Electron Density of Delocalized Bonds As a Universal Tool for Assessing Global and Local Effects of Chemical Resonance

Summary of Professional Accomplishments

Electron density of delocalized bonds as a universal tool for assessing global and local effects of chemical resonance

Dr. Dariusz Wojciech Szczepanik

Kraków 2021

1. Name Name and Surname: Dariusz Wojciech Szczepanik Degree: Doctor of Philosophy in Chemistry Current employment: Department of Theoretical Chemistry, Faculty of Chemistry, Jagiellonian University, Gronostajowa 2, 30-387 Kraków, Poland. ORCID: 0000-0002-2013-0617 ResearcherID: E-2787-2014 ScopusID: 36835515900 Personal website: http://www.eddb.pl/aboutme Presentation of the achievements: http://www.eddb.pl/hab

2. Diplomas, degrees conferred in specific areas of science or arts, including the name of the institution which conferred the degree, year of degree conferment, title of the PhD dissertation 2008 Master of Science Department of Computational Methods in Chemistry, Faculty of Chemistry UJ Thesis: „Entropic indices of the chemical bonds from information theory” Supervisor: dr. hab. Janusz Mrozek 2013 Doctor of Philosophy in Chemistry Department of Theoretical Chemistry, Faculty of Chemistry UJ Thesis: „Probabilistic models of the chemical bond in function spaces” (de- fended with honors) Supervisor: dr. hab. Janusz Mrozek

3. Information on employment in research institutes or faculties/departments or school of arts 2015 – 2020 Department of Theoretical Chemistry, Faculty of Chemistry, Jagiellonian Uni- versity. Position: Technician 2018 – 2020 Institute of Computational Chemistry and Catalysis, University of Girona, Posi- tion: EU-researcher (MSCA-IF, postdoc). 2020 – Department of Theoretical Chemistry, Faculty of Chemistry, Jagiellonian Uni- versity. Position: Adjunct

4. Description of the achievements, set out in art. 219 para 1 point 2 of the Act The basis for the scientific achievement entitled “Electron density of delocalized bonds as a universal tool for assessing global and local effects of chemical resonance” is a series of thematically linked publications composed of 9 scientific articles (H1-H9) published in 2014-2019 in peer-reviewed journals from the JCR list, and 1 monographic chapter (H10) of 2021. The subject of scientific articles 2 selected for the habilitation cycle (H1-H9) concerns development of the theoretical basis of my original computational method that enables one to “extract” from the total electron density of a molecule (obtained from quantum-chemical calculations) the so-called electron density of delocalized bonds (EDDB), which allows visualization and quantification of different effects of electron delocalization in chemical species, regardless of their size, topology and the electronic state. Publication (H10), in turn, can be regarded as a review article. All publications included in the habilitation cycle are based on my own original research ideas, and in all of these works I am the first and the corresponding author. I am the sole author of H3, H4, and H8, while in the case of H9 and H10 the second co-author is Prof. Miquel Solà, at whose invitation these works were prepared. The high percentage estimate of my contribution to the remaining articles is guided by the criterion of the amount of time spent planning research study, writing software and scripts automating calculations (partially carried out by PhD students), elaboration of the results, preparation and submission of the manuscript, improving them in the review process, etc. In addition, I am the sole author of the original model of the "migrating π-cycles" introduced in H7 (and being a generalization of the well-known in the literature model of “the migrating Clar’s π-sextet”).

4.1. Motivation1 It is commonly accepted that electron delocalization in aromatic rings is linked with unusual thermodynamic stability by means of the π-electron bookkeeping rules, like “4n+2” and “4n” (depending on the system topology and multiplicity), known from the age-old chemistry textbooks. Although these qualitative criteria of aromaticity (and antiaromaticity) adequately relate topology, symmetry, and degeneracy of molecular orbitals (MO) in the [n]annulene-like systems predominated by covalent resonance forms at their singlet or the lowest-lying triplet states, their use in a more general context regarding topologically diversified poly- and macrocyclic species (e.g. expanded porphyrins predominated by ionic forms), non-Kekulé molecules (e.g. radicals), etc., is not well-founded. Over the last decades an overwhelming number of quantitative ‘measures’ of aromaticity has been proposed in the literature, based on energetic, structural, magnetic, and electronic properties of molecules, thus providing a far more accurate account of aromatic stabilization than the electron-bookkeeping criteria. The most commonly used quantitative criteria of aromaticity within each of these groups are: 1) the aromatic stabilization energy (ASE), which is an energetic measure of π-aromaticity that emanates from the theory of valence bonds, and it can be evaluated by means of thermodynamic data for isodesmic, (hyper)homodesmotic or isomerization reactions (also, a multitude of schemes can be found in the literature that allows one to efficiently estimate the aromatic stabilization energies for a specific class of aromatic species); 2) the harmonic-oscillator model of aromaticity (HOMA), which is a π- aromaticity index based on structural properties of molecules (being a normalized measure of deviations of bond lengths in aromatic molecule from the corresponding optimum bond lengths in an idealized non-aromatic molecule as a reference) – HOMA is close to 0 for nonaromatic species, approaches 1 for highly aromatic ones, while for potentially antiaromatic rings it usually assumes neg-

1 Formal definitions, descriptions and scientific arguments adduced in this section comes from review articles in the special issue Chem. Soc. Rev 44 (2015), original publications H5, H7, and H10, as well as the following papers: A. Stanger, Chem. Comm. 2009 (2009) 1939; R. Hoffmann, Am. Sci. 103 (2015) 18; M. Solà, Front. Chem. 5 (2017) 22. 3 ative values; 3) the nucleus-independent chemical shift (NICS), which quantifies the effective magnetic shielding at the centroid (or above) of the aromatic ring in external magnetic field – the more negative (positive) value of NICS, the more aromatic (antiaromatic) is the molecular ring in question; 4) the multicenter index (MCI), which is a non-reference index of aromaticity that can be calculated from both the ab initio molecular wave function as well as the electron density; MCI has been shown to be superior to other aromaticity descriptors as the only one that passes a set of rigorous tests for aromaticity quantifiers designed by Prof. M. Solà (University of Girona, Spain).

Unfortunately, each of the above-mentioned aromaticity ‘measures’ has shortcomings and limitations that sometimes may lead to wrong predictions. The ASE seems to outwardly be the most adequate measure of global aromaticity since it can be evaluated by means of thermodynamical data and it emerges for the direct relationship between structural consequences of electron delocalization and the stability. However, designing of isodesmic and homodesmotic reaction scenarios is very difficult in practice and opens the door to a lot of arbitrariness. In contrast, an unquestionable advantage of HOMA is its computational and interpretative simplicity – it allows one to straightforwardly classify any molecular ring as aromatic, non-aromatic or (potentially) antiaromatic. Unfortunately, the principal problem with HOMA is the necessity of parametrization of bond lengths for an idealized reference molecule, which obviously cannot be chosen unambiguously. Consequently, the practical use of HOMA is limited to aromatic and heteroaromatic systems since the parameters for chemical bonds with metal atoms are not available. Furthermore, parametrization of HOMA should be performed using exactly the same quantum-chemical method as used in calculations of equilibrium geometries of the molecule under study, since routinely computed HOMA with the experimentally determined parameters is bound to suffer from large unsystematic errors and strong sensitivity to the choice of the basis set and the exchange-correlation functional in the density functional theory (DFT) based calculations. Aromaticity descriptors based on magnetic properties of molecules, especially NICS(0), NICS(1), and its axial component, NICS(1)zz, dissected NICS, etc., are one the most preferable measures of local aromaticity due to their relation to experiment; diatropic (aromatic) and paratropic (antiaromatic) ring currents indirectly manifest itself in the NMR spectra. However, the magnetic-based measures of aromaticity have also come under bitter criticism due to their complexity (NICS relies on the condensation of potentially complicated patterns of induced currents to a single number), and methodological shortcomings. Finally, the main disadvantage connected with the calculation of MCI is its numerical instability and computational cost which prevent using MCI to analyze systems containing more than 12-14 atoms; additionally, MCI suffers from the method dependence – in particular, the exchange-correlation functional selection at the density functional theory (DFT) level.

To summarize, most of the aromaticity descriptors suffer from serious issues such as the arbitrariness of choice of a reference system, lack of parametrization, ring-size extensivity issue, limited applicability, computational inefficiency, as well as methodological shortcomings and interpretative mistiness. However, introducing new aromaticity measures makes sense nowadays only if their performance provides an advantage over the already existing descriptors or they offer similar quality but at a significantly lower computational cost.

The main goal behind the development of the electron density of delocalized bonds (EDDB) method was to create possibly the most versatile tool that enables analysis of the electron delocalization in (but not only) aromatic species regardless of its type, size, topology, and the electronic state, free from all the above-mentioned issues and opening up new directions in the field, especially in the context of the excited-state aromaticity studies.

4.2. Theoretical background of the EDDB method Electron density of delocalized bonds is a part of the original method of one-electron density (ED) decomposition into ‘layers’ representing different levels of electron delocalization (H3):

ED(r) = EDLA(r) + EDLB(r) + EDDB(r), (1) with density of electrons localized on atoms (EDLA) representing inner shells, lone pairs, etc; electron density of localized bonds (EDLB) representing typical (2-center 2-electron) Lewis-like bonds; EDDB, which represents electron density that cannot be assigned to atoms or bonds due to its (multicenter) delocalized nature. In the basis of natural atomic orbitals (NAO), or any other representation of well-localized orthonormalized atomic orbitals, the spinless global electron density of delocalized bonds function, EDDBG(r), for a single-determinant molecular wavefunction is defined as follows (H2, H6):

(2) † ΩG EDDBG(푟) = ∑ 휒휇(푟)풟휇,휈 휒휈(푟), 휇,휈 where ΩG (3) 2 ΩG 휎 휎 ΩG,휎 휎 휎 † 휎 퓓 = 2 ∑ P [∑ 퓒푎,푏휺푎,푏 (흀푎,푏) 퓒푎,푏] P . 휎=훼,훽 푎,푏

In the above equation, Pσ (σ = α,β) stands for the σ spin-resolved charge and bond-order (CBO) ma- 휎 trix, 퓒푎,푏 is the matrix of linear coefficients of the appropriately orthogonalized σ spin-resolved two- center bond-order orbitals (2cBO) of the chemical bond Xa–Xb (obtained by diagonalization of the 휎 appropriate off-diagonal blocks of the CBO matrix), 흀푎,푏 represents the diagonal matrix collecting ΩG,휎 the corresponding 2cBO eigenvalues (occupation numbers), 휺푎,푏 is a diagonal matrix of the σ-spin bond-conjugation factors, and for an n-atomic molecular system, ΩG represents the set of all n(n-1)/2 possible atomic pairs (regardless of whether the atoms are formally bonded or not). The definition of ΩG,휎 the key matrix 휺푎,푏 is based on the bond-orbital projection (BOP) criterion developed by one of the authors, which relies on sophisticated orbital projection cascades involving 2cBOs, their 3-center counterparts (3cBO), and canonical MOs (H1, H4). According to BOP, for a typical well-localized ΩG,휎 (Lewis-like) bond Xa–Xb, all diagonal elements of the 휺푎,푏 matrix are close to zero, which means that the 2cBOs associated with this bond do not form effectively linear combinations with 2cBOs of all other bonds in a molecule. On the other hand, when the Xa–Xb bond is effectively conjugated with ΩG,휎 any other adjacent bond in the system, the 휺푎,푏 matrix has at least one element on its diagonal that approaches 1 (for systems with double and higher multifaceted aromaticity, the number of non-zero 5 diagonal elements is equal to the number of delocalization ‘channels’). The trace of such defined 퓓ΩG matrix can be straightforwardly interpreted as the population of electrons delocalized through the system of all conjugated bonds in a molecule, and as such, it can be used as a ‘measure’ of global aromaticity (H1, H3). One of the most distinctive features of the BOP technique is that one can easily restrict the set bonds/atomic pairs in ΩG giving rise to a series of different variants of global and local EDDB functions, as depicted in Fig. 1 (H10).

Figure 1. Isosurface contours and the corresponding populations of delocalized electrons in naphthalene by different global and local EDDB(r) functions. Source: H10.

In the case of the EDDBH(r) function, ΩH contains all possible atomic pairs in the molecule excluding hydrogen atoms from consideration. Thus, EDDBH(r) is also a global function that can be especially useful in the analysis of global π-aromaticity in organic molecules since the 2cBO involving hydrogen atoms tend to conjugate with the adjacent σ-bond orbitals noticeably increasing the delocalization in the σ-subsystem, which may sometimes lead to less precise conclusions. Moreover, the difference between EDDBG(r) and EDDBH(r) can be very useful in the identification of multivalent hydrogen interactions. For instance, as shown in Fig. 1, the net effect of eight C–H bonds in naphthalene on electron delocalization is 0.8054e, which means that only about 0.1e shared between carbon and hydrogen atoms is delocalized due to conjugation with other σ-bonds in the system; a much higher values (usually greater than 0.3e) are expected for bonds containing hydrogen and dihydrogen bonds. In contrast to EDDBG(r) and EDDBH(r), the next two EDDB functions can be regarded as local aromaticity measures (H10). In the case of the EDDBF(r) function, ΩF contains all possible atomic pairs in the selected molecular fragment (usually cyclic unit without H atoms), which in the case of a single 6-membered ring (6MR) in naphthalene gives rise to 6 ‧ 5 / 2 = 15 atomic pairs (the same number of linear combinations of 2cBO has to be considered within the BOP procedure). In turn, six chemical bonds are considered within the definition of the congeneric local EDDBP(r) function, which ‘measures’ electron delocalization along the selected pathway of adjacent bonds. Thus, in contrast to EDDBF(r), the EDDBP(r) function does not take into account the cross-ring delocalization effects, which for 6MR are associated with the resonance involving Dewar structures; in a sense, the relation between EDDBP and EDDBF corresponds to that between the multicenter index originally proposed

6 by Giambiagi et al. (IRing) and its averaged-over-permutations variant by Bultinck et al. (MCI). The EDDBP(r) function and the corresponding electron population (denoted simply by EDDBP) can be used to visualize and quantify local aromaticity in a wide range of molecular rings regardless of their size and topology, but also to assess macrocyclic aromaticity associated with particular delocalization pathways in larger structures like expanded porphyrins. Each of the EDDB(r) functions can easily be dissected into σ-, π- and higher-symmetry components by diagonalization of the corresponding density matrix defined in Eq. 2. The resulting eigenfunctions, called the natural orbitals for bond delocalization (NOBD), can be particularly useful in the identification of non-planar multifaceted aromatic compounds because they do not tend toward mixing of different symmetry types like the canonical MOs, even for twisted Möbius-type hetero- and metallacyclic aromatics (H9, H10). Within the current implementation of the EDDB method, the NOBDs are ordered from the high occupied to the low occupied ones, and, since the NOBDs responsible for aromaticity usually have the occupation numbers (ON) by order of magnitude greater than the rest of orbitals, the symmetry dissection of EDDB is straightforward. For instance, diagonalization of the EDDBH density matrix in naphthalene (Fig. 1) gives rise to 5 NOBDs (no 1-5) with occupation numbers from 1.5381e to 1.1907e, 29 NOBDs (no 6-34) with ONs from 0.0600e to 0.0011e, and the rest of NOBDs (no 35-228) have strictly zero ONs. By summation of the corresponding NOBD occupation numbers, one gets the following: π-EDDBH = 6.5100e and σ-EDDBH = 0.5721e; thus, even though the σ-delocalization effects are still noticeable, naphthalene is clearly π-aromatic with about 65% (6.5e/10.0e) of electrons effectively delocalized in the π-subsystem (for comparison, the corresponding effectiveness of delocalization of π-electrons in benzene is close to 90%). It has to be stressed that the bond-orbital projection technique underlying the EDDB method provides a strict criterion for bonding orbitals to form linear combinations with each other, which can be directly related to the effectiveness of conjugation of double bonds resulting from different resonance forms contributing to the wavefunction. For instance, naphthalene can be represented by 3 different resonance forms as presented in Fig. 2 (H10). Assuming equal contribution of each resonance form to the wavefunction (confirmed by the high-level NBO-NRT calculations), each of 11 2 1 1 bond positions in naphthalene can be thus represented either by the set {3 C–C + 3 C=C} or {3 C–C 2 + 3 C=C}; e.g., the central bond is single in two forms (red circles) and double in one form (blue 2 1 circle), which gives {3 C–C + 3 C=C}. One should realize that the superposition of the resonance form containing a double bond in the center (blue circle) with one of the resonance forms containing a single bond in the same position (red circle) gives rise to the Clar/Randic structure with a single π- sextet, while the superposition of two resonance forms containing a single bond in the center (red circles) results in a single π-dectet circuit (Fig. 2b). In the former, the central bond can be regarded as fully delocalized (one-and-a-half bond), whereas in the latter, it does not take part in delocalization. Since there are two possible combinations of the resonance structures (two different Clar π-sextets) responsible for delocalization of the central bond and only one combination (π-dectet) responsible for its localization, the effectiveness of delocalization of this bond is about 67%; alternatively, one can say that the central bond in naphthalene has a 67% aromatic and a 33% aliphatic character (the bond is indeed noticeably longer than that in benzene, RCC = 1.43Å compared to RCC = 1.40Å, respectively). The same degree of delocalization features all other bond positions in the system; e.g., the four ‘rim’ bonds (connecting carbon atoms in positions α and β) can be represented by 1 double and 2 one-and-a-half (delocalized) bonds, which means they have a 67% aromatic and a 33% olefinic

7 character (RCC = 1.37Å, so the bonds are noticeably shorter than the one in benzene). From the energetic point of view, the second- order perturbation theory involving the natural bond orbitals (NBO) predicts almost the same average stabilization effects (ca. 25 kcal/mol) associated with the delocalization of each double bond in all three resonance structures in naphthalene, which means that e.g. the delocalization of each ‘rim’ bond and the central bond contributes similarly to the aromatic stabilization energy. Therefore, it is clear that all chemical bonds in naphthalene have partially ‘aromatic’ character as they equally participate in the resonance, and the effectiveness of electron delocalization in naphthalene according to the EDDB predictions based on ab initio calculations (65%) is in full agreement with the classical resonance theory. In this context, it has to be emphasized that the representation of aromaticity in naphthalene exclusively by a single π- dectet circuit or the migrating Clar’s π-sextet is incorrect as it always discriminates delocalization effects associated with certain bond positions; thus, a reliable description of π-aromaticity in naphthalene requires both π- dectet and π-sextets circuits (Fig. 2b). More- Figure 2. a) The effectiveness of electron delocalization in over, in the presence of external magnetic naphthalene based on the classical resonance theory; b) Al- field, local diatropic ring currents associated ternative representations of the resonance forms involving with Clar’s π-sextets in naphthalene to a Clar’s π-sextet and π-dectet notation; c) Schematic representation of the π-circuit cancellation in naphthalene under the large extent cancel each other (Fig. 2c), thus external magnetic field; d) 3D-plots of π-EDDBH and π- favoring the envelope π-dectet diatropic cur- ACID in naphthalene. Source: H10. rent (which is depicted in Fig. 2d by utilizing the π-component of the anisotropy of current-induced density, π-ACID). The case of naphthalene clearly shows that the correspondence between aromaticity and the ring-current diatropicity may be not strict, and, in this context, the results of the ring-current analysis in polycyclic and topologically diversified aromatics should be always interpreted with caution. To summarize, the electron density of delocalized bonds allows one to quantify and visualize the population of electrons delocalized through the system of all (global) or selected (local) conjugated bonds in a wide range of aromatic species, and, in contrast to the induced ring-current methods, EDDB is derived from unperturbed one-electron density and, as such, it can be directly related to chemical resonance and its structural and energetic consequences.

4.3. Brief description of publications H1-H10 The scientometric data presented below comes from Google Scholar (accessed 20.04.2021). The impact factors (IF) for all articles and their total and average values are given according to the JCR 2019. The corresponding author is marked with the symbol ; the first (leading) author of the publication is highlighted. The total and average IF for works H1–H9 are 22.622 and 2.514, respectively, and the total number of citations equals to 238 (181 without self-cications).

4.3.1. Original scientific article H1 D.W. Szczepanik, E.J. Zak, K. Dyduch, J. Mrozek, „Electron delocalization index based on bond order orbitals”, Chemical Physics Letters 593 (2014) 154−159. IF: 2.029. MNiSW: 70. Citations: 31, without self-citations: 22. In this work, I proposed a new way of quantifying the effect of electron delocalization in planar aromatic rings, based on the criterion of effectiveness of forming linear combinations involving the two-center orbitals of adjacent chemical bonds. This is the first article to outline the concept of bond- orbital projections between 2-center bond orbitals and their 3-center counterparts, giving rise to the straightforward interpretation of the outcomes of the proposed 1st order population analysis: the resulting populations of the electrons correspond to the integral over the ‘layer’ of electron density that is represented by more than a single resonance form.

Figure 3. Decomposition of the valence- shell electron density of the benzene molecule into ‘layers’ representing electrons of localized bonds (~11.8e) and bonds delocalized in the ring (R) due to the chemical resonance (~5.5e).

The estimated percentage of my contribution to the work is 70%: original idea, implementation (full coding of the method), preliminary calculations, analysis and elaboration of the results, preparing the first draft of the manuscript, coordination. Students EJZ and KD carried out part of the quantum- chemical calculations; JM served expertise and discussions of the substance of the proposed formalism as well as the selection of the test sets of molecules, and contributed to the editorial work on the final version of the manuscript.

4.3.2. Original scientific article H2 D.W. Szczepanik, M. Andrzejak, K. Dyduch, E.J. Zak, M. Makowski, G. Mazur, J. Mrozek, „A uniform approach to the description of multicenter bonding”, Physical Chemistry Chemical Physics 16 (2014) 20514−20523. IF: 3.430. MNiSW: 100. Citations: 51, without self-citations: 37. In this work, the concept of the 2-center bond-orbital projection into the multicenter bond orbitals has been further developed and extended by me to enable ‘extraction’ from the total 1-electron density the density layer that represents delocalized electrons regardless of the topology of the molecular fragment and its size. The proposed method of the electron density of delocalized bonds (EDDB) was

9 used to analyze the complex effects of electron delocalization in mono- and polycyclic aromatic hydrocarbons, selected organometallic aromatics and molecular systems with atypical aromaticity.

Figure 4. Electron density of delocalized bonds (EDDB) in molecules with differentiated electronic structure, topology, and type of bond conjugation, such as charged rings, homo-aromatics, Möbius aromatics, etc.

The estimated percentage of my contribution to the work is 60%: original idea, implementation (full coding of the method), preliminary calculations, analysis and elaboration of the results, preparing figures and the first draft of the manuscript, coordination. Students EJZ and KD carried out part of the quantum-chemical calculations and prepared their preliminary elaboration; MA, MM, GM and JM served their expertise (especially as regards the selection of the test sets and discussion of the results) and language corrections during the work on the final version of the manuscript.

4.3.3. Original scientific article H3 D.W. Szczepanik, „A new perspective on quantifying electron localization and delocalization in molecular systems”, Computational and Theoretical Chemistry 1080 (2016) 33−37. IF: 1.605. MNiSW: 40. Citations: 26, without self-citations: 19. In this work, the original method of the bond-orbital projections between 2- and 3-center orbitals has been modified by introducing additional projections onto the space of the occupied canonical molecular orbitals. The newly proposed procedure gets rid of the non-orthogonality issue arising from solving the eigenproblems for 2-center bond operators independently for each atomic pair. As a result, the new formalism enables a strict dissection of the total 1-electron density into layers representing (1) electrons do not involved in bonding (e.g. free pairs, core electrons, etc.), EDLA, (2) electrons pairs shared in typical two-center Lewis-like bonds, EDLB, and (3) the electrons delocalized between different chemical bonds, EDDB. The capabilities of the new methodology are illustrated on the example of benzene, s-triazine, borazine and anthracene molecules. It is worth noting that the unique- ness of the new method is that it does not divide the molecule into molecular fragments (which always leads to ambiguity resulting from the definition of chemical bonding), but only separates the electron density layers by the level of delocalization (thus, EDLA, EDLB, and EDDB are global functions). The estimated percentage of my contribution to the work is 100%.

Figure 5. A strict decomposition of the valence-shell electron density of piridine into layers EDLA, EDLB i EDDB.

4.3.4. Original scientific article H4 D.W. Szczepanik, „On the three-center orbital projection formalism within the electron density of delocalized bonds method”, Computational and Theoretical Chemistry 1100 (2017), 13−17. IF: 1.605. MNiSW: 40. Citations: 12, without self-citations: 6. In this work, I investigated an alternative projection scheme for the bond-order orbitals taking into account the through-bridges and through-space interactions, leading to a significant reduction of the redundant electrons population that appear in the EDDB analyses involving charged molecules. The new technique was implemented and tested using two sets of molecules: (1) charged cyclic hydrocarbons and (2) a set of 25 different (hetero)aromatic molecules. The estimated percentage of my contribution to the work is 100%.

Figure 6. The new projection scheme between 3- and 2-center bond-order orbitals within the BOP formalism.

4.3.5. Original scientific article H5 D.W. Szczepanik, M. Solà, M. Andrzejak, B. Pawełek, J. Dominikowska, M. Kukułka, K. Dyduch, T.M. Krygowski, H. Szatylowicz, „The role of the long-range exchange corrections in the description of electron delocalization in aromatic species”, Journal of Computational Chemistry 38 (2017) 1640−1654. IF: 2.976. MNiSW: 100. Citations: 37, without self-citations: 29. In this work we examined reliability of the global and local description of aromatic rings by the EDDB method for the approximated density matrices obtained within the framework of the density functional theory involving diversified exchange-correlation functionals (XC). The results of the EDDB analysis were compared with those obtained using different aromatic indexes from magnetic (NICS and its variant), geometric (HOMA), and energetic criteria (HRCP), as well as selected electron- delocalization descriptors, such as IR, MCI, KMCI, FLU, PDI, ATI. The results clearly showed that that the description of the multicenter electron delocalization in molecules at the DFT level requires the use of XC functionals with long-range exchange corrections, which has so far been a very rare practice. Although the popular B3LYP hybrid XC functional works very well for molecules as regards e.g. reproduction of the experimental geometries (and is therefore suitable for calculations e.g. HOMA indices), it can lead to qualitatively incorrect evaluation of local aromatics according to electron criteria in polycyclic molecules.

Figure 7. In the case of polycyclics aromatics, the DFT calculations involving local XC functionals without the- long-range exchange corrections, such as B3LYP, systematically overestimate local aromaticity of particular rings, which is especially important in the case of linear acenes.

The estimated percentage of my contribution to the work is 50%: initial idea, preliminary study and calculations of selected aromaticity indices, analysis and elaboration of the results, preparation of the first draft of the manuscript, coordination. PhD students JD, BP, MK i KD carried out a part of the quantum-chemical calculations (including selected aromaticity descriptors) and collected and formatted materials for SI; the other co-authors served their expertise (especially as regards the selection of the test sets of molecules and discussion of the results) and participated in editorial works on the paper.

4.3.6. Original scientific article H6 D.W. Szczepanik, M. Andrzejak, J. Dominikowska, B. Pawełek, T.M. Krygowski, H. Szatylowicz, M. Solà, „The electron density of delocalized bonds (EDDB) applied for quantifying aromaticity”, Physical Chemistry Chemical Physics 19 (2017) 28970−28981. IF: 3.430. MNiSW: 100. Citations: 39, without self-citations: 31. In this work, I proposed a new index to quantify electron delocalization in the aromatic ring, based on the EDDB method, to provide an alternative to popular indices such as NICS, HOMA, PDI, FLU or MCI, and free from many disadvantages of these indices. The correlational and factorial analysis of the EDDB-based index was performed with 15 selected aromatic indexes involving different sets of test molecules. In addition, a comprehensive EDDB benchmark study was carried out to compare EDDB with the multicenter indices MCI and KMCI. The indices have been tested to describe changes in the trend of aromaticity in the course of acetylene trimerization reactions and Diels-Alder cycload- ditive reactions, and how the ring size and heteroatoms affect the description of electron delocalization by EDDB and KMCI. The results revealed a 100% agreement between aromaticity predictions by the newly proposed index and the multicenter index, which in many situations suffers from numerical instability and is incomparably more computationally more expensive: the calculation of the MCI for a 10-member ring can take up to several hours, while the index based on the EDDBF and EDDBP functions is calculated in a fraction of a second regardless of the ring size.

Figure 8. A detailed comparative study of EDDB with other popular chemical aromatic indices revealed that the EDDB method describes the effects of electron delocalization in a more universal way and that the EDDB predictions fully match the predictions of the MCI index, the calculation of which is incomparably more computationally demanding.

The estimated percentage of my contribution to the work is 60%: initial idea, preliminary calculations, preparation of the input files with optimized geometries for all systems under study, analysis and elaboration of the results, preparation of the fisrt draft of the manuscript, coordination. PhD students JD and BP carried out part of the calculations of aromaticity indices and collected materials for SI; the other co-authors served their expertise and linguistic advices, helped with the design of the test sets of molecular rings and with the interpretation of the results of factorial analyses; all the co-authors participated in editorial works on the manuscript and its revisions.

4.3.7. Original scientific article H7 D.W. Szczepanik, M. Solà, T.M. Krygowski, H. Szatylowicz, M. Andrzejak, B. Pawelek, J. Dominikowska, M. Kukulka, K. Dyduch, „Aromaticity of acenes: the model of migrating π-circuits”, Physical Chemistry Chemical Physics 20 (2018) 13430−13436. IF: 3.430. MNiSW: 100. Citations: 17, without self-citations: 14. In this work I proposed a generalization of the model of "migrating sextet” by Clar being routinely used in describing local aromaticity of polycyclic molecules, especially linear acenes. Considering of the possibility of migrating larger π-cycles (naphthacycles, anthracycles, etc.) allowed us to explain the source of the discrepancies in the descriptions of such systems (referred to as the so-called "anthracene problem"), and to better understand the relationship between aromatic nature of the electron density of a polycyclic molecule fragment and the frequency shifts observed experimentally in noncontact atomic force microscopy.

Figure 9. In contrast to the "migrating π-sextet" model by Clar, the extended model allowing migration of larger π-cycles enables a strictly quantitative prediction of local aromaticity in acenes, in accordance with experimental data (i.e. X-ray, NMR, nc-AFM, etc.).

The estimated percentage of my contribution to the work is 50%: initial idea, development of the theoretical model, carrying out part of the calculations, elaboration of the results, preparation of the first draft of the manuscript, coordination. PhD students BP, JD, MK, and KD performed calculations of selected aromaticity indices (only a small portion of those results was included in the manuscript); the other co-authors served their expertise and linguistic advices, shared the HPC resources and participated in editorial works on the final version of the manuscript.

4.3.8. Original scientific article H8 D.W. Szczepanik, „A simple alternative for the pseudo-π method”, International Journal of Quan- tum Chemistry 118 (2018) e25696. IF: 1.747. MNiSW: 70. Citations: 5, without self-citations: 4. In this work, I proposed the original method of calculating the multicenter electron delocalization index (as well as any other 1-electron quantity for the π-aromatic molecule), NPP, which offers ab initio accuracy at calculation cost of the Hückel method. The NPP method provides an alternative to the pseudo-π proposed in the literature, which has a number of disadvantages, such as the need to perform calculations twice (once within the original basis set of atomic orbitals, and once again in the basis of hydrogen atoms), the requirement of ring planarity, limited applicability (only aromatic hydrocarbons without heteroatoms or metal atoms), the size of the ring limited to 12-14 atoms, etc. The NPP method is free from these issues and, in combination with the EDDB method, enables au- tomated analysis of the electron delocalization pathways in the molecular systems of any size and topology; the current implementation allows the calculation of delocalization on pathways up to 10 000 atoms. The estimated percentage of my contribution to the work is 100%.

Figure 10. The NPP formalism enables calculations of the π-component of the multicenter electron delocalization index with excellent accuracy and at an incomparably lower computational cost; e.g. the single-threaded calculation of KMCI

in C7H7MnCl2 takes 3-15 minutes (depending on the basis set), and NPP reduces this time to about 0.05 s (regardless of basis set and the theory level).

4.3.9. Original scientific article H9 D.W. Szczepanik, M. Solà, „Electron delocalization in planar metallacycles: Hückel or Möbius aromatic?”, ChemistryOpen 8 (2019) 219−227. IF: 2.370. MNiSW: 70. Citations: 21, without self- citations: 19.

In this work, I proposed a method of fragmentation and orbital decomposition of the EDDBF(r) function to investigate the complex effects of electron delocalization in aromatic metallacycles. The results showed that, contrary to common knowledge and the qualitative criteria formulated by Mauksch and Tsogoyeva, the type of d-conjugation topology of the transition metal does not depend on the number of π electrons in the system, but on the number of π-conjuged atoms forming a ring, and the maximum contribution of each atom to cyclic π-localization must not be greater than ~1 electron; in other words, the cooperativity effect within the multicenter π-electron sharing in the aromatic ring of a metallacycle forces a kind of “rivalry” between Hückel or Craig-Möbius conjugations, which en- gage different nd orbitals of the transition metal. The results allowed us to divide the metallacycles by the role of metal in the delocalization into 4 groups: Hückel aromatic (mainly 6-membered rings), Möbius aromatic (mainly 8-membered rings), the hybrid Hückel-Möbius aromatics (mainly 7-membered rings), and quasi-aromatic hybrid aromatics (selected mono- and polycyclic systems containing metals such as Cu, Cd or Zn).

Figure 11. The EDDB method is particularly useful when structural, magnetic and qualitative rules of aromaticity based on topology and the π-electron bookkeeping fail.

The estimated percentage of my contribution to the work is 80%: original idea, carrying out of all the calculations, analysis and elaboration of the results, preparation of the first draft of the manuscript. MS invited me to publish in a special issue of ChemistryOpen, served his expertise (in particular, he helped in identification of the quasi-aromatic delocalization motif in the metallabicycles containing the Cu(I) atom), and participated in editorial work on the final version of the manuscript.

4.3.10. Original scientific article H10 D.W. Szczepanik, M. Solà, „The electron density of delocalized bonds (EDDB) as a measure of local and global aromaticity”; in: I. Fernandez (ed.) „Aromaticity: Modern Computational Methods and Applications”, Elsevier, 2021, Chapter 8, pp 259−283. IF: 0.000. MNiSW: 50. Citations: 0, without self-citations: 0. This work summarizes the most important theoretical development underlying the method of electron density of delocalized bonds (H1-H4), together with its subsequent modifications (an open-access invited paper in Chemistry-Methods, in preparation). The results of comparative analyses (H5-H7) has been presented and the capabilities of the new method of decomposition of EDDB into molecular fragments and orbital dissections (H9) has been briefly discussed. Also, the previously unpublished results of comparative analysis of EDDB with various electron density functions proposed in the literature to visualize electron delocalization has been presented. Additionally, the work includes a concise tutorial that enables a “quick start” for the beginners interested in using the EDDB method.

Figure 12. The 24π carboporphine within representations of EDDB, GIMIC, ACID, ELF, LOL, EDR and DORI.

The estimated percentage of my contribution to the work is 90%: the manuscript prepared mostly by me. MS invited me to joint publication, served his expertise especially in the context of selection of the material for the review, and participated in editorial work on the final version of the manuscript.

4.4. The EDDB method in scientific literature Over the past two years, dozens of works have been published (including the top chemical journals such as Nat. Chem., Angewandte, JACS, ACR, etc.), in which the EDDB method has been used to analyze the resonance structure and electron delocalization of new aromatic molecules; most of these papers have been prepared and published without me as a co-author, but with official thanks for the substantive evaluation of the manuscript in the acknowledgements section. At the same time, I have published several applicative works as a co-author in such prestigious journals as Chem. Sci., and Chem. Comm. Among others, the paper P31 published in 2020 deserves special attention, in which, together with the group of Prof. M. Solà (University of Girona) and the group of Prof. J. Zhu (Uni- versity of Xiamen), we demonstrated using advanced theoretical methods (including the EDDB method) the possibility of existence of the so-called Baird aromatics in metallic clusters. A news 15 about this work appeared in the ChemistryWorld magazine, and now the paper has the status of “hot article” (in just a few months from publication the work has already been cited a dozen times). In another important work from early 2021, P32, together with the group of Prof. M. Solà (University of Girona), the group of Prof. J. Zhu (University of Xiamen) and Prof. A. Muñoz-Castro (Universidad Autónoma de Chile), we have shown through theoretical calculations involving among others the EDDB method that even fullerenes with structural defects breaking local bonding conjugation can still sustain their aromatic character to a high degree; this work has already been cited 5 times within 2 months after publication. Finally, in the work P33 from early 2021, together with prof. H. Ottoson's group (Uppsala University) and prof. H. Bronstein's group (Cambridge University) we demonstrated among others that the paradigm shift in the study of electron structure and the properties of aromatic molecules associated with the EDDB method may be crucial for understanding the nature of electron excitation in organic molecules used in the design of materials for molecular electronics; the work has already been mentioned several dozen times (not to mention the preprints available since the end of last year). Finally, it is worth noticing that works P15, P22, P24, P25, P27, P29, and P34 utilizing the EDDB method and co-authored by me has been published in collaboration with experimentalists (2 other papers are under review and a total of 4 is in preparation). Although the results of the EDDB calculations were crucial for most of these works (in particular P22, where the EDDB method allowed us to discover a new type of aromaticity called the Möbius quasi-aromatic motif), they did not significantly contribute to the development of the method itself and hence they are not included in the habilitation cycle.

