentally studied chromophores, prototypical dipolar systems, is investigated. To that end, a generalized few-state model (GFSM) formula is derived for the 3PA transition strength for nonhermitian theories and employed at the coupled-cluster level of theory. Using various computational techniques such as molecular dynamics, linear and quadratic response theories, and GFSM, an in-depth analysis of various optical channels involved in 2PA and 3PA processes is presented. It is found that the four-state model involving the second and third excited singlet states as intermediates is the smallest model among all considered few-state approximations that produces 2PA and 3PA transition strengths (for S0 → S1 transition) close to the reference results. By analyzing various optical channels appearing in these models and involved in studied multiphoton processes, we found that the 2PA and 3PA activities in all the three chromophores are dominated and hence controlled by the dipole moment of the final excited state. The similar origins of the 2PA and the 3PA in these prototypical dipolar chromophores suggest transferability of structure-property relations from the 2PA to the 3PA domain.

Two-Photon Absorption Activity of BOPHY Derivatives: Insights from Theory.

We present a theoretical study of a two-photon absorption (2PA) process in dipolar and quadrupolar systems containing two BF2 units. For this purpose, we considered 13 systems studied by Ponce-Vargas et al. [ J. Phys. Chem. B 2017, 121, 10850-10858] and performed linear and quadratic response theory calculations based on the RI-CC2 method to obtain the 2PA parameters. Furthermore, using the recently developed generalized few-state model, we provided an in-depth view of the changes in 2PA properties in the molecules considered. Our results clearly indicate that suitable electron-donating group substitution to the core BF2 units results in a large red-shift of the two-photon absorption wavelength, thereby entering into the desired biological window. Furthermore, the corresponding 2PA strength also increases significantly (up to 30-fold). This makes the substituted systems a potential candidate for biological imaging.