Can Aromaticity be Evaluated Using Atomic Partitions based on the Hilbert-space?

Aromaticity is a fundamental concept in chemistry that explains the stability and reactivity of many compounds by identifying atoms within a molecule that form an aromatic ring. Reliable aromaticity indices focus on electron delocalization and depend on atomic partitions, which give rise to the concept of an atom-in-the-molecule (AIM). Real-space atomic partitions present two important drawbacks: a high computational cost and numerical errors, limiting aromaticity measures to medium-sized molecules with rings up to 12 atoms. This restriction hinders the study of large conjugated systems like porphyrins and nanorings. On the other hand, traditional Hilbert-space schemes are free of the latter limitations but can be unreliable for the large basis sets required in modern computational chemistry. This paper explores AIMs based on three robust Hilbert-space partitions --meta-Löwdin, Natural Atomic Orbitals (NAO), and Intrinsic Atomic Orbitals (IAO)-- which combine the advantages of real-space partitions without their disadvantages. These partitions can effectively replace real-space AIMs for evaluating the aromatic character. For the first time, we report multicenter index (MCI) and \iring values for large rings and introduce ESIpy, an open-source Python code for aromaticity analysis in large conjugated rings.

Unveiling the quantum secrets of triel metal triangles: a tale of stability, aromaticity, and relativistic effects.

Low lying electronic states of Al3-, Ga3-, In3-, and Tl3- have been characterized using high level multiconfigurational quasi degenerate perturbation theory on the multiconfigurational self-consistent field. Among these species, the singlet states emerge as the predominant energy minima, displaying remarkable stability. However, within the Tl3- series, our investigation leads to the identification of the high-spin , as the most stable spin state, a result corroborated by previous experimental detection via photoelectron spectroscopy. Similarly, we have also identified the singlet state of In3- as the signal detected previously experimentally. By applying Mandado's rules and an array of aromaticity indicators, it is conclusively demonstrated that both the singlet and quintet states exhibit multiple-fold aromaticity, while the triplets exhibit conflicting aromaticity. Furthermore, this investigation highlights the significant impact of relativistic effects, as they enhance the stability of the state relative to its singlet counterpart. These findings shed new light on the electronic structures and properties of these ions, offering valuable insights into their chemical behavior and potential applications.

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.

Understanding Aromaticity in [5]Helicene-Bridged Cyclophanes: A Comprehensive Study.

This study explores the aromaticity of doubly [5]helicene-bridged (1,4)cyclophane and triply [5]helicene-bridged (1,3,5)cyclophane via calculations of the magnetic response and of electronic aromaticity indices. The primary objective is to assess the π-electron delocalization to determine whether they sustain global ring currents associated with π aromaticity. The molecules show local ring currents in the presence of an external magnetic field. The ring currents flow diatropically in the stacked six-membered rings and in the helicene arms. However, these π currents are not interconnected due to the discontinuity of the π delocalization at the C-C single bonds connecting the central six-membered rings to the helicene arms. Electronic indices suggest that the helicene-arm systems have significantly smaller electron delocalization than benzene. The reduction in the delocalization does not compromise their ability to exhibit ring currents in the presence of an external magnetic field. The analysis provides further evidence that the magnetic criteria yield a different degree of aromaticity for the helicene arms than obtained in the calculation of the electronic aromaticity indices. However, both approaches confirm that the studied molecules are not globally aromatic.

All-Purpose Measure of Electron Correlation for Multireference Diagnostics.

We present an analytical relationship between two natural orbital occupancy-based indices, IND¯ and INDmax, and two established electron correlation metrics: the leading term of a configuration interaction expansion, c0, and the D2 diagnostic. Numerical validation revealed that IND¯ and INDmax can effectively substitute for c0 and D2, respectively. These indices offer three distinct advantages: (i) they are universally applicable across all electronic structure methods, (ii) their interpretation is more intuitive, and (iii) they can be readily incorporated into the development of hybrid electronic structure methods. Additionally, we draw a distinction between correlation measures and correlation diagnostics, establishing MP2 and CCSD numerical thresholds for INDmax, which are to be used as a multireference diagnostic. Our findings further demonstrate that establishing thresholds for other electronic structure methods can be easily accomplished using small data sets.

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.

Electric-field induced second harmonic generation responses of push-pull polyenic dyes: experimental and theoretical characterizations.

The second-order nonlinear optical properties of four series of amphiphilic cationic chromophores involving different push-pull extremities and increasingly large polyenic bridges have been investigated both experimentally, by means of electric field induced second harmonic (EFISH) generation, and theoretically, using a computational approach combining classical molecular dynamics (MD) and quantum chemical (QM) calculations. This theoretical methodology allows to describe the effects of structural fluctuations on the EFISH properties of the complexes formed by the dye and its iodine counterion, and provides a rationale to EFISH measurements. The good agreement between experimental and theoretical results proves that this MD + QM scheme constitutes a useful tool for a rational, computer-aided, design of SHG dyes.