4.5. What is and what is not the EDDB method In conclusion, EDDB is currently the only known method that describes in a strict way (at a given level of the theory) electron delocalization resulting from the effect of superposition of the resonance structures contributing to the wavefunction of a molecule. Thus, EDDB quantifies the direct cause of aromatic stabilization, not its indirect manifestations, such as reduction of the bond length alternation, anomalous chemical shifts, etc. A strict relation to the chemical resonance and the fact that EDDB is a global function of total electron density (i.e. a well-defined physical quantity), makes the method superior to other popular and commonly-used methods based on ring currents in the external magnetic field, which, despite being associated with experimental observables (like chemical shifts), to very limited extent describe the effect of electron delocalization (or rather the current of electron density) associated mainly with frontal 2- or 4-electron virtual excitations, so that for the systems with a complex topology their relationship to the actual effect of the resonance/aromatic stabilization disappears. It should be stressed that EDDB is not yet another index of aromaticity, but it is a universal method for assessment and quantification of the electron delocalization effects which feature the vast majority of known organic molecules and determine their structure and physico-chemical properties. In particular, the EDDB does not answer the question whether a molecule is (anti)aromatic or not; with a huge number of types of aromaticity introduced in the literature and the ways of its identification and quantification, such a question remains purely academic in nature, and the value of the possible answer, in the context of the real impact of aromatic (de)stabilization on the structure and properties of a molecule is often only apparent (although it certainly increases the attractiveness of publication).2

2 R. Hoffmann, „The Many Guises of Aromaticity”, Am. Sci. 103 (2015) 18. 16

5. Presentation of significant scientific or artistic activity carried out at more than one university, scientific or cultural institution, especially at foreign institutions The main field of my current scientific activity is mathematical chemistry, machine learning and the development of computational methods in quantum chemistry. During my master's and doctoral studies, I was primarily involved in the first of these branches, in particular the application of information theory to investigate the electron structure of chemical species. In my dissertation entitled “Probabil- istic models of the chemical bond in function spaces" I studied the properties of the purely bayesian descriptions of the electron structure and reactivity of molecules based on the theory of homogeneous Markov’s chains and the so-called orbital communication theory. My long-term scientific advisor, mentor and supervisor of my MSc and PhD theses dr. hab. J. Mrozek gave me a carte blanche in choosing the subject of my researches and developing my scientific interests from the very beginning. Consequently, out of a total of 16 works published jointly with my supervisor (M1, P1-P14, and P18), I am the first and the sole corresponding author in14 of them (P1-P4, P6-P14, and P18); some of these works have been published in highly specialized journals with relatively low impact factor (e.g. Journal of Mathematical Chemistry), and they have been cited 307 times so far (GoogleScholar, accessed 20.04.2021). The most important achievement during the PhD period I consider the new method of calculating the correlation coefficient of chemical bonds (CCCB, i.e., the quantity that provides information on the effect on specific chemical bonds in a molecule of the formation/breaking of other bonds), which is an extremely useful tool in predicting chemical reaction products and among others identifying weak local non-carbon interactions (P8). The use of Shannon’s algebra communication channels and the so-called reduced probabilities of communication of states in four-atom homogeneous Markov’s chains to assess the “correlation” of the electron populations in a molecule, led to significant reduction of the total cost of calculation of CCCB by 3-4 orders of magnitude compared to its original definition by Prof. W.A. Goddard III (based on the covariance cascade of charge-and-bond- order operators).3 The alternative definition of the correlation coefficient of chemical bonds proposed by was utilized among others to explain the local/global (de)stabilizing effects of dihydrogen bonding on the structure of the biphenyl molecule, reconciling the contradictory findings by other researchers (P8); another article addressing this topic in collaboration with Prof.M.P. Mitoraj is in preparation. After defending my doctoral thesis, I was employed as a technician at the Department of The- oretical Chemistry (Jagiellonian University); this was partly due to a chronic disease that during this period significantly reduced my scientific activity and mobility. For the first two years after PhD, I have been developing theoretical basis and partial implementation of the EDDB method (in particular, EDDBG), which is a basis for the proposed scientific achievement. The EDDB method has been rec- ognized from the very beginning by the scientific community (work H2 received very flattering reviews, e.g. “An excellent step forward in the analysis and study of electron delocalization features. Probably the most important in the last 20 years”, “The EDDB method may become a tool of reference in the field of aromaticity measurements. If so, the impact could be enormous”, etc.), and it marked the beginning of the collaboration among others with:

3 T. Yamasaki, W.A. Goddard III, “Correlation Analysis of Chemical Bonds”, J. Phys. Chem. A 102 (1998) 2919. 17

• a computational group led by Prof. M. Solà (University of Girona) – which I have visited 14 times (89 days) since 2016, and 4 times I have been a member of the group within the framework of different mobility programs and stipends funded by the Government of Catalonia (3 months), National Agency for Academic Exchange (12 months), and European Commission (24+2 months), and the results of this collaboration are dozens of joint publications and 1 monographic chapter; • a computational group by Prof. H. Ottosson (Uppsala University) and experimental group by Dr. H. Bronstein (Cambridge University) with whom I have been collaborating since 2018 within the framework of the research on new materials for among others the singlet fission; • an experimental group by Dr. P. Zahl (Brookhaven University) and Dr. Y. Zhang (ExxonMo- bil) with whom I have been collaborating since 2018; our research focus on deepening the understanding of the electronic resonance structure of poly- and macrocyclic aromatics based on the images from the non-contact high-resolution atomic force microscopy; • a computational group by Prof. J. Zhu (Xiamen University) with whom I have been collaborating since 2019 in the researches on the so-called Baird’s and the adaptive aromaticity of organometallic compounds, but also, with indigenous scientists including Prof. T.M. Krygowski (University of Warsaw, collaboration since 2017), Prof. H. Szatylowicz (Technical University of Warsaw, collaboration since 2017), Prof. M.P. Mitoraj (Jagiellonian University, collaboration since 2017), Dr. J. Dominikowska (University of Lodz, collaboration since 2017) or GawelGroup (IChO PAN, Warsaw, collaboration since 2020). In the period from 2016 to 2021, I led a total of 4 scientifically independent and competitive research projects, including: NCN Sonata IX (36 months), H2020 HPC-Europa3 (3 months, mini-project), H2020 MSCA-IF (24 months), and NAWA Bekker (12 months), which were directly or indirectly related to the development of new methodologies and computational techniques based on the EDDB approach and their practical applications. My scientific achievements after PhD were awarded among others the START scholarship of the Foundation for Polish Science (2015), two travel schol- arships funded by the Government of Catalonia (2015-2016) and the European Commission (2016- 2017), the Maria Sklodowska-Currie individual fellowship by European Commission (2018-2020), the MNiSW scholarship for young outstanding researchers (2018-2021), and Bekker’s stipend by the National Agency for Academic Exchange (2020-2021). I have also been invited several times by research centers in Poland and abroad to give a lecture, and in 2018, as one of about 50 specialists from around the world (including Nobel laureates), I was invited to attend the IUPAC congress in Mexico organized in order to update the definition of the chemical aromaticity and antiaromaticity concepts. I am currently applying for two research grants, "Electrons move in mysterious (path)ways: from monocycles to nanorings" (NCN Opus) and "The information-entropy origins of molecular aromaticity" (NCN Sonata Bis), in which I planned to collaborate closely with the experimental groups by Dr. P. Zahl (Brookhaven University, USA) and Dr. Y. Zhang (ExxonMobil, USA), as well as the experimental group by Dr. H. Bronstein (University of Cambridge, UK); also, I have been declared as an external collaborator in the project applications by Prof. M. Solà at the University of Girona, Spain, and Dr. P. Zahl at the University of Brookhaven, USA. In accordance with the latter, I am also preparing the application for the individual mobility fellowship within the Polish-American Fulbright Commission programme. In the coming years, I will also consider to apply for the ERC Consolidator grant to fund research on the neural-network models that could be used to facilitate the automate 18 design of the materials for molecular electronics, based on the so-called extreme resonance entropy principle for the electron excitations in organic molecules (which I have originally proposed in my NCN Sonata Bis project proposal):

The most distinguished feature of aromatic compounds is their ability to counterbalance ‘destruc- tive’ effects of the electron excitations on the ground-state system of π-conjugated bonds by re- distribution of resonance forms in such a way that maximizes the resonance-entropy production or minimizes the resonance-entropy loss to preserve as much information contained in their π- systems as possible. ❑

Figure 13. The extreme resonance entropy principle could be used among others in the interpretation of the STM and nc-HRAFM images, in controlling the course of (hyper)homodesmotic reactions, in predicting the electronic structure of the low-lying excited states, or in the design of neural networks supporting modelling of the materials with the tuneable transport and optical properties. In the proposed project (NCN Sonata Bis), implementation of the extreme resonance entropy principle will be based on the extended bond-orbital projection technique (eBOP) analogous to the one used in the formulation of the method of electron density of delocalized bonds (EDDB).

List of scientific or artistic achievements which present a major contribution to the development of a specific discipline

I. INFORMATION ON SCIENTIFIC OR ARTISTIC ACHIEVEMENTS SET OUT IN ART. 219 PARA 1. POINT 2 OF THE ACT

2. Cycle of scientific articles related thematically, pursuant to art. 219 para 1. point 2b of the Act. After PhD: H1. D.W. Szczepanik, E.J. Zak, K. Dyduch, J. Mrozek, „Electron delocalization index based on bond order orbitals”, Chemical Physics Letters 593 (2014) 154−159. H2. D.W. Szczepanik, M. Andrzejak, K. Dyduch, E.J. Zak, M. Makowski, G. Mazur, J. Mrozek, „A uniform approach to the description of multicenter bonding”, Physical Chemistry Chemical Phys- ics 16 (2014) 20514−20523. H3. D.W. Szczepanik, „A new perspective on quantifying electron localization and delocalization in molecular systems”, Computational and Theoretical Chemistry 1080 (2016) 33−37. H4. D.W. Szczepanik, „On the three-center orbital projection formalism within the electron density of delocalized bonds method”, Computational and Theoretical Chemistry 1100 (2017), 13−17. H5. D.W. Szczepanik, M. Solà, M. Andrzejak, B. Pawełek, J. Dominikowska, M. Kukułka, K. Dyduch, T.M. Krygowski, H. Szatylowicz, „The role of the long-range exchange corrections in the description of electron delocalization in aromatic species”, Journal of Computational Chemistry 38 (2017) 1640−1654. H6. D.W. Szczepanik, M. Andrzejak, J. Dominikowska, B. Pawełek, T.M. Krygowski, H. Szatylowicz, M. Solà, „The electron density of delocalized bonds (EDDB) applied for quantifying aromaticity”, Physical Chemistry Chemical Physics 19 (2017) 28970−28981. H7. D.W. Szczepanik, M. Solà, T.M. Krygowski, H. Szatylowicz, M. Andrzejak, B. Pawelek, J. Dominikowska, M. Kukulka, K. Dyduch, „Aromaticity of acenes: the model of migrating π-circuits”, Physical Chemistry Chemical Physics 20 (2018) 13430−13436. H8. D.W. Szczepanik, „A simple alternative for the pseudo-π method”, International Journal of Quantum Chemistry 118 (2018) e25696. H9. D.W. Szczepanik, M. Solà, „Electron delocalization in planar metallacycles: Hückel or Mö- bius aromatic?”, ChemistryOpen 8 (2019) 219−227. H10. D.W. Szczepanik, M. Solà, „The electron density of delocalized bonds (EDDB) as a measure of local and global aromaticity”; w: I. Fernandez (ed.) „Aromaticity: Modern Computational Methods and Applications”, Elsevier, 2021, Chapter 8, pp. 271−295.

II. INFORMATION ON SCIENTIFIC OR ARTISTIC ACTIVITY

2. List of published chapters in scientific monographs. Before PhD: M1. R.F. Nalewajski, D.W. Szczepanik, J. Mrozek, „Bond differentiation and orbital decoupling in the orbital-communication theory of the chemical bond”; in: J.R. Sabin, E. Brandas (ed.) „Ad- vances in Quantum Chemistry vol. 61”, Elsevier, 2011, Chapter 1, pp. 1−48. After PhD: M2 (H10). D.W. Szczepanik, M. Solà, „The electron density of delocalized bonds (EDDB) as a measure of local and global aromaticity”; in: I. Fernandez (ed.) „Aromaticity: Modern Computational Methods and Applications”, Elsevier, 2021, Chapter 8, pp. 271−295.

4. List of articles published in scientific journals (including the articles not mentioned in section I.2). Before PhD: P1. D.W. Szczepanik, J. Mrozek, “Entropic bond descriptors from separated output-reduced communication channels in AO-resolution”, Journal of Mathematical Chemistry 49 (2011) 562-575. P2. D.W. Szczepanik, J. Mrozek, “Probing the interplay between multiplicity and ionicity of the chemical bond”, Journal of Theoretical and Computational Chemistry 10 (2011) 471-482. P3. D.W. Szczepanik, J. Mrozek, “Symmetrical orthogonalization within linear space of molecular orbitals”, Chemical Physics Letters 521 (2012) 157-160. P4. D.W. Szczepanik, J. Mrozek, “Electron population analysis using a reference minimal set of atomic orbitals”, Computational and Theoretical Chemistry 996 (2012) 103-109. P5. R.F. Nalewajski, D.W. Szczepanik, J. Mrozek, “Basis set dependence of molecular information channels and their entropic bond descriptors”, Journal of Mathematical Chemistry 50 (2012) 1437- 1457. P6. D.W. Szczepanik, J. Mrozek, “On several alternatives for Löwdin orthogonalization”, Compu- tational and Theoretical Chemistry 1008 (2013) 15-19. P7. D.W. Szczepanik, J. Mrozek, “Ground-state projected covalency index of the chemical bond”, Computational and Theoretical Chemistry 1023 (2013) 83-87. P8. D.W. Szczepanik, J. Mrozek, “Through-space and through-bridge interactions in the correlation analysis of chemical bonds”, Computational and Theoretical Chemistry 1026 (2013) 72-77. P9. D.W. Szczepanik, J. Mrozek, “Stationarity of electron distribution in ground-state molecular systems”, Journal of Mathematical Chemistry 51 (2013) 1388-1396. P10. D.W. Szczepanik, J. Mrozek, “On quadratic bond-order decomposition within molecular orbital space”, Journal of Mathematical Chemistry 51 (2013) 1619-1633. P11. D.W. Szczepanik, J. Mrozek, “Minimal set of molecule-adapted atomic orbitals from maximum overlap criterion”, Journal of Mathematical Chemistry 51 (2013) 2687-2698.

P12. D.W. Szczepanik, J. Mrozek, “Nucleophilicity index based on atomic natural orbitals”, Jour- nal of Chemistry 2013 (2013) 684134. After PhD: P13 (H1) D.W. Szczepanik, E.J. Zak, K. Dyduch, J. Mrozek, „Electron delocalization index based on bond order orbitals”, Chemical Physics Letters 593 (2014) 154−159. P14 (H2) D.W. Szczepanik, M. Andrzejak, K. Dyduch, E.J. Zak, M. Makowski, G. Mazur, J. Mrozek, „A uniform approach to the description of multicenter bonding”, Physical Chemistry Chem- ical Physics 16 (2014) 20514−20523. P15. M. Andrzejak, D.W. Szczepanik, L. Orzeł, “The lowest triplet states of bridged cis-2,2'-bithiophenes - theory vs experiment”, Physical Chemistry Chemical Physics 17 (2015) 5328-5337. P16 (H3). D.W. Szczepanik, „A new perspective on quantifying electron localization and delocalization in molecular systems”, Computational and Theoretical Chemistry 1080 (2016) 33−37. P17 (H4). D.W. Szczepanik, „On the three-center orbital projection formalism within the electron density of delocalized bonds method”, Computational and Theoretical Chemistry 1100 (2017), 13−17. P18. D.W. Szczepanik, E.J. Zak, J. Mrozek, “From quantum superposition to orbital communication”, Computational and Theoretical Chemistry 1115 (2017) 80-87. P19 (H5). D.W. Szczepanik, M. Solà, M. Andrzejak, B. Pawełek, J. Dominikowska, M. Kukułka, K. Dyduch, T.M. Krygowski, H. Szatylowicz, „The role of the long-range exchange corrections in the description of electron delocalization in aromatic species”, Journal of Computational Chemistry 38 (2017) 1640−1654. P20 (H6). D.W. Szczepanik, M. Andrzejak, J. Dominikowska, B. Pawełek, T.M. Krygowski, H. Szatylowicz, M. Solà, „The electron density of delocalized bonds (EDDB) applied for quantifying aromaticity”, Physical Chemistry Chemical Physics 19 (2017) 28970−28981. P21 (H7). D.W. Szczepanik, M. Solà, T.M. Krygowski, H. Szatylowicz, M. Andrzejak, B. Pawelek, J. Dominikowska, M. Kukulka, K. Dyduch, „Aromaticity of acenes: the model of migrating π-circuits”, Physical Chemistry Chemical Physics 20 (2018) 13430−13436. P22. M.P. Mitoraj, G. Mahmoudi, F.A. Afkhami, A. Castineiras, I. Garcia-Santos, A.V. Gurbanov, M. Kukulka, F. Sagan, D.W. Szczepanik, D. A. Safin, “Quasi-aromatic Möbius metal chelates”, Inorganic Chemistry 57 (2018) 4395−4408. P23 (H8). D.W. Szczepanik, „A simple alternative for the pseudo-π method”, International Journal of Quantum Chemistry 118 (2018) e25696. P24. G. Mahmoudi, F.A. Afkhami, A. Castineiras, G. Giester, I.A. Konyaeva, F.I. Zubkov, F. Qu, A. Gupta, M.P. Mitoraj, F. Sagan, D.W. Szczepanik, D.A. Safin, “Effect of solvent on the structural diversity of quasi-aromatic Möbius cadmium(II) complexes”, Crystal Growth and Design 19 (2019) 1649−1659. P25. M.P. Mitoraj, M.G. Babashkina, K. Robeyns, F. Sagan, D.W. Szczepanik, Y. Garcia, D.A. Safin, “The chameleon-like nature of anagostic interactions and metalloaromaticity in square-planar nickel complexes”, Organometallics 38 (2019) 1973−1981.

P26 (H9). D.W. Szczepanik, M. Solà, „Electron delocalization in planar metallacycles: Hückel or Möbius aromatic?”, ChemistryOpen 8 (2019) 219−227. P27. F. A. Afkhami, G. Mahmoudi, A. Khandar, A. Gurbanov, F. Zubkov, R. Waterman, M. Babash- kina, M.P. Mitoraj, D.W. Szczepanik, D. Safin, “Structural versatility of the quasi-aromatic Mö- bius type zinc(II)-pseudohalide complexes – experimental and theoretical investigations”, RSC Ad- vances 9 (2019) 23764−23773. P28. M.P. Mitoraj, F. Sagan, D.W. Szczepanik, J.H. de Lange, A.L. Ptaszek, D.M.E. van Niekerk, I. Cukrowski, “Origin of Hydrocarbons Stability from Computational Perspective – A Case Study of Xylene Isomers”, ChemPhysChem 21 (2020) 494–502. P29. M.P. Mitoraj, D.S. Shapenova, A.A. Shiryaev, M. Bolte, M. Kukułka, D.W. Szczepanik, J. Hooper, M.G. Babashkina, G. Mahmoudi, D.A. Safin, “Resonance assisted hydrogen bonding phenomenon unveiled from both experiment and theory – an example of new family of ethyl n‐salicylideneglycinate dyes”, Chemistry - A European Journal 26 (2020) 12987−12995. P30. D. Chen, D.W. Szczepanik, J. Zhu,, M. Solà, „Probing the origin of adaptive aromaticity in 16-valence-electron metallapentalenes”, Chemistry - A European Journal 26 (2020) 12964−12971. P31. D. Chen, D.W. Szczepanik, J. Zhu,, M. Solà, „All-metal Baird aromaticity”, Chemical Com- munications 56 (2020) 12522−12525. P32. D. Chen, D.W. Szczepanik, J. Zhu, A. Muñoz-Castro, M. Solà,  „Aromaticity survival in hydrofullerenes: the case of C66H4 with its π-aromatic circuits”, Chemistry - A European Journal 27 (2021) 802−808. P33. W. Zeng, O. El Bakouri, D.W. Szczepanik, H. Bronstein, H. Ottosson,  ”Excited state character of CIBA-type compounds interpreted in terms of Hückel-aromaticity: a rational for singlet fission chromophore design”, Chemical Science, accepted (10.26434/chemrxiv.13515989). P34. G. Mahmoudi, M. Babashkina, W. Maniukiewicz, F.A. Afkhami, B.B. Nunna, F.I. Zubkov, A.L. Ptaszek, D.W. Szczepanik, M.P. Mitoraj, D.A. Safin,, “Solvent-induced formation of novel Ni(II) complexes derived from bis-thiosemicarbazone ligand: an insight from experimental and theoretical investigations”, International Journal of Molecular Sciences 12 (2021) 5337. Full and up-to-date list of publications is available on: www.eddb.pl/publications

7. Information on presentations given at national or international scientific or arts conferences, including a list of lectures delivered upon invitation and plenary lectures. Before PhD: K1. (Poster) “Chemical bond indices from communication theory.” D.W. Szczepanik, Current Trends in Theoretical Chemistry V (CTTC5), Kraków, Poland, July 6-10, 2008. International conference. K2. (Poster) “Communication theory of the chemical bond.” D.W. Szczepanik, Information Tech- nologies for Chemists, Kraków, Poland, September 19-20, 2008. Local conference. K3. (Poster) “Applications of the orbital communication theory of the chemical bond.” D.W. Szczep- anik, Central European Symposium on Theoretical Chemistry VIII (CESTC8), Dobogoko, Hungary, September 25-28, 2009. International conference.

K4. (Poster) “IT-ionicity concepts within the orbital communication theory.” D.W. Szczepanik, HITY - Zastosowanie teorii w badaniach molekularnych, Kraków, Poland, May 18-20, 2011. Local confer. K5. (Poster) “An information-theoretic approach to the chemical bond.” D.W.Szczepanik, Central European Symposium on Theoretical Chemistry X (CESTC10), Torun, Poland, September 25-28, 2011. International conference. K6. (Poster) “Probabilistic models of the chemical bond.” D.W. Szczepanik, Current Trends in The- oretical Chemistry VI (CTTC6), Kraków, Poland, September 1-5, 2013. International conference. Additionally, 6 local seminar talks. After PhD: K7. (Oral) “The effectiveness of bond conjugation - a new criterion of aromaticity.” D.W. Szczep- anik, Central European School of Physical Organic Chemistry (CES2016), Przesieka, Poland, June 6-10, 2016. International conference. K8. (Poster) “The influence of cations on inclusion of anthracene to β-cyclodextrin.” A. Stachowicz- Kuśnierz, D.W. Szczepanik, J. Korchowiec, Current Trends in Theoretical Chemistry VII (CTTC7), Krakow, Poland, September 4-8, 2016. International conference. K9. (Invited Lecture, Poster) “Aromaticity of metallacycles: Hückel or Möbius?” D.W. Szczepanik, M. Solà, Aromaticity 2018, Riviera Maya, Mexico, November 28 - December 1, 2018. International conference. K10. (Oral) “Electron delocalization in metallacycles: Hückel or Möbius aromatic?” D.W. Szczep- anik, M. Solà, European Meeting on Physical Organic Chemistry, Spala, Poland, June 3 - 7, 2019. International conference. K11. (Poster) “From linear to circular polycyclic compounds: aromaticity study on singlet and triplet states.”, S. Escayola, D.W. Szczepanik, A. Poater, M. Solà, Tools for Chemical Bonding, Bremen, Germany, July 14 – 19, 2019. International conference. K12. (Poster) “Global and local aromaticity reversals between singlet and triplet states in expanded porphyrins.” S. Escayola, D.W. Szczepanik, A. Poater, M. Solà, First International Conference on Excited State Aromaticity and Antiaromaticity, Sigtunastiftelsen, Sweden, July 30 – August 2, 2019. International conference. K13. (Lecture) “Electron delocalization and the magnetically induced ring current in poly- and macrocyclic aromatics.” D.W. Szczepanik, M. Solà, First International Conference on Excited State Ar- omaticity and Antiaromaticity, Sigtunastiftelsen, Sweden, July 30 – August 2, 2019. International conference. Additionally, 10 seminar talks, including 4 invited seminar talks abroad.

8. Information on participation in organizational and scientific committees at national or international conferences, including the applicant’s function. Before PhD: O1. Executive committee member, Current Trends in Theoretical Chemistry VI (CTTC6), Kraków, Poland, September 1-5, 2013. International conference.

After PhD: O2. Executive committee member, Current Trends in Theoretical Chemistry VII (CTTC7), Krakow, Poland, September 4-8, 2016. International conference. O3. Executive committee member, Current Trends in Theoretical Chemistry VIII (CTTC8), Kraków, Poland, September 1-5, 2019. International conference.

9. Information on participation in the works of research teams realizing projects fi- nanced through national and international competitions, including the projects which have been completed and projects in progress, and information on the function performed in the team. Research projects completed before PhD: G1. (2011-2013) “Probabilistic models of the chemical bond in the function spaces and physical space”, Ministry of Science and Higher Education, WFPD UJ: K/DSC/000133+000987+001469 (PI). Research projects completed after PhD: G2. (2015-2016) „Assessing the degree of Baird and Hückel aromaticity in Hückel-Baird hybrid species by means of electronic indices”, Generalitat de Catalunya: 2014SGR931 (participant, PI: M. Solà). G3. (2016-2019) „The application of the EDDB method in the analysis of structure and reactivity of molecular systems” National Science Centre, Poland, Sonata IX: 2015/17/D/ST4/00558 (PI). G4. (2016-2017) „Theoretical description of the quasi-aromatic stabilization effects in metallacycles with different topology”, European Comission, H2020 RIA-INFRAIA-2016-1, GA: 730897, contract nr HPC17158J2 (participant – PI of a minigrant). G5. (2018-2020) „Theoretical description of the multifaceted aromaticity and resonance effects in ground- and excited-state molecular systems”, European Comission, H2020 MSCA-IF-ST-2017, GA: 797335 (PI). Ongoing research projects: G6. (2020-2021) „Theoretical study of the ground- and excited-state electronic structure of aromatic molecules with resonance-assisted hydrogen bonds”, Polish National Agency for Academic Ex- change, Bekker programme: PPN/BEK/2019/1/00219 (PI).

10. Membership in international or national organizations and scientific societies, including the functions performed by the applicant. T1. (2018-) Polish Chemical Society (Physical Organic Chemistry Section), MN: 993036, Warsaw, Poland. T2. (2018-) American Chemical Society (Physical Chemistry Division), MN: 31494884, Blacksburg, Virginia, USA. T3. (2018-) Marie Curie Alumni Association (Theoretical and Computational Chemistry Panel), Brussels, Belgium.

11. Information on internships completed in scientific or artistic institutions, also abroad, including the place, time and duration of the internship and its character. S1. (2015-2016) 3-month stay in the Institute of Computational Chemistry and Catalysis, in Di- MoCaT group led by Prof. M. Solà, University of Girona, Catalonia, Spain. S2. (2018-2020) 24-month stay (H2020-MSCA-IF) in the Institute of Computational Chemistry and Catalysis, in DiMoCaT group led by Prof. M. Solà, University of Girona, Catalonia, Spain. S3. (2020-2021) 12-month stay (NAWA-Bekker) in the Institute of Computational Chemistry and Catalysis, in DiMoCaT group led by Prof. M. Solà, University of Girona, Catalonia, Spain. Additionally, in the period 2015-2020 I visited University of Girona 14 times (in total 89 days).

13. Information on scientific or artistic works reviewed, in particular those published in international journals. After PhD, in the period 2016-2020, I reviewed 47 scientific papers for 22 JCR journals, including:

R1. (RSC) Physical Chemistry Chemical Physics: 13 reviews. R2. (Elsevier) Chemical Physics Letters: 6 reviews. R3. (Wiley) Chemistry - A European Journal: 3 reviews. R4. (Elsevier) Physica A: Statistical Mechanics and Its Applications: 2 reviews. R5. (Wiley) Chemistry - An Asian Journal: 2 reviews. R6. (Wiley) ChemistrySelect: 2 reviews. R7. (Wiley) International Journal of Quantum Chemistry: 2 reviews. R8. (Wiley) Journal of Physical Organic Chemistry: 2 reviews. R9. (MDPI) Molecules: 2 reviews. R10. (ACS) Organometallics: 1 review. R11. (ACS) The Journal of Organic Chemistry: 1 review. R12. (Elsevier) Computational and Theoretical Chemistry: 1 review. R13. (Elsevier) Journal of Molecular Graphics and Modelling: 1 review. R14. (Elsevier) Journal of Molecular Liquids: 1 review. R15. (Elsevier) Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy: 1 review. R16. (RSC) CrystEngComm: 1 review. R17. (RSC) RSC Advances: 1 review. R18. (Springer) Journal of Molecular Modelling: 1 review. R19. (Taylor & Francis) Molecular Physics: 1 review. R20. (Wiley) Chemistry and Biodiversity: 1 review. R21. (Wiley) Journal of Computational Chemistry: 1 review. R22. (MDPI) International Journal of Molecular Sciences: 1 review.

Author's personal copy

Chemical Physics Letters 593 (2014) 154–159

Contents lists available at ScienceDirect

Chemical Physics Letters

journal homepage: www.elsevier.com/locate/cplett

Electron delocalization index based on bond order orbitals ⇑ Dariusz W. Szczepanik a, , Emil Zak_ a, Karol Dyduch a, Janusz Mrozek b a Department of Theoretical Chemistry, Jagiellonian University, Ingardena 3, 30-060 Cracow, Poland b Department of Computational Methods in Chemistry, Jagiellonian University, Ingardena 3, 30-060 Cracow, Poland article info abstract

Article history: A new index of electron delocalization in atomic rings is introduced and briefly discussed. The newly pro- Received 5 December 2013 posed delocalization descriptor is defined as an atom averaged measure of the effectiveness of forming In final form 6 January 2014 linear combinations from two-center bond-order orbitals for a given sequence of bonded atomic triplets, Available online 13 January 2014 and corresponds directly to electron population analysis; it allows one to get very compact and intuitive description of p-conjugation effects without additional parametrization and calibration to the reference molecular systems. The numerical results of illustrative calculations for several typical aromatic and homoaromatic compounds seem to validate the presented methodology and definitions. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction It is the main goal of this short paper to introduce step-by- step the algorithm of calculation of a new electron delocalization Superior stability of aromatic compounds is usually considered index for cyclic and polycyclic compounds. Unlike other delocal- as a manifestation of structure resonance and cyclic delocalization ization measures, our index directly referes to condensed atomic of electrons [1–3] (historically, it was Kekule who first solved the population of electrons and therefore it provides very intuitive problem of aromatic stabilization of molecules by introducing and compact description of p-conjugation systems within the the model of superimposing resonance forms [4]). As generally framework of population analysis. The presented methodology well-known, for planar and cyclic/polycyclic molecular structures takes advantage of some previous research on bond-order orbi- with arrangement of alternating single (r) and double bonds tals done by Jug [15] as well as the recently proposed (r þ p), the effective conjugation of 2b þ 1 p-bonds through-bridge communication formalism [16,17], formulated (b ¼ 0; 1; 2; ...) gives rise to a characteristic ring of delocalized within the framework of information-theoretic [18,19] methods electrons; therefore, for symmetric species, the overall population of exploration of molecular electronic structure and reactivity of bonding p-electrons is equally distributed between constituent [20–28]. bonds of aromatic ring. There is a multitude of sophisticated measures of electron delocalization and aromaticity itself, concerning various aspects of 2. Computational scheme molecular properties obtained from calculations as well as experiments. Interesting but obviously not exhaustive overviews and The complete computational procedure of evaluation of elec- benchmarks of diverse aromaticity measures can be found in [5– tron delocalization index can be described in the following five 14]). As pointed out by Katritzky et al. [6] and confirmed and gen- steps: eralized shortly afterwards by Jug and Köster [7], an aromaticity index without reference to a measurable property is essentially 1. Calculate the density matrix D within representation of orthog- irrelevant since aromaticity itself is a multidimensional phenom- onalized atom-centered basis functions jvi (e.g. polarized or ena usually emerging in magnetic, energetic and geometric proper- unpolarized sets of minimal-basis atomic orbitals (AO) [29– ties of molecules. On the other hand, the structure resonance and 36]), electron delocalization are the most commonly regarded as a pri- D ¼ CnCy; ð1Þ mary determinant of distinctive properties of aromatics [8] and can be evaluated directly from quantum-chemistry calculations where C is a matrix of LCAO coefficients for a set of natural orbitals within the framework of Molecular Orbital (MO) theory as well jui [37] and n stands for a diagonal matrix of the corresponding as the theory of Atoms-In-Molecules by Bader (e.g. [9–14]). occupation numbers. Subsequently, for a set of two adjoining chemical bonds, e.g. A–X and X–B, construct the following submatrices ⇑ Corresponding author. from the corresponding off-diagonal atomic blocks of elements of E-mail address: [email protected] (D.W. Szczepanik). matrix D,

D.W. Szczepanik et al. / Chemical Physics Letters 593 (2014) 154–159 155 1=2 2 2 0DA;X 0DX;B C~ ¼ DðDyDÞ ; D ¼ xðnbÞ hfbj#bixðnbÞ ; ð11Þ DAX ¼ ; DXB ¼ ; ð2Þ 2 2 3 3 DX;A 0 DB;X 0 where the function x is defined as follows: and 0 1 1 for ni;i < s; xðni;iÞ¼ ; s a threshold value: 0DA;X 0 1 B C s ni;i for ni;i P s @ A DAXB ¼ DX;A 0DX;B : ð3Þ ð12Þ 0DB;X 0 More details and properties of this ’’physical’’ orthogonalization In above equations scheme one can find in [32–34]. y 4. Calculate the diagonal matrix b of elements defined as follows: DA;X ¼ Dl;m : l 2 A; m 2 X ¼ðDX;AÞ ; "# ! 4 y ð Þ XAX XXB Xi1 DB;X ¼ Dl;m : l 2 B; m 2 X ¼ðDX;BÞ : bi;i ¼ Cl;iCm;i Cl;iCm;i þ 2 Cl;kCm;k ; ð13Þ l m According to the discussion given in [26], do not take into account k¼1 the direct (through-space) interaction between atoms A and B, i.e. and determining the relative phase factors of three-center bond y DA;B ¼ðDB;AÞ 0. orbitals. The preceding equation is based on the information-theo- 2. Diagonalize matrices DAX; DXB and DAXB to find the correspond- retic criterion of maximum separation of nearly degenerated eigen- ing sets of two-center bond orbitals and their occupation values of a density matrix [27]. In accordance with [27], eigenvalues numbers, b of n3 that correspond to negative values of factor (13) usually ap- y pear in a density matrix spectrum right next to their phase-reversed jf i¼jviCAX; nAX ¼ðCAXÞ DAXCAX; AX 5 countertypes and always assume nearly the same absolute values. y ð Þ jfXBi¼jviCXB; nXB ¼ðCXBÞ DXBCXB; Therefore, their contributions to the delocalized bond density effectively cancels out. The diagonal matrix of such signed (phase-spec- as well as the set of three-center bond orbitals and their diagonal ified) ’’populations’’ of delocalized electrons in a set of three-center occupation matrix, bond orbitals (6) is straightforwardly determined by the product y j#AXBi¼jviCAXB; nAXB ¼ðCAXBÞ DAXBCAXB: ð6Þ b del k ¼ bn3; and N3 ¼ trk: ð14Þ As pointed out by Jug [15], within the spectrum of eigenvalues of Here, Ndel stands for the effective population of electrons delocal- s s matrices of type (2), there are always bs=2c nonnegative and 3 ized in the particular triplet of bonded atoms, A–X–B. To cancel exactly the same number of nonpositive eigenvalues. The former out negative values of k with their corresponding countertypes are related to bonding (in character) two-center bond orbitals, del b b and get the positive matrix k one cas use a simple populational jfAXi and jfXBi (e.g. r2; p2), whereas the latter refer to anti-bonding a a criterion of maximum overlap; i.e, for a given negative ki;i, the posi- ones, jfAXi and jfXBi (e.g. r2; p2). Moreover, þ tive countertype ki0;i0 always satisfy the following condition: X bXs=2c bXs=2c no 2 b 2 a 2 2 X 2 jðDAXÞl;mj ¼ 2 jðnAXÞj;jj ¼ 2 jðnAXÞj;jj ¼ tr ðnAXÞ ; ~ 2 þ ~ 2 l;m k jC j þ k 0 0 jC 0 j ¼ minimum: ð15Þ j j i;i l;i i ;i l;i l X bXs0 =2c bXs0 =2c no 2 b 2 a 2 2 jðD Þ j ¼ 2 jðn Þ 0 0 j ¼ 2 jðn Þ 0 0 j ¼ tr ðn Þ ; del l;m XB l;m XB j ;j XB j ;j XB The resulting matrix k of effective contributions to population of 0 0 j j electrons delocalized in a chemical bond of type A–X–B can be used ð7Þ to calculate the corresponding contributions to population of well- and the from algebraic properties of (3) we get localized two-center bonds, X X X loc del loc loc jðD Þ j2 ¼ jðD Þ j2 þ jðD Þ j2; k ¼ nb k ; and N ¼ trk : ð16Þ l;m AXB l;m AX l;m XB l;m 3 3 l;m l;m nonono ð8Þ Obviously, the overall population Ndel from (14) is invariant due 2 2 2 3 tr ðnAXBÞ ¼ tr ðnAXÞ þ tr ðnXBÞ : to the canceling procedure described above. As a matter of fact, for our purpose this procedure can be skipped without impacting Hence, within the spectrum of eigenvalues of s s matrices of type further step, since the electron populations kloc are not required (3), there are also bs=2c nonnegative and exactly the same number in the calculation of electron delocalization index. It follows from of nonpositive eigenvalues giving rise to the corresponding bonding properties of the Jug-type bond orbitals [15] that the electron b a and anti-bonding three-center bond orbitals, # and # , b j AXBi j AXBi population of particular three-center bond orbital, ðn Þ , is al- 3 i;i respectively (r3; r3; p3; p3, etc.). ways equally distributed between atomic orbitals centered on 3. Combine both subsets of eigenvectors of matrices (2) corre- atom X and the rest of AOs, collectively. Hence, the population sponding to bonding two-center orbitals, i.e. ! of electrons localized/delocalized ’’through’’ atom X in the atomic nb 0 sequence A–X–B reads fb fb ; fb ; nb AX ; 9 j 2iðj AXi j XBiÞ 2 b ð Þ loc loc del del 0nXB NXj3 ¼ N3 =2 and NXj3 ¼ N3 =2: ð17Þ ~b and construct the new set of orthonormal bond orbitals jf2i that 5. Calculate the electron delocalization index for the p-atomic ring resemble the most their origins (with emphasis on highly occupied Rp, given by the atomic sequence A–X1––Xp-2–B, defined as bond orbitals) and allows one to expand the subset of three-center ! Yp 1=p orbitals as p CR ¼ nt ; ð18Þ b b ~b C~ b b t¼1 j#3ij#AXBi¼jf2i ; n3 nAXB: ð10Þ p C~ where nt represents the tth element of the following p-tuple: The LCBO matrix can be straightforwardly calculated using the following modified version of the constrained orthogonalization del del del del n ¼ N ; N ; ...; N ; N : ð19Þ procedure recently proposed and discussed in [34], AjBAX1 X1jAX1X2 Xp2jXp3Xp2B BjXp2BA Author's personal copy

156 D.W. Szczepanik et al. / Chemical Physics Letters 593 (2014) 154–159

In the preceding equation labels A,fg Xt0 and B are given explicitly cyclohexadiene molecule p-delocalization is practically not for each triplet of bonded atoms (instead of label ’’3’’) to avoid observed; every r- and p-bond has a typical structure of well local- any confusion. ized two-center orbital. Therefore, overall electron populations from Figure 1 suggest the presence of 8 chemical bonds (6 r

According to the definition (18), one can regard CR as an average and 2 p) in the R6 unit, as expected. On the other hand, benzene populational measure of the effectiveness of forming linear combi- molecule represents distinctly different situation. Here, the calcu- nations from bond-order orbitals of particular sequences of bonded lated population of p-electrons delocalized along the R6 ring comes triplets of atoms in the atomic ring R. Using the geometric mean in to about 5:5e. One should realize, that the residual population of (18) is essential to ensure the nonlocal character of such electron 0:5e is associated with relatively weak ’’across the ring’’ interac- del delocalization measure. Consequently, if there is at least one atom tions. Indeed, additional calculations of NXj3 for atomic triplets in the ring R that prevents the unrestrained flow of delocalized involving the p-coupling between carbons atoms in para positions electrons it thereby becomes the limiting one and has dramatic im- give rise to the population of nearly 0:45e, which, if combined with pact on the overall delocalization index. Thus, the definition (18) ’’along the ring’’ delocalization satisfactorily reproduces the overall seems to be superlative more adequate in quantitative description number of six pi-electrons in benzene. of delocalization in aromatic rings, especially if compared to the Figure 2 presents five the highest occupied three-center orbitals loc loc overall delocalized population in atomic ring R or even a simple and the corresponding populations NXj3 and NXj3, respectively, for arithmetic mean, selected triplet of carbon atoms in benzene and 1,3-cyclohexadiene. As can be easily seen, for both systems main bonding contri- Xp Xp del p del 1 p bution is due to first three orbitals (10). In the case of absence of NR ¼ nt ; and NXjR ¼ p nt ; ð20Þ t¼1 t¼1 electron delocalization in C6H8 ring, two- and three-center orbitals are merely distinguishable, whereas in C6H6 the latter form effec- respectively. In order to demonstrate the usability of the newly pro- tive combinations of two-centered functions. It should be noticed posed measures of electron delocalization in cylclic (and policyclic) that, in the case of benzene molecule, the procedure of cancelling molecules, in the next section we will present and briefly discuss re- of delocalized populations of orbitals r3 and r3 is so far effective, sults of illustrative calculations for several selected aromatic hydro- that the residual population of about 0:003e is of the order of mag- carbons (AH) and polycyclic aromatic hydrocarbons (PAH). nitude smaller than electron populations of orbitals #3 and #3. The latter should be considered as an artefacts of orthogonalization 3. Illustrative examples procedure (11) and are typical for calculations involving extended sets of basis functions. Indeed, as demonstrated elsewhere [26], the All calculations were carried out for equilibrium geometries quantitative separation of bond orbitals without any significant determined at B3LYP/cc-pVDZ/NAO theory level [38–40,29], using hybridization–orthogonalization artefacts is viable only within ab initio quantum chemistry package GAMESS [41] and NBO 6.0 representation of the effective minimal set of atomic orbitals (e.g. software [42]. [30–36]). At first, qualitative and quantitative differences in bond-order Table 1 collects the results of calculations of indices (18) and orbitals and bond densities of two representative molecular sys- (20) performed at the B3LYP/cc-pVDZ/NAO theory level for equilib- tems, benzene and 1,3-cyclohexadiene, will be shown. Figure 1 rium geometries of the following representative aromatic hydrocarbons (AH) and polycyclic aromatic hydrocarbons (PAH), presents contours of electron densities along carbon R6-rings of shown in Figure 3: cyclopropenylium cation (I), cyclobutadienyli- both molecules, qRðrÞ, obtained by appropriate combination of density-matrix layers corresponding to elements of (19) with the um dication (II), cyclobutadienide dianion (III), cyclopentadienide loc del anion (IV), benzene (V), tropylium cation (VI), homotropylium cat- canceling procedure applied; electron densities qR ðrÞ and qR ðrÞ, loc del ion (VII), cis and trans isomers of 1,3-bishomotropylium cation giving rise to localized (NR ) and delocalized (NR ) electron populations, were obtained in a very similar way, but incorporating (VIIIa and VIIIb, respectively), naphthalene (IX), phenanthrene (X) also relations (14) and (16). As follows from Figure 1, in 1,3 and coronene (XI). In the PAHs case, the R-unit naming convention is as follows: every cyclic subsystem is labeled with capitals A,B,..., while the combinations of type A–B, A–B–C, ..., always set up the largest possible rings enclosing all their R-subunits; e.g., in case of

molecule (XI), the notation A–B stands for a subsystem of R10-type ring (naphthalene-like), A–B–C–G represents a subsystem of

R14-type (pyrene-like), A–B–C–D–E–F is a subsystem of R18 (coronene-like) and, the largest one, A–B–C–D–E denotes a subsystem of

R22-type (benzo[c,g]phenanthrene-like). Additionaly, for polycyclic del molecules IX-XI atomic delocalization indices NXj3 were calculated for selected carbon atoms and are presented in Figure 3.

3.1. The simplest AHs

According to the Hückel’s rule [43], aromatic structures I and II contain 2 delocalized p electrons, whereas compounds III-VIII are typical 6 electron p-conjugated cyclic systems. However, beyond

Figure 1. Contours and overall populations of electron bond densities qRðrÞ along expectation, values from Table 1 palpably identify cyclopropenyli- R6-rings of benzene (a) and 1,3-cyclohexadiene (b), obtained by appropriate um cation as an electron system with almost 3 delocalized combination of density-matrix layers corresponding to elements of (19), with the electrons. The explanation for this outcome refers to the loc canceling procedure applied. Electron densities giving rise to localized (qR ðrÞ) and del loc del co-occurrence of r-, (out-of-plane) piz- as well as the concomitant delocalized (qR ðrÞ) electron populations, NR and NR , respectively, were obtained in a very similar way, but by incorporating also relations (14) and (16). Method: (in-plane) pixy-delocalization effects, connected with very high B3LYP/cc-pVDZ/NAO, equilibrium geometries. accumulation of positive charge on hydrogen atoms; indeed, the Author's personal copy

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loc del Figure 2. The ﬁrst ﬁve highest occupied three-center orbitals and the corresponding populations NXj3 and NXj3, for selected triplet of carbon atoms in benzene (a) and 1,3- cyclohexadiene (b). Method: B3LYP/cc-pVDZ/NAO, equilibrium geometries.

Table 1 Electron delocalization descriptors (18) and (20) for a number of cyclic and polycyclic aromatic systems shown in Figure 3. Only one of the symmetry equivalent cyclic units denoted by capital letters are listed. More details in the text. Method: B3LYP/cc-pVDZ/NAO, equilibrium geometries.

del del Molecule Rp p del CR Molecule Rp p del CR NR NCjR NR NCjR I 1–2–3 3 2.7630 0.9210 0.9210 VIIIb 1–8–9 3 0.3433 0.1144 0.0941 II 1–2–3 3 1.5567 0.5189 0.5188 1–2–3–4–5–7–8 7 1.8981 0.2712 0.1736 1–2–3–4 4 2.1231 0.5308 0.5308 1–2–3–4–5–6–7–8 8 2.0121 0.2515 0.1603 III 1–2-3 3 1.4476 0.4825 0.4812 1–2-3–4-5–6-7–8–9 9 2.1244 0.2360 0.1506 1–2–3–4 4 2.1270 0.5317 0.5317 IX A 6 3.5294 0.5882 0.5872 IV 1–2–3 3 0.9238 0.3079 0.0838 A–B 10 5.9127 0.5913 0.5897 1–2–3–4 4 1.8067 0.4517 0.1652 X A 6 4.2813 0.7136 0.7126 1–2–3–4–5 5 4.3607 0.8721 0.8721 B 6 2.2690 0.3782 0.3675 V 1–2–3 3 0.9235 0.3078 0.0122 A–B 10 5.1837 0.5184 0.4823 1–2–3–4 4 1.9963 0.4991 0.2670 A–B–C 14 8.0986 0.5785 0.5419 1–2–3–4-5 5 2.7866 0.5573 0.1631 XI A 6 3.2798 0.5466 0.5425 1–2–3–4–5–6 6 5.5245 0.9208 0.9208 G 6 3.3866 0.5644 0.5644 VI 1–2–3 3 0.8697 0.2899 0.0236 A–B 10 5.3492 0.5349 0.5307 1–2–3–4 4 1.7528 0.4382 0.1116 A–G 10 5.5374 0.5537 0.5512 1–2–3–4–5 5 2.6147 0.5229 0.1679 A–B–G 12 6.4787 0.5399 0.5362 1–2–3–4–5–6 6 3.4793 0.5799 0.2272 A–B–C 14 7.4191 0.5299 0.5258 1–2–3–4–5–6–7 7 6.0335 0.8619 0.8619 A–G–D 14 7.6896 0.5493 0.5457 VII 1–7–8 3 0.2091 0.0697 0.0679 A–B–C–G 14 7.4191 0.5299 0.5258 1–2–3–4–5–6–7 7 4.4826 0.6404 0.5991 A–B–C–D–G 16 8.3602 0.5225 0.5181 1–2–3–4–5–6–7–8 8 4.1125 0.5141 0.3491 A–B–C–D 18 9.4885 0.5271 0.5230 VIIIa 1–8–9 3 0.3737 0.1246 0.0859 A–B–G–D–E 18 9.5707 0.5317 0.5272 1–2–3–4–5–7–8 7 1.9938 0.2848 0.1752 A–B–C–D–E–G 18 9.3008 0.5167 0.5122 1–2–3–4-5–6–7–8 8 2.1363 0.2670 0.1665 A–B–C–D–E–F 18 9.0308 0.5017 0.4976 1–2–3–4–5–6–7–8–9 9 2.2787 0.2532 0.1602 A–B–C–D–E 22 9.5204 0.4760 0.3901

natural population analysis of constituent atoms of I gives rise to conﬁned electron delocalization along the R5 atomic ring is nearly +0.31 charge on each hydrogen atom and about +0.02 on affected by the presence of merely weak interaction between meta each carbon atom. Similar charge accumulation can be found in carbon atoms. However, the arithmetic average of delocalized elec- compound II. In this case, substantial decrease in delocalization in- tron population (20) in R5 unit recalls a typical value for small AHs dex for R4 with respect to structure I is due to strong cross-ring (cf. II, III or IX) which is mainly due to relatively strong ’’local’’ delo- delocalization of electrons between carbon atoms. This phenome- calization of electrons along the C1–C2–C3–C4 atomic chain. This del non is plainly manifested by the comparable value of index (18) observation allows one to draw the conclusion that NR as well del for R3 and R4 ring units in II and III. Slightly weaker cross-ring inter- as NXjR cannot be considered as adequate delocalization measures action is observed in III; here, doubly occupied anti-bonding LUMO since they are not capable of comparing AHs with different num- orbital effects with the overall number of about 2 delocalized elec- bers of atoms in rings. trons instead of 6. In similar manner the R5 ring delocalization in IV is slightly below the level for benzene molecule; other possible 3.2. Tropylium and its homologs rings, i.e. R3 and R4 do not exhibit any signiﬁcant p-conjugation. As one could expect, in the case of benzene molecule (V), the An interesting class of resonance-stabilized organic compounds net effective delocalization spreads over entire R6 ring, whilst form homoaromatic molecules, in which p-conjugation is inter- 3 within smaller units only R4 performs noticeable value of CR, pro- rupted by one or more sp -hybridized carbon atoms (usually meth- vided predominantly by interaction of para carbons. Concurrently, ylene groups). Such cyclic hydrocarbons, regardless of their Author's personal copy

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III III IV V

1 1 1 4 1 4 1 6 2 5 _ 2 + 2+ 2- 3 2 3 2 5 3 3 2 34 4

VI VII VIIIa VIIIb

8 9 9

1 1 1 1 7 2 7 2 8 2 8 2 + + + + 6 3 6 3 7 3 7 3

45 456 456 45

IX X XI 0.45 0.45 0.76 0.45 0.45 0.54 0.54 0.72 0.76 AB C 0.62 0.62 0.72 0.45 0.45 0.68 AB 0.76 F G C 0.62 0.62 A B 0.45 0.45 0.54 0.54 0.76 0.27 E 0.68 0.27 D 0.45 0.45 0.45 0.45

Figure 3. Skeletal formulas for several selected organic molecules: cyclopropenylium cation (I), cyclobutadienylium dication (II), cyclobutadienide dianion (III), cyclopentadienide anion (IV), benzene (V), tropylium cation (VI), homotropylium cation (VII), cis and trans isomers of 1,3-bishomotropylium cation (VIIIa and VIIIb, respectively), naphthalene (IX), phenanthrene (X) and coronene (XI). More details in the text. diversiﬁed geometry and electronic structure, reveal chemical, two methylene groups. As follows from Table 1, both isomers, cis thermodynamic, magnetic and spectroscopic properties quite sim- (VIIIa) and trans (VIIIb), differ only marginally in terms of p-delo- ilar to the typical arenes. Probably the best studied examples of calization effectiveness in the respective ring units (but, according homoaromatic compounds are homologs of tropylium cation (VI), to expectations, the cis isomer reveals slightly higher resonance- i.e. homotropylium (VII) and 1,3-bishomotropylium cations (VIIIa stabilization in R7; R8 and R9 units). As follows from comparison and VIIIb). A reference to Table 1 indicates that in the case of tro- of delocalization indices (18) for R7 in both isomers with the corre- pylium cation (VI) pz-delocalization is slightly less effective than in sponding value for R7 in (VII) we can see that the insertion of an- the benzene molecule (V). One should notice, however, that the in- other CH2 group to homotropylium cation gives rise to dramatic crease of values of the Ct–Ct+1–Ct+2 bond angles in the R7 ring in (VI) fall of CR, i.e. down to 70%; this is somewhat understandable, since, relative to the R6 in (V) give rise to somewhat ampliﬁed interaction unlike with (VII), in both isomers of 1,3-bishomotropylium cation between carbon atoms Ct and Ct+2. Consequently, unlike the case of the R7 ring is not planar. benzene molecule, in (VI) the electron delocalization index (18) monotonically increases in R3 to R7. 3.3. Three selected PAHs

Inserting a single methylene group between atoms C1 and C7 in (VI) gives rise to the homoaromatic compound (VII). In accordance The last three structures depicted in Figure 3 represent a sub- with Table 1, electron delocalization in the R3 unit of (VII) is incon- class of polycyclic aromatic compounds involving only R6-type cyc- siderably small due to disrupting influence of sp3-hybridized lic units (denoted by capital letters). A reference to Table 1 carbon atom in CH2 group on the continuous overlapping of indicates that p-delocalization in both cyclic units of naphthalene pz-orbitals. On the other hand, the value of CR for R3 in (VII) is evi- (IX), R6-type A (B) and the composite R10-type ring (A–B), is for dently larger than for the corresponding units of (V) or (VI), which about 35% less effective relative to the benzene molecule (V). p can be connected with hiperconjugation of r-bond C–H in methy- Moreover, a detailed analysis of nt from Eq. (19) (calculated for lene group with both neighbouring carbon atoms, C1 and C7. The constituent carbon atoms of each PAHs and presented in Figure comparison of delocalization indices for R7 and R8 units clearly 3) allows one to draw the conclusion that carbon atoms in posi- indicates that p-conjugation is more effective in the former one tions b are slightly more resonance-stabilized than those in posi- 3 (DCR 0:25). In other words, the sp -carbon atom in the R8 ring tions a. Somewhat different situation one can observe in the case decreases the electron delocalization relative to the R7 unit for of phenanthrene (X); here, the difference between delocalization about 42%. Concurrently, the comparison of CR for R7 in (VII) with effectiveness in atomic rings A,C and the delocalization in B is cru- the corresponding value for the largest ring in (VI) leads to the con- cial, DC 0:35 and reflects the relative reactivity of these R6-type clusion that the insertion of group CH2 into aromatic ring of type R7 units. Mainly, two carbon atoms in less resonance-stabilized atom- results in significant decrease of electron delocalization index, ic ring B distinctly easier undergo such reactions as organic oxida-

DCR 0:26, i.e. for about 30%; similar conclusions about the rel- tion to phenanthrenequinone with chromic acid, organic reduction ative aromaticities of (VI) and (VII) can be drawn on the basis of the to dihydrophenanthrene with hydrogen gas and Raney nickel, elec- bond separation energy approach [44]. trophilic halogenation to bromophenanthrene with bromine or Bishomoaromatic compounds (VIIIa) and (VIIIb) arise from sep- even ozonolysis to diphenylaldehyde. On the other hand, the most del arative interruption of conjugation of the aromatic system (VI) by aromatic carbon atoms in A and C (NXj3 ¼ 0:76) preferably undergo Author's personal copy

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Figure 4. An illustrative example of prediction of the electrophilic attack position based on the analysis of resonance-stabilization effects in Wheland intermediates using electron delocalization index (18). The values of CA and CB are inscribed into particular rings while the values of CAB are placed above the corresponding R10-type units. Method: B3LYP/cc-pVDZ/NAO, optimized geometries. electrophilic aromatic substitution, e.g. aromatic sulfonation to congeneric indeces based on the multicenter bond-order concept phenanthrenesulfonic acids with sulfuric acid [45]. Finally, in the [9–11] as well as the Jug’s ring-current aromaticity criterion [50]. case of coronene molecule (XI) one can observe that overhelming majority of cyclic units from R6 to R18 feature quite comparable le- References vel of delocalization, similar to that in naphthalene (X). Such equalization of values of (18) over almost all atomic rings results directly [1] P.R. Schleyer, Chem. Rev. 101 (2001) 1115. [2] P.R. 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Phys. 2 (2000) 3381. However, as presented in Figure 4, when the reaction is carried out [10] P. Bultinck, R. Ponec, S. Van Damme, J. Phys. Org. Chem. 18 (2005) 706. at about 80C (kinetic control mechanisms), a-naphthalenesulfonic [11] P. Bultinck, M. Rafat, R. Ponec, B. van Gheluwe, R. Carbó-Dorca, P. Popelier, J. acid is the major product. Indeed, according to the Hammond’s Phys. Chem. A 110 (2006) 7642. [12] X. Fradera, M.A. Austen, R.F.W. Bader, J. Phys. Chem. A 103 (1999) 304. Postulate [46] (which states that the relative energies of the inter- [13] J. Poater, X. Fradera, M. Duran, M. Solà, Chem. Eur. J. 9 (2003) 400. midiates should approximate the energies of the transition states [14] X. Fradera, J. Poater, S. Simon, M. Duran, M. Solà, Theor. Chem. Acc. 108 (2002) leading to the formation of the particular products), the a-arenium 214. [15] K. Jug, J. Am. Chem. 99 (1977) 7800. ion is for about 3.5 kcal/mol more stable than the corresponding b [16] R.F. Nalewajski, Int. J. 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A uniform approach to the description of multicenter bonding Cite this: Phys. Chem. Chem. Phys., 2014, 16, 20514 Dariusz W. Szczepanik,*a Marcin Andrzejak,a Karol Dyduch,a Emil Z˙ak,a Marcin Makowski,a Grzegorz Mazurb and Janusz Mrozekb

A novel method for investigating the multicenter bonding patterns in molecular systems by means of the so-called Electron Density of Delocalized Bonds (EDDB) is introduced and discussed. The EDDB method combines the concept of Jug’s bond-order orbitals and the indirect (‘‘through-bridge’’) interaction formalism and opens up new opportunities for studying the interplay between diﬀerent atomic Received 4th July 2014, interactions as well as their impact on both local and global resonance stabilization in systems of Accepted 10th August 2014 conjugated bonds. Using several illustrative examples we demonstrate that the EDDB approach allows DOI: 10.1039/c4cp02932a for a reliable quantitative description of diverse multicenter delocalization phenomena (with special regard to evaluation of the aromatic stabilization in molecular systems) within the framework of a consistent www.rsc.org/pccp theoretical paradigm.

1 Introduction of Bond Orders (LOBO),8,9 representing core orbitals (1c–2e), lone pairs (1c–2e) and chemical bonds (2c–2e). The overwhelming majority of chemical interactions in mole- However, many molecules cannot be adequately described cules can be described by a set of well localized two-center two- by such localized one- or two-center orbitals and the formalism electron bonds (2c–2e), i.e. standard chemical bonds. Within of multicenter bonding has to be utilized.10–15 In this context, the framework of the age-old qualitative theory of chemical intense investigations are focused on conjugated p bonds in bonding by Lewis and Langmuir, they represent pairs of aromatics, chelatoaromatics and all-metal clusters, hypervalent electrons shared by two atoms so that each attains the electron species, boranes, molecular systems with hydrogen/dihydrogen configuration of the nearest noble gas (‘‘the octet rule’’).1,2 The bonds, agostic bonds, planar tetra- and pentacoordinated carbon development of quantum-mechanical theories of the electronic atoms, etc. The concept of the Generalized Population Analysis 16–20

Published on 11 August 2014. Downloaded by UNIWERSYTET JAGIELLONSKI 24/09/2014 13:56:57. structure over the decades gave rise to deeper insights into bond (GPA) has successfully been used to develop the entire forming processes and provided a multitude of sophisticated panoply of the so-called Electron Sharing Indices (ESI)23–26 tools quantifying chemical bonding patterns. One of the most congeneric with the Multicenter Indices (MCI),21,22 which well-known and resoundingly successful theories is the theory of depend on the n-order Reduced Density Matrix (n-RDM).27 Molecular Orbitals (MOs),3 within which chemical bonds in The MCI approach has turned out to be especially useful in diatomic species are described by linear combinations of atom- evaluation of multicenter electron delocalization in aromatic centered functions – Atomic Orbitals (AOs). Admittedly, in the species.28–33 One should realize, however, that this multicenter general case of polyatomics the molecular orbitals do not refer to descriptor is designed to deal only with local molecular cyclic well localized 2c–2e bonds anymore being usually delocalized units of predefined size (which opens up the door to some over the whole molecule and reflecting molecular symmetry. degree of arbitrariness). As such it does not provide a compre- Fortunately, in many polyatomic molecules the chemical language hensive tool for the description of multicenter bonding in more connected with the Lewis model can be simply adopted at the extended systems. Furthermore, being the RDM-derived quan- level of modern theory by introducing the doubly-occupied Loca- tities, MCIs are related to the many-orbital joint probabilities34–40 lized Molecular Orbitals (LMOs)4–7 or so-called Localized Orbitals and so they do not correspond directly to simple electron numbers (even though different ways of normalization of MCIs have also been introduced41). Therefore they cannot take into account the influence of other multicenter interactions, depending on a Department of Theoretical Chemistry, Jagiellonian University, Ingardena 3, the choice of the basis set (especially within the MO-approach 30-060 Cracow, Poland. E-mail: [email protected]; 42–45 Tel: +48 12 663 22 13 involving the classical Mulliken scheme ). They are also b Department of Computational Methods in Chemistry, Jagiellonian University, subject to interpretative problems e.g. if one compares the Ingardena 3, 30-060 Cracow, Poland degree of delocalization in cyclic units of different sizes

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(renormalization allows one to remedy this problem, but it can be straightforwardly distributed between all atoms in the

always demands for predetermination of ring size). Moreover, molecule, {Xa}, for large and complex molecular systems calculations of MCIs X X X can be difficult and time consuming, especially if extended N ¼ trD ¼ traD ¼ trDa;a ¼ Na: (2) a a a basis sets are used. 46 It has recently been argued that the three-center chemical In the case of one-determinant wavefunction the density matrix interactions are by far the most significant in detecting multi- is duodempotent, i.e. D =21kDk for k 4 1, which allows one to center electron delocalization in molecular systems. A large generalize the population analysis scheme to comprise the number of studies that can be found in the literature point at whole hierarchy of multicenter electron population indices.16–20 47–50 the relatively low importance of higher-order interactions One of the most important among them is the Wiberg-type bond in quantifying multicenter chemical bonds within the language covalency index:66 of the population analysis, i.e. using the numbers of electrons. X Herein we briefly introduce the original method of multicenter XXa Xb N 1 D 2 bonding analysis (with special regard to evaluation of the ab ¼ 2 m;n ; (3) m n aromatic stabilization in molecular systems) that takes advan- 46 tage of three-center delocalization and does not suffer from directlyreferringtotheconceptofchemicalbondorder,67–71 shortcomings and limitations of the GPA-based methods. The deeply embedded in chemical intuition. new method is formulated within the framework of molecular It has originally been pointed out by Jug8 that, within the orbital theory and, just like other familiar methods, e.g. the representation of the minimal basis of atomic orbitals, the 7,51,52,54,55 Natural Bond Order (NBO) or the Adaptive Natural bond covalency index (3) can be simply decomposed into s, p 56 Density Partitioning (AdNDP) analysis, it makes use of the and higher components by solving the following eigenproblem: age-old concept of the so-called bond order orbitals, originally 0 1 proposed by Jug.8 Unlike the already existing formalisms, 0Da;b @ A y however, the proposed approach harnesses the power of indirect Dab ¼ ¼ CablabCab: (4) Dy 0 (‘‘through-bridge’’) interaction formalism57–62 that considerably a;b simplifies the analysis of multicenter bonding patterns and Indeed, the subset of eigenvectors associated with positive opens up new opportunities for the investigation of the interplay eigenvalues of D (denoted by superscript ‘‘b’’) gives rise to between different interactions and their impact on resonance ab the two-center bonding orbitals (2cBOs): stabilization. E X X Our method is to some extent inspired by the method b b b 2 zab ¼jwiCab; and lab Nab;i ¼ Nab: (5) 46 i;i originally proposed by Bridgeman and Empson. However, con- i i trary to their model which involves colored lines and triangles to % b describe delocalization, we use a visualization tool directly related In eqn (5) matrix Cab constitutes of an extension of the b to Electron Density (ED). Note that the new approach provides the rectangular matrix Cab that expands 2cBOs on the basis of all overall picture of electron delocalization and the detailed descrip- AOs. Except that Jug’s bonding orbitals are on their own very useful in probing bonding patterns of molecular systems, they

Published on 11 August 2014. Downloaded by UNIWERSYTET JAGIELLONSKI 24/09/2014 13:56:57. tion of delocalized electron populations in atomic resolution. It can therefore be used for quick detection of regions of increased can also be used to ‘‘reconstruct’’ the bonding part of one- b aromaticity in very large systems as well as for quantitative electron density, r (r): studies of electron delocalization, aromatic stabilization, reactivity X ð b y b b b etc. in selected molecular fragments. r ðrÞ¼ wnðrÞDm;nwmðrÞ; r ðrÞdr ¼ N : (6) m;n 2 Theoretical background Here, Nb stands for the overall number of electrons delocalized in all chemical bonds in the molecule, and the bonding density 2.1 Bonding electron density matrix Db is defined as a simple sum of density-matrix layers Firstly, let us express the one-electron density of closed-shell corresponding to all possible pairs of atoms: molecular systems, r(r), by means of basis functions {w (r)} and m X X 2 63 b 1 b b by the corresponding one-electron density matrix as follows: D ¼ 2 Cab lab Cab: (7) a X ð a b a rðrÞ¼ wy ðrÞD w ðrÞ; rðrÞdr ¼ N: (1) n m;n m Obviously, it follows directly from the normalization condition m;n b b b b b in (6) that trD = N as well as traD = Na (in the literature Na is On the basis of well atom-assigned localized orthonormal func- usually referred to as the chemical valence of atom Xa). tions, e.g. Natural Atomic Orbitals (NAOs),64,65 one can split the spinless density matrix D into diagonal and oﬀ-diagonal atomic 2.2 Electron density of delocalized bonds 72 blocks, Da,a ={Dm,n: m, n A Xa}andDa,b ={Dm,n: m A Xa, n A Xb}, It has recently been argued that an eigenproblem analo- respectively. Consequently, the overall electron population N gous to (4) can be formulated for atomic-block oﬀ-diagonal

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density matrices representing indirect interactions of type with the following normalization condition: ð Xa–Xb–Xg: X X 0 1 d d d d r ðrÞdr ¼ traD ¼ Na ¼ N : (14) 0Da;b 0 B C a a B C B y C y Dabg ¼ B Da;b 0Db;g C ¼ CabglabgC : (8) The EDDB matrix used in above equations takes the @ A abg following form: 0Dy 0 b;g X X b 2 by Dd ¼ 21 Bd ; Bd ¼ C ld C ; (15) The corresponding subset of three-center bonding orbitals ab ab ab ab ab a baa b % b (3cBOs), |zabgi =|wiCabg, is crucial for determining the multicenter delocalized electron density. Without going into details where described in our previous paper,72 it should be noted that ld ={(N d )1/2d }. (16) the projection of 3cBOs onto the set of orthogonalized 2cBOs, ab ab,i i,j ~b ~b ~b |z2i(|zabi,|zbgi), followed by the procedure of canceling of d The Bab matrix describes this part of the electron density of non-bonding and mutually phase-reversed 3cBOs, allows one to the chemical bond Xa–Xb that is delocalized in a multicenter b transform labg into a diagonal matrix collecting numbers of sense with all other bonds in the molecular system. Thus, one d electrons delocalized in a 3-center sense, labg.Furthermore,for can regard the EDDB matrix as assembled from density layers any particular triatomic sequence of conjugated bonds, Xa–Xb–Xg, relating to all possible diatomic interactions in the molecule the number of electrons delocalized ‘‘through’’ atom Xb can be under consideration. calculated straightforwardly as follows: What should be noticed is that, in the case of planar molecules/ d 1 d 2 % b d 2 % b† molecular fragments, the EDDB can also be strictly dissected into N b|abg =2 tr(labg) =trb[Cabg(labg) Cabg]. (9) the ‘‘in-plane’’ and ‘‘out-of-plane’’ EDDB-layers or the density For our purposes it is of special interest to evaluate to what layers corresponding to respective symmetry components, degree electrons assigned to the chemical bond Xa–Xb partici- EDDB = EDDB + EDDB + ..., (17) pate in the overall delocalized electron population of 3cBOs s p corresponding to both subsystems of conjugated bonds, Xa–Xb–Xg by solving the corresponding eigenproblem of the density and Xg–Xa–Xb. The simplest way to get this information is to involve matrix (15). Strict separation of the symmetry components s direct projections of 3cBOs onto the subset of orthogonalized and p of the EDDB follows mainly from the facts that degen- 2cBOs corresponding to Xa–Xb as follows: eration within the spectrum of eigenvalues of the EDDB matrix DE X X 2 practically never occurs (in contrast to the ED matrix). It has to d d ~b b d 2 Nabjabg ¼ Nab;ijabg ¼ zab;ijzabg;k labg ; (10) k;k be stressed, however, that the exact separation of higher sym- i i;k metry components may not be possible always (e.g.,adissection and of s and d bonding contributions is not necessarily possible in DE an exact way53). X X 2 d d ~b b d 2 Nabjgab ¼ Nab;ijgab ¼ zab;ijzgab;k lgab ; (11) k;k i i;k 2.3 Global and local characters of EDDB-populations Published on 11 August 2014. Downloaded by UNIWERSYTET JAGIELLONSKI 24/09/2014 13:56:57. d d First we would like to stress that the outlined scheme of the where Nab,i|abg and Nab,i|gab stand for the populations of electrons EDDB construction is far more efficient than ‘‘multicenter originally assigned to the ith 2cBO of the bond Xa–Xb and deloca- 54–56 lized through the corresponding atomic triplets. Obviously, electron scanning’’ techniques available in other formalisms, espe- populations from preceding equations take different values cially in the case of highly accurate wavefunctions of large-sized molecular systems. In our method the multicenter bonding depending on the choice of atom Xg. Therefore, to evaluate the electron population of the ith 2cBO that is effectively delocalized density is reconstructed by means of two-atomic fragments and d regards only local resonance triatomic hybrids, Xa–Xb–Xg and in a three-center sense, N ab,i, it is necessary to calculate orbital Xg–Xa–Xb, representing the corresponding indirect interactions populations (10–11) for each possible choice of atom Xg covalently b b (‘‘Xa with Xg through Xb’’ and ‘‘Xg with Xb through Xa’’, bonded with atom Xa or Xb (i.e. g a a, b and Nag, Nbg Z t ,wheret d respectively). is an arbitrary threshold value). Then, we can define Nab,i as follows: It should be emphasized that the use of such indirect d d d a N ab,i = max{N ab,i|abg, N ab,i|gab: g a, b}. (12) interaction formalism allows one to investigate the influence of particular chemical interactions and their mutual coupling Thus, one can interpret N d as the highest number of ab,i on the effectiveness of multicenter bonding in particular mole- electrons of bonding orbital |zb i that effectively participate ab,i cular fragment by ‘‘enabling’’ or ‘‘disabling’’ the appropriate in a three-center bonding with all other atoms in the molecule. subspace of interacting atoms X for each two-atomic density Finally, making use of eqn (6)–(11) we can define the Electron g layer. Therefore, beyond a routine study of multicenter bonds Density of Delocalized Bonds (EDDB) as X and aromatic stabilization, the EDDB method provides precise d y d EDDB r ðrÞ¼ wnðrÞDm;nwmðrÞ; (13) and valuable information about the coupling between two adjacent m;n rings in polycyclic aromatics, the impact of cross-ring interactions

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It has to be emphasized that the use of the representation of natural atomic orbitals is crucial for the EDDB-based population analysis since the NAO-based populations automatically satisfy Pauli constraints. Furthermore, the stability of the weighted orthogonalization procedure used in the construction of NAOs65,81,82 automatically ensures appropriate convergence profiles and numerical stability of atomic charges and bond orders with respect to basis set enlargement.64,65,83 This contrasts sharply with electron populations and the corresponding multicenter indices obtained within the framework of Mulliken’s population analysis scheme, which are known to exhibit unphysical negative values and numerical instabilities when the extended basis sets are used.84–86

3.1 Simple aromatic hydrocarbons Fig. 2 presents isosurfaces of ED and EDDB with the corresponding electron populations, global (black numbers) and the Kekule´-like (bold burgundy numbers), for the following simple aromatic hydrocarbons (AH): cyclopropenyl cations d d Fig. 1 Global and local (‘‘Kekule´an’’) EDDB-populations N and NC (along- + 2+ (C3H3 ), cyclobutadienyl dications (C4H4 ), cyclopentadienyl side atoms); contributions from hydrogen atoms are neglected. Method: anions (C5H5 ), benzene (C6H6) and cycloheptatrienyl cations B3LYP/6-31G*/NAO, equilibrium geometry. + (C7H7 ); the calculations were performed at the CAM-B3LYP/

aug-cc-pVTZ/NAO theory level and at tb = 0.001 (bonding threshold)

on the eﬀectiveness of along-ring multicenter delocalization, etc. The latter is clearly illustrated in Fig. 1 with the benzene molecule used as an example. Even a cursory look at global (including delocalized electron contributions from all possible triatomic resonance hybrids) and Kekule´-like EDDB-populations allows one to draw the conclusion that local resonances between cross-ring interactions (mainly Dewar’s para carbon–carbon bonds) and along-ring carbon–carbon bonds contribute noticeably to the global EDDB (up to 8% of the overall delocalized electron population assigned to carbon atoms). Therefore their influence on the electron delocalization within such a 6-member

Published on 11 August 2014. Downloaded by UNIWERSYTET JAGIELLONSKI 24/09/2014 13:56:57. molecular cyclic unit should not be neglected, especially in accurate calculations. Indeed, it is well known that the cross- ring interactions are more important in benzenoid-like units in which the para-delocalization effect was the basis of methods such as the PDI.28,73–75

3 Several illustrative examples

To demonstrate the performance of the EDDB approach, several illustrative examples are presented and briefly discussed. All the ab initio calculations were performed using Gamess76,77 and Gaussian78 packages at the DFT level with the B3LYP/CAM- B3LYP87–89 exchange–correlation functional as well as two correlation-consistent basis sets: cc-pVDZ and aug-cc-pVTZ.90 All

electron population descriptors introduced in the text were calcu- Fig. 2 Isosurfaces of ED (blue) and EDDB (green), generated at tb = 0.001 64,65 lated within the NAO-representation obtained from the NBO6 (bonding threshold) and tr = 0.015 (isosurface values), with the corres- d 91 ponding electron populations N (second column), global (black numbers) software by means of several computer scripts originally devel- d and the Kekule´-like (bold burgundy numbers), populations NC (colored oped by the first author; ED and EDDB contour maps were numbers alongside atoms) and natural atomic charges (colored numbers 79 80 obtained using visualization programs Molden and MacMolPlt below molecule) for several simple AHs. Method: CAM-B3LYP/aug-cc- with a number of manually prepared special input files. pVTZ/NAO, equilibrium geometries.

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A more detailed discussion of these points can be found in our previous paper.72 It is worth noticing that, as follows directly from the analysis of along-ring EDDB-populations (bold burgundy numbers inscribed in cyclic structures), resonances between along-ring (Kekule´-like) and cross-ring (Dewar-like) atomic interactions are particularly important in charged aromatic hydrocarbons. Preliminary results of more insightful analysis indicate that, contrary to the para-p-delocalization effect observed in the benzene molecule, in other charged AHs also the meta- p-delocalization as well as s-delocalizations (including even hydrogen atoms) play important roles.

3.2 Polycyclic aromatic hydrocarbons

Isosurfaces of EDDB, EDDBp and EDp with the corresponding electron populations for selected polycyclic aromatic hydrocarbons (Fig. 3 and 4) were calculated using the B3LYP/cc-pVDZ/

NAO method (equilibrium geometries) at tb = 0.001 and tr = 0.015. Black numbers denote global populations while bold burgundy numbers refer to populations of electrons delocalized

Fig. 3 Isosurfaces of EDDB, generated at tb = 0.001 (bonding threshold)

and tr = 0.015 (isosurface values), with the corresponding electron d populations N (first column), global (black numbers) and the Kekule´-like d (bold burgundy numbers), populations NC (second column) for several simple PAHs. Method: B3LYP/cc-pVDZ/NAO, equilibrium geometries.

t Published on 11 August 2014. Downloaded by UNIWERSYTET JAGIELLONSKI 24/09/2014 13:56:57. and r = 0.015 (isosurface values). In the last column formal charges of ions (inscribed in molecular rings), total natural atomic charges of all carbon atoms as well as atomic populations of delocalized electrons obtained from eqn (14) are displayed. It should be noticed that, in contrast to the example presented in Fig. 1, Fig. 2, 3 and 5 the sum of all EDDB populations assigned to carbon atoms slightly diﬀers from total EDDB populations reported below each structure. This is mainly due to very minor but noticeable contribution of hydrogen atoms to multicenter bonding (usually 0.01–0.03e per atom). It is evident even from a cursory analysis of numbers in Fig. 2 that, to a greater or lesser extent, populations N d diﬀer + from the expected Hu¨ckel’s numbers: ‘‘2’’ for C3H3 , and 2+ + C4H4 and ‘‘6’’ for C5H5 ,C6H6, and C7H7 . Essentially, there are three reasons for these discrepancies: (1) Nd counts for electrons from both, p- as well as s-delocalization, (2) the cross- ring interactions between carbon atoms and (3) C–H bonds of Fig. 4 (a) Isosurfaces of EDp (left column) and EDDBp, generated at charged AHs are much more polarized revealing tendency to tb = 0.001 (bonding threshold) and tr = 0.010 (isosurface values), with the corresponding electron populations Np and Nd,p and the HMO resonance somewhat overgenerous accumulation of electrons on carbon energies for benzene and several small PAHs. (b) Correlations between p d,p atoms (it follows directly from comparison of total atomic charges resonance energies and the overall populations N (EDp), N (EDDBp)and of all carbon atoms with formal charges of molecules in Fig. 2). Nd (EDDB). Method: B3LYP/cc-pVDZ/NAO, equilibrium geometries.

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only along each cyclic unit. The second column presents atomic An insightful investigation of possible resonance structures d populations Na (colored numbers near atoms); resonance ener- of the coronene molecule indicates that, according to the well- 98,99 gies (Fig. 4) for EDp were calculated at the Hu¨ckel Molecular known Clar’s rule, the center benzenoid is less aromatic Orbital (HMO) theory level.92–95 than external rings. This evidently contradicts the delocaliza- Fig. 3 clearly shows that in several cases even qualitative tion pattern that follows from the EDDB-based population analysis of EDDB contours enables one to predict the relative analysis; the latter accords to some extent with the picture of aromatic stabilization of respective cyclic units that is in coronene proposed by Popov et al.100 As a matter of fact, agreement with the common knowledge about the reactivity analysis of EDDBs of all PAHs larger than naphthalene allows of these species. In particular, it is evident from comparing for the conclusion that the Clar’s rule is fulfilled only for EDDBs and the corresponding global populations for anthra- species having a single unambiguous Clar structure (the same cene and phenanthrene molecules that multicenter electron conclusion has been drawn previously by G. Portella et al.101). delocalization is more eﬀective by about 0.5e in the latter one. Moreover, it follows from a comprehensive analysis of a larger Accordingly, resonance energies (Fig. 4a) for these molecules group of PAHs that at the level of the Hu¨ckel MO method local are 3.60 eV and 3.95 eV, respectively. Quantitative analysis of aromaticities of the overwhelming majority of species satisfy EDDB-based populations of electrons delocalized only along Clar’s rule regardless of the number of equivalent Clar struc- each cyclic unit (bold burgundy numbers) leads to the con- tures per each molecule.22,101 One should realize, however, that clusion that the most highly resonance-stabilized rings are: the HMO method is only a crude approximation that assumes the middle one in anthracene and two side rings in phen- the same idealized geometries for benzenoids and does not anthrene. An in-depth study reveals that 8 atoms in the take into account any s-delocalizations. This, in our opinion, is anthracene terminal rings and 2 atoms in the phenanthrene more than enough to cast doubt on the relative resonance d middle ring have significantly lower values of Na,which stability of cyclic units in coronene predicted by the Clar’s rule. means that their contributions to multicenter bonding in Note that qualitative analysis of the p-layer of electron

both molecules are of minor importance. Consequently, in density (1), EDp, must not necessarily lead to the same conclu-

the anthracene molecule the electrophilic aromatic substitu- sions as the analysis of EDDBp itself. Comparison of EDp and

tion usually involves only the inner ring (two equivalent atoms EDDBp contours for several simple aromatic hydrocarbons d with N a = 1.049) while in the phenanthrene molecule it prefers (Fig. 4a) shows that not the entire p-electron population is d d fouratomsofouterrings(withN a = 0.882 and N a = 0.907). delocalized along aromatic rings, as one might expect. In fact, However, it should be stressed that, in general, local aromati- the eﬀectiveness of p-delocalization varies between 70% (PAHs) city is not simply related to reactivity since the eﬀectiveness of and 92% (benzene). It should be stressed here that this result delocalization in a particular molecular fragment says nothing has been obtained from the first principles and is free from any about the HOMO–LUMO gap or the stability of the transition arbitrariness and references to any idealized system. Moreover, state (e.g., the Diels–Alder reactions always take place in the as follows directly from the analysis of p-electron populations central ring of the anthracene molecule in spite of the fact that and their correlations with the corresponding resonance ener-

it is a more resonance-stabilized cyclic unit than the terminal gies (Fig. 4b), the EDp-populations fail in predicting of relative benzenoids). Nevertheless, the relative aromaticity of the aromatic stabilization of iso-p-electronic systems and only the

Published on 11 August 2014. Downloaded by UNIWERSYTET JAGIELLONSKI 24/09/2014 13:56:57. central and external rings in anthracene is still a matter of EDDB- and EDDBp-populations are able to reliably evaluate controversy in the literature.96 global aromaticity of all molecules. Another important conclusion can be drawn from the analysis of the EDDB and the corresponding population 3.3 d-Aromaticity and atypical aromatics numbers referring to electrons delocalized along a particular Multicenter delocalized electron density contours and the ring (bold burgundy numbers) in the case of a fluoranthene corresponding atomic populations can be very helpful in probing molecule (Fig. 3). As follows from these numbers, only the molecular systems with electron delocalization involving d-block benzenoid-like cyclic units are found to be aromatic, and, transition metals. Fig. 5 presents populations and isosurfaces of 2 what is more important, the whole molecule can be regarded EDDBs for the porphine dianion (Por ), the cobalt(II) porphine as built up from the naphthalene unit (3.536e per ring complex (Co-Por) as well as maltol complexes with a vanadyl

compared to 3.529e per ring in a separate naphthalene mole- dication (VO(Ma)2) and aluminium (Al(Ma)3). Calculations were cule) being cross-linked to the benzene unit (4.997e compared performed using the B3LYP/cc-pVDZ/NAO method (equilibrium

to 5.295e in a separate benzene molecule). Indeed, it is well- geometries) at tb = 0.001 and tr = 0.015; the last column presents d known that application of the Hu¨ckel rules sometimes leads atomic populations N a or the total number of electrons deloca- to the conclusion that particular polycyclic compounds (e.g. lized over a particular molecular fragment. Comparing porphine fluoranthene or pyrene) should be anti-aromatic, which dis- with its cobalt complex it is clear that the central atom partici- agrees with their known chemical properties. The examination pates in electron delocalization. Quantitative analysis exhibits of such PAHs as conjugated cyclic polyenes which are intern- some outflow of delocalized electron population from the ring ally cross-linked and/or linked to other cyclic polyenes was due to the presence of the cobalt atom (bridging character of the historically the first commonly accepted solution for this central atom). This back-donation arises only within systems with problem.97 d-electron central atoms. For the next two examples in Fig. 5,

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Fig. 5 Isosurfaces of EDDB, generated at tb = 0.001 (bonding threshold) Fig. 6 Isosurfaces of EDDB, generated at tb = 0.001 (bonding threshold)

and tr = 0.015 (isosurface values), with the corresponding electron and three diﬀerent isosurface values, tr = 0.010, 0.015, 0.020, respectively, populations Nd (first column) and constituent atomic/fragment popula- with the corresponding electron populations Nd for the homotropylium tions (second column) for porphine dianions, cobalt(II) porphine com- cation and the cyclononatetraenyl cation. Additionally, the most highly

plexes, and vanadyl(IV) and aluminium maltol complexes. Method: B3LYP/ occupied EDDB-derived natural orbitals (generated at tb = 0.001 and tr = cc-pVDZ/NAO, equilibrium geometries. 0.015) and the corresponding occupation numbers are displayed. Method: B3LYP/cc-pVDZ/NAO, equilibrium geometries.