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.

Quest for the Most Aromatic Pathway in Charged Expanded Porphyrins.

Despite the central role of aromaticity in the chemistry of expanded porphyrins, the evaluation of aromaticity remains difficult for these extended macrocycles. The presence of multiple conjugation pathways and different planar and nonplanar π-conjugation topologies makes the quantification of global and local aromaticity even more challenging. In neutral expanded porphyrins, the predominance of the aromatic conjugation pathway passing through the imine-type nitrogens and circumventing the amino NH groups is established. However, for charged macrocycles, the question about the main conjugation circuit remains open. Accordingly, different conjugation pathways in a set of neutral, anionic, and cationic expanded porphyrins were investigated by means of several aromaticity indices rooted in the structural, magnetic, and electronic criteria. Overall, our results reveal the predominance of the conjugation pathway that passes through all nitrogen atoms to describe the aromaticity of deprotonated expanded porphyrins, while the outer pathway through the perimeter carbon atoms becomes the most aromatic in protonated macrocycles. In nonplanar and charged macrocycles, a discrepancy between electronic and magnetic descriptors is observed. Nevertheless, our work demonstrates AVmin remains the best tool to determine the main conjugation pathway of expanded porphyrins.

Three-centre electron sharing indices (3c-ESIs) as a tool to differentiate among (an)agostic interactions and hydrogen bonds in transition metal complexes.

The agostic bond plays an important role in chemistry, not only in transition metal chemistry but also in main group chemistry. In some complexes with M⋯H-X (X = C, N) interactions, differentiation among agostic, anagostic, and hydrogen bonds is challenging. Here we propose the use of three-centre electron sharing indices to classify M⋯H-X (X = C, N) interactions.

Excited state dynamics and conjugation effects of the photoisomerization reactions of dihydroazulene.

Herein, we present an investigation of the excited state dynamics of the dihydroazulene photoswitch and its photoinduced reaction to vinylheptafulvene. The focus is on how the introduction of a benzannulated ring in different sites of the structure can modify the excited state topology and thus the kinetics of the ring opening reaction of DHA by alteration of the excited state conjugation of the system. The dynamics of the systems is obtained utilizing ab initio density functional theory calculations in different solvents coupled with unimolecular reaction theory. To accompany these results, the electron delocalization is investigated using the quantum theory of atoms in molecules partitioning to follow the trends induced by the benzannulated ring. It is observed that the introduction of a benzannulated ring can both enhance and diminish the rate of the photoinduced ring opening of dihydroazulene and that certain patterns of conjugation are consistent with the rate constants. Lastly, we find good agreement with earlier experimental studies indicating that the chosen approach could be used to predict whether photochromic systems lose their photoswitchability upon being optimized for specific applications via functionalization.

Reply to the Correspondence on "How Aromatic Are Molecular Nanorings? The Case of a Six-Porphyrin Nanoring".

A recent article by Anderson and co-workers challenges our conclusions on the aromaticity of the four oxidation states of a butadyine-linked six-porphyrin nanoring, based on the experimental 1 H-NMR data and some recent calculations they have performed using the BLYP35 functional. Here, we show that BLYP35 should be taken with caution and demonstrate that the indirect evidence of a ring current from experimental 1 H-NMR data is not a definite proof of aromaticity.

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.

Natural range separation of the Coulomb hole.

A natural range separation of the Coulomb hole into two components, one of them being predominant at long interelectronic separations (hcI ) and the other at short distances (hcII ), is exhaustively analyzed throughout various examples that put forward the most relevant features of this approach and how they can be used to develop efficient ways to capture electron correlation. We show that hcI , which only depends on the first-order reduced density matrix, can be used to identify molecules with a predominant nondynamic correlation regime and differentiate between two types of nondynamic correlation, types A and B. Through the asymptotic properties of the hole components, we explain how hcI can retrieve the long-range part of electron correlation. We perform an exhaustive analysis of the hydrogen molecule in a minimal basis set, dissecting the hole contributions into spin components. We also analyze the simplest molecule presenting a dispersion interaction and how hcII helps identify it. The study of several atoms in different spin states reveals that the Coulomb hole components distinguish correlation regimes that are not apparent from the entire hole. The results of this work hold out the promise to aid in developing new electronic structure methods that efficiently capture electron correlation.

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.

Impact of van der Waals interactions on the structural and nonlinear optical properties of azobenzene switches.