Published on 11 August 2014. Downloaded by UNIWERSYTET JAGIELLONSKI 24/09/2014 13:56:57. VO(Ma)2 and Al(Ma)3, the total number of electrons delocalized over the maltol unit is greater in the former case for about 0.2e, In turn, solving the eigenproblem of the EDDB matrix for the which is in agreement with the well-known facts about the cyclononatetraenyl cation gives rise to the well-known Mo¨bius- resonance stabilization of maltol in its chelatoaromatic like orbitals.105 complexes.102,103 Sometimes, especially in the case of non-planar and atypical aromatic molecules, natural orbitals that diagonalize the EDDB 4 Conclusions matrix can give additional insight into the electronic structure of the studied systems. Fig. 6 shows isosurfaces of EDDB, To summarize, there are several important features that set the calculated using the B3LYP/cc-pVDZ/NAO method (equilibrium newly proposed method apart from other measures of multi-

geometries) at three diﬀerent values of density, tr = 0.010, center delocalization in aromatic rings. (1) Universality – the 0.015, 0.020, with the corresponding electron populations for EDDB-based populations can be easily calculated for planar the homotropylium cation and the cyclononatetraenyl cation. and non-planar molecular rings and therefore they can be Additionally, the most highly occupied natural orbitals of the successfully used for the study of a wide range of aromatic

EDDB matrix selected for both structures are presented (tb = species including both the Hu¨ckel- and Mo¨bius-type aro- 106 104 0.001 and tr = 0.015). The analysis of eigenvectors and eigen- matics, homoaromatics and even non-cyclic aromatic values of the EDDB matrix for the homotropylium cation molecules.107 (2) Intuitiveness and interpretative simplicity – indicates that, besides the evident p-homoconjugation, some quantifying multicenter bonds involves the language of the residual s-delocalization through the methylene carbon atom first-order population analysis. (3) The lack of arbitrariness exists; this is a new result that might have important implica- connected with the necessity of predefining the size of molec- tions for the origin of homoaromatic stabilization eﬀects.104 ular cyclic units (like in the MCI-based techniques) when

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Computational and Theoretical Chemistry

journal homepage: www.elsevier.com/locate/comptc

A new perspective on quantifying electron localization and delocalization in molecular systems ⇑ Dariusz W. Szczepanik

K. Gumin´ski Department of Theoretical Chemistry, Faculty of Chemistry, Jagiellonian University, Ingerdena 3, 30-060 Cracow, Poland article info abstract

Article history: The original method of electron density partitioning is introduced that allows one to probe electron local- Received 15 January 2016 ization and delocalization within one theoretical paradigm. The newly proposed method makes use of the Received in revised form 29 January 2016 age-old concept of bond-order orbitals as well as the recently developed bond-orbital projection formal- Accepted 2 February 2016 ism to decompose the one-electron density into density layers representing electrons localized on atoms Available online 8 February 2016 (inner shells, lone pairs), shared between atoms (chemical bonds) and delocalized between adjacent bonds (multi-center bonding). The details of the current implementation are brieﬂy discussed and several Keywords: illustrative examples are provided. Electron localization Ó 2016 Elsevier B.V. All rights reserved. Electron delocalization Density partitioning Aromaticity

Our understanding of the chemical structure and reactivity is The definition of the EDDAðrÞ component is crucial for the entire usually built up from and dependent upon such intuitive concepts method. In contrary to other approaches regarding the electron as atom in molecule, chemical bond, lone pair, Lewis structure, density/population distributions within physical space [7–9], here bond covalency and ionicity, etc. [1,2]. Although the semantics of we use the Hilbert-space partitioning scheme based on the one- these fundamental entities is not sharply defined in modern quan- electron density matrix,noD, within the basis of the natural atomic tum mechanics, a multitude of different formalisms has been pro- orbitals (NAO) [10], vlðrÞ , but any other set of orthonormal and posed in the literature either to determine them as functions of the still well atom-assigned functions can also be used [11,12]. Here, electron density (ED), like within the framework of the quantum it has to be noticed that the presented formalism is restricted to theory of atoms in molecules (QTAIM) [3,4], or provide them a one-determinant wave functions only and thus it can be used at localized-orbital representation, like the natural bond orbital HF and DFT theory levels (bear in mind, however, that the Kohn– (NBO) approach does [5]. Despite great popularity and success, Sham determinant relies on the auxiliary set of orbitals represent- both these approaches have certain drawbacks that sometimes ing non-interacting electrons and as such it provides only an may cut back their applicability, eg. the shortcomings regarding approximation to the first order density matrix). The definitions criteria for the chemical bond to exist [6], extremely high compu- given in the text are correct for closed-shell systems. The spin- tational cost of the localized-orbital description for large molecules resolved calculations for open-shell systems should be carried out with multi-center bonding, and many others. In this paper, for the separately for both a- and b-spin density matrices and then, in first time these two methodologies are unified to provide a scheme accordance with the Gopinatham-Jug definition of the bond- for the exact decomposition of the one-electron density into the covalency index [13], the resulting sum of spin-components needs density layers representing electrons well-localized on atoms to be multiplied by 2. (EDLA), like core electrons, lone pairs and ionic bonds, and the elec- The construction of the EDDAðrÞ function requires solving a set trons delocalized between all atoms in a molecule, (EDDA), ie. of the eigenproblems for all possible Jug’s matrices in the molecu- covalent bonds, lar system [14]. The classical Jug’s matrix is determined by the EDðrÞ¼EDLAðrÞþEDDAðrÞ: ð1Þ a; b off-diagonal ( )-diatomic blocks of the one-electron density matrix ( Dab ) [15], ! ⇑ 0Da;b y Tel.: +48 12 663 20 25. y ¼ CabkabCab !jfabi¼jviCab; ð2Þ E-mail address: [email protected] Da;b 0 URL: http://www.chemia.uj.edu.pl/~szczepad http://dx.doi.org/10.1016/j.comptc.2016.02.003 2210-271X/Ó 2016 Elsevier B.V. All rights reserved. 34 D.W. Szczepanik / Computational and Theoretical Chemistry 1080 (2016) 33–37 no no X X DA DA b n N ¼ aDDA ¼ N : ð Þ and its spectrum contains positive ( ka;b ), zero ( ka;b ) and nega- tr a 9 no a a tive eigenvalues ( ka ), associated with the corresponding eigen- a;b In accordance with Eq. (1), the density layer representing elec- b n vectors representing bonding (jfabi), non-bonding (jfabi) and anti- trons well-localized on atomic centers, EDLA(r), can be straightfor- a bonding (jfabi) two-center bond orbitals (2cBO), respectively, i.e. wardly obtain by the subtraction of EDDA(r) from the one-electron density, ED(r). But, furthermore, as shown very recently by the b n a b n a kab ¼ kabjkabjkab ; Cab ¼ CabjCabjCab : ð3Þ author [14,19], the EDDA(r) layer can be regarded as a sum of two components representing electron density of localized and As originally pointed out by Jug [15], the sum of squared elements delocalized bonds, EDLB(r) and EDDB(r), respectively. The latter b of the diagonal matrix kab is identical to the bond-order definition is determined through the bond-orbital projection technique, by Wiberg [16], Wab, which, alongside involving the subset of appropriately orthogonal- a b ized [20] 2cBOs, requires also their three-center counterparts [21] 2 X X ¼ kb ¼ 1 2 : ð Þ to represent all the possible bond-conjugations in the molecular Wab tr ab Dab l;m 4 2 l m system. Thus, taking into account additional step of the EDDA(r) This property (characteristic only for idempotent density matrices decomposition, the electron density can be partitioned as follows: [17]) is used to define the quasi-metric of all bonding 2cBOs within the representation of NAOs, EDðrÞ¼EDLAðrÞþEDLBðrÞþEDDBðrÞ: ð10Þ X 2 y To illustrate how the newly proposed ED-partitioning proce- Sb ¼ Cb kb Cb ; ð5Þ f ab ab ab dure works in practice, it has been used to probe the electronic a;b structure of the valence shells of benzene, s-triazine and borazine, When the summation in (5) includes only atomic pairs that repre- calculated at the B3LYP/6-311++G⁄⁄ theory level, as well as the ⁄⁄ b sents typical covalent interactions then the trace of the Sf matrix anthracene molecule calculated using the CAM-B3LYP/6-311+G can be regarded as a good approximation to the population of all method. The quantum-chemical software including Firefly [22], shared-electron pairs in the molecular system (i.e. the population MultiWFN [23] and Molden [24] has been used to perform compu- ð Þ of electrons delocalized through the net of all chemical bonds). tations and visualize the results. The EDDB r density layers have r p Technically, this is the case only if the highest occupied 2cBOs form been dissected into and components to provide a more detail description of the bond-conjugation effects [18]. a subset of the NBO-like orthonormalno orbitals associated with the The results, as presented in Fig. 1, clearly shows that the nearly degenerated eigenvalues ðkb Þ . However, if one takes ab l;l EDLAðrÞ function easily copes with detection of lone-pairs in into account also the weak non-covalent interactions or if the s-triazine as well as regions of highly localized electrons due to multi-center bonding occurs, the 2cBOs cannot be expanded as lin- charge polarization (borazine, but also to a small extent the ben- ear combinations of the doubly-occupied molecular orbitals (MO) zene molecule). However, in contrast to the expectations depen- b only. To solve this problem and give the diagonal of the Sf matrix dent upon the qualitative model of bonding by Lewis, in the case a strict populational character, it is proposed to project the eigen- of borazine the six p-electrons do not form a typical aromatic ring b but tend to remain as unhybridized electron-pairs centered on vectors of Sf onto the ground-state molecular wave function or, which is fully equivalent, transform the quasi-metric (5) under nitrogen atoms. This fact explains remarkable electron localization the following orthogonal similarity transformation: on nitrogen atoms, but also rationalizes significantly less effective multi-center delocalization quantified by the EDDBp component, ~ b 1 y b b ~ b Sf ¼ 2 D Sf D; N ¼ tr½Sf : ð6Þ which is in full agreement with findings by other researchers [25]. In turn, the effectiveness of p-delocalization in both benzene Here, one should realize that the total population of electrons delo- and s-triazine is very high, but still slightly lower than expected for b calized between atoms in a molecule, N , is markedly underesti- an ideal aromatic 6p-electron system. Here, the reason is that the mated if the bond-conjugation effect occurs [18]. This is especially cross-ring interactions, which are particularly important for important for aromatic species, in which the shared p-electron pairs para-related atoms, are reluctant to conjugate with the covalent are delocalized between more than two atomic centers and thus bonds along molecular ring, remaining weak but well-localized they cannot be exactly described by any subset of localized bond ‘bonds’ in nature. Although this fact has already been noticed in orbitals. It should be stressed, however, that in such cases the the literature [19], no comprehensive research has been reported bond-order representability condition for the matrix (5) determines so far regarding the influence of cross-ring interactions on the significant contribution of the virtual MOs to the shape of bonding effectiveness of bond-conjugation in aromatics. Here, it should be 2cBOs (this property of the 2cBOs will be the subject of a separate mentioned that the para-delocalization effect is the basis of the study). If so, the complementary set of anti-bonding 2cBOs can be very popular aromaticity index by Poater et.al. (PDI) [26]. Since it used to ‘recover’ the lacking electron population in (6). Therefore, has been reported that in some cases this descriptor is not suitable in analogy to (5) and (6), the GS-projected quasi-metric of all for the description of local aromaticity due to overestimation of the anti-bonding 2cBOs within the representation of NAOs is given by "# contribution of Dewar’s resonance structure to the electron delo- X 2 calization [27], a comparative study of PDI with EDLB and EDDB S~ a ¼ 1 y Ca ka Cay : ð Þ f 2 D ab ab ab D 7 components of the electron density seem to be even more strongly a;b justified. In contrast to the p-type bonds, which have a much Then, the entrywise sum of the matrices (6) and (7) gives rise to the higher tendency to conjugation in the unsaturated species, the NAO-based density matrix of the EDDAðrÞ function, and hence r-bonding represents rather typical localized two-center bonds X in all three molecular systems. Again, the EDDBrðrÞ function reaf- ð Þ¼ vy ð ÞDDA v ð Þ; DDA ¼ S~ b þ S~ a; ð Þ EDDA r l r l;m m r f f 8 firms this fact setting the magnitude of r-delocalization up to l;m about 5% of total population of r-electrons. The corresponding total population of electrons delocalized The newly proposed ED-partitioning scheme can also be very between atoms in a molecule can be easily partitioned into atomic useful in rationalizing the electronic structure of polycyclic sys- contributions: tems with conjugated p-bonds. In particular, it is well-known that D.W. Szczepanik / Computational and Theoretical Chemistry 1080 (2016) 33–37 35

Fig. 1. Contours of the valence-shell electron density and its components visualized at isovalue s ¼ 0:015, and the corresponding electron populations obtained by integration of the appropriate function over the whole space. Method: B3LYP/6-311++G⁄⁄ (equilibrium geometries). the reactivity of anthracene is determined by its electron delocal- neglect or underestimate the bond-bond delocalization effects. ization pattern in the transition state, predicting the external rings Indeed, as depicted in Fig. 2, the cyclic delocalization patterns in to be more aromatic [28]. However, in the ground state the picture anthracene provided by such popular tools as the electron localiza- of p-delocalization in anthracene is not clear and only a few aro- tion function (ELF) [30], localized orbital localizator (LOL) [31] or maticity quantiﬁers support the ﬁnding based on the structural the laplasian of the electron density, seem to be topologically and magnetic criteria of aromaticity that the p-conjugation is more closer to the EDLB function. If one takes into account that the effective in the central ring [29]. It has recently been pointed out EDLB-component represents only about 35% of the total that the ‘‘anthracene problem” introduced by the electronic criteria p-electron population in anthracene, while the EDDB population of aromaticity may be a consequence of using quantities and tools is nearly twice bigger, it become quite obvious that the EDDB that favors the effect of electron delocalization between atoms and component is dominating and determines the local aromaticity of 36 D.W. Szczepanik / Computational and Theoretical Chemistry 1080 (2016) 33–37

functions, but requires the use of additional data files generated by the NBO software [32] and containing the matrix of linear coefficients for MOs in the NAO basis, as well as the AO ! NAO transformation matrix. By default, the program decomposes the one-electron density regarding delocalization effects between all atoms in a molecule. However, if required, one can specify particular molecular fragment, and even select individual bonds that the ED-partitioning should be restricted to. The foregoing results seem to be promising enough to legitimize further and more extensive studies, especially in the field of chemical aromaticity in excited states [33].

Acknowledgments

The author is very grateful to Prof. Tadeusz M. Krygowski, Prof. Miquel Solà and Dr. Marcin Andrzejak for multiple stimulating discussions. This research was supported in part by the Foundation for Polish Science (FNP START 2015 stipend), National Science Centre, Poland (NCN, Grant No. 2015/17/D/ST4/00558), as well as the PL- Grid Infrastructure, with the calculations performed on Zeus: HP Cluster Platform of the Academic Computer Centre CYFRONET.

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Contents lists available at ScienceDirect

Computational and Theoretical Chemistry

journal homepage: www.elsevier.com/locate/comptc

On the three-center orbital projection formalism within the electron density of delocalized bonds method

Dariusz W. Szczepanik

K. Gumin´ski Department of Theoretical Chemistry, Faculty of Chemistry, Jagiellonian University, Ingerdena 3, 30-060 Cracow, Poland article info abstract

Article history: A new development of the Electron Density of Delocalized Bonds formalism (EDDB) is proposed that pro- Received 21 November 2016 vides marked improvement in the description of electron delocalization in aromatic rings. Special atten- Received in revised form 1 December 2016 tion is paid to charged aromatic hydrocarbons of different size, for which the total population of electrons Accepted 1 December 2016 delocalized between adjacent bonds from the original formulation of the EDDB method signiﬁcantly Available online 2 December 2016 overestimates the multicenter p-electron sharing effects. The revised bond-orbital projecting scheme gives rise to systematic improvement of the results of the EDDB analysis, which now supports ﬁndings Keywords: by other researchers. Bond conjugation Ó 2016 Elsevier B.V. All rights reserved. Orbital projection Electron delocalization EDDB Aromaticity

1. Introduction quantities can be easily calculated for both planar and nonplanar molecular rings and do not suffer from the ring-size exten- The electron density of delocalized bonds (EDDB) [1,2] formal- sivity issues. Therefore, they can be successfully used for investiga- ism has recently been proposed for comprehensive description of tions of a wide range of aromatic species. It should be noted in electron delocalization in molecules with conjugated bonds. The passing that, apart from some very specific situations (e.g. [13]), EDDB approach provides both the overall picture of multicenter the EDDB scheme is able to strictly separate r; p and higher com- electron delocalization for the entire molecule and the detailed ponents of multicenter delocalization (even in the case of non- description of the delocalized electron populations in atomic reso- planar ring units) as well as probe their mutual interplay. lution for particular molecular fragment. Visualization of the elec- The EDDB formalism is rooted in the Molecular Orbital (MO) tron density of delocalized bonds facilitates quick detection of theory and, just like other familiar techniques, e.g. the Natural regions of increased aromaticity in very large systems, while the Bond Order (NBO) [14–16] or the Adaptive Natural Density Parti- detailed information on the delocalized populations enables quan- tioning (AdNDP) [17], it is based on the bond-order orbitals formal- titative studies of bond conjugation in a selected molecular frag- ism, which was originally introduced by Jug [18]. In contrast to ment [1–5]. other methods, however, the electron density of delocalized bonds The most important features setting the EDDB analysis apart is constructed only from two-atomic and three-atomic bond-order from other techniques of evaluation of multicenter delocalization orbitals, which considerably reduce computational cost and sim- are the non-referential character, intuitiveness and interpretative plifies the analysis of multicenter bonds in large molecular simplicity. Indeed, unlike with other quantum-chemical measures systems. of the multicenter electron sharing effects [6–10], the electron The originally proposed formulation of the EDDB analysis is population determined by the trace of the EDDB density matrix based on the approximate relationship between two- and three- represents a total number of electrons delocalized through the sys- center Bonding Orbitals (2cBO and 3cBO, respectively), which has tem of conjugated bonds [2]. Depending on the strategy chosen to recently been studied using some information-theoretic tools construct the electron density matrix of delocalized bonds, it can [19–23]. This approximation, however, may sometimes lead to possess local (only conjugations of selected bonds) or global (all quantitatively incorrect predictions of the number of electrons possible bond conjugations in a molecule) character. Also, unlike delocalized in molecular systems. A detailed investigation has other aromaticity descriptors (e.g. NICS [11,12]), the EDDB-based revealed that in some cases, especially for charged aromatic hydrocarbons (AH), the original EDDB scheme systematically overestimates the number of electrons assigned to conjugated bonds of E-mail address: [email protected] p URL: http://www.chemia.uj.edu.pl/~szczepad -type [4]. It is the main objective of this paper to introduce a http://dx.doi.org/10.1016/j.comptc.2016.12.003 2210-271X/Ó 2016 Elsevier B.V. All rights reserved. 14 D.W. Szczepanik / Computational and Theoretical Chemistry 1100 (2017) 13–17 modification to the orbital-projecting procedure that remedies this where, by definition, the corresponding LC coefficients read: DE DE issue and improves the EDDB analysis. Tab ¼ ~fb fb ; Tbc ¼ ~fb fb ð Þ abc ab;i abc;m abc bc;j abc;m 8 i;m j;m 2. Theoretical background In the crucial step of the EDDB matrix construction, the coefficients from (8) are used to determine the population of electrons origi- The electron density of delocalized bonds can be straightfor- nally assigned to particular chemical bond, e.g. Xa–Xb, and delocal- wardly extracted from a one-determinant wavefunction of molec- ized through the sequence of bonded atoms Xa–Xb–Xc: ular system obtained by means of standard methods of quantum X 2 2 chemistry. It involves the use of well atom-assigned basis func- d ab d Nabjabc ¼ Tabc kabc : ð9Þ tions, which can be the Natural Atomic Orbitals (NAO) [24,25], i;m m;m no i;m vlðrÞ , or any other representation including the effective d Here, the diagonal matrix kabc collects populations of multicenter minimal-basis subspace of orthonormal atomic orbitals, e.g. [26– delocalized electrons assigned to each 3cBO; more details about 33]. Then, the one-electron EDDB function can be expressed as d XX the definition of kabc can be found in [5]. ð Þ¼ v ð ÞyDd v ð Þ; Nd ¼ Dd: ð Þ EDDB r l r l;m m r tr 1 As indicated in the preceding section, the problem with the l m 3cBO ! 2cBO projection scheme (3)–(9) lies in the fact that it com- Here, Nd stands for the total population of electrons delocalized in pletely neglects the through-space (direct) interaction between d terminal atoms in a system of two conjugated bonds. A simple all conjugated bonds in a molecule while the density matrix Dl;m is solution to this issue is to define the three-center bonding orbitals defined as a sum of ‘‘density layers” representing all possible inter- by solving the eigenproblem of the 3 3 Jug’s matrix that implic- actions between atoms in pairs: XX itly includes the interaction between Xa and Xc, i.e. d d 0 1 D ¼ Bab; ð2Þ 0Da;b Da;c a b–a B C E B y C b Da;b 0Db;c ! n : ð10Þ Bd @ A abc where ab represents this part of the electron density of the chem- y y Da;c Db;c 0 ical bond Xa–Xb that is delocalized in a multicenter sense with all Bd other bonds in molecular system. The construction of ab is a com- For the resulting three-center bonding orbitals, however, the plex task falling outside the framework of this work and it has been appropriately orthogonalized two-center bonding orbitals (5) do comprehensively discussed in two recent papers [2,5]. What should not form a complete basis set. Therefore, the improved be emphasized at this point is that the determination of each layer 3cBO ! 2cBO projection scheme must involve an additional subset of the density E matrix (2) involves the use of three-center bonding of orbitals obtained by solving the eigenproblem of the Jug’s matrix orbitals fb , representing the Xa–Xc bonding: abc ! E E 0Da;c b b ! ~fb : ð Þ fabc ¼jviCabc; ð3Þ y ac 11 Da;c 0 obtained by solving the eigenproblem of the following matrix, 0 1 Then, according to Eq. (6), each 3cBO can be regarded as a linear 0Da;b 0 combination (LC) of orthogonalized 2cBO from three different B C B y C y bonds (either covalent or noncovalent), Xa–Xb,Xb–Xc and Xa–Xc, Dabc ¼ @ Da;b 0Db;c A ¼ CabckabcCabc; ð4Þ respectively, y E E E 0Db;c 0 X X nb ¼ ~fb ab þ ~fb bc abc;m ab;i Tabc bc;j Tabc i;m j;m and selecting the subspace of eigenvectors associated only with i j X E positive eigenvalues [2]; here the elements of the matrix Dabc rep- þ ~fb ac ; ð Þ ac;j Tabc 12 resent the off-diagonal atomic blocks of the one-electron density k;m k b a matrix, and Cabc ¼ðCabcjCabcÞ, where superscripts b and a denotes subspaces of bonding and anti-bonding orbitals, respectively. The Here, the new 3cBO-scheme involves the LC coefficients given by the following direct projections: resulting three-center bonding orbitals (3) are then projected onto DE DE c the set of appropriatelyE E orthogonalized [34,35] two-center bonding ab ¼ ~fb nb ; b ¼ ~fb nb ; Tabc ab;i abc;m Tabc bc;j abc;m ~b ~b i;m j;m orbitals, fab and fbc , determined by solving the corresponding DE eigenproblems of the classical 2 2 Jug’s matrices [18] (in the same ac ¼ ~fb nb : ð Þ Tabc ac;k abc;m 13 way as in the preceding equations), i.e. k;m ! ! E E 0Da;b 0Db;c with the normalization condition ! ~fb ; ! ~fb : ð Þ y ab y bc 5 X X X Da;b 0 Db;c 0 ab 2 bc 2 ac 2 Tabc þ Tabc ¼ 1 Tabc : ð14Þ i;m j;m k;m Therefore, each 3cBO can be regarded as a linear combination of i j k orthogonal 2cBOs, It follows directly from (14) that the population of electrons origi- E E E X X fb ¼ ~fb Tab þ ~fb Tbc ; ð Þ nally assigned to the Xa–Xb and delocalized through the atomic tri- abc;m ab;i abc bc;j abc 6 i;m j;m plet Xa–Xb–Xc, calculated using the modified 3cBO-scheme (10)– i j (13), is usually lower than the corresponding electron population with the following normalization condition: from Eq. (9): XX ab 2 bc 2 X 2 2 d ab d0 d Tabc þ Tabc ¼ 1; ð7Þ ¼ k 6 N : ð Þ ; ; Nabjabc Tabc abc abjabc 15 i m j m i;m m;m i j i;m D.W. Szczepanik / Computational and Theoretical Chemistry 1100 (2017) 13–17 15

d0 Again, the diagonal matrix kabc collects the orbital populations of projecting schemes seems to increase monotonically as the size multicenter delocalized electrons assigned to each bonding orbital of molecular ring is increased. Furthermore, for the majority of spe- (10) and calculated using the procedure of canceling of non- cies, total populations of delocalized electrons calculated using the bonding and mutually phase-reversed 3cBOs, introduced in the original 3cBO-scheme (yellow bars) have substantially greater val- recent paper [5]. ues than the corresponding populations from the improved EDDB analysis (blue bars). A more detailed analysis reveals that for all the molecular systems containing one p-bond as well as for species þ 2þ 4þ Nd 3. Results and conclusions C7H7 ,C8H8 and C8H8 , the p based on the originally proposed 3cBO-scheme significantly exceeds the total number of p- 6þ 2þ To demonstrate how the reformulated 3cBO orbital-projecting electrons (red bars)! Admittedly, in the cases of C8H8 and C8H8 scheme improves the results of the EDDB analysis, let us consider the corresponding populations obtained from the improved EDDB a test set of 17 model cyclic hydrocarbons containing from three to analysis also exceed the respective formal numbers, but only by eight carbon atoms and up to three p-bonds. The EDDB density no more than 0:1%, which is actually on the very edge of the matrices (2) for each molecular system were constructed by con- assumed accuracy limit (see [5]). On the other hand, comparison sidering all possible conjugations of chemical bonds between car- of the EDDB populations from both 3cBO projection schemes with bon atoms (including the weak cross-ring carbon–carbon the corresponding total Wiberg-type bond orders [38–41] (green interactions). The values of p-component of Nd were obtained lines), calculated at the same theory level, indicates that the over- by solving the corresponding eigenproblem for Dd [2] at the estimations may come rather from the weakness of the one- B3LYP/6-31G⁄ theory level using two computational-chemistry determinant definition of the quadratic bond-order itself, especially with regard to the cross-ring interactions involving carbon tools, GAMESS v5.1.2013 [36] and NBO v6.0 [37], as well as several p p scripts developed by the author. The calculated EDDB-populations atoms in meta positions [4]. Indeed, for aromatic 2 - and 6 - from both 3cBO-schemes are depicted in Fig. 1. systems the total bond-orders are generally in line with the corre- At first glance, in almost all Hückel’s aromatic systems the sponding EDDB-populations from the original formulation. The effectiveness of p-electron delocalization predicted by both 3cBO only exception is the benzene molecule, for which the magnitude of the meta-CC interaction is well-known to be very small [6,42]. This observation allows one to draw the conclusion that, in accordance with (15), both 3cBO-schemes provide comparable results only if the interaction between terminal atoms in each possible pair of adjacent conjugated bonds can be neglected, which is not the case for highly ionized rings, regardless of their size [4]. Interestingly, the use of the improved EDDB scheme shows clearly that, for the Hückel’s aromatic systems, the effectiveness of bond conjugation increases logarithmically as the size of molecular ring is increased, while in the case of Hückel’s antiaromatics it displays

an exponential relationship, but only after excluding the C4H4 molecule. Due to symmetry of both occupied molecular p- orbitals, in the case of cyclobutadiene both EDDB formulations properly predict the lack of any delocalized bonding. As a matter of fact, this archetypical antiaromatic molecule at its equilibrium geometry does not exhibit neither p- nor r-delocalization and its extreme destabilization energy has been proved to be caused by a combination of angle strain, torsional strain, and Pauli repulsion [43–45]. It has to be stressed once again that for benzenoid-like systems containing three molecular orbitals of p-type the resulting effectiveness of p-delocalization is nearly the same regardless of the version of the 3cBO-scheme used in the EDDB deﬁnition. Indeed, for the test set of 26 six-membered polycyclic aromatic hydrocarbons (PAH), listed in Table 1 and containing 82 unique benzenoid

Table 1 The list of 26 polycyclic aromatic hydrocarbons.

No Molecule No Molecule 1 Benzene 14 1,2–3,4–5,6–7,8-Tetrabenzoanthracene 2 Naphthalene 15 1,2–4,5-Dibenzopyrene 3 Anthracene 16 1,2-Benzopyrene 4 Tetracene 17 1,2-Tetraphene 5 Pentacene 18 1,2,3,4-Dibenzanthracene 6 Hexacene 19 1,2–6,7-Dibenzopyrene 7 Heptacene 20 1,2,5,6-Dibenzanthracene Fig. 1. Total populations of p-electrons delocalized through a system of conjugated d 8 Phenanthrene 21 3,4-Benzopyrene bonds, Np, obtained from both the original EDDB method (yellow bars) and the 9 Pyrene 22 1,2,7,8-Dibenzanthracene improved one (blue bars) for model monocyclic hydrocarbons and their ions. For 10 Chrysene 23 1,2-Benzotetracene comparison, the sums of the Wiberg-type bond orders has been added (green solid ⁄ 11 Triphenylene 24 Picene lines). Method: B3LYP/6-31G , equilibrium geometries. (For interpretation of the 12 Pentaphene 25 2,3,7,8-Dibenzophenanthrene references to color in this figure legend, the reader is referred to the web version of 13 Coronene 26 2,3,5,6-Dibenzophenanthrene this article.) 16 D.W. Szczepanik / Computational and Theoretical Chemistry 1100 (2017) 13–17 units, the improvement of the 3cBO-scheme has no noticeable effect on the results of the EDDB analysis (B3LYP/6-31G⁄), as presented in Fig. 3a. That is to say, due to relatively weak cross-ring interactions between carbon atoms in meta positions, total populations of electrons delocalized in each benzenoid-like unit of particular PAH calculated using the modified EDDB formalism take only slightly greater values than those obtained using the original one, but the correlation between them is still very tight (R2 ¼ 0:99). In contrast, for non-benzenoid cyclic systems the EDDB-populations based on the original 3cBO-scheme tend to be somewhat overestimated, regardless of the number of p bonds in a molecular ring. Consequently, one should avoid comparing the EDDB-derived total populations for aromatic rings of different size when the previous 3cBO projection procedure is used. For demonstration purposes, let us consider the results of the EDDB analysis (at the CAM-B3LYP/ def2-TZVPP theory level) for the test set of 25 monocyclic species from Fig. 2, which includes mainly five- and seven-membered molecular rings (this test set comes from the original work [46]). Although the correlation between total populations of delocalized electrons from both 3cBO-schemes is maybe not the worst (R2 0:81), as shown in Fig. 3b, the use of the original formulation of the EDDB clearly fails to correctly describe the effectiveness of bond conjugation simultaneously in five- (points over the trendline) and seven-membered rings (points under the trendline).

Fig. 3. Correlation analyses between total EDDB-populations calculated using the original 3cBO ! 2cBO projection scheme and the newly proposed one, for (a) the 4. Summary test set of 82 symmetry-unique benzenoids from molecules listed in Table 1 and (b) the test set of 25 monocyclic species presented in Fig. 2. Methods: (a) B3LYP/6-31G⁄ In this paper a new development of the electron density of delo- and (b) CAM-B3LYP/def2-TZVPP, equilibrium geometries. calized bonds formalism has been proposed to remedy some serious deficiencies emerging within its original formulation [2] and It has to be mentioned that very recently an alternative scenario regarding the description of electron delocalization in aromatic of removing the electron overcounting in the EDDB method has rings. It has been demonstrated that taking explicitly into account been proposed in which additional projection onto the subspace all the trough-space interactions within the definition of three- of occupied molecular orbitals is used to eradicate non- center bond orbitals gives rise to systematic improvement of the orthogonalities in the EDDB density matrix from (2) [47]. For most results of the EDDB analysis for aromatic rings in which the of organic aromatics this particular technique gives very similar cross-ring interactions tend to play a vital role. In particular, the results as presented in this work. However, in some difficult cases EDDB-populations from the reformulated formalism do not exceed of multicenter bonds (e.g. small all-metal clusters in triplet states) the total number of p-electrons in a molecular system, which was it systematically underestimates the populations of electrons delo- sometimes observed for the corresponding populations from the calized between adjacent bonds. Since this ‘undercounting’ is prob- previous formulation of the theory. ably caused with neglecting the through-space interactions between terminal atoms in the definition (4), we are planning to combine both scenarios in the next implementation of the EDDB method.

Acknowledgments

The author is very grateful to Prof. Miquel Solà, Prof. Tadeusz M. Krygowski, and Prof. Halina Szatyłowicz for support and multiple stimulating discussions. This research was supported in part by the Foundation for Polish Science (FNP START 2015, stipend 103.2015), National Science Centre, Poland (NCN SONATA, grant 2015/17/D/ST4/00558), as well as the PL-Grid Infrastructure, with the calculations performed on Zeus: HP Cluster Platform of the Academic Computer Centre CYFRONET.

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The Role of the Long-Range Exchange Corrections in the Description of Electron Delocalization in Aromatic Species Dariusz W. Szczepanik ,*[a] Miquel Sola ,[b] Marcin Andrzejak,[a] Barbara Pawełek,[c] Justyna Dominikowska,[d] Mercedes Kukułka,[a] Karol Dyduch,[a] Tadeusz M. Krygowski,[e] and Halina Szatylowicz[f]

In this article, we address the role of the long-range exchange presented results indicate that the noncorrected exchange- corrections in description of the cyclic delocalization of elec- correlation functionals significantly overestimate cyclic delocali- trons in aromatic systems at the density functional theory zation of electrons in heteroaromatics and aromatic systems level. A test set of diversified monocyclic and polycyclic aro- with fused rings, which in the case of acenes leads to conflict- matics is used in benchmark calculations involving various ing local aromaticity predictions from different criteria. VC 2017 exchange-correlation functionals. A special emphasis is given Wiley Periodicals, Inc. to the problem of local aromaticity in acenes, which has been a subject of long-standing debate in the literature. The DOI: 10.1002/jcc.24805

Introduction calculations.[34] Hartree–Fock (HF) calculations systematically overestimate the charge transfer between bonded atoms due Quantification of the electron delocalization is crucial for ratio- to lack of Coulomb electron correlation in the one-electron nalizing electronic structure and molecular properties within density.[11,34] The calculation of delocalization indices at the the language of such intuitive terms as partial atomic charge, correlated post-HF levels of theory like the configuration inter- chemical valence, covalency and ionicity, bond multiplicity, action method is rarely performed because it requires corre- and so forth[1–6] A multitude of different electron delocaliza- lated density matrices of higher-orders, which are hardly ever tion descriptors can be found in the literature, the majority of available in computational chemistry programs. Instead, the which are based on the coordinate-space or the Hilbert-space partitioning of the electron density (ED) from ab initio calcula- [a] D. W. Szczepanik, M. Andrzejak, M. Kukułka, K. Dyduch tions. In this context, such quantities as the Wiberg’s,[7,8] Faculty of Chemistry, K. Guminski Department of Theoretical Chemistry, Mayer’s,[9] and the fuzzy-atom bond orders,[10] Bader’s diatomic Jagiellonian University, Ingardena 3, Cracow 30-060, Poland E-mail: [email protected] delocalization index,[11] and Fulton’s electron-sharing index[12] [b] M. Sola deserve special attention owing to their reliability and univer- Institut de Quımica Computational i Catalisi, Universitat de Girona, Maria sality.[6,13] Using these descriptors is not limited only to typical Aure`lia Capmany 6, Girona 17003, Catalonia, Spain two-center bonds. They can easily be extended to cover also [c] B. Pawełek Institute of Botany, Department of Plant Cytology and Embryology, the multicenter electron sharing effects (like the three-center Jagiellonian University, Gronostajowa 9, Cracow 30-387, Poland A A two-electron B H B bond in diborane) as well as the cyclic [d] J. Dominikowska delocalization of electrons in aromatic molecules.[14] In rela- Faculty of Chemistry, Department of Theoretical and Structural Chemistry, tively small rings, the cyclic delocalization of electrons can be University of Lodz, Pomorska 163/165, Lodz 90-236, Poland quantified directly by the so called multicenter indices [e] T. M. Krygowski Faculty of Chemistry, Department of Theoretical Chemistry and [15–19] (MCI). However, in cases of larger molecular circuits calcu- Crystallography, University of Warsaw, Pasteura 1, Warsaw 02-093, Poland lations, MCI is computationally inefficient and alternative mea- [f] H. Szatylowicz sures of cyclic delocalization can be used instead, such as the Faculty of Chemistry, Department of Physical Chemistry, Warsaw University of Technology, Noakowskiego 3, Warsaw 00-664, Poland AV1245 index[20] and the recently proposed electron density of [21–23] Contract grant sponsor: Foundation for Polish Science; Contract grant delocalized bonds (EDDB). Other popular aromaticity number: FNP START 2015, stipend 103.2015 (to D.S.); Contract grant descriptors involving delocalization indices include the para- sponsor: National Science Centre, Poland (NCN SONATA); Contract grant delocalization index (PDI),[24,25] and the fluctuation index of number: 2015/17/D/ST4/00558 (to D.S.); Contract grant sponsor: PL-Grid Infrastructure of the Academic Computer Centre CYFRONET, with the aromaticity (FLU).[25,26] In fact, these indices are often superior calculations performed on cluster platforms Zeus and Prometheus; to other “ground-state” (structural and energetic) as well as Contract grant sponsor: Ministerio de Economıa y Competitividad of the “response” (magnetic) criteria of aromaticity,[27–29] espe- Spain (Project CTQ2014-54306-P to M.S.); Contract grant sponsor: cially for heteroaromatic systems, chemical reactions, and the Generalitat de Catalunya (project number 2014SGR931, Xarxa de [30–33] Refere`ncia en Quımica Teorica i Computacional, and ICREA Academia all-metal and semimetal clusters. prize); Contract grant sponsor: European Fund for Regional In general, the description of electron delocalization Development; Contract grant number: UNGI10-4E-801 depends on the level of the theory used in quantum-chemical VC 2017 Wiley Periodicals, Inc.

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one-determinant HF-type formalism is routinely used with the 1 12erfðlr12Þ erfðlr12Þ 5 1 ; (1) Kohn–Sham (KS) orbitals at the density functional theory (DFT) r12 r12 r12 level; such DFT-based delocalization descriptors have been widely reported to be convenient and efficient tools, which where r12 5 |r1 – r2|, while l determines the extent of the provide useful chemical insight.[34–38] Although in DFT even short-range region. Applying the standard HF exchange inte- the first-order density matrix is not strictly defined, in many sit- gral to the second term in eq. (1) he got the following expres- uations it can be reasonably well represented by the Slater sion for the long-range exchange interaction energy: determinant obtained using approximations from the upper ðð X Xocc 1 † † rungs of the DFT Jacob’s ladder (the resulting KS-MOs to some ELR52 w ðr Þw ðr Þ x 2 ir 1 jr 2 extent have physical meaning, so they can be used in HF-like r i;j (2) expressions).[39,40] However, since the electron-pair density is erfðlr12Þ 3 3 not currently available in DFT, the instantaneous Coulomb 3 wjrðr1Þwirðr2Þd r1d r2; r12 electron–electron correlation as well as the correlation between electrons of different spin cannot be taken into where Wir represents the ith KS r-spinorbital. In turn, the LDA account,[41] which results in an overestimation of the electron exchange functional was combined with the first term in eq. sharing effects between covalently bonded atoms.[34,42] Fur- (1) giving rise to the following expression for the short-range thermore, description of the electronic structure at the DFT exchange interaction energy: level of theory faces yet another obstacle: the incorrect long- ð P 4=3n h range behavior due to the local character of the approximated SR 8 1=2 1 [43] Ex;LDA5c r qrðrÞ 12 ar p erf exchange-correlation (XC) functional. For a long time this 3 2ar issue has been regarded as the principal cause of failure of the (3) io KS-DFT approach in terms of the reproduction of such molecu- 1 [44] 3 3 3 lar properties as barrier heights and reaction enthalpies, 1ð2ar24arÞexp 2 23ar14ar d r; 4ar excitation energies,[45] van der Waals interactions,[46] nonlinear optical response properties,[47] ionization potentials,[48] and so where forth. Over the last years a great progress has been made to remedy these issues by the development of the long-range 3 3 1=3 l3 1=3 c52 and a 5 : (4) corrected exchange-correlation functionals (LC-XC), in which r 2 2 4p 48p qrðrÞ the exchange part of XC is separated into short-range (DFT exchange) and long-range (further referred to as the HF In the above equation qr(r) is the r-spin electron density. The exchange) parts using a standard error function.[49–53] corresponding LC scheme for XC functionals from the general Although the local character of the exchange part of the con- gradient approximation (GGA) was developed by Iikura and ventional XC functionals has not been found to affect the coworkers.[51] They modified the original Savin’s scheme by description of electron delocalization between two covalently putting in the averaged relative Fermi momentum included in bonded atoms, in aromatic rings, in which p-electrons are eq. (4) the gradient terms of the GGA functional. The resulting cyclically delocalized through the net of conjugated bonds LC-GGA short-range exchange interaction energy reads over quite large distances, utilizing the LC-XC functionals ð [54,55] X 4=3 n h might be of vital importance. SR GGA 8 1=2 1 Ex;GGA5c qrðrÞ Kr 12 ar p erf In this article, we address the performance of the LC-XC r 3 2ar functionals in accounting for electron delocalization in selected io 3 1 3 3 aromatic and heteroaromatic rings as well as polycyclic aro- 1ð2ar24arÞexp 2 23ar14ar d r; 4ar matic hydrocarbons. Since there has been a lot of controversy regarding the problem of local aromaticity in acenes (“the where anthracene problem”),[56–59] in this study, we take a closer look at the effect of long-range exchange corrections on the l2KGGA 1=2 a 5 r q ðrÞ21=3: (6) description of cyclic delocalization of electrons in these r 36p r species. GGA Here the GGA exchange-correlation kernel Kr is determined by a particular exchange functional used such that Methodology ð 4=3 GGA 3 Long-range correction schemes Ex;GGA5 qrðrÞ Kr d r: (7)

The long-range correction scheme for XC functionals within The ratio between the short-range and long-range terms in the local density approximation (LDA) was originally proposed eq. (1), l 5 0.33, was determined to reproduce bond distances by Savin.[49] He made use of the standard error function erf to of homonuclear diatomics up to the third period.[51] separate the two-electron operator into the short-range (SR) Over the last decade increasing attention has been paid to and long-range (LR) parts: develop the LC schemes for hybrid density functionals, which

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are the most widely used due to their proven performance for consequences of deviations of bond lengths {Ra} in the n- thermochemistry, kinetics, noncovalent interactions, and so membered aromatic ring from the corresponding optimum [60,61] o forth. One of the most popular LC-XC hybrids is CAM- values fRag for an idealized aromatic system: B3LYP proposed by Yanai and coworkers, who modified the Xn two-electron operator partitioning scheme from eq. (1) using 1 o 2 [62] HOMA512 kaðRa2RaÞ ; 21 < HOMA 1; (8) the Coulomb-attenuating method (CAM). By combining the n a51 CAM potential with the B3LYP hybrid[63] they found the opti-

mum portions of the HF exchange to amount to 19% at the where ka stands for the geometry-to-energy conversion factor. short-range and about 65% at long-range (the intermediate The HOMA index varies from close to 1 for highly aromatic region is smoothly described through the standard error func- rings, through 0 for nonaromatic rings to negative values for tion with l 5 0.33). Later Head-Gordon and coworkers came potentially antiaromatic rings. up with their xB97x hybrid[60] by systematic optimization of Becke’s B97 functional.[64] They found that retaining 100% of HRCP. The density of total electron energy at the ring critical the Hartree–Fock exchange for long-range electron–electron [76] point (HRCP) is one of the atoms-in-molecule (AIM) parame- interactions reduces to a large extent the self-interaction prob- ters that has been proven to serve as a quantitative measure lems associated with global hybrid functionals; at short-range of aromaticity in a large group of molecular rings.[77] For they used the portion of 16% HF exchange with the range- example, for a set of 33 phenolic rings of varying aromaticity, separation ratio equals 0.30. In this study the long-range cor- and a set of 20 quasi-rings formed by intramolecular hydrogen rected hybrids CAM-B3LYP, xB97x, and xB97xD (variant with and lithium bonds, the correlation with HOMA and NICS was [61] Grimme’s D2 dispersion model) are used together with the 2 2 [76] found to be very tight: R (HOMA)50.99, R (NICS(1)zz)50.95. LC-XC GGA functionals from the LC-scheme by Iikura and One should bear in mind, however, that for more diversified coworkers. sets of molecular rings such a tight correlation between NICS and topological characteristics of RCPs may no longer hold Aromaticity indices true.[78] Moreover, while collecting data for this study, however, The cyclic delocalization of p-electrons gives rise to unusual we have found that the density of total electron energy at 1 stability and characteristic structural (reduced bond length angstrom above the RCP provides even more reliable descrip- alternation) and magnetic (induced ring current in external tion of aromaticity in monocyclic and polycyclic species than

magnetic field) properties of aromatic species. Thus, apart the original index. In particular, standard deviations of HRCP(1)

from the electronic measures of aromaticity based on delocali- are usually smaller by an order of magnitude than HRCP(0); zation indices, in this article, we also investigate the influence interestingly, similar observations we have made regarding of long-range corrections on the aromaticity predictions from NICS(0) and NICS(1) but in this case differences between the structural and magnetic criteria. A brief outline of the descrip- corresponding standard deviations are much less marked—see tors utilized in the present study is given below. Please note Tables S2–28 in the Supporting Information. Therefore, in the that all the provided definitions of electronic measures of aro- present study both variants of this descriptor are used. maticity apply to the closed-shell systems only, although analogous expressions exist for open-shell species. EL. EL[79] is an aromaticity index based on bonding electron NICS. The nucleus-independent chemical shift (NICS)[65–67] density deformations, reflected in bond ellipticities. Bond ellip- quantifies diatropicity/paratropicity of the induced ring current ticity (e) is defined as follows: by means of the effective magnetic shielding measured usually k at the centroid of aromatic ring in the external magnetic field, e5 1 21; (9) NICS(0), or x angstroms above/below it, NICS(x). The axial com- k2 ponent of the nucleus-independent chemical shift 1 angstrom where k1 and k2 are the two negative eigenvalues of the Hes- above the ring centroid, NICS(1) is one of the most popular zz sian matrix of the electronic density in the bond critical point measures of local p-aromaticity since it usually gives aromatic- (BCP). EL is given as follows: ity predictions consistent with results based on other criteria. The more negative value of NICS, the more aromatic is the c Xn EL512 je 5e j: (10) molecular ring in question. Contrariwise, highly positive values n i ref of NICS usually indicate antiaromatic species. Although aroma- i51 ticity descriptors based on magnetic properties of molecules In the above equation c is a normalization constant, n is the are probably the most preferable measures of aromaticity due number of bonds forming the ring, ei is the ellipticity of the to its relation to experiment (diatropic and paratropic ring cur- ith bond, and eref is the ellipticity of the CC bond in benzene rents indirectly manifest itself in the NMR spectra), they have (computed at the same level of theory as the studied system). also come under increasing criticism.[27,68–71] EL equals 1 for a fully aromatic system, it is close to 0 for non- HOMA. The harmonic oscillator model of aromaticity aromatic species and negative for potentially antiaromatic (HOMA)[72–75] is a normalized measure of the energetic ones.

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FLU. The aromatic fluctuation index (FLU)[25,26] is a measure study, PDI refers by default to the aromaticity index defined of uniformity of the kekulean mode of electron delocalization within the FAS-partitioning scheme, while PDI(AO) and PDI(- and its bonding difference with respect to an idealized aro- NAO) denote the corresponding average two-center indices. matic system. It is defined as follows: KMCI. The KMCI variant of the multicenter index measures Xn j 2 directly the kekulean mode of the multicenter electron sharing dðA Þ dðA ; A Þ FLU5n21 a a a21 21 ; (11) in aromatic ring.[14–19] Within the FAS representation it is calcu- dðA Þ d ðA ; A Þ a51 a21 ref a a21 lated as follows: where the localization and diatomic delocalization indices read Xocc n21 KMCI52 Si1i2 ðA1ÞSi2i3 ðA2ÞSini1 ðAnÞ; (16) X Xocc i1;i2;...;in dðAaÞ5 dðAa; AbÞ; dðAa; AbÞ54 SijðAaÞSijðAbÞ; (12) b6¼a i;j thus being a generalization of the diatomic delocalization index from EQ. (12). The multicenter delocalization index, con- respectively. Here, the summation runs over all (i,j)-pairs of generic with KMCI but defined within the framework of the occupied molecular orbitals (MO) and Sij(Aa) represents the Hilbert-space partitioning scheme, reads corresponding MO-projections integrated within the basin of atom A . The j index in eq. (11) ascertains that the first term XA1 XA2 XAn a KMCIðAO=NAOÞ52n21 M M M : (17) is never lower than 1, that is k1;k2 k2;k3 kn;k1 k1 k2 kn ( 1ifdðAaÞ > dðAa21Þ; In the literature, the multicenter index obtained from the j5 (13) 21ifdðAaÞdðAa21Þ: coordinate-space ED-partitioning schemes is usually referred to as IR. However, for the sake of clarity, in this article, we use the Atomic overlap integrals, Sij(Aa), of eq. (12) were computed KMCI label for all three variants of the index, specifying in within the fuzzy-atom space (FAS)[10] partitioning scheme. By brackets the particular ED-partitioning formalism used (if not definition, FLU is near zero for strongly aromatic systems and specified, KMCI refers to the FAS-based version by default, just increases as the molecule departs from the reference (idealized) like in the case of PDI). aromatic system. It has been demonstrated elsewhere that the EDDB. The electron density of delocalized bonds (EDDB)[21–23] square root of aromaticity fluctuation index usually displays bet- is a part of the recently proposed method of partitioning the ter correlation with other “ground-state” and the “response” one-electron density into components representing different aromaticity descriptors.[31] However, to make the trends of FLU levels of electron delocalization.[23] The EDDB formalism makes changes comparable with all the other indices in this study, the use of the recently developed bond-orbital projection tech- FLU21/2 has been used in most cases of less aromatic rings. nique[82–86] to approximate the multicenter electron sharing effects by means of the three-atomic local resonances repre- PDI senting conjugations between the adjacent bonds only. Conse- The para-delocalization index (PDI)[24,25] measures the average quently, the EDDB method is computationally far more effect of electron delocalization between three para-related efficient than the multicenter indices. In the basis of natural atoms in a benzenoid unit. For a molecular ring given by the atomic orbitals (or, formally, in any other representation of [87,88] sequence of bonded atoms A1, A2, ..., A6 PDI reads orthonormalized atomic orbitals), {vl(r)}, the electron density of delocalized bonds is defined as 1 X PDI5 ½dðA1; A4Þ1dðA2; A5Þ1dðA3; A6Þ : (14) † DB 3 EDDBðrÞ5 vlðrÞDlm vmðrÞ; (18) l;m The average two-center index (ATI)[16] has the definition very similar to PDI but instead of diatomic delocalization indices, where the EDDB metric in the basis of NAOs reads "# d(a,b), it involves the quantum-mechanical bond orders, Ba,b, XX 1 † defined as DDB5 P C e k2 C P: (19) 2 ab ab ab ab X X a;b6¼a Ba;b52 ðPSÞk;lðPSÞl;k: (15) k2a l2b In the above equation P stands for the CBO matrix, Cab is a matrix of linear-combination coefficients of the appropriately [84,87] [83] In the above equation P and S stand for the charge and orthogonalized two-center bond orbitals (2cBO), kab is bond-order matrix (CBO) and the overlap matrix.[80] Within the a diagonal matrix of the corresponding 2cBO occupation num- representation of nonorthogonal atomic orbitals eq. (15) gives bers, and eab represents a diagonal matrix of the bond- rise to the bond order definition by Mayer,[9] while within the conjugation factors.[22] For ideally localized (two-center) bond [81] basis of natural atomic orbitals (NAO), for which S is an Aa–Ab eab is a null matrix, while in the case of delocalized (mul- identity matrix, eq. (15) leads to the quadratic bond-order con- ticenter) bonds there is at least one diagonal element in eab [7] generic with the one originally proposed by Wiberg. In this close to 1. Formal definition of the eab matrix is deeply rooted

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in the formalism of the orbital communication theory by NBO6 software[107] and the script program written by one of Nalewajski[89–91] and as such it falls outside the framework of the authors (DS).[108] To calculate aromaticity index EL the this brief description. It should be noted in passing, however, analysis of electron density distribution within the framework that the trace of the EDDB metric matrix from eq. (19) deter- of quantum theory of atoms in molecules (QTAIM)[109] was per- mines the population of electrons delocalized through the sys- formed using AIMAll program.[110] tem of conjugated bonds denoted by X. Two variants of the EDDB function are used in this study: EDDBk(r) and EDDBg(r). The first one describes cyclic delocalization of electrons due to Results and Discussion the resonance of the kekulean forms; for example, for a 5- Energetic and structural properties membered ring (5-MR) we have X 5 {(A1–A2),(A2–A3),(A3– k At first we investigated some structural and energetic charac- A4),(A4–A5), (A5–A1)}. The resulting EDDB population does not account for the cross-ring electron delocalization and as such teristics of molecules with cyclically delocalized electrons can be regarded as a counterpart of KMCI. The EDDBg(r) func- obtained at the KS-DFT theory level using all the benchmarked tion, in turn, considers conjugations between all the chemical exchange-correlation functionals. In particular, we calculated bonds in a molecule, and, as such, it can be used to evaluate bond lengths at equilibrium geometries and compared them global aromatic stabilization in large molecular systems[22] or, to the corresponding experimental values. The complete for example, to study the nonlocal resonance effects in conju- results with references to experimental data can be found in gated aromatic rings (in both ground- and excited states).[92] Supporting Information—see Figure S1 and Table S1; here Fig- ure 1 summarizes the results only for a few selected systems Computational details by displaying experimental (black numbers) and the DFT- calculated bond lengths averaged over both groups of func- The test set of molecules used in the benchmark KS-DFT calcula- tionals, XC (red numbers) and LC-XC (blue numbers). Green tions was designed to contain aromatic rings diversified with numbers in brackets refer to the average bond lengths from respect to: ring size and the number of p-electrons— calculations involving only the long-range exchange corrected cyclopropenyl cation (3-MR), cyclopentadienyl anion (5-MR), ben- hybrids: CAM-B3LYP, xB97x, and xB97xD. In turn, numbers zene (6-MR), cycloheptatrienyl cation (7-MR), cyclooctatetraenyl inside rings provides the corresponding percentage errors of dication (8-MR), cyclononatetraenyl anion (9-MR); type of conjuga- the calculated bond lengths considering each aromatic ring tion with heteroatom—pyridine and s-triazine (resonance of p- separately. Next, the benchmark calculations of vertical ioniza- bonds), pyrrole, furan, and thiophene (resonance of p-bonds and tion potentials (IP ) obtained from the DFT-Koopmans theelectronpaironheteroatom),borazine(resonanceofp-bonds vert approximation[111] was performed for all the aromatic systems, with some contribution of the ionic form with negative and posi- for which the respective experimental data are available.[112] tive charges on nitrogen and boron, respectively); topology of ring condensation—phenanthrene, chrysene, and picene (phena- The calculated percentage errors of IPvert for benzene, bora- cenes), naphthalene, anthracene, tetracene, and pentacene zine, thiophene, and pentacene are presented in Figure 2, (acenes). The DFT calculations with full geometry optimizations while the results for other systems are collected in Figure S2 were performed using Gaussian 09.[93] The study covered 10 XC in Supporting Information. Finally, we compared the Kohn– and 10 LC-XC functionals of different types, as listed below: Sham p-MO isocontours determined using BLYP and its long- range corrected counterpart in pentacene. The results are XC functionals: B3LYP,[63] BLYP,[94,95] BP86,[94,96] M06L,[97] depicted in Figure 3. PBE0,[98] PW91,[99] SVWN,[100,101] tHCTH,[102] TPSS[103], Although the long-range exchange corrections have been TPSSh;[103] successful in predicting excited state properties and chemical LC-XC functionals: CAM-B3LYP,[62] LC-BLYP, LC-BP86, LC- reactivity, so far they have not been reported to be superior M06L, LC-PW91, LC-tHCTH, LC-TPSS, LC-xPBE,[104] to the conventional XC functionals with respect to molecular xB97x,[60] xB97xD.[61] structure calculations. In fact, such popular GGA functionals as PBE or the hybrid B3LYP one are still regarded as methods of The standard cc-pVTZ basis set by Dunning[105] was used in the first choice for geometry optimizations of the most of all the computations. The resulting equilibrium geometries are organic species despite their incorrect asymptotic behavior in [113,114] included in Supporting Information. The NICS(0), NICS(1), and the long-range region. The results presented in Figure 1

NICS(1)zz values for all molecular rings were determined using support this fact but only with respect to the monocyclic aro-

Gaussian 09 (keyword NMR). HOMA, HRCP(0), HRCP(1), FLU, PDI(- matics: the average errors of the bond lengths calculated FAS), PDI(AO), KMCI(FAS), and KMCI(AO) were calculated using using the LC-XC functionals are more than twice as large as the MultiWFN program;[106] standard parameters for HOMA the XC-based ones. In practice, the conventional functionals (based on experimental bond lengths) and FLU (involving the perform better by about 0.01–0.02 A˚ per bond. However, reference diatomic delocalization indices determined at the when only the LC-XC hybrids are concerned, the calculated B3LYP/6-31G* theory level) were used as implemented in bond lengths are clearly superior to the ones obtained using the program. PDI(NAO), KMCI(NAO), as well as the EDDBg- and the pure GGA LC-XC functionals. Here, the differences between EDDBk-based electron populations were obtained using the bond lengths from both groups of functionals are much less

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two times lower than from the calculations involving standard XC functionals. Furthermore, according to Table S1d in Sup- porting Information, the total average errors of the calculated bond lengths for the entire molecule are as follow: 0.8% (XC), 1.1% (LC-XC), and 0.7% (hybrid LC-XC). Thus, both groups of exchange-correlation functionals characterizes similar accuracy of the predicted bond lengths in pentacene, with Handy and coworkers’ as well as Head-Gordon and coworkers’ LC-XC functionals being at a minor advantage over the rest. A thorough analysis of the results for anthracene and tetracene included in Tables S1b and S1c in Supporting Information indicates that the performance of the LC-XC hybrids is only slightly worse than the conventional ones (the differences are at most 0.1– 0.3%) while substantially larger errors produce the pure GGA LC-XC functionals. In contrast to pentacene, in the case of its angular isomer picene average errors of the calculated bond

Figure 1. Experimental and calculated average bond lengths with the corresponding percentage errors per ring (numbers inside cycles); results for each XC separately and references to experimental data are included in Supporting Information. Method: KS-DFT/cc-pVTZ, equilibrium geometries. [Color figure can be viewed at wileyonlinelibrary.com] than 0.01 A˚ per bond for most of the monocyclic aromatic systems. The situation is quite different in the case of polycyclic aromatic systems, as exemplified in Figure 1 using pentacene and picene. In the former the average error of the calculated bond lengths per benzenoid unit strongly depends on the position of the ring: for the terminal rings the LC-XC functionals seem to outdo the conventional ones while in both the inner and the central ring we get the opposite trend. Again, the LC-XC Figure 2. Percentage errors of vertical ionization potentials from the DFT- Koopmans approximation using different XC (violet bars) and LC-XC func- hybrids provide much better predictions of bond lengths espe- tionals (yellow bars). Method: KS-DFT/cc-pVTZ, equilibrium geometries. cially for the outer rings where the average error is more than [Color figure can be viewed at wileyonlinelibrary.com]

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corrected hybrid xB97xD, which in more than half of the cases underestimates the experimental vertical ionization potentials by only 1–2%. In fact, all the LC-XC hybrids are characterized

by the best performance in terms of reproduction the IPvert for all aromatic systems in this study. The dramatic impact of the long-range exchange corrections on the energetics of the Kohn–Sham HOMO orbitals (as well as the lower lying ones), which now to a certain extent satisfies the DFT-Koopmans theorem (“fictitious constructs but good approximations all the same”),[115] does not automatically entail changes in the patterns of cyclically delocalized electrons. Indeed, a close inspection of the KS-orbital shapes in pentacene depicted in Figure 3 reveals rather subtle changes introduced by the LC scheme (marked by green rectangles). However, it will be shown in the next section that these seemingly insignificant changes are of vital importance for the description of the cyclic delocalization of electrons.

Cyclic delocalization by means of aromaticity indices

As a preliminary, the EDDBg populations were calculated using BLYP and LC-BLYP functionals for all molecules from the test set, and then dissected into different symmetry components to give an account of the total numbers of r- and p-electrons delocalized through the net of all the chemical bonds. Table 1 collects the absolute and percentage (in brackets) changes of EDDBg and its components, EDDBg(p) and EDDBg(r), due to the long-range corrections. A full set of the XC/LC-XC benchmark calculation results of all the other aromaticity indices is Figure 3. p-MO energies (in a.u.) and the corresponding orbital isocontours provided in Tables S2-30 in Supporting Information. Here, (at s 5 0.02) from calculations utilizing BLYP and LC-BLYP functionals. Table 2 summarizes those results by presenting the percent- Method: BLYP,LC-BLYP/cc-pVTZ, equilibrium geometries. [Color figure can age changes of mean values (MV) and the corresponding stan- be viewed at wileyonlinelibrary.com] dard deviations (SD, in brackets) of KMCI, PDI, FLU21/2, EDDBk,

lengths per ring do not depend on the position of the benzenoid unit and to a large extent also on the type of the Table 1. Absolute and percentage changes (in brackets) in delocalization exchange-correlation functional (the differences are less than of r- and p-electrons due to the LC scheme for the BLYP functional. 0.3%). Indeed, total average errors of the calculated bond lengths regarding entire molecule are as follow: 1.4% (XC), Aromatic ring DEDDBg DEDDBg(p) DEDDBg(r) 1.4% (LC-XC), and 1.3% (hybrid LC-XC). The corresponding Monocyclic aromatics results for phenanthrene and chrysene presented in Tables S1e Benzene 20.01 (0) 0.00 (0) 20.01 (–5) Cyclopropenyl cation 0.01 (1) 0.00 (0) 0.01 (16) and S1f in Supporting Information show that the performance Cyclopentadienyl anion 20.02 (0) 0.00 (0) 20.02 (–7) of the CAM-B3LYP, xB97x, and xB97xD functionals is princi- Cycloheptatrienyl cation 20.04 (–1) 0.00 (0) 20.04 (–9) pally the same as the conventional ones while meaningfully Cyclooctatetraenyl dication 20.06 (–1) 0.00 (0) 20.06 (–10) less accurate bond lengths we obtain from the LC scheme for Cyclononatetraenyl anion 20.13 (–1) 0.00 (0) 20.13 (–22) Pyridine 20.02 (0) 20.01 (0) 20.01 (–2) GGA functionals. s–Triazine 20.10 (–2) 20.05 (–1) 20.05 (–5) Figure 2 clearly shows that also vertical ionization potentials Borazine 20.17 (–4) 20.30 (–8) 0.13 (20) approximated by the LC-XC HOMO energies (taken with an Pyrrole 20.51 (–12) 20.53 (–14) 0.02 (11) Furan ––0.49 (–16) 20.45 (–17) 20.04 (–15) opposite sign) are closer to the corresponding experimental Thiophene 20.45 (–14) 20.41 (–14) 20.04 (–17) values, that is, the relative errors of IPvert are 2–3 times smaller Acenes and phenacenes than those predicted by the HOMO energies obtained with Naphthalene 21.24 (–16) 21.20 (–17) 20.04 (–6) the conventional XC functionals. For all the studied molecules, Anthracene 21.84 (–17) 21.76 (–18) 20.08 (–8) Tetracene 22.57 (–19) 22.49 (–20) 20.08 (–6) the LC Iikura scheme overestimates IPvert by on average 10– Pentacene 22.84 (–17) 22.77 (–18) 20.07 (–4) 15%; but there are aromatics in which the error is much Phenanthrene 21.21 (–11) 21.15 (–11) 20.05 (–6) smaller, like 2% for pyrrole (see Fig. S2h in Supporting Informa- Chrysene 22.39 (–16) 22.28 (–17) 20.11 (–8) tion). This slight overshading of the correlation effects by the Picene 22.73 (–15) 22.60 (–16) 20.13 (–8) LC-XC GGA functionals seems to be fixed in the long-range Method: BLYP,LC-BLYP/cc-pVTZ, equilibrium geometries.

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Table 2. Summary of Tables S2–27 from Supporting Information: the numbers before brackets stand for the percentage average changes of aromaticity

indices due to the long-range exchange corrections, %DLC 5 (MVLC-XC 2 MVXC)/MVXC3100% (MV – mean value), while the numbers in brackets represent the percentage changes of the corresponding standard deviations.

k 21/2 Aromatic ring EDDB KMCI PDI FLU HRCP(1) EL -NICS(1) HOMA Monocyclic aromatics Benzene 0 (0) 0 (0) 0 (–1) 0 (–) 0 (1) 0 (0) 2 (0) 22 (1) Cyclopropenyl cation 1 (0) 21 (0) – (–) 0 (–) 31 (5) – (–) 5 (0) 232 (14) Cyclopentadienyl anion 0 (0) 0 (0) – (–) 0 (–) 21 (–2) – (–) 21 (0) 14 (–5) Cycloheptatrienyl cation 21 (0) 0 (0) – (–) 0 (–) 3 (2) – (–) 1 (3) 22 (1) Cyclooctatetraenyl dication 21 (0) 0 (0) – (–) 0 (–) 7 (1) – (–) 2 (1) 10 (–4) Cyclononatetraenyl anion 21 (0) 0 (0) – (–) 0 (–) 5 (–1) – (–) 3 (1) 4 (–3) Pyridine 0 (0) 22 (0) 22 (–1) 0 (–) 0 (0) – (–) 2 (0) 22 (2) s-Triazine 21 (0) 25 (0) 24 (0) 0 (–) 2 (–2) – (–) 0 (0) 22 (1) Borazine 24 (–2) 216 (–3) 224 (–3) 24 (–3) 0 (–1) – (–) 215 (2) – (–) Pyrrole 212 (0) 29 (0) – (–) 27 (–2) 23 (–2) – (–) 23 (0) 23 (–1) Furan 218 (–1) 213 (–1) – (–) 215 (–1) 23 (0) – (–) 23 (0) 44 (–40) Thiophene 218 (0) 220 (–1) – (–) 219 (–2) 26 (0) – (–) 23 (0) 24 (–2) Acenes and phenacenes Naphthalene 216 (3) 22 (0) 1 (0) 24 (0) 24 (–1) 11 (–2) 21 (0) 3 (–5) Anthracene[a] 228 (5) 212 (2) 24 (0) 211 (1) 21 (1) 10 (–3) 27 (2) 23 (–5) Anthracene[b] 12 (1) 17 (1) 9 (0) 16 (1) 1 (1) 18 (–2) 3 (–1) 25 (–7) Tetracene[a] 227 (5) 222 (3) 28 (1) 215 (1) 21 (1) 5 (–1) 214 (3) 213 (–3) Tetracene[c] 5 (0) 14 (1) 7 (0) 9 (0) 1 (1) 12 (–2) 1 (–1) 28 (–9) Pentacene[a] 238 (7) 228 (5) 212 (1) 217 (2) 21 (1) 21 (2) 220 (5) 222 (–1) Pentacene[c] 29 (4) 6 (0) 3 (0) 2 (0) 0 (1) 5 (–1) 23 (0) 24 (–10) Pentacene[b] 15 (1) 21 (2) 8 (0) 14 (1) 1 (1) 24 (–2) 4 (–1) 41 (–10) Phenanthrene[a] 23 (1) 4 (0) 4 (0) 6 (0) 0 (1) 9 (–1) 0 (0) 5 (–4) Phenanthrene[b] 238 (6) 212 (1) 1 (1) 25 (0) 0 (1) 30 (–5) 26 (1) 9 (–9) Chrysene[a] 29 (2) 2 (0) 3 (0) 0 (0) 0 (1) 8 (–2) 0 (0) 4 (–4) Chrysene[c] 230 (5) 23 (1) 4 (0) 23 (0) 0 (1) 16 (–5) 22 (0) 11 (–8) Picene[a] 27 (2) 3 (0) 3 (0) 2 (–1) 0 (1) 9 (–2) 0 (0) 4 (–4) Picene[c] 233 (5) 27 (1) 2 (–1) 24 (0) 0 (1) 21 (–5) 24 (1) 9 (–8) Picene[b] 218 (3) 6 (0) 9 (0) 2 (0) 0 (1) 16 (–4) 0 (0) 13 (–7) Method: KS-DFT/cc-pVTZ, equilibrium geometries. [a] Terminal ring. [b] Central ring. [c] Inner ring.

HRCP(1), EL, -NICS(1), and HOMA, due to the long-range bond lengths in 6p-aromatic rings, going from exchange corrections. These quantities are, in our opinion, the benzene (DRCC520.024 A˚) to cyclooctatetraenyl dication most representative indices for different aromaticity criteria and (DRCC 520.026 A˚). In contrast to r-delocalization, which is perform generally better than their other variants (significantly responsible for holding the ideally symmetric structure of the smaller values of standard deviations). Additionally, in Figure 4 ring,[116–119] no changes of EDDBg(p) are observed. This partic- the results for picene and pentacene were visualized to provide ularly should not be surprising since the cyclic delocalization the qualitative framework and illustrate to what extent the LC of p-electrons is known to be resistant to any change of the scheme can determine the picture of local aromaticity in these bond lengths that preserve the symmetry of the ring.[120] In systems. In the cases where the effect of LC was much below pyridine and s-triazine the LC scheme slightly adjusts the the accuracy of FLU calculation (affected mainly by the numeri- effects of p-electron delocalization but the magnitude of the cal integration accuracy and the theory-level inconsistency of resulting EDDBg changes remains small. In turn, more signifi- 21/2 parametrization) the average FLU values were assumed to cant reduction of the electron delocalization is observed for be zero and the standard deviations were not calculated. The borazine, pyrrole, furan, and thiophene as well as for all poly- results for EL aromaticity index were collected in Table S29 from cyclic systems. In all of these systems about 80–90% of the Supporting Information. EL was calculated only for uncharged total delocalization is attributed to the p-electron effects, polycyclic systems due to the fact that EL was defined solely for which has a direct bearing also on the EDDBg(p) changes due the p-conjugated carbon–carbon bonds and it was not tested to the long-range corrections, especially in polycyclic aromatic [79] for a set of charged species. hydrocarbons. For instance, in pentacene the local character of p-versus r-delocalization. Even a cursory glance on the results the exchange part of the BLYP functional leads to overestima- in Table 1 leads to the conclusion that the long-range tion of the p- and r-delocalization effects for about 2.8e- and exchange corrections do not significantly affect cyclic delocali- 0.1e-, respectively. Therefore, for the aromatic systems in this zation of electrons in aromatic hydrocarbons. In fact, a small study, there is no need to dissect delocalization indices into decrease of the EDDBg(r) components has actually no impact symmetry components and in practice the total values of the on the relative changes of EDDBg, but interestingly it is in line indices reflect the p-electron cyclic delocalization effects with with also very small but systematic reduction of the CC good accuracy.

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Figure 4. A comparison of mean values (with the corresponding standard deviations) of selected local aromaticity indices calculated using XC (left column) and LC-XC functionals (right column) for picene and pentacene. The values were multiplied by the power of 10 to be in the range of 0 to 1. Method: KSDFT/ cc-pVTZ, equilibrium geometries. [Color figure can be viewed at wileyonlinelibrary.com]

Monocyclic aromatics. The results of KS-DFT benchmark calcu- NICS(1) and HRCP(1) clearly fail to predict the expected reduc- lations summarized in Table 2 indicate that for monocyclic tion of electron delocalization due to corrected long-range species EDDBk, KMCI, PDI, and FLU21/2, follow generally the behavior of the XC functionals (minor extenuation of both same trends as the total EDDBg populations from Table 1. In indices can be observed in pyrrole, furan, and thiophene). The particular, for aromatic hydrocarbons they consistently predict corresponding changes of aromaticity index based on struc- the kekulean mode of electron delocalization to be invariant tural criterion are quite difficult to assess. On the one hand, with respect to any change of the exchange-correlation func- for most of the aromatic rings we get the results that qualita- tional. In heteroaromatic rings the effect of long-range correc- tively resemble predictions based on delocalization indices. It tions is roughly the same for all four indices except the should be mentioned here that since HOMA is determined borazine molecule, for which KMCI and PDI seem to be much directly by the bond lengths it is not surprising that the k 21/2 more sensitive to the LC scheme than EDDB and FLU . decrease of r-delocalization effects is better marked in the Although it is rather difficult to strictly explain this discrepancy results (compare with the last column in Table 1). However, k one should bear in mind that in principle EDDB and FLU are there are four cases in which HOMA fails to give expected predefined by delocalization indices referring to separately- dictions: cyclopentadienyl anion, cyclooctatetraenyl dication, regarded bonds. Consequently, the opposite changes of elec- cyclononatetraenyl anion, and furan. This can partially be tron populations on boron and nitrogen atoms caused by LC attributed to the theory-level inconsistencies in the bond to a certain extent compensate each other. In contrast, KMCI length parametrization[74] and the arbitrariness of the choice quantifies the effect of cooperativity of all bonds in a ring and of appropriate reference aromatic system, which is known to as such is slightly more sensitive to the electron outflow at be a major issue in heteroaromatic rings.[75] Moreover, in the any atomic center. PDI is even more receptive to the long- case of furan the dramatic decrease of the standard deviation range corrections than KMCI despite the fact that both indices of HOMA (by about 40%) indicates huge discrepancies [16] are known to be tight correlated for a wide range of 6-MRs. between bond lengths calculated using various conventional One should realize, however, that PDI is defined by delocaliza- XC functionals, which may significantly affect the impact of tion indices of weak interactions between para-related atoms, the long-range exchange corrections. which in nature are rather poorly reproduced by the conventional XC functionals.[46] Linear and angular acenes. As follows from the analysis of g HRCP(1), NICS(1), and HOMA are not direct measures of cyclic DEDDB collected in Table 1, using the standard XC functionals delocalization of p-electrons, which is somewhat reflected by leads to significant overcounting of the total population of p- their quite disorderly and inexplicable values. In particular, for electrons delocalized between all the chemical bonds in angu- all the systems except borazine and the 5-MR heterocycles larly and linearly fused benzenoid rings. This effect is very clear

1648 Journal of Computational Chemistry 2017, 38, 1640–1654 WWW.CHEMISTRYVIEWS.COM WWW.C-CHEM.ORG FULL PAPER for anthracene, where the predicted overestimation of the calculated using the standard XC functionals consistently rec- global aromatic stabilization is by about 50% greater than for ognize the inner and terminal cycles in picene as the least and phenanthrene. As the number of the fused rings increases, the the most aromatic ones, respectively, while aromaticity of the difference between the LC effects in acenes and phenacenes central ring ranks somewhere between them. Admittedly, the becomes less pronounced. Conversely, Table 2 clearly shows large deviations of the values of HRCP(1) and HOMA consider- that the impact of the long-range exchange corrections on ably impede the evaluation of relative aromaticity, but, when local aromaticity is more difficult to assess as it strongly considering each XC functional separately, they reproduce gen- depends on the position of the particular 6-MR. The only erally the same cyclic delocalization patterns as other indices. [121–124] exception is the HRCP(1) index, for which the influence of the This result can be easily rationalized by Clar’s rule, LC scheme is too small to be detectable, and EL, which quite according to which the covalent form with three p-sextets unexpectedly predicts increase of aromaticity due to the long- localized in the central and terminal rings contributes most to range corrections regardless of the cycle position in a system. the overall resonance hybrid. In turn, even a cursory look at In the case of phenacenes the only discernible regularity in Figure 4b reveals that introducing the long-range exchange aromaticity reduction due to the LC scheme considers the cen- corrections does not entail any significant changes in the pic- tral ring in phenanthrene and the inner ones in chrysene and ture of local aromaticity in picene. The only detectable differ- picene. Particularly, EDDBk, KMCI, FLU21/2, and NICS(1) consis- ences consider slightly smaller standard deviations of values tently predict loss of aromaticity although its magnitude (especially in the case of HOMA) and noticeable reduction of highly depends on the type of descriptor used; for example, aromaticity in the inner rings, which makes the differences for the central ring in phenanthrene the aromaticity reduction between aromaticity of individual cycles more clear cut. A varies from 5% for FLU21/2 and NICS(1) to almost 40% for closer study of Tables S17, S18, S21, and S22 in Supporting EDDBk. PDI and HOMA do not follow this trend displaying Information allows one to draw generally the same conclu- increase of local aromaticity up to 13% regardless of the ring sions with regard to local aromaticity in phenanthrene and position (one should realize however that the changes of chrysene: all the indices predict terminal rings to be the most HOMA might be statistically insignificant due to the standard aromatic ones regardless of the density functional approxima- deviation changes of the same magnitude as the index values). tion used. It is worth noticing that the EDDBk populations indicate a rela- Acenes, have no localized p-sextets and according to Clar’s tively small reduction of the cyclic delocalization also for benze- rule their benzenoid units are indistinguishable even though noid units in positions identifying localized Clar’s sextets,[121–124] they are not symmetry unique. As follows from Figure 4c, in that is, the terminal rings in all phenacenes plus the central one pentacene this qualitative picture of local aromaticity seems to 21/2 in picene. This is in contrast to all other aromaticity indices con- be partially supported by such indices as PDI, FLU , HRCP(1), sidered in this study. and HOMA, for which the differences in cyclic delocalization of Contrariwise, for acenes all the local aromaticity indices give particular rings are much smaller than the corresponding devi- qualitatively the same estimations of the LC effect: significant ations due to approximated character of the exchange- reduction of the electron delocalization in terminal rings is correlation functionals. Conversely, this result is in contrast not accompanied by the enhanced cyclic delocalization in the cen- only with the EDDBk, EL, and NICS(1) predictions that aromatic tral ones. Furthermore, most of the indices consistently predict stabilization systematically increases when going from outer- the aromaticity reduction in terminal rings to increase in mag- most to the central ring, but also with KMCI advocating for nitude with the number of fused benzenoid units. In other the exactly opposite trend. However, even a first look at the words, the larger the acene the larger the systematic error results in Figure 4d reveals that these contradictory predictions caused by the local character of the approximated XC func- vanish for all the aromaticity indices when the LC scheme is tional. Although the predicted by HOMA increase of cyclic involved: the electronic, magnetic, and structural criteria of delocalization in inner and central rings seem to be quite local aromaticity consistently predict decreasing aromatic sta- excessive (which is probably caused by relatively large average bilization effect from central ring to the outer ones. Further- errors of the calculated bond lengths in these rings—see Fig. more, aside from the results by HRCP(1), the resulting order 1), the results for terminal units are nearly quantitatively the seems to be independent on the type of the long-range same as the corresponding changes of the NICS(1) indices. exchange corrected density functional used in calculations,

Even the HRCP(1) index gives qualitatively the same predictions since the differences between local aromaticity of particular for aromaticity changes in terminal rings in acenes but again, rings are statistically significant and well detectable. the effects of the LC-scheme are too small to be regarded as These findings hold true also for anthracene and tetracene reliable. (see Fig. 5 as well as Tables S15, S16, S19, and S20 in Support- Next, let us examine how the changes of cyclic delocaliza- ing Information). Moreover, the effect of the LC scheme on the tion caused by the long-range exchange corrections translate electronic indices does not depend even on the choice of the into the picture of relative aromaticity of benzenoid units in electron density partitioning method, what is particularly illus- acenes. In Figure 4, the appropriately renormalized mean val- trated in Figure 5 using KMCI and PDI as an example. Admit- ues (with the corresponding standard deviations) of local aro- tedly, the LC scheme gives rise to significant deviations of the maticity indices for particular rings in picene and pentacene PDI based on the Mayer’s bond-order definition, but this is are displayed. As follows from Figure 4a, all the indices more likely the effect of the (relatively) large basis set used in

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anthracene and tetracene PDI determined using standard XC density functionals gives essentially the same predictions as KMCI, which is somewhat in contrast to pentacene (see Fig. 6 in the text, Fig. S3 and Tables S23-25 in Supporting Information). As indicated in previous subsection, experimental bond lengths in acenes are slightly better reproduced in the DFT calculations involving the GGA XC density functionals without the LC scheme. Although the absolute average errors are usually worse only by less than 0.01 A˚ when the corresponding LC-XC functionals are used, discrepancies regarding atomic distances in the inner rings of acenes are explicitly better marked. To investigate if and to what extent this minor deterioration of the equilibrium geometries affects the performance of KMCI and PDI, the indices were recalculated using density matrices approximated by the GGA XC functionals but at the geometries from the corresponding LC-XC calculations, and vice versa. The complete results considering all the GGA XC functionals separately are collected in Figure S3 in Supporting Informa- tion, while the average values are presented here in Figure 6. The results involving the XC and LC-XC density functionals and the corresponding equilibrium geometries (Figures 6a and 6d)

Figure 5. A comparison of mean values (with the corresponding standard deviations) of selected electronic indices of aromaticity calculated using XC and LC-XC functionals for anthracene and tetracene. The values of all indices were multiplied by the power of 10 to be in the range of 0 to 1. Method: KSDFT/ cc-pVTZ, equilibrium geometries. [Color figure can be viewed at wileyonlinelibrary.com]

calculations as the AO-partitioning is known to give reliable Figure 6. A Separation of structural and electron-density effects of the long-range exchange corrections on local aromaticity in linear acenes by results only for the sets of well-localized effective minimal- means of the average KMCI and PDI values (multiplied by 104). Method: basis orbitals.[125–127] It should also be noticed that for KS-DFT/cc-pVTZ.