The geometrical structures, relative Z-E energies, and second-order nonlinear responses of a collection of azobenzene molecules symmetrically substituted in the meta-position with functional groups of different bulkiness are investigated using various ab initio and density functional approximations. We show that RI-MP2 and RI-CC2 approximations provide very similar geometries and relative energies and evidence that London dispersion interactions existing between bulky meta-substituents stabilize the Z conformer. The ωB97X-D exchange-correlation functional provides an accurate description of these effects and gives a good account of the nonlinear optical response of the molecules. We show that density functional approximations should include no less than 50% of Hartree-Fock exchange to provide accurate hyperpolarizabilities. A property-structure analysis of the azobenzene derivatives reveals that the main contribution to the first hyperpolarizability comes from the azo bond, but phenyl meso-substituents can enhance it.

How Aromatic Are Molecular Nanorings? The Case of a Six-Porphyrin Nanoring*.

Large conjugated rings with persistent currents are novel promising structures in molecular-scale electronics. A six-porphyrin nanoring structure that allegedly sustained an aromatic ring current involving 78π electrons was recently synthesized. We provide here compelling evidence that this molecule is not aromatic, contrary to what was inferred from the analysis of 1 H-NMR data and computational calculations that suffer from large delocalization errors. The main reason behind the absence of an aromatic ring current in these nanorings is the low delocalization in the transition from the porphyrins to the bridging butadiyne linkers, which disrupts the overall conjugated circuit. These results highlight the importance of choosing a suitable computational method to study large conjugated molecules and the appropriate aromaticity descriptors to identify the part of the molecule responsible for the loss of aromaticity.

How Many Electrons Does a Molecular Electride Hold?

Electrides are very peculiar ionic compounds where electrons occupy the anionic positions. In a crystal lattice, these isolated electrons often form channels or surfaces, furnishing electrides with many traits with promising technological applications. Despite their huge potential, thus far, only a few stable electrides have been produced because of the intricate synthesis they entail. Due to the difficulty in assessing the presence of isolated electrons, the characterization of electrides also poses some serious challenges. In fact, their properties are expected to depend on the arrangement of these electrons in the molecule. Among the criteria that we can use to characterize electrides, the presence of a non-nuclear attractor (NNA) of the electron density is both the rarest and the most salient feature. Therefore, a correct description of the NNA is crucial to determine the properties of electrides. In this paper, we analyze the NNA and the surrounding region of nine molecular electrides to determine the number of isolated electrons held in the electride. We have seen that the correct description of a molecular electride hinges on the electronic structure method employed for the analyses. In particular, one should employ a basis set with sufficient flexibility to describe the region close to the NNA and a density functional approximation that does not suffer from large delocalization errors. Finally, we have classified these nine molecular electrides according to the most likely number of electrons that we can find in the NNA. We believe this classification highlights the strength of the electride character and will prove useful in designing new electrides.

Guidelines for Tuning the Excited State Hückel-Baird Hybrid Aromatic Character of Pro-Aromatic Quinoidal Compounds*.

Pro-aromatic molecules have higher-energy diradicaloid states that are significantly influenced by resonance structures in which conjugated rings take on Hückel-aromatic character. Recently, it has been argued that there are also pro-aromatic molecules that adopt central units with 4nπ-electron Baird-aromatic character in the T1 state, although detailed analysis suggests that these compounds are better labelled as T1 Hückel-Baird hybrid molecules where Hückel-aromaticity dominates. Herein, we consider a series of symmetrically substituted conjugated rings with potential Baird aromaticity in the lowest excited triplet and singlet states. Our computational results allow us to establish general guidelines for the rational design of molecules with excited state Hückel/Baird aromaticity in pro-aromatic quinoidal compounds. We found two main strategies to promote high Baird aromatic character: 1) anionic and small conjugated rings with electron donating groups as substituents and small exocyclic groups with electron withdrawing substituents, or 2) electron deficient conjugated rings with exocyclic electron-donor substitution.

Performance of DFT functionals for calculating the second-order nonlinear optical properties of dipolar merocyanines.

The second-order nonlinear optical responses of a series of recently designed dipolar merocyanines are investigated using the 2006 Minnesota family of hybrid exchange-correlation functionals (XCFs), as well as the LC-BLYP, ωB97XD and CAM-B3LYP long-range (LR) corrected XCFs. The performance of these different levels of approximation is discussed in regard to reference second-order Møller-Plesset calculations and experimental data obtained from Hyper-Rayleigh Scattering (HRS) measurements. Particular focus is given to the influence of the amount of exact Hartree-Fock exchange included in the XCF on the magnitude of the static HRS responses, as well as to the impact of tuning the range-separation parameter in LR-XCFs, according to a system-specific nonempirical procedure. Frequency dispersion effects are also investigated, as well as their crucial role in the comparison between theoretical and experimental data.