1650 Journal of Computational Chemistry 2017, 38, 1640–1654 WWW.CHEMISTRYVIEWS.COM WWW.C-CHEM.ORG FULL PAPER assert again, but more clearly, the total effect of the long- In the monocyclic aromatic systems all the LC-corrected range exchange corrections: noticeable reduction of cyclic delocalization indices, except HRCP, consistently predict a delocalization in terminal rings, up to 33% (KMCI) and 19% decrease of aromaticity up to about 20% for the molecu- (PDI) in pentacene, and simultaneous enhancement of aroma- lar rings with heteroatoms; the aromaticity indices based ticity in inner rings, up to 76% (KMCI) and 15% (PDI) in anthra- on the magnetic and structural criteria do not follow this cene. Figure 6b indicates that, admittedly, the change of trend and their response to the long-range corrections is geometry itself enhances cyclic delocalization in inner rings, quite disordered and inexplicable. but the reduction of aromaticity in terminal cycles is much In the phenacenes the wrong long-range behavior of the smaller. In fact, the effect of geometry is insufficient to reverse conventional XC density functionals lead to noticeable the local aromaticity trends predicted by KMCI, but seems to overestimation of the cyclic delocalization mainly in the be adequate enough for the corresponding change of cyclic central or inner rings. However, the distances between delocalization patterns from PDI predictions (7% reduction for local aromaticities of particular rings are clear cut enough terminal rings in pentacene and only 3% enhancement for the that the LC scheme does not introduce any significant central one in anthracene). As follows from Figure 6c, the changes to the overall electron delocalization patterns effect of the long-range exchange corrections itself (i.e., with- obtained using the noncorrected XC functionals. In the acenes, the local character of the approximated out reoptimization of molecular geometries) is clearly superior exchange-correlation functionals entails significant over- to the effect of geometry as it is strong enough to change estimation of electron delocalization in the terminal rings, local aromaticity trends predicted by KMCI. In particular, it and underestimation in the central/inner ones. This leads leads to reduction of cyclic delocalization in terminal rings, up to conflicting local aromaticity estimations even within to 20% (KMCI) and 9% (PDI) in pentacene, and simultaneous the framework of the same aromaticity criterion (Fig. 4c); enhancement of aromaticity in inner rings, up to 64% (KMCI) the long-range exchange corrections significantly change and 11% (PDI) in anthracene. Thus, although both electronic the performance of almost all the aromaticity indices, and geometry effects are responsive to the LC scheme, it is which now consistently predict a decrease of the local the former that determines the expected order of aromatic aromaticity when going from the inner rings to the out- rings in acenes, and with this regard KMCI seem to be much ermost ones. more sensitive than PDI. The results regarding acenes are of special importance in Conclusions the context of the so-called “anthracene problem,” which has been a subject of heated debate in the literature since it dem- A multitude of aromaticity descriptors from the electronic, onstrates the inadequacy of reactivity as a local aromaticity cri- structural and magnetic criteria have been used to address the terion.[56–59] Although anthracene and phenanthrene features role of the long-range exchange corrections in description of completely different Clar structures (i.e., one migrating p- the cyclic delocalization of electrons in aromatic systems at sextet in the former against two localized p-sextets in the lat- the density functional theory level. A test set of diversified ter) the central ring of both molecules is the most reactive monocyclic and polycyclic aromatics has been used in bench- towards addition; this can be easily rationalized by the fact mark DFT calculations involving various exchange-correlation that in both cases the Clar structures of the corresponding functionals. The presented results clearly show that the wrong adducts are aromatically stabilized by two p-sextets well local- long-range behavior of the conventional XC density functionals ized in the terminal rings.[121,128–134] Seemingly, many theoreti- may lead to significant overestimation of the cyclic delocaliza- cal calculations involving different types of delocalization tion of p-electrons, which in some cases dramatically changes indices predict terminal rings in anthracene (and higher the overall picture of the aromatic stabilization in a molecular acenes) to be the most aromatic ones hereby supporting to system. The essential conclusions of our study can be summa- some extent the claims of a direct link between the reactivity rized as follows: and the ground-state aromaticity.[16,24,25,58,77,135–142] One should bear in mind, however, that most of them were rou- Regarding reproduction of the experimental bond tinely determined at the DFT level of theory using approxi- lengths, conventional XC density functionals seem to be mated density functionals without the long-range exchange superior to their LC counterparts, but only for the mono- corrections. Thus, in the light of the results from our investiga- cyclic aromatics; in phenacenes and acenes the LC-XC tion, these particular local aromaticity predictions must be hybrids like xB97x give comparable (or even slightly bet- taken with caution. Conversely, the contradictive predictions ter) results as the noncorrected functionals. following magnetic considerations used to be criticized due to The long-range exchange corrections affect predomi- the coexistence of several ring currents (benzenic, naphtha- nantly the cyclic delocalization of p-electrons, which lenic, and anthracenic), which artificially enhances the induced seems to be particularly well accounted for by the LC-XC field at the center of the middle ring and thus undermines the hybrids, especially xB97x and xB97xD, which also quanti- credibility of NICS as a local (single-circuit) measure.[58,143–145] tatively satisfy the DFT-Koopmans’s theorem for all the As far as the structural criterion is concerned, discrepancies studied aromatics. between HOMA and delocalization indices were usually

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[143] S. Fias, P. Fowler, J. L. Delgado, U. Hahn, P. Bultinck, Chemistry 2008, [152] J. I. Aihara, J. Phys. Org. Chem. 2008, 21, 79. 14, 3093. [153] M. Mandado, J. Chem. Theory. Comput. 2009, 5, 2694. [144] S. Fias, S. V. Damme, P. Bultinck, J. Comput. Chem. 2008, 29, 358. [154] P. W. Fowler, S. Cotton, D. Jenkinson, W. Myrvold, W. H. Bird, Chem. [145] S. Fias, S. V. Damme, P. Bultinck, J. Comput. Chem. 2010, 31, 2286. [146] A. R. Katritzky, P. Barczynski, G. Musummara, D. Pisano, M. Szafran, J. Phys. Lett. 2014, 597, 30. Am. Chem. Soc. 1989, 111,7. [155] P. W. Fowler, W. Myrvold, D. Jenkinson, W. H. Bird, Phys. Chem. Chem. [147] K. Jug, A. M. Koster, J. Phys. Org. Chem. 1991, 4, 163. Phys. 2016, 18, 11756. [148] A. R. Katritzky, M. Karelson, S. Sild, T. M. Krygowski, K. Jug, J. Org. [156] J. I. Aihara, Bull. Chem. Soc. Jpn. 2016, 89, 1425. Chem. 1998, 63, 5228. [149] M. K. Cyranski, T. M. Krygowski, A. R. Katritzky, P. v. R. Schleyer, J. Org. Chem. 2002, 67, 1333. Received: 27 January 2017 [150] J. I. Aihara, J. Am. Chem. Soc. 2006, 128, 2873. Revised: 22 March 2017 [151] M. Mandado, M. J. Gonzalez-Moa, R. A. Mosquera, J. Comput. Chem. Accepted: 23 March 2017 2007, 28, 1625. Published online on 24 April 2017

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The electron density of delocalized bonds (EDDB) applied for quantifying aromaticity† Cite this: Phys. Chem. Chem. Phys., 2017, 19,28970 Dariusz W. Szczepanik, *a Marcin Andrzejak,a Justyna Dominikowska, b Barbara Pawełek,c Tadeusz M. Krygowski,d Halina Szatylowicz e and f Miquel Sola`

In this study the recently developed electron density of delocalized bonds (EDDB) is used to define a new measure of aromaticity in molecular rings. The relationships between bond-length alternation, electron delocalization and diatropicity of the induced ring current are investigated for a test set of representative molecular rings by means of correlation and principal component analyses involving the Received 7th September 2017, most popular aromaticity descriptors based on structural, electronic, and magnetic criteria. Additionally, Accepted 14th October 2017 a qualitative comparison is made between EDDB and the magnetically induced ring-current density DOI: 10.1039/c7cp06114e maps from the ipsocentric approach for a series of linear acenes. Special emphasis is given to the

Creative Commons Attribution-NonCommercial 3.0 Unported Licence. comparative study of the description of cyclic delocalization of electrons in a wide range of organic rsc.li/pccp aromatics in terms of the kekulean multicenter index KMCI and the newly proposed EDDBk index.

Introduction delocalization of electrons and as such it can be quantified by diﬀerent delocalization indices (DI).5,6 Indeed, the precise Aromaticity is an important and extensively used concept in relation between energy and delocalization indices, originally chemistry. It plays a fundamental role in predicting and proposed by Rafat and Popelier,7 has been shown to be very rationalizing the structure, spectroscopy, reactivity, and mag- useful for measuring the aromatic stabilization energy of This article is licensed under a netic properties of countless number of chemical species that aromatic molecules;8,9 very recently the exact algebraic relation- have closed 2D or 3D circuits. Like many other concepts in ship between DI and the interatomic exchange–correlation chemistry, aromaticity has not been precisely defined.1 In energies has also been established.10 practice, however, it is determined enumeratively on the basis Among all the ways to quantify aromaticity through the Open Access Article. Published on 17 October 2017. Downloaded 2/20/2020 12:12:29 PM. of distinctive properties of aromatic species. These properties ‘‘ground-state’’ criteria (energetic, structural, and electronic), regard inter alia increased stability with respect to the (linear) there are descriptors that have gained enormous recognition unsaturated counterparts without cyclic delocalization of and wide acceptance: the aromatic stabilization energy p-electrons (energetic criterion),2 vanishing or significantly (ASE),2 harmonic oscillator model of aromaticity (HOMA),11–13 reduced alternation of bond lengths (structural criterion),3 and and different types of the multicenter delocalization indices large magnetic anisotropies accompanied by abnormal chemical (MCDI).14–18 In turn, one of the most popular aromaticity shifts (magnetic criterion).4 From the electronic-structure descriptors based on the response properties is the nucleus- point of view, aromatic stabilization is usually caused by cyclic independent chemical shift (NICS) and its various derivatives.4,19,20 Despite the unquestionable success and popularity of these a K. Gumin´ski Department of Theoretical Chemistry, Faculty of Chemistry, aromaticity indices, some of their imperfections still cuts back Jagiellonian University, Gronostajowa 2, 30-387 Krako´w, Poland. their applicability to relatively small and simple systems. E-mail: [email protected] In particular, design of isodesmic and homodesmotic reaction b Department of Theoretical and Structural Chemistry, Faculty of Chemistry, scenarios to determine ASE is very difficult in practice and it University of Lodz, Pomorska 163/165, 90-236 Ło´dz´, Poland 2 c Department of Plant Cytology and Embryology, Institute of Botany, leaves room to a lot of arbitrariness. The principal problem Jagiellonian University, Gronostajowa 9, 30-387 Cracow, Poland with HOMA, in turn, is the necessity of parametrization of bond d Department of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warszawa, lengths for an idealized reference molecule, which obviously Poland cannot be chosen unambiguously. Consequently, the practical e Faculty of Chemistry, Warsaw University of Technology, Noakowskiego 3, use of HOMA is limited to aromatic and heteroaromatic systems 00-664 Warszawa, Poland f Institut de Quı`mica Computacional i Cata`lisi, Universitat de Girona, since the parameters for chemical bonds with metal atoms are not 3 C/Maria Aurelia Capmany, 69, 17003 Girona, Catalonia, Spain available. Furthermore, the parametrization of HOMA should be † Electronic supplementary information (ESI) available. See DOI: 10.1039/c7cp06114e performed using the same quantum-chemical method as used in

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calculations of equilibrium geometries of the molecule under delocalization in molecular systems.40 It makes use of the age- study, since routinely computed HOMA with the experimentally old concept of bond-order orbitals as well as the recently determined parameters is bound to suffer from large and unsyste- developed bond-orbital projection formalism41–45 to probe matic errors.13 The magnetic-based measures of aromaticity, like different levels of electron delocalization by decomposition of NICS, have also been criticised for their complexity (NICS relies on ED into density layers representing electrons localized on the condensation of potentially complicated patterns of induced atoms (inner shells, lone pairs), EDLA(r), electrons localized currents to a single number, especially in the case of fused between atomic pairs (typical two-center bonds), EDLB(r), and aromatic rings),21–26 methodological shortcomings (aromaticity electrons delocalized between conjugated bonds (multicenter evaluation using NICS is limited mainly to planar units of similar electron sharing), EDDB(r): size),6,20,27 and interpretative mistiness (quoting Prof. P. Bultinck, ED(r) = EDLA(r) + EDLB(r) + EDDB(r). (1) ‘‘...a multicenter delocalization is a necessary condition for a diatropic ring current to exist, provided that there are proper virtual Fig. 1a presents the results of such electron density partitioning molecular orbitals to excite to.’’).28–32 Finally, the DI-based indices in the case of the pyridine molecule (only valence electrons (especially MCDI), unlike the aromaticity measures described included). The last component called electron density of above, enable one to study most of the types of aromaticity delocalized bonds is of our special interest in the context of that can be found in literature.6 Also, the MCDIs are the only aromatic stabilization eﬀect, especially if we consider aromati- descriptors that passed a set of rigorous tests for aromaticity city as a property of the ground-state electron density in the quantifiers.33–35 Unfortunately, the main disadvantage connected spirit of the first Hohenberg–Kohn theorem of the conceptual with calculation of MCDI is their computational cost – at the level density functional theory (DFT).46 Formally, the electron of one-determinant wavefunction with moderate basis set one density of delocalized bonds is defined in the basis of natural 47 cannot use multicenter delocalization indices to evaluate aroma- atomic orbitals (NAO), {wm(r)}, but any other representation of ticity of the molecular fragments containing more than a dozen of the well-localized orthonormalized atomic orbitals can be used 6 48–50

Creative Commons Attribution-NonCommercial 3.0 Unported Licence. atoms. Thus, for large molecular rings, the analysis of aromatic as well: stabilization using MCDI is applicable only within the framework X EDDB r wy r DDBw r ; of simple approximations like Hu¨ckel’s or pseudo-p methods and ð Þ¼ mð Þ mn nð Þ (2) m;n for limited range of cases.36 Recently the original method of the electron density of where the corresponding EDDB matrix reads delocalized bonds (EDDB) has been proposed to facilitate quick "# 1 XO qualitative analysis of different bond-conjugation patterns and DDB ¼ P C e k 2Cy P: (3) 2 ab ab ab ab to provide a bird’s-eye view on the global aromaticity and a;baa resonance effects in molecular systems that due to their size This article is licensed under a and complex structure are the major challenge for the currently In the above equation P represents the standard charge and 51 used tools.37,38 However, in a number of preliminary tests bond-order matrix, Cab is a matrix of linear-combination EDDB turned out to be highly capable of providing also a quantitative evaluation of electron delocalization in many Open Access Article. Published on 17 October 2017. Downloaded 2/20/2020 12:12:29 PM. diversified aromatic rings and simultaneously free from the aforementioned shortcomings of the commonly used aromaticity indices.39 This work provides a comprehensive view on the performance of the electron density of delocalized bonds in quantification of local aromaticity by comparative study with other well-known descriptors based on structural, magnetic and electronic criteria. Special emphasis is given to the analysis of cyclic delocalization of p-electrons in terms of the newly proposed EDDB-based index and the kekulean multicenter index. Since the former does not strictly take into account the cooperativity of all atomic centers in cyclic delocalization of electrons, as the latter does, the results will show if and to what extent the multicenter sharing effects are important for reliable description of organic aromatics.

Electron density of delocalized bonds Fig. 1 (a) Diﬀerent levels of electron delocalization from the one-electron density decomposition scheme by eqn (1) for pyridine (only valence Electron density of delocalized bonds derives from the original electrons); (b) global EDDB(r) function for the LEU-LYS-GLU-GLN-PRO- method of the electron density (ED) partitioning that has been ARG-HIS-PHE-TYR-TRP decapeptide (colored fragments indicate aromatic introduced to provide a uniform approach to quantify electron rings). Method: HF/6-311G**//PM6.

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coefficients of the appropriately orthogonalized52,53 two-center using the natural minimal basis (NMB) instead of the default 42 bond-order orbitals (2cBO), lab stands for a diagonal matrix (full) NAO-representation.

collecting the corresponding 2cBO occupation numbers, eab represents a diagonal matrix of the bond-conjugation factors,37 and O denotes the system of conjugated bonds. For a typical Computational details Lewis-like (localized) bond Aa–Bb all the elements of eab are close to zero, while in the case of delocalized (conjugated) Two test sets of molecules have been used in our study. Test set T1, partially based on the set proposed by Andrzejak et al.13 and bonds there is at least one diagonal element in eab close to 1. presented in Fig. 2, contains both Hu¨ckel’s aromatic and The definition of eab involves a series of projections of localized 2cBO onto their three-center counterparts, followed by the antiaromatic systems (including the polycyclic aromatic hydro- projection onto the delocalized (in nature) occupied molecular carbons of diﬀerent topology) that rely on cyclic conjugation of orbitals (MO).40 Formal definition of this projection cascade is 2p orbitals and involve only carbon atoms. For such homo- Q deeply rooted in the formalism of the orbital communication geneous group of molecular rings with only C–C and C C theory by Nalewajski54–58 and as such it falls outside the bonds it is reasonable to expect the differences between framework of this work. But it should be mentioned that the aromaticities reflected by different criteria to be more or less trace of the resulting DB-density matrix, DDB, can be straight- related. forwardly interpreted as the population of electrons delocalized We have chosen the following descriptors as the most through the system of conjugated bonds, O. For the purpose of representative and commonly used aromaticity measures that this work, O is restricted to represent only the cyclic delocaliza- can be found in literature: 11–13 tion of electrons due to the resonance of the kekulean forms. Harmonic oscillator model of aromaticity (HOMA), For instance, in the case of the 5-membered rings (5-MR) O which is a normalized measure of the energetic consequences contains five chemical bonds as follows: of deviations of bond lengths in the molecular ring from the corresponding optimum values for an idealized aromatic system. Creative Commons Attribution-NonCommercial 3.0 Unported Licence. O ={(A A ),(A A ),(A A ),(A A ),(A A )}. The axial component of the nucleus-independent chemical 1 2 2 3 3 4 4 5 5 1 4,19,20 (4) shift calculated at 1 Å above the ring centroid, NICS(1)zz,

The resulting electron population, in this paper denoted simply by EDDBk, does not account for the cross-ring electron delocalization, which for most of the organic aromatics in this study do not play an important role (or at least does not change the qualitative picture of the electron delocalization pattern) This article is licensed under a and can be simply ignored (see Table S1 in ESI†). Although the EDDBk index is used here as a local aromaticity descriptor, one should realize that, by default (i.e. without specified O), the EDDB(r) function considers conjugations between all the Open Access Article. Published on 17 October 2017. Downloaded 2/20/2020 12:12:29 PM. chemical bonds in a molecule, and, as such, it can be used to evaluate global aromatic stabilization38 or to study the non- local resonance effects in conjugated aromatic rings (in both ground- and excited states).59 It has to be emphasized that EDDB is far more efficient than MCDI, especially in the case of highly accurate wavefunctions of large molecular systems. For instance, the calculation of MCDI takes from a dozen of seconds to several hours depending on the size of the ring and the computational method used, while the EDDB calculation takes less than 1 s for all the aromatic rings considered in this study regardless of their size.39 Even the determination of global aromaticity/resonance effects in molecules containing hundreds of atoms, like the decapeptide depicted in Fig. 1b, is very fast and takes less than 40 s if the appropriate threshold for the 2cBO occupations is used and much less than 10 minutes otherwise.39 Such speed-up is possible because within the EDDB formalism the multicenter electron sharing is approximated by means of decoupled three- atomic local resonances representing conjugations between the 60 adjacent bonds only (cf. the Bridgeman–Empson method). Fig. 2 The T1 test set of molecular rings used to benchmark the perfor- Further reduction of computational time can be achieved by mance of different aromaticity descriptors.

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which quantifies diatropicity/paratropicity of the induced ring Shannon aromaticity (SA),63 which measures the Kullback– current by means of the eﬀective magnetic shielding. Leibler distance of the bonding electron density distribution in The kekulean multicenter index (KMCI),14–18 which directly aromatic ring from uniformity; SA is a non-referential index. quantifies cyclic delocalization of electrons in aromatic rings. Fluctuation index of aromaticity (FLU),5,64 which has a Our preliminary studies indicate that the KMCI values for all similar interpretation as the SA index but depends upon para- aromatic species from both test sets tightly correlate (R2 = 1.00) meters determined for an idealized aromatic system.64 with those of MCI – the multicenter index originally proposed Ellipticity index of aromaticity (EL),66 which measures by Bultinck et al.15 that implicitly takes into account the bonding electron density deformations reflected in negative cross-ring delocalization of electrons. However, since MCI is eigenvalues of the Hessian matrix of the electronic density in far more computationally expensive and offers virtually the bond critical point (BCP). no advantage over KMCI (at least for the studied systems) its Density of the total electron energy at the ring critical point 67 use in this benchmark is not necessary; the MCI values (HRCP), which is one of the AIM parameters that has been are included in Table S1 in ESI.† In view of the well- proven to serve as a quantitative measure of aromaticity in a known problems of the multicenter indices with the ring-size number of molecular rings.68 extensivity, in order to perform the correlation and principal Formal definitions of all the above listed indices can be component analysis (PCA) involving the entire T1 set of 5-, 6-, found in the source papers as well as in the manual of the and 7-MR molecules we used the nth root of the original index, MultiWFN program,69 which has been used to calculate HOMA denoted by KMCI1/n (see Fig. 3a and 4a), as suggested in the (with parameters for the C–C bond calculated consistently literature.6,16 according to ref. 9) as well as ATI and KMCI (both in the NAO 15 Average two-center index (ATI), which measures the basis). To calculate HRCP, FLU, SA, and EL the analysis of average eﬀect of the electron delocalization between three electron density distribution within the framework of quantum para-related atoms in a benzenoid unit. Its use is thus limited theory of atoms in molecules has been performed using AIMAll 70

Creative Commons Attribution-NonCommercial 3.0 Unported Licence. to 6-membered rings only. ATI has the same theoretical foun- program. Although the electronic indices calculated using dations as the well-known para-delocalization index (PDI) by different partitioning schemes may sometimes give rise to Poater et al.5,62 but it involves a Hilbert-space partitioning different aromaticity predictions,71 our preliminary studies within the basis of atomic orbitals instead of Bader’s atoms- show that the NAO-based indices like KMCI and ATI are fully 65 61 in-molecule (AIM) or the fuzzy-atomic space (FAS). equivalent to their AIM- and FAS-based counterparts, i.e. IRing This article is licensed under a Open Access Article. Published on 17 October 2017. Downloaded 2/20/2020 12:12:29 PM.

Fig. 3 The arrays of the R2-values multiplied by 100 for linear (below the diagonal) and exponential, if available (above the diagonal) correlations between EDDBk and diﬀerent aromaticity indices calculated using (a) the entire set of molecules from T1 as well as (b, c, and d) within the corresponding

subsets of aromatic rings of the same size, i.e. 5-, 6-, and 7-MRs, respectively; for polycyclic hydrocarbons ATI is used instead of NICS(1)zz. The numbers in the column on the right side of each array stand for the percentage of explained variance by first component in PCA (more details in the text). Method: CAM-B3LYP/def2-TZVPP, equilibrium geometries.

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and PDI, respectively (the r-squared coefficients are very close are much smaller and, therefore, neglected in further analysis.

to 1.00 in all cases); the IRing and PDI values are included Boxed numbers refer to the entire set of indices while the numbers

in Table S1 in ESI.† The NICS(1)zz values have been determined below represent the effect of the cummulative exclusions (in the for all systems (using the Gaussian 09 package72) except for order given by Roman numerals in brackets) of the corresponding polycyclic aromatic hydrocarbons, for which NICS is ill-defined, indices from the analysis; for detailed explanation see below. Fig. 4 owing to its proven non-local character caused by mixing of illustrates in detail the results of the correlation analyses between different ring currents.21–25 The EDDBk indices have been EDDBk and other aromaticity indices considered in this study. obtained using the NBO6 software73 and the script program Even a cursory glance at Fig. 3a (as well as Fig. 4b and g) 74 6 written by one of the authors (DS). The corresponding indicates that the ring-size extensivity issue makes HRCP and

EDDB(r) maps have been generated by means of the standard NICS(1)zz incomparable with the rest of aromaticity descriptors tools from the Gaussian 09 package (formchk and cubegen when molecular rings of diﬀerent size are considered.78 On the programs). The CAM-B3LYP75/def2-TZVPP76 calculations other hand, reasonable linear correlation (with R2 coeﬃcients with full geometry optimizations have been performed using close to or greater than 0.90) can be observed within the set of Gaussian 09. The correlation and principal component analyses SA, EL, FLU and HOMA indices, while their correlation with based on appropriately scaled aromaticity indices (scaling EDDBk is mainly non-linear in character, but still relatively tight factors can be found in Table S1 in ESI†) have been carried (up to R2 = 0.96 for HOMA). This is clearly shown in Fig. 4c–f. out using the R-package77 and their results are summarized in The fact that aromaticity changes predicted by different aro- Fig. 3 and 4, while the complete results of the benchmark maticity measures may not necessarily be linearly proportional calculations are collected in Tables S1 and S2 in ESI.† to each other is quite obvious. It should be noticed, however, Test set T2 contains more diverse types of aromatic systems that here the exponential relation between EDDBk (a quantity and has been used to assess if and to what extent the cyclic based on bond-order orbitals) and HOMA (an index involving delocalization patterns in aromatic rings predicted by KMCI bond lengths) explicitly refers to the bond-distance/bond-order k 79

Creative Commons Attribution-NonCommercial 3.0 Unported Licence. can be reproduced by the newly proposed index EDDB ; a more correlation originally established by Pauling. As follows from comprehensive study involving the AIM-based counterpart of Fig. 3a, in turn, the renormalized multicenter index, KMCI1/n, KMCI and other indices based on diﬀerent criteria of aroma- seems to fall slightly behind EDDBk, SA, EL, FLU, and HOMA ticity can be found elsewhere.33–35 The T2 set is based on the with r-squared coefficients in the range of 0.70 to 0.90 (excluding 33 collection of tests originally designed by Sola` et al. to evaluate HRCP and NICS(1)zz), but it still performs dramatically better than new aromaticity indicators proposed in the literature and it the original KMCI for size-differentiated aromatic rings (see accumulates chemical experience about the expected trends in Fig. 4a and Table S2a in ESI†). aromaticity changes in the following systems: distorted ben- The results of PCA indicate that PVE1C for the entire set of zene, substituted benzene, metal complexes, penta- and hepta- aromaticity indices calculated for all aromatic rings from T1 This article is licensed under a fulvenes, claromatic systems, heteroaromatic systems, as well equals 78% (see Fig. 3a). However, consecutive eliminations of as the aromatic transition states in selected chemical reactions. indices with the smallest contributions to the first component All the test molecules are depicted in Fig. 6–8. KMCI and EDDBk (and significant contributions to the second one) from the set indices were calculated at the B3LYP/6-311++G** theory level systematically improve PVE1C. In particular, by eliminating the Open Access Article. Published on 17 October 2017. Downloaded 2/20/2020 12:12:29 PM.

(equilibrium geometries) using Gaussian 09 and are listed in HRCP measure from the set PVE1C grows to a value of 87%, then

Fig. 6 and 7 (results for distorted benzene and two chemical by exclusion of NICS(1)zz it reaches 91%, next, elimination of EL reactions); Fig. 8 presents a summary of the results for other increases PVE1C up to 93%, and so on; red numbers indicate species from the test set, which are displayed and briefly PVE1Cs for the last two indices remaining after exclusion of all described in Fig. S1–S5 in ESI.† other aromaticity measures (here FLU and SA). In fact, it should not be surprising that two mutually linked indices (in a sense both quantify uniformity of the electron distribution in aromatic Results and discussion ring) remain at the end of the procedure. But for our purposes the final effect is not as important as the partial result of Correlation and principal component analysis elimination of a specific descriptor: the smaller the PVE1C Let us first consider the results of the correlation and principal growth, the better. In this context, EDDBk (eliminated as the component analyses performed using the T1 set of molecules. fourth in order) gives rise to relatively small increase of PVE1C by Fig. 3 collects arrays of the R2 coefficients multiplied by 100 for 2 percent points (pp), i.e. from 93% to 95%; basically the same is linear (below the diagonal) and exponential, if available (above true for the T1 subsets of 5- (+2pp), 6- (+2pp) and 7-membered the diagonal) correlations between different aromaticity indices rings (0pp). Generally, if one regards size-differentiated rings calculated using the entire set of molecules from T1 (Fig. 3a) as separately (Fig. 3b–d) PVE1Cs for the entire set of aromaticity well as within the corresponding subsets of aromatic rings of indices are close to 90%, which means that all the criteria of the same size (Fig. 3b–d for 5-, 6-, and 7-MR, respectively). aromaticity used in this study give a more or less consistent The numbers in the column on the right side of each array picture of aromatic stabilization for the set T1. Moreover, a represent the percentage of variance explained by the first closer inspection of the antecedent PVE1C values reveals that principal component (PVE1C); the second and higher components in each case EDDBk is in the subset of only a few descriptors with

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Fig. 4 Correlation between EDDBk and diﬀerent aromaticity indices (a–h) for the entire test set T1 (black) as well as the subsets of 5- (blue), 6- (green), and 7-membered rings (red). The r-squared coefficients in brackets refer to the unnormalized KMCI. Method: CAM-B3LYP/def2-TZVPP, equilibrium geometries.

the rate of at least 96% of variance explained in the case of analyzed. Apparently, KMCI1/n is superior to KMCI (R2-values in monocyclic aromatics (Fig. 3b and d) and at least 95% in the brackets) if one wants to compare aromaticity of rings of polycyclic aromatic hydrocarbons (Fig. 3c). different size; at the same time renormalization of KMCI seems The effect of the ring-size extensivity issue of the original to give only a slight adjustment to the r-squared coefficients (unnormalized) kekulean multicenter index is clearly shown when we consider rings of different size separately. One should in Fig. 4a, in which the correlation with the EDDBk index is realize, however, that the situation dramatically changes e.g. in

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the case of NICS(1)zz, where the correlation coeﬃcients for the the central ring (27) in triphenylene from the 6-MR subset of T1 entire test set T1 speaks in favor of the original KMCI. Indeed, significantly improves the correlation with EDDBk (as well as accordingly to Table S2a in ESI,† renormalization of KMCI with other indices like HOMA and FLU) giving rise to R2 = 0.95. decreases R2 from 0.67 (exponential correlation) to 0.36 (linear On the other hand, it follows from Fig. 4a, b, d and h that in the correlation). This particular should not be surprising, since the case of polycyclic aromatic hydrocarbons the correlation with 1/n nucleus-independent chemical shift is known to share the lack KMCI , HRCP, EL and ATI is significantly weaker. This eﬀect is of ring-size extensivity with the unnormalized multicenter associated with a more general problem of the definition of index.6,20 In contrast, when regarding 5- and 7-MRs separately local aromaticity in polycyclic systems.21,22 A good example renormalization of KMCI has negligible impact on the correla- here are linear acenes as they have been a subject of heated

tion with NICS(1)zz (see Tables S2b and d in ESI†). It should also debate in literature, owing to dramatic discrepancies between be mentioned that excluding 1,3-cyclopentadiene from the local aromaticity descriptions provided by diﬀerent aromaticity subset of 5-MRs significantly improves the correlation with criteria (‘‘the anthracene problem’’).21,22,80–83

NICS(1)zz of both cyclic delocalization measures, KMCI and The aromaticity indices based on magnetic and structural EDDBk, i.e. the corresponding correlation coeﬃcients rise from criteria predict increasing aromaticity going from terminal to 0.79 to 0.97 (KMCI) and from 0.86 to 0.94 (EDDBk), i.e. by 23% the central ring, while the electronic aromaticity indices like 33,34,84 and 9%, respectively. When the AIM-based counterpart of IRing, PDI, or HRCP predict the opposite trend. It has 2 KMCI (i.e. the IRing index) is considered, the R value increases recently been demonstrated, however, that in the latter group by nearly 60% (unpublished results). Thus, it seems that EDDBk of indices the results dramatically depend on the choice of the performs slightly better than KMCI regarding the correspon- exchange–correlation functional at the DFT theory level and as dence between magnetic and electronic criteria of aromaticity, such they are somewhat less reliable at least in the case of especially when both Hu¨ckel’s aromatic and antiaromatic systems polyacenes.84 In this context, the EDDBk index gives virtually are taken into account. the same predictions as HOMA and FLU (regardless of the level

Creative Commons Attribution-NonCommercial 3.0 Unported Licence. A closer look at Fig. 4e and f reveals that the correlation of of the theory). Furthermore, even the qualitative comparison EDDBk with FLU and HOMA is relatively tight regardless of the global EDDB(r) maps with the magnetically induced of whether we consider rings of diﬀerent size separately or ring-current density maps from the ipsocentric approach4,23,85 collectively. Additionally, in the case of SA (Fig. 4c), excluding presented in Fig. 5 clearly show that, despite fundamental This article is licensed under a Open Access Article. Published on 17 October 2017. Downloaded 2/20/2020 12:12:29 PM.

Fig. 5 Contours of the (global) electron density of delocalized bonds and the corresponding maps of the ring-current density from the ipsocentric approach for the first four acenes (only the p-type molecular orbitals were taken into account). Method: B3LYP/6-311G**, equilibrium geometries. CRD(r) maps reproduced from ref. 85 with permission from Elsevier.

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methodological differences (induced ring-current density is pyramidalization and the boat-like deformation are found to a response property that occurs only in the presence of an affect p-bond delocalization to a very limited extent (less than external magnetic field while the cyclic delocalization of 5%). Interestingly, in the case of BLA the EDDBk index is clearly electrons is a ground-state property that is present irrespective more sensitive than KMCI as it predicts reduction of aromati- of the presence or absence of an external magnetic field), both city for about 85% in comparison to the about 45% reduction approaches lead to the same conclusions about local aromaticity predicted by KMCI. Since for the bond-alternation parameter of acenes as those from HOMA and FLU calculations. DR = 0.25 Å we actually get the hypothetical structure of 1,3,5- cyclohexatriene with highly localized double bonds, the result Aromaticity of distorted benzene rings by EDDB seem to be even more reliable than those predicted by As an archetypical aromatic molecule, benzene can be used to KMCI. To rationalize the difference between sensitiveness of assess the performance of aromaticity descriptors in a series of EDDB and KMCI in this particular case one has to realize that in-plane deformation modes such as bond length alternation in the former we approximate the effect of bond resonance (BLA) and clamping (CLA), as well as the out-of-plane distortions from the perspective of each C–C bond in a ring (decoupled such as pyramidalization (PYR), boat-like (BOA) and chair-like local resonances), while in the latter we actually do not measure (CHA) deformations (see Fig. 6). These types of distortions are very the chemical resonance at all but rather the effect of coopera- often observed in large and strained benzene-based molecular tivity of all atomic centers in electron delocalization. Indeed, systems like graphene, nanotubes, and fullerenes, and they from the multicenter electron sharing perspective the reduction generally alter the cyclic delocalization of electrons leading to of cyclic delocalization of p-electrons is somewhat balanced reduction of the aromatic character of benzenoid units.86 by the enhanced electron delocalization within the Lewis-like In principle, the reference-based electronic aromaticity descrip- (two-center) p-bonds. Thus, in this particular case EDDBk seems tors (e.g. FLU) perform reasonably well, since, by definition, to account strictly for the bond-resonance stabilization rather they measure the deviation of particular electronic-structure than cyclic delocalization of electrons. It should be mentioned,

Creative Commons Attribution-NonCommercial 3.0 Unported Licence. properties from benzene. Some of the electronic criteria of however, that according to our unpublished results, for the aromaticity, however, are quite insensitive to most of the archetypical Hu¨ckel’s antiaromatic molecule, 1,3-cyclobutadiene, distortion modes (e.g. PDI).33,34 As Fig. 6 clearly shows, both nodiscrepancyisobservedbetweenEDDBk and KMCI, and both KMCI and the recently proposed EDDBk index perfectly and in indices predict no resonance/cyclic delocalization of electrons. full compliance reproduce the expected decrease of cyclic delocalization when a distortion is applied. Both indices con- Aromaticity changes along the reaction path sistently identify the bond length alternation as the most The interplay between aromaticity and reactivity is of vital ‘‘resonance killing’’ deformation in a benzenoid unit while importance in organic chemistry since a considerable number of chemical reactions involve species with a clear aromatic or This article is licensed under a antiaromatic character. The concept of transition state-aromaticity plays a key role in pericyclic, pseudopericyclic, and non-pericyclic reactions, properly determining and allowing one to understand the reaction mechanism.87 For the purpose of this comparative Open Access Article. Published on 17 October 2017. Downloaded 2/20/2020 12:12:29 PM. study, let us consider two simple reactions involving aromatically stabilized transition states: the standard Diels–Alder (DA) cycloaddition and the acetylene trimerization. In the case of the former, it is well-known that the reaction takes place through a boat-like aromatic transition state thus giving rise to a peak of cyclic electron delocalization in the ring at the vicinity of the TS along the reaction path. For the latter, an increase of aromaticity is expected when going from reactants to transition state. After this point significant reduction of the aromatic character is observed until a final increase due to formation of benzene as a product. The trends of aromaticity changes in both reactions are known to be perfectly reproduced by all electronic indices except for FLU, which depends critically on the model aromatic molecules chosen as a reference. Therefore, it cannot be used to study reactivity.33,34 Admittedly, aromaticity indices based on magnetic criteria properly reproduce the shape of the curve in the trimerization reaction, but they incorrectly identify the transition state with s-delocalized bonds to be even more aromatic than benzene itself.33,34 k Fig. 6 Diﬀerent benzene distortions together with the corresponding Fig. 7 presents plots of KMCI and EDDB vs. the reaction KMCI and EDDBk values as well as the EDDB(r) contour maps. KMCI values coordinate for both reactions. Analysis of these plots leads to have been multiplied by 103. Method: B3LYP/6-311++G**. the conclusion that despite some minor diﬀerences in the

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observed when a metal atom is complexed to benzene, as in the 6 33,34,91 case of (Z -C6H6)Cr(CO)3. All the electronic indicators of aromaticity like PDI, FLU, KMCI/MCI, etc. were shown to perfectly reproduce these trends, in contrast to the indices based on structural and magnetic criteria, which for some substituted benzenes predict higher aromatic stabilization than for benzene itself.33,34 For the effects of atom- and ring- size dependence, it was also shown that electronic indices of aromaticity, especially the appropriately normalized multicenter index, are superior to the rest of aromaticity descriptors.16,33,34 Three additional groups of aromatic systems complete the T2 set of molecules: heteroaromatics, claroaromatics and fulvenes. The first one was originally proposed to predict the proper trend of aromaticity changes along a well-established heteroaromatic series including five aza-derivatives of benzene and five heterocyclic compounds of type C4H4X (where X = CH , + NH, O, CH2, BH, CH ); the second group was designed to test the effect of fusing aromatic rings represented by five Clar systems, while the third subset was used to assess the expected trend of aromaticity in 5-MR and 7-MR fulvenes with different substituents.33 In general, all the electronic aromaticity indices were reported to pass the three tests.33,34 The only exceptions

Creative Commons Attribution-NonCommercial 3.0 Unported Licence. concern the series of pentafulvenes, which are particularly diﬃcult to assess by aromaticity quantifiers as they display a tunable aromatic character (sometimes being termed ‘‘aromatic chameleons’’92). Magnetic and structural indices are in most cases also in line with the expected trends for all three tests, except a

single incorrect prediction that C4H4NH is more aromatic than 33,34 C5H5 . Fig. 7 Plot of KMCI and EDDBk vs. the reaction coordinate for (a) the How does the EDDBk index deal with the above mentioned Diels–Alder cycloaddition and (b) the acetylene trimerization. For TS and tests? The answer to this question is provided by Fig. 8, which

This article is licensed under a products the EDDB(r) contour maps are displayed with the corresponding 3 displays a summary of calculations for all 47 test systems used values of both indices. KMCI values have been multiplied by 10 . Method: k B3LYP/6-311++G**. to compare the performance of EDDB and KMCI; detailed results with comments are collected in Fig. S1–S5 in ESI.† The presented results leave no doubt: the index based on the Open Access Article. Published on 17 October 2017. Downloaded 2/20/2020 12:12:29 PM. shape of curves, both aromaticity indices precisely identify the electron density of delocalized bonds predicts exactly the same s-aromatic transition state and the p-aromatic product of the trends of aromaticity changes as the kekulean multicenter index. acetylene trimerization (here the product is correctly identify Furthermore, in the overwhelming majority of cases these predic- as more aromatically stabilized than the corresponding TS) tions are in full agreement with general expectations.16,33,34 There as well as the aromatic TS in the Diels–Alder cycloaddition. are only three systems for which significant discrepancies The EDDB(r) contour maps show very clearly that TS in the DA between the expectations and the predictions based on both reaction has a boat-like structure with characteristic cyclic indices are observed: 23, 32, and 42. In fact, discrepancies delocalization pattern in the plane between butadiene and between different aromaticity criteria are very common for ethylene fragments.88 Moreover, although both TS and the heteroaromatic species, because in such systems proportionality product of the trimerization reaction are to a similar extent of the energetic effects of aromatic stabilization and the electron stabilized by resonance (according to the EDDBk values, the delocalization may depend on the heteroatoms present in the difference is less than 10%) it is quite obvious even from the system.12,93–97 Nevertheless, EDDBk index perfectly reproduce the first look at the EDDB(r) contours that TS is stabilized by cyclic predictionsbyKMCI,whichclearlyshowsthatthebond-orbital delocalization of s-electrons while the final product represents projection formalism behind the EDDB approach provides a a typical for organic aromatics p-delocalization pattern. widely applicable and reliable tool for quantitative evaluation of the multicenter electron sharing and aromatic stabilization effects Heteroaromatics, claromatics, fulvenes, and others (at least in the case of organic species). It should be pointed out, It has been pointed out by Krygowski et al. that substitution however, that the situation might be different in the cases, for to an aromatic ring, either electron-donating or electron- which cyclic delocalization of electrons cannot be represented accepting groups, decreases aromaticity of the ring as it leads with the resonant covalent forms (e.g. in small charged aromatic to partial localization of p-electrons.89,90 A similar behavior is rings, metal clusters, etc.) or when the strong cross-ring

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Fig. 8 Summary of the five tests used to compare the performance of EDDBk and KMCI as an aromaticity quantifiers. The complete results with brief comments are collected in Fig. S1–S5 in ESI.†

interactions appear. Then, according to our preliminary aromaticity measures in the first test proves that the proposed (unpublished) studies, EDDBk and KMCI can predict slightly ground-state electron density partitioning in eqn (1) can be different trends of aromaticity changes; a detailed analysis of considered reliable, and the extracted density layer corres- this issue will be published elsewhere. ponding to the delocalized bonds (EDDB) in fact seems to be

Creative Commons Attribution-NonCommercial 3.0 Unported Licence. closely related to the properties being characteristic for aromatic rings. In turn, a detailed comparison of EDDB to KMCI Conclusions shows that for a majority of studied aromatic rings the cyclic delocalization of electrons can be reliably approximated within Throughout recent decades, several dozen types of chemical the framework of the decoupled local resonances at incomparably aromaticity have been reported in the literature.98 As far as the lower computational cost. physicochemical and electronic-structure properties are con- The current implementation of the EDDB method, called cerned, many of those concepts differ much from the arche- RunEDDB, works with both RHF and UHF one-determinant typical p-aromaticity of the benzene molecule. It stimulated wavefunctions and involves the Hilbert-space partitioning This article is licensed under a research towards quantification of the aromatic stabilization within representation of natural atomic orbitals, which is effect. Every now and then new quantitative criteria of chemical widely available for most of the popular quantum-chemistry aromaticity are introduced in the literature. Different criteria, packages through NBO73 and JaNPA101 interfaces. RunEDDB is on which the quantifiers are based and their sometimes still under active development and is freely available on the Open Access Article. Published on 17 October 2017. Downloaded 2/20/2020 12:12:29 PM. inconsistent predictions has led to aromaticity concept reputa- author’s website.74 tion being somewhat tarnished and often criticised, both in its conceptual and methodological layer.99,100 One may add – the criticism often being undeserved since, even on the level of a Conflicts of interest qualitative theory, the usefulness of the aromaticity concept, There are no conflicts to declare. in the context of structure and reactivity prediction of a whole bunch of organic molecules, cannot be overestimated.100 One must admit, however, that among others introducing Acknowledgements new aromaticity measures makes sense nowadays if their performance has the advantage over the already existing The research was supported in part by the Faculty of Chemistry descriptors or they enable one to study molecular systems that at Jagiellonian University (grant K/DSC/001469, DS), Founda- due to their size and structure are the major challenge for tion for Polish Science (FNP START 2015, stipend 103.2015, DS), currently used tools.6,100 National Science Centre, Poland (NCN SONATA, grant 2015/17/ The results presented in this work clearly show that the D/ST4/00558, DS) as well as the PL-Grid Infrastructure of the EDDB method safely fulfils the above conditions. For a wide Academic Computer Centre CYFRONET with the calculations spectrum of organic compounds, the EDDBk index predicts the performed on the cluster platform ‘‘Prometheus’’. MS thanks for same trends of aromaticity changes as most of the aromaticity the support of the Ministerio de Economa y Competitividad of indices from different criteria. What is more, EDDB(r) maps Spain (Project CTQ2014-54306-P), Generalitat de Catalunya (project allow for easy identification of resonance-stabilised regions and number 2014SGR931, Xarxa de Refere`ncia en Qumica Teo`rica i for tracing of aromaticity changes due to chemical reactions. Computacional, and ICREA Academia prize), and European The comparison of electronic populations from EDDB to other Fund for Regional Development (FEDER grant UNGI10-4E-801).

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This journal is © the Owner Societies 2017 Phys. Chem. Chem. Phys., 2017, 19, 28970--28981 | 28981 Received: 13 February 2018 | Revised: 1 April 2018 | Accepted: 21 May 2018 DOI: 10.1002/qua.25696

RAPID COMMUNICATION

A simple alternative to the pseudo-p method

Dariusz W. Szczepanik

K. Guminski Department of Theoretical In this work, we introduce an approximate method for the multicenter index calculation that is Chemistry, Jagiellonian University, very simple in implementation and has the same computational cost as the pseudo-p approach. In Gronostajona 2, 30-387, Krakow, Poland contrast to the latter, however, the newly proposed method does not require additional single- Correspondence point calculations and is capable of quantifying multicenter electron sharing in aromatic rings con- Dariusz W. Szczepanik, K. Guminski taining heteroatoms and transition metals. Department of Theoretical Chemistry, Jagiellonian University, Gronostajona 2, 30-387 Krakow, Poland. KEYWORDS Email: [email protected] aromaticity, electron delocalization, Huckel€ method, multicenter index, pseudo-p method Funding information European Commission through the HPC- Europa3 project, Grant/Award Number: H2020 RIA INFRAIA-2016-1, GA: 730897, Contract: HPC17158J2; National Science Centre, Poland (NCN SONATA), Grant/ Award Number: 2015/17/D/ST4/00558

1 | INTRODUCTION

Over the past two decades a great progress has been made in the development of quantum-mechanical measures of electron delocalization in aromatic species.[1] In this context, a special attention should be given to the so called multicenter index (MCI),[2–5] which measures the cooperativity effect in the multicenter electron sharing between all atomic members of the aromatic ring, and, unlike other aromaticity descriptors based on energetic, structural, or magnetic criteria, it enables one to study almost all known types of aromaticity, conjugation and hyperconjugation effects, multicenter bonding, and so forth.[6–10] Unfortunately, calculation of MCI is a computationally demanding task and for large aromatic circuits and macrocyclic frames it can be performed only within the framework of the rough-and-ready approximations like the pseudo-p (PP) method (for aromatic hydrocarbons only).[11] In this work, we propose an alternative to the PP approach that is exceptionally simple in implementation and can be used to study a wide range of monocyclic and polycyclic p-aromatics, their heteroatomic analogues as well as selected metallacycles. Within the atomic orbital (AO) representation, the n-center delocalization index is defined as[2]

XX1 XX2 XX3 X XXn X hi n21 C ; MCI MCIn52 i ðPSÞlmðPSÞmr ðPSÞgl (1) l m r ... g i where the summations over AOs are restricted to particular ring members only labeled by A1,A2 ...An, P is the charge-and-bond-order (CBO) matrix, S is the AO-metric (overlap) matrix, and the C operator generates all ðn21Þ! cyclic permutations with exactly one fixed element, that is, the so called ðn21Þ-cycles. For aromatic rings there is at least one permutation that represents the kekulean mode of cyclic delocalization, KMCI, and for cycles including more than 4 atoms it usually provides an excellent approximation to MCI.[3] The computational cost of MCI for a n-membered homoatomic aromatic scales as ðn21Þ!rn, where r is the number of AOs centered on each atom. Consequently, at the level of HF/DFT with moderate basis sets one can use the multicenter index to evaluate aromaticity of molecular rings and circuits containing no more than 10–12 atoms.[1] Admittedly, the MCI calculations in large basis sets are slightly less computationally expensive within the framework of the physical-space partitioning, but here the numerical accuracy problems prevent calculations for rings with more than 10 members.[1] In the case of simple monocyclic and polycyclic p-aromatic hydrocarbons one can use the Huckel€ molecular orbital (HMO) method to get the approximated one-function-per-atom CBO matrix and

Int J Quantum Chem. 2018;e25696. http://q-chem.org VC 2018 Wiley Periodicals, Inc. | 1of5 https://doi.org/10.1002/qua.25696 2of5 | SZCZEPANIK

FIGURE 1 Relative errors of KMCINPP with respect to the full-basis KMCI and its p-component for the test set of planar p-aromatic species from Ref. [6]

[12] then easily calculate KMCIHMO for large circuits. The HMO approach, however, has a limited application due to the lack of any geometry, and thus it could not be used to determine local aromaticity even in relatively small systems like anthracene.[13] A more efficient and comprehensive approach, called the pseudo-p (PP) method, was proposed by Fowler and Steiner,[14] who took advant- [15] age of the finding by Paolini et al. that there is no preference for the use of 2pp orbitals over the nodeless 1s orbitals of a hydrogen-like atom for the description of planar p-conjugated monocyclic and polycyclic hydrocarbons; it should be mentioned, however, that the remarkable analogy between the p system in benzene and the corresponding cyclic unit built from hydrogen atoms was actually reported much earlier by London.[16] Within the pseudo-p approach, the multicenter index is calculated in the following three-step procedure: (1) the standard ab initio optimization of the molecular geometry is performed; (2) all the hydrogen atoms are removed and carbon atoms are replaced by the STO-3G s- type Gaussian basis functions; (3) a single-point ab initio computations are performed once again and the resulting CBO matrix is used to calculate the approximated multicenter index, KMCIPP. It has been demonstrated that this scheme allows one to successfully reproduce the multicenter indices from real molecular calculations.[11] Moreover, it has also been shown that the pseudo-p approach can be used to investigate the induced ring currents in monocyclic and polycyclic aromatic hydrocarbons.[17–19] However, the PP method has two major disadvantages, that is, it always involves hand crafting of the hydrogen-like model systems and additional single-point SCF calculations, and it can be used only for p-conjugated monocyclic and polycyclic hydrocarbons. In this work, we propose an alternative to the pseudo-p method that does not share the aforementioned shortcomings. Our approach, hereafter referred to as the NPP method (here, the NPP acronym simply stands for the npp orbital, albeit the method itself can be easily extended to cover also d-type orbitals), relies on several important features of planar p-aromatic rings: (1) for such systems the CBO matrix can be strictly separated [20] into symmetry components, that is, CBOr,CBOp, and so forth; (2) transformation of the AO basis into the effective minimal-basis (MB) representation[21] followed by removal of the core orbitals significantly reduces the rank of the CBO matrix;[22] (3) within the representation of valence MB- orbitals subtraction of the CBOr “layer” from the total charge and bond-order matrix leaves nonzero diagonal elements associated only with orbitals npp,ndp,ndd, and so forth; (4) there is exactly one p-type MB-orbital per atom for nonmetallic ring members, at most two d-type MB-orbitals per metal atoms (representing p-andd-type conjugations), and so forth. By putting together all these points we get a very simple in implementation scheme for calculation of the multicenter index that has the same computational cost as the pseudo-p method but is much more universal and does not require additional single-point calculations (see Methods for the implementation details). SZCZEPANIK | 3of5

2 | RESULTS AND DISCUSSION

In principle, the NPP method allows for an exact reproduction of p-component of the multicenter index in the minimal-basis calculations. However, when the split-valence basis set with polarization and diffuse functions is involved, transformation into the MB representation introduces a systematic error with respect to the full-basis multicenter index. This is well illustrated in Figure 1 presenting relative errors of KMCINPP with respect to the exact KMCI (left column) and its p-component (right column) for the test set of planar p-aromatic rings selected from the original study by Sola et al.[6] Admittedly, in the case of pentafulvenes (17–21), 5-membered heteroaromatic rings (33–37), and to some extent also azabenzenes (27–32), a significant contribution from r-delocalization appears, especially for antiaromatic heterocycles 36 and 37 (up to about 11%). But this effect seem to be proportional to the p-conjugation since the linear correlation between the full-basis KMCI and its NPP-based counterpart remains extremely 2 tight (R 5 0.9997). Moreover, if one compares KMCINPP to the p-component of the exact KMCI the average percentage error goes down to about 0.3% (R2 5 1.0000); thus, the systematic error due to the AO!MB transformation is really small and has no significant effect on the multicenter index. Accordingly, the KMCINPP index can straightforwardly be used to benchmark the PP and HMO approximations for large p-electron circuits. Figure 2 presents the results of such comparison for the test set of 18 different delocalization pathways in coronene (from 6- up to 22-membered circuits). In view of the well-known problems of the multicenter indices with the ring-size extensivity, here, we used a renormalized multicenter index, KMCIr, defined as the nth root of the original quantity.[1] It is particularly not surprising that the PP method gives much better approximation r to the reference KMCINPP values than the age-old qualitative HMO theory (average errors of 8% and 28%, respectively); in both cases, however, the error systematically increases with the size of p-circuit, up to 16% (PP) and 80% (HMO). As shown in Figure 2B, poor correlation between r r 2 2 KMCIHMO and KMCINPP (R 5 0.8354) worsens even more along with the consecutive eliminations of small circuits (R 5 0.7700 after elimination of

r r r FIGURE 2 A, Relative errors of KMCIPP and KMCIHMO with respect to KMCINPP for different delocalization pathways in coronene; B, r Correlation between KMCINPP and the corresponding PP- and HMO-based indices 4of5 | SZCZEPANIK

FIGURE 3 A, Variation of MCINPP and the full-basis MCI as well as its symmetry components along two series of model aromatic clusters based on Al, P, Ge and Se; B, A comparison of KMCI and KMCINPP for selected 7- and 8-membered metallacycles taken from Ref. [25] benzenoid cycles), which virtually disqualifies the HMO approximation for evaluation of the multicenter electron sharing effects. Contrariwise, r r 2 2 KMCIPP correlates very well with KMCINPP (R 5 0.9972), even after elimination of the smallest 6-membered rings (R 5 0.9931). Interestingly, all three approaches identify delocalization pathways (1) and (15) as the least and the most aromatic, respectively, which is in agreement with findings of several other studies.[23,24] However, only the PP method enables one to reproduce exactly the same order of p-electron circuits by increasing € aromaticity as KMCINPP. Finally, unlike the pseudo-p and Huckel MO approaches, the newly proposed method is capable of quantifying cyclic delocalization of electrons in aromatic rings containing at the same time atoms from different periods (Figure 3A), as well as transition metals (Figure 3B). r 22 22 Although for small aromatic clusters -delocalization may significantly contribute to MCI (especially in the Al4 ! Ge4 series) the NPP method allows one to successfully reproduce the MCIp component with the relative error up to about 6%-7%. A very good reproduction of the full-basis

KMCI we get also for planar metallacycles C6H6CrCl2 and C7H7MnCl2, in which metal atoms contribute to electron delocalization through the p- [25] and d-type conjugation (orbitals ndp and ndd, respectively); here, the average error of the NPP approximation is below 1%. To summarize, the NPP method has the same computational cost as the pseudo-p approach, but, contrariwise, it does not require additional single-point calculations and is capable of quantifying multicenter electron sharing in aromatic rings containing heteroatoms and transition metals.

Since our method allows for an exact reproduction of the MCIp component in the minimal-basis calculations, it can be used to benchmark different methodologies of approximation of the multicenter electron sharing in expanded porphyrins and macrocycles. This particular is a part of the ongoing [26–30] study involving the recently proposed EDDB, AV1245, and AVmin indices.

3 | METHODS

In this study the exact (full basis) KMCI and MCI indices were calculated within the representation of natural atomic orbitals (NAO)[31] while the corresponding NPP-based multicenter indices were determined using the NAO-subset of atomic orbitals called the natural minimal basis (NMB); both NAO and NMB subsets of orthogonal atomic orbitals are available for most of the popular quantum-chemistry software through NBO[32] and JaNPA[33] interfaces. Within the NAO representation all orbitals are automatically labeled by the NBO/JaNPA program as Cor (the core orbitals), Val (the valence orbitals), and Ryd (the Rydbergs’ ones); by definition, NMB5fCor; Valg while NAO5fNMB; Rydg. Thus, within the NPP method the Val-NMB AOs of the p-electron system can be very easily extracted from the CBO matrix by the presence of only a few nonzero coefficients of the corresponding npp,ndp, and ndd orbitals. The current implementation of the NPP approach is available on author’s personal website (the MCI‐NPP script program)[34] and it works with input files generated by the following NBO6 keywords: $NBO AONAO 5 W49 NAOMO 5 W49 $END. Electronic structure calculations were carried out at the B3LYP/6–31111G(d,p) level of the density functional theory using the Gaussian 09 code.[35] Equilibrium geometries for all studied systems were taken entirely from Supporting Information associated with Refs. 10,13,25.

ACKNOWLEDGMENTS

The author is very grateful to Prof. Miquel Sola for his critical reading of the manuscript and constant encouragement. Special thanks also to Dr. Karol Dyduch for stimulating discussions. This research was supported in part by the European Commission (H2020-RIA-INFRAIA-2016-1, Grant 730897,” HPC-Europa3”, contract: HPC17158J2), National Science Centre, Poland (NCN SONATA, Grant 2015/17/D/ST4/00558) as well as Faculty of Chemistry at Jagiellonian University in Krakow and Institute of Computational Chemistry and Catalysis at University of Girona. Calculations were partially carried out in the PL-Grid Infrastructure of the Academic Computer Centre (CYFRONET) and the Barcelona Supercomputing Center (BSC-CNS). SZCZEPANIK | 5of5

ORCID

Dariusz W. Szczepanik http://orcid.org/0000-0002-2013-0617

REFERENCES [1] F. Feixas, E. Matito, J. Poater, M. Sola, Chem. Soc. Rev. 2015, 44, 6434. [2] M. Giambiagi, M. S. de Giambiagi, C. D. dos Santos, A. P. de Figueiredo, Phys. Chem. Chem. Phys. 2000, 2, 3381. [3] P. Bultinck, R. Ponec, S. V. Damme, J. Phys. Org. Chem. 2005, 18, 706. [4] J. Cioslowski, E. Matito, M. Sola, J. Phys. Chem. A 2007, 111, 6521. [5] W. Heyndrickx, P. Salvador, P. Bultinck, M. Sola, E. Matito, J. Comput. Chem. 2011, 32, 386. [6] F. Feixas, E. Matito, J. Poater, M. Sola, J. Comput. Chem. 2008, 29, 1543. [7] M. Sola, F. Feixas, J. O. C. Jimenez-Halla, E. Matito, J. Poater, Symmetry 2010, 2, 1156. [8] F. Feixas, J. O. C. Jimenez-Halla, E. Matito, J. Poater, M. Sola, J. Chem. Theory Comput. 2010, 6, 1118. [9] F. Feixas, E. Matito, J. Poater, M. Sola, J. Phys. Chem. A 2011, 115, 13104. [10] D. W. Szczepanik, M. Andrzejak, J. Dominikowska, B. Pawełek, T. M. Krygowski, H. Szatylowicz, M. Sola, Phys. Chem. Chem. Phys. 2017, 19, 28970. [11] P. Bultinck, M. Mandado, R. Mosquera, J. Math. Chem. 2008, 43, 111. [12] F. Feixas, E. Matito, M. Sola, J. Poater, J. Phys. Chem. A 2008, 112, 13231. [13] D. W. Szczepanik, M. Sola, M. Andrzejak, B. Pawełek, J. Dominikowska, M. Kukułka, K. Dyduch, T. M. Krygowski, H. Szatylowicz, J. Comput. Chem. 2017, 38, 1640. [14] P. W. Fowler, E. Steiner, Chem. Phys. Lett. 2002, 364, 259. [15] L. Paoloni, M. S. De Giambiagi, M. Giambiagi, Atti Soc. Nat. Matematici Modena 1969, 100, 89. [16] F. London, J. Phys. Radium 1937, 8, 397. [17] G. Monaco, R. G. Viglione, R. Zanasi, P. W. Fowler, J. Phys. Chem. A 2006, 110, 7447. [18] S. Fias, S. V. Damme, P. Bultinck, J. Comput. Chem. 2008, 29, 358. [19] S. Fias, S. V. Damme, P. Bultinck, J. Comput. Chem. 2010, 31, 2286. [20] D. W. Szczepanik, J. Mrozek, J. Math. Chem. 2013, 51, 1619. [21] D. W. Szczepanik, J. Mrozek, J. Math. Chem. 2013, 51, 2687. [22] D. W. Szczepanik, J. Mrozek, J. Math. Chem. 2013, 51, 1388. [23] I. A. Popov, A. I. Boldyrev, Eur. J. Org. Chem. 2012, 2012, 3485. [24] A. Kumar, M. Duran, M. Sola, J. Comput. Chem. 2017, 38, 1606. [25] M. Mauksch, S. B. Tsogoeva, Chemistry 2010, 16, 7843. [26] D. W. Szczepanik, E. J. Zak, K. Dyduch, J. Mrozek, Chem. Phys. Lett. 2014, 593, 154. [27] D. W. Szczepanik, M. Andrzejak, K. Dyduch, E. J. Zak, M. Makowski, G. Mazur, J. Mrozek, Phys. Chem. Chem. Phys. 2014, 16, 20514. [28] M. Andrzejak, D. W. Szczepanik, Ł. Orzeł, Phys. Chem. Chem. Phys. 2015 , 17, 5328. [29] E. Matito, Phys. Chem. Chem. Phys. 2016, 18, 11839. [30] I. Casademont-Reig, T. Woller, J. Contreras-García, M. Alonso, M. Torrent-Sucarrat, E. Matito, Phys. Chem. Chem. Phys. 2018, 20, 2787. [31] A. E. Reed, R. B. Weinstock, F. Weinhold, J. Chem. Phys. 1985, 83, 735. [32] E. D. Glendening, J. K. Badenhoop, A. E. Reed, J. E. Carpenter, J. A. Bohmann, C. M. Morales, C. R. Landis, F. Weinhold, NBO 6.0, Theoretical Chemistry Institute, University of Wisconsin, Madison 2013. [33] T. Y. Nikolaienko, L. A. Bulavin, D. M. Hovorun, Comput. Theor. Chem. 2014, 1050, 15. [34] D. W. Szczepanik, The MCI-NPP script program. http://www2.chemia.uj.edu.pl/~szczepad/runeddb (accessed March 31, 2018). [35] M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Izmaylov, J. L. Sonnenberg, D. Wil- liams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Thros- sell, J. A. Montgomery, Jr, J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, T. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, D. J. Fox, Gaussian 09 (Revision D.01), Gaussian, Inc., Wallingford, CT 2010.

How to cite this article: Szczepanik DW. A simple alternative to the pseudo-p method. Int J Quantum Chem. 2018;e25696. https://doi.org/ 10.1002/qua.25696 PCCP

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Aromaticity of acenes: the model of migrating p-circuits Cite this: Phys. Chem. Chem. Phys., 2018, 20, 13430 a b c Dariusz W. Szczepanik, * Miquel Sola`, Tadeusz M. Krygowski, Halina Szatylowicz, d Marcin Andrzejak, a Barbara Pawełek,e Justyna Dominikowska, f Mercedes Kukułkaa and Karol Dyducha

In this work we extend the concept of migrating Clar’s sextets to explain local aromaticity trends in linear acenes predicted by theoretical calculations and experimental data. To assess the link between resonance and reactivity and to rationalize the constant-height AFM image of pentacene we used the electron density of delocalized bonds and other functions of the one-electron density from conceptual Received 16th February 2018, density functional theory. The presented results provide evidence for migration of Clar’s p-sextets and Accepted 25th April 2018 larger circuits in these systems, and clearly show that the link between the theoretical concept of DOI: 10.1039/c8cp01108g aromaticity and the real electronic structure entails the separation of intra- and inter-ring resonance eﬀects, which in the case of [n]acenes (n = 3, 4, 5) comes down to solving a system of simple rsc.li/pccp linear equations.

Introduction be well-explained by Clar’s aromatic p-sextet rule.3 Contrariwise, species from the third group are usually described by a super- The aromatic p-sextet rule, originally proposed by Erich Clar,1,2 position of equivalent Clar structures, giving rise to the so called allows one to rationalize the reactivity of a great number of ‘migrating’ p-sextet. In such cases the original formulation of polybenzenoid hydrocarbons by qualitative assessment of their the aromatic p-sextet rule may not give a clear answer to which global and local aromatic character.3,4 It states that the electronic ring is more stabilized than the other (each position of the structure of a polycyclic benzenoid system is predominated by migrating p-sextet is equally likely).3 Probably the best example the so called Clar structure, i.e. the Kekule´ resonance form is the class of [n]acenes (n = 3, 4, 5) as they have been a subject of containing the largest possible number of disjoint aromatic heated debate in the literature for a long time (‘the anthracene p-sextets (i.e. fully conjugated benzene-like moieties).2 Accordingly, problem’)5–9 owing to discrepancies between the ground-state Published on 27 April 2018. Downloaded by UNIWERSYTET JAGIELLONSKI 2/20/2020 12:14:17 PM. benzenoid species can be divided into three groups: (I) those aromaticity descriptions provided by Clar’s rule (equally aromatic containing only aromatic p-sextets and Clar’s ‘empty’ rings, e.g. rings),3 quantitative studies involving different local aromaticity triphenylene; (II) those that have p-sextets and rings with only one criteria (inner rings more aromatic than the outer ones or the double bond, e.g. phenanthrene; (III) and those containing rings contrary),5–17 and the experimentally assessed reactivity (inner rings with two double bonds, e.g. anthracene.3 Benzenoid systems from more reactive towards addition and Diels–Alder reactions).18–24 In the first two groups are characterized by a unique Clar structure this work we reconcile these seemingly conflicting descriptions by with well-localized p-sextets and their physical and chemical introducing a simple extension of the qualitative concept of Clar’s properties have been proved experimentally and theoretically to migrating p-sextets and we use it to rationalize the relationship between local aromaticity and the actual reactivity of anthracene, a K. Gumin´ski Department of Theoretical Chemistry, Faculty of Chemistry, tetracene, and pentacene. Jagiellonian University, Gronostajowa 2, 30-387 Krako´w, Poland. One of the fundamental limitations of Clar’s rule is that E-mail: [email protected] it excludes from considerations any other fully-conjugated b Institut de Quı`mica Computacional i Cata`lisi, Universitat de Girona, p p C/Maria Aure`lia Capmany, 69, 17003 Girona, Catalonia, Spain circuits except -sextets (6 ), thus it is unable to describe the c Department of Chemistry, University of Warsaw, Pasteura 1, resonance eﬀects between adjacent benzenoid units, i.e. the 02-093 Warszawa, Poland inter-ring resonance. Consequently, the predicted global aromaticity d Faculty of Chemistry, Warsaw University of Technology, Noakowskiego 3, of the polybenzenoids may be systematically underestimated 00-664 Warszawa, Poland (especially for systems from the third group of Clar’s classification).25 e Department of Plant Cytology and Embryology, Institute of Botany, Jagiellonian University, Gronostajowa 9, 30-387 Krako´w, Poland Moreover, it has been shown many times by statistical analyses that f Theoretical and Structural Chemistry Group, Faculty of Chemistry, by taking into account higher p-electron circuits, i.e. p-dectets (10p), University of Lodz, Pomorska 163/165, 90-236 Ło´dz´, Poland p-tetradectets (14p), etc., one can explain inconsistencies between

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qualitative pictures of local aromaticity based on magnetic and patterns of relative local aromaticity {[1...1]m}equalsexactlythe electronic-structure criteria.26–28 Generally, there is no universal number of symmetry unique ring positionsineachparticular method to determine contributions from higher p-electron acene. For instance, in the case of pentacene with exactly three circuits and thus to investigate the inter-ring resonance effects symmetry unique rings we have three different patterns, i.e. 25 in polycyclic benzenoid systems. Linear acenes, however, are [11111]6, [12221]10/18, and [12321]14, resulting from the migration very specific in this context, because here p-sextets ‘migrate’ of the 6p-, 10p/18p-, and 14p-circuits, respectively. Here, we throughout the entire system of fused rings giving rise to principally do not take into account the largest 22p-circuit since

uniform-ratio patterns of relative local aromaticity: [111]6 for it does not represent any positional isomerism. Moreover, it can be

anthracene, [1111]6 for tetracene, and [11111]6 for pentacene shown that the p-conjugation along the 22-membered pathway is (see the first column in Fig. 1). To include explicitly the inter- much less effective than in the 6-membered units at any ring ring resonance effects in acenes, a natural extension of Clar’s position. Indeed, within the framework of the recently proposed original concept has to be introduced that considers similar NPP method,29 the predicted p-conjugation in the central ring in positional isomerism for higher-order circuits – see columns pentacene is about five times more effective than that in the 2–5 in Fig. 1. In contrast to the benzene-like moieties, super- perimetric circuit;30 for the same reason one can also exclude from position of the isomeric structures of fully p-conjugated m-circuits further considerations the largest p-circuits in anthracene and (m =10,14,...), except the largest (perimetric) ones, introduces tetracene. Admittedly, in the light of the findings by Bultinck diversity into the relative ring aromaticity. The number of possible et al.,26–28 who demonstrated that it is possible to nearly Published on 27 April 2018. Downloaded by UNIWERSYTET JAGIELLONSKI 2/20/2020 12:14:17 PM.

Fig. 1 Schematic representation of migrating p-sextets and higher circuits in [n]acenes, n = 3, 4, 5, with the resulting local aromaticity ratios.

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quantitatively reproduce the ring-current maps of a large intra-ring (6p) resonance. As regards the anthracene and number of aromatic polycycles by involving 6p-, 10p-, and tetracene p-systems, the model of migrating p-circuits relies 14p-circuits only, one could also remove the 18p-circuits from on the following equations: ! ! ! the model. But for our purposes it is actually not necessary 1 1 2 W6 RI to distinguish between 10p- and 18p-circuits as they both ¼ ; and (2) contribute to the inter-ring resonance only. 11 W10 1 ! ! ! 1 1 2 W6 RI Methodology ¼ ; (3) 11 W10;14 1 Within the proposed model diﬀerent positional-isomeric structures of fully conjugated p-circuits give rise to essentially two types of respectively. patterns with fixed local aromaticity ratios: (a) a uniform pattern Solving eqn (1)–(3) for given R-values from local aromaticity representing intra-ring (Kekule´-like) resonance and (b) one or more calculations allows one to determine if and to what extent the diversified patterns representing inter-ring eﬀects. In principle, the inter-ring resonance effects contribute to the global aromaticity contribution of each pattern to the picture of global aromaticity of of acenes within the framework of a particular aromaticity

linear acenes is not implicitly determined, but within the framework criterion. That is, the greater W6, the lower the magnitude of of a particular aromaticity criterion it can be easily calculated by the inter-ring resonance effects and hence the more reliable the solving a system of linear equations. original p-sextet model by Clar. To show how this simple For illustration, let us focus again on the pentacene p-system extension of Clar’s concept works in practice we selected six

with three diﬀerent patterns: [11111]6, [12221]10/18, and [12321]14. quantities representing electronic (IR, PDI, HRCP, and EDDB), After normalization with respect to the most aromatic ring (in structural (HOMA), and magnetic (NICS) criteria of aromaticity; this case – the central one) each of these three patterns can be a brief description of these quantities with full references can 31 represented by the row vector of numbers {Rm}referringonlyto be found in the review by Feixas et al. and in our recent the symmetry unique ring positions from the outermost to the comparative studies.32,33 To determine R-values we used data center ones, [11111]6 (1,1,1)6, [12221]10/18 (1/2,1,1)10/18,and from an independent benchmark study for all of the above- [12321]14 (1/3,2/3,1)14. One can directly compare these mentioned descriptors at the density functional theory level numbers with the corresponding (normalized) relative local (available in the comprehensive Supplementary Information aromaticity ratios based on a particular aromaticity criterion. associated with the paper);32 specifically, we chose the oB97X/ E.g., the para-delocalization index (PDI)31 predicts the local cc-pVTZ method as it provides the most reliable description of aromaticities in pentacene (going from the terminal to the electron delocalization and the best (of all tested methods) central ring) to be:32,33 0.057, 0.063, and 0.066, respectively. reproduction of experimental geometries and ionization potentials So, after normalization (i.e. after dividing these numbers by of all three acenes.32 0.066) we get the following R-vector: (0.87, 0.95, 1.00)PDI. The observed increase of the deviation from uniformity going from the central to the terminal ring indicates a non-negligible Results and discussion

Published on 27 April 2018. Downloaded by UNIWERSYTET JAGIELLONSKI 2/20/2020 12:14:17 PM. contribution of inter-ring effects, especially those associated The results presented in Fig. 2 and Table 1 indicate that, with the migration of the 14p-circuit (the [12321]14 pattern). despite the apparent qualitative similarities in local aromaticity However, to determine quantitatively the contributions {Wm}of trends shown by most of the indices, the calculated contributions particular patterns to the global p-electron resonance in penta- of the corresponding p-electron circuits diﬀer dramatically cene we have to solve a system of linear equations that in matrix (Table 1) and divide the aromaticity descriptors into two groups. notation can be compactly represented as: 0 10 1 0 1 1 1 1 W6 RI B 2 3 CB C B C B CB C B C B 2 CB W C B R C: @ 113 A@ 10;18 A ¼ @ II A (1)

111 W14 1

In the above equation, the columns of the fitting matrix collect

the R-vectors associated with patterns [11111]6, [12221]10/18

and [12321]14 from our model, the W-vectors collect the corresponding weights, while the R-values on the right side of the equation represent renormalized local aromaticities from

calculations, in our case RI = 0.87 and RII = 0.95. By solving the above equation we get W6 = 0.79, W10/18 = 0.06, and W14 = 0.15, Fig. 2 Relative ring aromaticities (Ri 100%) from DFT calculations which clearly show that PDI seems to be much in favor of the involving diﬀerent aromaticity criteria; computational data were taken classical (local) model by Clar with a predominant role of the from the Supplementary Information associated with ref. 32.

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Table 1 The percentages of individual p-circuits (Wm 100%) obtained by The experiment seems to be in favor of the second group solving eqn (1)–(3) with R-values taken from Fig. 2 of aromaticity indicators. For instance, let us focus on the Anthracene Tetracene Pentacene pentacene molecule. Apart from the structural and energetic properties discussed elsewhere,32 the experimental 1H NMR Index W W W W W W W 6 10 6 10/14 6 10/18 14 spectrum obtained by Nagano et al.52 reveals that the aromatic IR 86 14 80 20 77 2 21 protons in pentacene are shifted downfield by about 3.7 ppm PDI 88 12 82 18 79 6 15 (central ring), 3.3 ppm (inner rings) and 1.9 ppm (terminal HRCP 98 2 98 2 98 2 0 53 NICS(1)zz 40 60 20 80 9 46 45 rings) from the reference chemical shift of non-aromatic HOMA 44 56 26 74 15 46 39 protons in 1,3-cyclobutadiene (on average, d = 5.5 ppm).54 EDDB 36 64 32 68 21 46 33 H,av Solving eqn (1) with these particular numbers (appropriately

normalized to the corresponding R-values) gives W6 = 13%,

W10/18 = 54%, and W14 = 33%, which perfectly agrees with the

The first group of quantities, i.e. IR,PDI,andespeciallyHRCP, theoretical predictions from NICS(1)zz, HOMA, and EDDB. But preserves to a large extent the local (benzene-like) nature of even more impressive evidence for the overwhelming non-local aromatic stabilization and seems to only echo the conjugation character of the resonance eﬀects in pentacene is provided by eﬀects between adjacent rings in all cases (up to about 20% for an image of pentacene from constant-height atomic force pentacene). These indices give the largest weight to 6p-circuits. microscopy (AFM) obtained by Gross et al.55 Admittedly, it This result is in line with the Glidewell–Lloyd rule,19 an extension has been demonstrated many times that the AFM as well as of Clar’s rule, stating that the total population of p-electrons in the scanning tunneling microscopy (STM) images of polycyclic fused conjugated polycyclic systems that have a closed-shell aromatic hydrocarbons show patterns that resemble very much singlet ground state tends to form the smallest 4n + 2 groups.4 the unique Clar aromatic structures (i.e., rings with localized Moreover, the result of the first group of indices may be regarded Clar p-sextets are indeed better marked in the STM/AFM 56–62 as reasonable as aromaticity indicators like IR and PDI principally images). But as shown in Fig. 3, within the AFM image of refer only to the cyclic delocalization of electrons solely within a pentacene the most visible are the outer edges of the terminal particular ring (the intra-ring resonance),31 and they only indirectly rings suggesting the strong localization of electrons in these take into account inter-ring resonance effects, since equilibrium fragments (cf. the AFM image of hexabenzenocoronene).61 It geometries (for which the indices are computed) are determined turns out that this characteristic pattern can be rationalized by the balance between electrostatics, Pauli repulsion, charge transfer, s-system contributions, and other effects that may principally have a non-local nature.34 All these factors introduce structural differentiation into the consecutive ring positions in acenes making the C–C bond lengths more equalized in the inner rings rather than in the outer ones.32 This is particularly confirmed by HOMA, which directly quantifies aromaticity in terms of bond-length alternation.35 As shown in Fig. 2 and Table 1, HOMA indeed predicts the significant diversity of ring Published on 27 April 2018. Downloaded by UNIWERSYTET JAGIELLONSKI 2/20/2020 12:14:17 PM. aromaticity in acenes (e.g. the terminal rings in pentacene are only half as aromatic as the central one). Thus, according to the presented model, inter-ring resonance effects seem to rule the roost in this case (up to 85% for pentacene). In fact, HOMA has already been shown by graph-theoretical methods to exhibit a non-local characteristic in polycyclic aromatic systems.36–39 But the most surprising fact is that, despite methodological differences,

also NICS(1)zz and EDDB show nearly the same magnitude of non- local resonance contributions. On average, aromaticity indices from the second row in Fig. 2 predict the predominating role of inter-ring resonance effects from about 60% in anthracene through 75% in tetracene up to 85% in the case of pentacene.

Admittedly, indices like NICS(1)zz tend to be criticized severely as aromaticity measures owing to their proven non-local character caused by mixing of different ring currents.40–46 But, since HOMA and EDDB (by definition, a global measure of aromaticity and resonance)47–51 quantitatively confirm the predominant role of

inter-ring effects in acenes, the non-locality manifested by Fig. 3 ‘‘Images’’ of pentacene obtained from experiment and quantum- NICS(1)zz cannot be further regarded only as a methodological chemical calculations involving p-electron density. The AFM image is artifact,42–46 at least in the case of acenes. reproduced from ref. 55 with permission from AAAS.

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within the framework of the recently proposed electron density identified so far. This is mainly because most of the previous partitioning scheme,50 in which one extracts from the total studies were routinely determined at the DFT theory level with- molecular density ED(r) the components representing localized out long-range exchange corrections and thus, in accordance bonds, EDLB(r), and delocalized ones, EDDB(r). The latter with our recent findings,32 their quantitative conclusions about function is congeneric with the EDDB index and provides a the role of non-local resonance effects in acenes should be taken global view on the resonance effects in molecular systems from with caution. The results presented in this work are based on the covalent bond-orbital projection perspective.63–69 long-range corrected DFT calculations involving the exchange– Even a cursory look at the EDLB(r) and EDDB(r) maps correlation functional that provides the best reproduction of (calculated at the same theory level as the aromaticity descriptors) most of the molecular properties characteristic of aromatic presented in Fig. 3 shows that all the inter-ring contributions to species.32,33 Surprisingly, our results clearly show all but quantitative resonance concern mainly the central anthracyclic unit, while the compatibility between magnetic (NICS), structural (HOMA) and outermost rings preserve to a large extent the electronic structure electronic (EDDB) aromaticity descriptors as well as the experi- of two well localized double bonds (the corresponding CQC mental NMR data, especially for the pentacene molecule. It is bond length is 1.35 Å, from both calculations and experiment, worth mentioning that practical use of the model of migrating which in turn is very close to the experimental bond length of the p-circuits is limited to molecular fragments containing at most localized double bond in trans-butadiene, i.e. 1.34Å).32 A striking five linearly fused benzenoid units, because [n]acenes with n 4 5are resemblance between the experimental AFM image and the known to be unstable in their closed-shell singlet state.73 calculated EDLB(r) map, supported by the results collected in Table 1, leaves no doubt that inside the pentacene molecule electrons are delocalized over much larger distances than a single Conflicts of interest hexagon. Furthermore, along with the expansion of the volume There are no conflicts to declare. occupied by the electrons delocalized through the inter-ring resonance between the central ring and the inner ones, one should also expect an increase of polarizability in this area.70 Acknowledgements Indeed, theoretical calculations based on the DFT concept of the so called local softness S(r),71,72 depicted in Fig. 3, clearly show The research was supported in part by the European Commission that in pentacene carbon atoms in the central anthracycle (and (H2020-RIA-INFRAIA-2016-1, grant 730897, ‘‘HPC-Europa3’’, con- especially in the central ring) are the most reactive centers in the tract: HPC17158J2, DS), the National Science Centre, Poland (NCN entire molecule, at least in the context of the maximum hardness SONATA, grant 2015/17/D/ST4/00558, DS) as well as the PL-Grid principle.72 In the light of experimental data, this is directly Infrastructure of the Academic Computer Centre CYFRONET with confirmed by the fact that the central ring in pentacene (but also the calculations performed on the cluster platform ‘‘Prometheus’’. in anthracene and tetracene) is protonated, adds bromine, and Calculations were also partially carried out using resources undergoes Diels–Alder reactions readily.18–24 Other authors have provided by the Wroclaw Centre for Networking and Super- attributed the larger reactivity of the central ring of pentacene to computing (http://wcss.pl), grant No. 118 (JD). MS thanks for the the generation of an extra p-sextet (and also an extra p-dectet in support of the Ministerio de Economıà y Competitividad of our model) in the final adduct.3,19,21,23 Spain (Project CTQ2017-85341-P), Generalitat de Catalunya Published on 27 April 2018. Downloaded by UNIWERSYTET JAGIELLONSKI 2/20/2020 12:14:17 PM. (project number 2014SGR931, Xarxa de Refere`ncia en Quı`mica Teo`rica i Computacional, and ICREA Academia prize), and European Fund for Regional Development (FEDER grant Conclusions UNGI10-4E-801). HS thanks the Warsaw University of Technology To summarize, in this work we have demonstrated using the for supporting this work. JD acknowledges financial support model of migrating p-circuits that the local aromaticity concept, from the National Science Centre of Poland (Grant No. 2015/19/ intuitively connected with the cyclic delocalization of electrons, B/ST4/01773). is actually of minor importance in the context of the structural properties and reactivity of [n]acenes (n = 3, 4, 5). The results of Notes and references theoretical calculations supported by experimental data provide evidence for the migration of Clar’s p-sextets and larger circuits 1 E. Clar, Polycyclic Hydrocarbons, Academic Press, London, in these systems, and clearly show that the link between the 1964. theoretical concept of aromaticity and the real electronic structure 2 E. Clar, The Aromatic Sextet, Wiley, New York, 1972. (revealed by the constant-height AFM imaging technique) entails 3 M. Sola`, Front. Chem., 2013, 1, 22. the separation of intra- and inter-ring resonance effects, which in 4 O. El Bakouri, J. Poater, F. Feixas and M. 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13436 | Phys. Chem. Chem. Phys., 2018, 20, 13430--13436 This journal is © the Owner Societies 2018 DOI: 10.1002/open.201900014 Full Papers

1 2 3 Electron Delocalization in Planar Metallacycles: Hückel or 4 5 Möbius Aromatic? 6 [a, b] [a] 7 Dariusz W. Szczepanik* and Miquel Solà* 8 9 10 In this work the relationship between the formal number of π- electron density of delocalized bonds (EDDB) method can 11 electrons, d-orbital conjugation topology, π-electron delocaliza- successfully be used not only to quantify and visualize 12 tion and aromaticity in d-block metallacycles is investigated in aromaticity in such difficult cases, but also – in contrast to 13 the context of recent findings concerning the correlation of π- magnetic aromaticity descriptors – to provide a great deal of 14 HOMO topology and the magnetic aromaticity indices in these information on the real role of d-orbitals in metallacycles 15 species. It is demonstrated that for π-electron rich d-metalla- without the ambiguity of bookkeeping of electrons in the π- 16 cycles the direct link between aromaticity, the number of π- subsystem of the molecular ring. Interestingly, some of the 17 electrons and the frontier π-orbital topology does not strictly metallacycles studied cannot be classified exclusively as Hückel 18 hold and for such systems it is very difficult to unambiguously or Möbius because they have a hybrid Hückel-Möbius or even 19 associate their aromaticity with the “4n+2” (Hückel) and “4n” quasi-aromatic nature. 20 (Möbius) rules. It is also shown that the recently proposed 21 22 1. Introduction chemical shift (NICS),[4] should be taken with caution.[5] On the 23 other hand, aromaticity descriptors that depends on the 24 Over the last decades the transition-metal metallacycles has definition of a reference system, such as the harmonic-oscillator 25 been receiving a substantially increasing attention as important model of aromaticity (HOMA)[6] or the aromaticity fluctuation 26 catalyst precursors and intermediates in organometallic index (FLU),[7] cannot straightforwardly be used due to lack of 27 chemistry.[1,2] From the molecular-structure perspective, a d- parameterization for bonds with metal atoms. Calculations of 28 metallacycle can be regarded as a derivative of the carbocyclic aromatic stabilization energies (ASE),[8] in turn, usually requires 29 system with at least one carbon atom replaced by the d-block design of different isomerization reaction scenarios, which may 30 metal atom. Although d-metallacycles were many times con- lead to a lot of arbitrariness and makes it limited in 31 firmed experimentally to be more reactive than their aromatic applications.[1f] One of the most promising aromaticity descrip- 32 hydrocarbon precursors,[2] the cyclic delocalization of π-elec- tors in the context of organometallic systems is the multicenter 33 trons has been considered to play a crucial role in determining index (MCI).[9] MCI quantifies the effect of cooperativity of all 34 their physicochemical properties from the very beginning.[3] ring members in the cyclic delocalization of π-electrons (a 35 Unfortunately, semiquantitative (aromaticity is a concept and multicenter sharing of π-electrons); it is a non-reference 36 not an observable in a strict quantum-mechanical sense) quantity that can be calculated from both the molecular wave 37 analysis of aromatic stabilization in metallacycles faces serious function as well as the n-electron density.[9] Unlike with other 38 difficulties.[1f] In particular, anisotropy of the metal center, non- descriptors, the multicenter index enables one to study almost 39 local character and influence of ligands on diatropic ring all types of aromaticity that can be found in the literature, 40 current are the main reasons why the aromaticity predictions including the all-metal aromaticity,[10] and it is one of the few 41 based on magnetic criteria, such as the nucleus-independent descriptors that successfully passed a set of rigorous tests for 42 aromaticity indices, designed by our group.[11] Unfortunately, 43 the calculations of MCI face several challenges such as high 44 [a] Dr. D. W. Szczepanik, Prof. M. Solà computational cost, numerical accuracy problems, ring-size 45 Institut de Quìmica Computacional i Catàlisi and Departament de Química, extensivity issue,[5l] and considerable sensitivity to the level of 46 Universitat de Girona, the theory (and even the choice of the exchange-correlation 47 C/ Maria Aurèlia Capmany, 69, 17003 Girona, Catalonia, Spain E-mail: [email protected] functional at the DFT level).[10,12] 48 [email protected] Despite the difficulties with quantification of aromaticity, 49 [b] Dr. D. W. Szczepanik also the nature of the aromatic stabilization in d-block metal- 50 K. Guminski Department of Theoretical Chemistry Faculty of Chemistry, Jagiellonian University loaromatic systems itself remains under continuous debate,[1f] 51 Gronostajowa, 2, 30-387 Kraków, Poland. because it can involve contribution of different d-orbitals of the 52 An invited contribution to a Special Collection dedicated to Computational transition metal to the π-electron delocalization.[3a] As shown in 53 Chemistry ©2019 The Authors. Published by Wiley-VCH Verlag GmbH & Co. KGaA. Figure 1a, chemical bonding between transition metal and the 54 This is an open access article under the terms of the Creative Commons adjacent atoms allows two different topologies of d-orbital 55 Attribution Non-Commercial License, which permits use, distribution and conjugation within the system of molecular π-orbitals (MO): 1) 56 reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes. the π-type (Hückel) topology without phase inversion, involving 57

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we will use the test set of metallamonocycles (in silico designed 1 by Mauksch and Tsogoeva in their seminal proof-of-concept 2 work)[13] and several representative d-metallabicyclic aromatics. 3 4 5 2. Methodology 6 7 To quantify cyclic delocalization of π-electrons in metalloaro- 8 matic rings we used the electron density of delocalized bonds 9 (EDDB).[16] The EDDB method has originally been proposed to 10 facilitate visualization and quantitative study of chemical 11 resonance and multicenter bonding in molecular systems with 12 different topology and size.[16b] It has recently been demon- 13 strated, however, that also the quantitative predictions of 14 aromaticity by EDDB are in excellent agreement with a wide 15 range of descriptors based on structural, magnetic, and 16 electronic-structure criteria of aromaticity.[17] There are several 17 important features, that set the EDDB method apart from other 18 aromaticity descriptors: 1) EDDB does not suffer from the ring- 19 Figure 1. a) Schematic representation of two different topologies of π-MOs size extensivity issue and can be used to study electron 20 involving d-orbitals. b) The approximate relationship between orbital delocalization in any type of aromatic system regardless of its 21 occupation number ni, valency Vi, and the binary (Shannon) entropy Hi within the closed-shell π-electron system. size and topology (in contrast to NICS); 2) EDDB does not 22 depend upon parametrization to the reference model system 23 (in contrast to HOMA and FLU); 3) EDDB provides aromaticity 24 the d metal orbital, and 2) δ-type (Möbius) topology, in which predictions very similar to MCI but it is much less computation- 25 yz d metal orbital acts as a “phase switch” allowing cyclic ally expensive and does not share the numerical-accuracy 26 xz delocalization to fall into the opposite phase side over the cyclic problems;[10a] 4) EDDB enables one to quantify cyclic delocaliza- 27 unit.[13] In contrast to typical twisted Möbius hydrocarbon tion of electrons within the framework of the first-order 28 aromatics, both the MO topologies co-exist within the same π- population analysis (the number of electrons delocalized 29 system and deciding which of them determines aromaticity in through the system of conjugated bonds), so the results are 30 particular case is not straightforward. Interestingly, Mauksch much easier to interpret than those from other approaches.[16b] 31 and Tsogoeva have recently shown that there is a relationship The EDDB(r) function is defined in the basis of Weinhold’s 32 between the magnetic aromaticity by NICS and the topology of natural atomic orbitals (NAO)[18] as 33 the highest occupied molecular π-orbital (π-HOMO), which 34 X comes down to the following rule:[14] EDDBðrÞ ¼ cyðrÞDDBc ðrÞ, 35 m mn n ð1Þ “The metallacycle is aromatic (antiaromatic) when the number m;n 36 of π-MOs is even and the π-HOMO is of Möbius (Hückel) 37 topology-and vice versa when the number of π-MOs is odd”. where the corresponding DB-density matrix reads: 38 This frontier π-orbital topology rule, however, depends " # 39 1 XW y upon bookkeeping of electrons in the π-system (which itself DDB ¼ D C e l C D: ð2Þ 40 2 ab ab ab ab can be very difficult even for planar rings due to the extensive a;b6¼a 41 π-MO delocalization nature) and it does not explicitly take into 42 account the effect of cyclic delocalization of electrons. It should In the above equation D represents the standard one- 43 be emphasized that, from the electronic point of view, electron density matrix, Cαβ collects linear-combination coef- 44 aromaticity is always a result of specific interference between ficients of the orthogonalized two-center bond orbitals 45 [16a,19] molecular orbitals and, according to the balance-equivalence (2cBO), λαβ is a diagonal matrix collecting the correspond- 46 [15] theorem by Klein and Balaban, the link between size of the π- ing 2cBO squared occupation numbers, ɛαβ is a diagonal matrix 47 system and its aromaticity (antiaromaticity) holds strictly only of the bond-orbital delocalization factors (ɛαβ is close to 1 for 48 for the annulene-like systems. This is because annulene delocalized bonds and approaches 0 for the localized ones), 49 represents a perfect balance between the number of occupied and Ω={(Xα, Xβ)} denotes a set of all the atomic pairs in a 50 [16a,b] bonding and unoccupied antibonding valence π-MOs or, molecular ring. The definition of ɛαβ involves a series of 51 equivalently, the number of π-electrons equals the number of projections of 2cBO onto their three-center counterparts (3cBO), 52 2p orbitals. followed by the projection onto the occupied molecular orbitals 53 z In this work the link between the number of electrons in (MO), which significantly extends the electron delocalization 54 the π-system, d-orbital conjugation topology, electron delocal- over all the ring members; such projection cascade is deeply 55 ization and aromaticity in d-metallacycles will be studied in the rooted in the formalism of the so called orbital communication 56 [20] DB context of the frontier π-orbital topology rule.[14] In particular, theory. The trace of the D matrix from Eq. (1) can be 57

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interpreted as the population of electrons delocalized through 1 the system of conjugated bonds in the molecular ring Ω; in 2 contrast, summation (over NAOs centered on the ring members 3 only) of diagonal elements of D gives the corresponding overall 4 (natural) population of electrons in the cycle. In this work, these 5 electron populations will be denoted simply as EDDB and ED, 6 respectively. 7 For planar aromatics the subset of occupied π-MOs can be 8 used to get the corresponding π-ED and π-EDDB components. 9 Furthermore, since within the NAO representation both D and 10 DDB have the same atomic-block structure, it is possible to 11 separate aromatic ring from the rest of the molecule (simply by 12 zeroing the corresponding off-diagonal elements of the density 13 matrix) and partition both the electron density, π-ED(r), and the 14 electron density of delocalized bonds, π-EDDB(r), into compo- 15 nents representing metal atom and the rest of ring members: 16 17 p-EDðrÞ ¼ p-EDMeðrÞ þ p-EDRðrÞ, ð3Þ 18 19 and 20

21 p-EDDBðrÞ ¼ p-EDDBMeðrÞ þ p-EDDBRðrÞ: ð4Þ 22 Figure 2. a) Contour plots of the occupied molecular π-orbitals in [Fe 23 Finally, diagonalization of the metal atomic-blocks of such (CH)6H2]. b,c) Results of the analysis of d-orbital contributions to π-ED(r) and 24 π-density matrices gives rise to the effective pz, dxz, and dyz π-EDDB(r) with the electron populations corresponding to the 8π- (black) [21] 25 atomic orbitals polarized within the molecular environment. and 10π-system (grey). 26 It should be noticed that the orbital decomposition of π-ED 27 leads to the overall orbital populations (occupation numbers), 28 ni, while in the case of π-EDDB we get the corresponding orbital heptatriene ring can be regarded as a 8π system. Indeed, the 29 contributions to the cyclic delocalization of π-electrons in the electron population analysis based on the LCAO π-MO 30 aromatic ring, θi. One should realize that for an idealized coefficients (Figure 2b) shows that 5.970jej is assigned to the 31 aromatic ring the latter is very close to the orbital valency, Vi, carbon fragment (ED ) while 1.922jej comes from the metal 32 which in the case of closed-shell π-systems at the one- R atom (ED ). The orbital-decomposition of ED reveals that it is 33 determinant theory level reads: Me Me predominated by the 3d orbital (1.697jej, i.e. ~88%) with 34 xz q � V ¼ 2n À n 2: ð5Þ only a small admixture of 3d (0.225jej, i.e. 12%), which makes 35 i i i i yz this particular metallacycle a perfect candidate for being a 36 The above relation follows directly from duodempotency of Möbius aromatic system. In fact, aromaticity in [Fe(CH) H ] 37 6 2 the π-density matrix and strictly holds for unhybridized orbitals manifests itself through significant reduction of the carbon- 38 only. As illustrated in Figure 1b, orbital valency is closely related carbon bond length alternation, HOMA=0.930, and negative 39 [19] to the Shannon (binary) entropy from information theory, Hi, value of the axial component of the nucleus-independent 40 and it reaches maximum for the atomic orbitals occupied chemical shift at 1 Å above the ring plane, NICS(1) =À 15.0.[13] 41 zz exactly by one electron (maximum uncertainty of electron= Moreover, the highest occupied molecular π-orbital is of 42 maximum valency). In other words, the contribution of each of Möbius type (see Figure 2a), which, in accordance with the 43 these orbitals to the π-delocalization in a closed-shell aromatic recent findings on the relationship between NICS and π-HOMO 44 ring is always lower than or at most equal to 1 electron. This topology,[14] seems to confirm the existence of Möbius 45 fact is of crucial importance in the context of π-conjugation aromaticity in [Fe(CH) H ]. It should be noticed that, although 46 6 2 topology in aromatic d-metallacycles and it will we discussed in including HOMO-3 in the π-electron population analysis 47 the next section. increases the 3d orbital occupation number for about four 48 yz times (grey numbers in Figure 2b), it does not change the final 49 conclusion about predominating role of the δ-conjugation 50 3. Results and Discussion topology in this particular d-metallacycle. 51 Figure 2c, in turn, presents the results of the corresponding 52 Let us begin with the model of hypothetical ferracyclohepta- partition and orbital-decomposition of the π-EDDB(r) function. 53 triene, [Fe(CH) H ],[13] with the π-system consisting of 5 doubly The resulting EDDB populations indicate that about 69% (i.e. 54 6 2 occupied MOs (Figure 2a); however, since HOMO-3 is in 99% 5.409jej) of the total population of π-electrons in the aromatic 55 localized on the metal atom, it does not contribute to the cyclic ring is delocalized and the contribution from 3d to the cyclic 56 xz delocalization of π-electrons and, consequently, the ferracyclo- delocalization is up to 0.574jej (i.e. only 33% of the 57

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corresponding orbital occupation number). Although these 1 Table 1. Averaged NICS(1)zz (in ppm), HOMA and the results of the EDDB- electron populations significantly differ from the ED-based based partition and orbital-decomposition (in jej) for d-metallacycles 1– 2 a ones, they seem to be quite reasonable as long as EDDB is 15. 3 considered a quantitative criterion of aromaticity. In fact, NICS(1) HOMA EDDB EDDB EDDB 4 zz R Me metallacycles are commonly known to be less aromatic than σ π δ 5 their classical counterparts, and it has recently been shown 1 À 4.9 0.984 4.415 3.634 0.342 0.327 0.098 6 using the EDDB method that in the case of benzene about 90% 2 À 23.0 0.955 5.562 4.710 0.115 0.326 0.406 7 3 À 27.6 0.971 5.976 4.811 0.142 0.479 0.539 of electrons are delocalized within the π-system.[17a] In this 8 4 À 4.9 0.925 5.541 4.355 0.138 0.433 0.611 context, the cyclic delocalization of electrons in [Fe(CH) H ] is at 5 À 15.0 0.930 6.017 5.068 0.112 0.254 0.578 9 6 2 6 À 25.6 0.970 5.465 4.722 0.210 0.232 0.297 least 20% less effective than in C6H6. On the other hand, the 10 7 À 40.3 0.935 5.620 4.584 0.398 0.225 0.391 orbital valency approximated by Eq. (5), V =2×1.7–1.72 �0.5, 11 3dxz 8 À 9.4 0.973 5.297 4.524 0.221 0.276 0.196 is very close to the corresponding eigenvalue of the π-EDDB 9 À 9.5 0.947 5.425 4.684 0.543 0.057 0.137 12 Me 10 À 27.4 0.980 6.781 5.544 0.248 0.272 0.714 density matrix, #3dxz = 0.574, which clearly shows that the 3dxz 13 11 À 57.0 0.946 7.119 5.966 0.084 0.104 0.962 orbital contributes to electron delocalization to very limited 14 12 À 19.7 0.958 7.076 5.996 0.266 0.144 0.665 extent (and hence it preserves to some degree the non-bonding 13 À 65.3 0.958 7.808 6.389 0.144 0.255 0.974 15 nature). Moreover, since the π-electron delocalization in 14 À 8.2 0.957 7.782 6.242 0.151 0.415 0.932 16 15 À 1.2 0.945 7.220 5.955 0.189 0.221 0.849 aromatics relies on the cooperativity of all ring members in the 17 a electron sharing, one should expect the sum of # and # The NICS(1)ZZ values taken from Ref. [13] 18 3dxz 3dyz to be more or less equal to the average EDDB population per 19 carbon atom; indeed, the former is 0.722jej while the latter is 20 0.781jej (for comparison, in the case of benzene, the π-MOs may be not possible. To demonstrate the performance 21 corresponding π-EDDB population per each ring member is of the EDDB-based orbital-decomposition method in such cases, 22 0.889jej). The reduced aromaticity in the ring is in line with let us consider a test set of 15 d-metallacycles in silico designed 23 noticeable alternation of the calculated NAO-based Wiberg π- by Mauksch and Tsogoeva,[13] but re-optimized without symme- 24 bond orders,[20] i.e. 0.296, 0.479, 0.386, and 0.458 (going from try and geometrical constrains forcing planarity. The results of 25 the FeÀ C bond). Furthermore, the EDDB analysis involving the EDDB partition and orbital-decomposition are collected in 26 separate subsets of the Möbius- and Hückel-type π-MOs reveals Table 1 and depicted in Figure 3; the corresponding values of 27 that their mutual interference leads to ~23% drop of the metal HOMA (based only on the C–C bonds) and NICS(1) (taken from 28 zz contribution to aromaticity, i.e. from 0.938jej (π-EDDB +π- Ref. [13]) has been added for comparison. As regards the orbital 29 Me,δ EDDB ) to 0.722jej (π-EDDB ); in particular, it decreases decomposition, only the first five highest occupied eigenvectors 30 Me,π Me, δ+π # for about 31% (i.e. from 0.833jej to 0.574jej) and of the EDDB matrix were analyzed reproducing in most cases 31 3dxz Me increases # for nearly 41% (i.e. from 0.105jej to 0.148jej). (except 8) up to 99% of the overall EDDB population (see 32 3dyz Me The opposite effect is observed for the carbon fragment where Figure 3b). It should be noted that in several cases (especially 33 the interference of the π-MO subsets of different types molecular rings significantly distorted from planarity) metal 34 reinforces the cyclic delocalization of electrons for nearly 70% contributes to electron delocalization mainly through the σ- 35 (from 2.756jej to 4.687jej). All in all, the results presented in delocalization involving d orbitals, while the π-delocalization 36 x2-y2 Figure 2c show that in [Fe(CH) H ] 3d prevails over 3d as involves additionally d and p orbitals along with the d 37 6 2 xz yz z2 z yz regards participation in cyclic delocalization of electrons, which ones.[1h] 38 allows one to draw the conclusion that [Fe(CH) H ] is indeed a At first glance, it is clear from Table 1 that all the indices 39 6 2 Möbius aromatics (with minor ‘assistance’ of the π-conjugation predict metallacycles 1–15 to be aromatic (negative NICS 40 topology). Moreover, although the aromatic ring in ferracyclo- values, HOMA close to unity, and the EDDB populations in the 41 heptatriene seems to represent a perfect balance between the range of 4.4–7.8jej), although the dramatic deviations of NICS 42 number of electrons and atomic orbitals within the π-system (1) – even for metallacycles of the same type and size (e.g. 13 43 zz (8:8), it is much less aromatic than benzene because of the π/δ and 14) – are difficult to explain; thus, the results of aromaticity 44 anti-cooperativity and, consequently, limited participation of quantification based on NICS(1) should be taken with 45 zz the metal 3d and 3d orbitals in the cyclic delocalization of caution.[5] Admittedly, the direct comparison of HOMA and 46 xz yz electrons. In fact, both d-orbitals together are in total less EDDB misses the point since the systems consist of different 47 R effectively conjugated with other AOs in the metallacyclic ring numbers of atoms and π-electrons, however, the averaged 48 than a single 2p orbital in benzene. EDDB populations calculated per carbon atom for 6-, 7-, and 8- 49 z R It should be noticed that, in contrast to the ED-based results membered d-metallacycles from Table 1 represent respectively 50 depicted in Figure 2b, the EDDB analysis gives rise to practically about 82%, 90%, and 98% of the corresponding EDDB per 51 R the same picture of cyclic delocalization of π-electrons regard- ring-member population in benzene (0.889jej per each C). 52 less of the assumption about the size of the π-MO subspace Thus, the overall population of π-electrons delocalized in 53 (i.e. with or without HOMO-3) – the total EDDB populations metallacycles 1–15 increases with the number of ring members, 54 differ by less than 0.5%. But the EDDB method can easily be although the CÀ C bond-length equalization seems to not follow 55 used also for d-metallacycles with aromatic rings significantly this trend; this, however, should not be surprising since the 56 distorted from planarity, for which a strict separation of σ- and latter is regarded as the effect of σ-system rather than π- 57

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Figure 3. a) Ball-and-stick models of the optimized structures of d-metallacycles 1–15. b,c) Graphical representation of the EDDB results presented in Table 1; labels H (Hückel) and M (Möbius) refer to the topology of π-HOMO. 25 26 27 delocalization.[23] Also, the ratio between different metal d- especially 6, 7, and 8, contain noticeable contribution from d 28 z2 orbital contributions to delocalization changes significantly as orbitals,[13] which, together with substantial delocalization of π- 29 the ring size increases (although the average EDDB over the MOs over the ligand atoms, hinders one from classifying them 30 Me entire test set is ~1jej). In particular – leaving aside electron as ‘pure’ Hückel aromatics. Finally, 11–15 with metallacycloocta- 31 delocalization within the σ-system – the iridabenzene core in tetraene cores represent predominantly Möbius aromatics 32 the archetypical metallabenzene 1 is predominantly Hückel (EDDB reaches a maximum of ~1jej in 11 and 13) with up 33 Me,δ aromatic with only a small dope of δ-conjugation topology by to 27% support of the π-topology (14). Admittedly, 13 and 14 34 the 5d orbital (~13%); since the π-HOMO is of Hückel-type have the π-HOMOs of Hückel type, but again, they consist of d 35 xz z2 topology, aromaticity in 1 follows the frontier π-orbital top- rather than d orbitals and thus (following suggestions by 36 yz ology rule. Admittedly, this result pleads in favor of the 6π Mauksch and Tsogoeva to rely on the π-MOs containing only 37 metallabenzene model system as reported by Thorn and d and d orbitals)[14] both these systems could be regarded 38 xz yz Hoffmann,[3a] but to some extent it also reconciles the influence Möbius aromatics according to their HOMO-1 topology. 39 of the doubly-occupied 5d orbital, as suggested by Schleyer To sum up, the results presented in Table 1 and Figures 3b,c 40 xz and supported later by Jia et al.[24] It has to be noticed that indicate that the exclusive classification of Hückel or Möbius 41 Hückel aromaticity has also been confirmed for octahedral Fe, aromaticity in d-metallacycles misses the point as both d and 42 yz Mn, Os, and Rh-benzenes with neutral electron-pair donors d contribute to the cyclic delocalization of electrons (but to a 43 xz such as PH or CO.[1f] In contrast, within 7-membered systems 2– variable extent). Interestingly, δ-type conjugation involving the 44 3 10 δ-conjugation topology is slightly more favorable, albeit the d orbitals becomes more important as the ring size increases, 45 xz ratio between d and d orbital contributions to π-delocaliza- which can be explained in terms of the overlapping between 46 yz xz tion is in many cases (especially in 2–4, and 6) much more a atomic orbitals of the metal center and two adjacent carbon 47 one-to-one like, which suggests a hybrid Hückel-Möbius nature. atoms (see Figure 1a). E.g., going from 1-chloroferrabenzene 48 Interestingly, the highest occupied molecular π-orbitals in 3–5 through its 7- (2) to 8-membered (13) homologues the 49 and 10 are of Möbius type, while for the rest of the 7- percentage of the 3d orbital-contribution to the cyclic π- 50 xz membered rings they have Hückel topology; this shows certain delocalization increases from 27% through 48% up to 69% 51 incompatibility between topological criterion of aromaticity following the corresponding change of the CÀ MeÀ C bond angle 52 based on the π-HOMO shape and the cyclic delocalization of π- from 96° through 109° up to 114°; accordingly, the opposite 53 electrons (see Figure 3c). It should be noticed, however, that in trend is observed for the 3d orbital, i.e. its contribution to 54 yz 2, 3, 5, and 9 two of the highest occupied molecular π-orbitals EDDB systematically decreases from 57% through 38% to 55 Me are nearly degenerated (Δe�0.01 a.u.) having the opposite 18%, respectively. 56 topology types; moreover, some of the Hückel-type π-HOMOs, 57

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It has to be emphasized that the EDDB orbital-decomposi- antiaromatic (6 π-MOs, π-HOMO of Hückel topology in isomer I 1 tion scheme principally does not rely on bookkeeping of and 5 π-MOs, π-HOMO of Möbius topology in isomer II), but 2 electrons within the π-subsystem of the metallacyclic ring; in this is inconsistent with their structural, thermochemical and 3 other words, no assumption on the formal number of π- magnetic properties.[25] Indeed, large negative values of π-NICS 4 electrons (“4n+ 2” or “4n”) is needed to assess aromaticity and (1) supported by the π-ACID plots unquestionably shows that 5 ZZ the orbital-conjugation topology in d-metallacycles. In this both isomers are aromatic. It should be noticed, however, that 6 context, the proposed methodology can easily be used for due to influence of the local currents associated with metal- 7 metallacycles with extensively delocalized π-MOs involving ligand bonds it is very difficult to assess the topology of d- 8 more than one cyclic unit. For instance, let us consider the orbital conjugation based exclusively on the π-ACID surfaces (in 9 metallabicyclic core of one of the osmapentalenes recently this context, also the π-NICS(1) values should be taken with 10 ZZ reported by Zhu et al.[25] Figure 4 collects π-MOs and the results caution). In contrast, even a nodding glance at the π-EDDB(r) 11 of the EDDB analysis with the corresponding populations based contours in Figure 4b leaves no doubt that significant delocali- 12 on the LCAO-MO coefficients for two isomeric structures of zation of π-electrons through the osmium atom involves mainly 13 osmapentalene; the anisotropy of the current induced π-density the 5d orbital (especially in II) advocating for Möbius 14 xz (π-ACID)[26] isosurfaces and the averaged π-NICS(1) values are aromaticity. The corresponding EDDB-based populations indi- 15 ZZ added for comparison. Figure 4a indicates that both isomers cate that in each case about 6.0–6.5jej are delocalized within 16 have the π-MOs delocalized over the entire molecule (including the π-system (thus, the average π-EDDB population per ring 17 metal atom), but they differ in the size of the π-system and member in isomers I and II reaches respectively about 91% and 18 topology of π-HOMO. In fact, according to the Mauksch- 85% of the corresponding π-EDDB value for benzene), and the 19 Tsogoeva rule,[14] one could expect that both isomers are metal contribution is indeed predominated by 5d (up to 85% 20 xz in isomer II). Interestingly, participation of 5d in cyclic 21 yz delocalization is much better marked in isomer I (0.3jej vs 22 ~0.1jej), suggesting that the π-type topology involving the 23 bridgehead Os atom is necessary to correctly represent electron 24 delocalization over the smaller 5-membered cycles. Moreover, 25 just like in the case of [Fe(CH) H ], the mutual interference of 26 6 2 the subsets of Hückel- and Möbius-type molecular orbitals 27 significantly decrease the overall metal contribution to the 28 cyclic delocalization (for up to ~50% in isomer II), thus 29 confirming the anti-cooperativity of π- and δ-conjugation 30 topologies. Admittedly, the 5d metal orbital builds π-HOMO 31 z2 and two of the lowest lying π-MOs in isomer I, but it does not 32 participate in electron delocalization. In fact, since the OsÀ H 33 bond is perpendicular to the ring plane it makes 5d an 34 z2 inherent part of the π-system (with the occupation number 35 equals to 1.152jej). With that in mind, one could subtract the 36 5d orbital population from the total population of π-electrons 37 z2 in the ring to obtain 9.995jej, i.e. a formally 10π Hückel 38 aromatic system. Furthermore, taking into account that the 39 orbital population of 5d (1.905jej) is determined mainly by 40 yz the HOMO-1 (localized in 84% on the metal atom) allows one 41 to identify isomer I as the 8π Möbius aromatics (the highest 42 occupied and delocalized π-MO containing either 5d or 5d is 43 xz yz HOMO-2). Similarly, excluding from consideration HOMO-5, 44 which is in 81% localized on the [H Os] fragment (although it is 45 2 not so obvious from the first glance at the isosurface contour 46 plot), makes isomer II the 8π Möbius aromatic system as well. 47 The example of osmapentalenes clearly shows that without 48 careful analysis of the LCAO-MO coefficients and, even more 49 important, without any initial assumption about the role of 50 different d-orbitals, the qualitative criterion based on the formal 51 number of π-electrons and the π-MO topology may lead to 52 conclusions that are inconsistent with quantitative criteria of 53 aromaticity. But the situation becomes even more complicated 54 Figure 4. a) Contour plots of the occupied π-MOs in two different isomers of in the case of planar d-metallaheterobicyclic aromatics like the 55 osmapentalene. b) Results of the π-EDDB analysis with the corresponding electron populations based on the LCAO MO coefficients (blue numbers). c) one presented in Figure 5. According to the original report by 56 [27] The π-ACID isosurfaces with the corresponding averaged π-NICS(1)ZZ values. Rimola et al., this formally 16π system is regarded Möbius 57

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aromatic mainly due to significant equalization of the CÀ C, CÀ N, 1 and CÀ O bond lengths (HOMA=0.939), the normalized multi- 2 center index[9c] being half as large as in benzene (I =0.020), 3 NG and characteristic shape of the lowest lying occupied π-MO. 4 One should realize, however, that HOMO-15 is actually of 5 Hückel type because here the copper atom does not allow 6 cyclic delocalization to fall into opposite phase over a single 7 cyclic unit. Interestingly, also the highest occupied molecular π- 8 orbital (here HOMO-1) possess the Hückel-type topology, which, 9 following the frontier π-orbital rule by Mauksch and 10 Tsogoeva,[14] indicates Hückel antiaromaticity. What is more, the 11 π-component of the anisotropy of the current induced density 12 together with the marginal value of π-NICS(1) complicates the 13 ZZ case even further suggesting non-aromaticity of the system: at 14 the (standard) isovalue of 0.05 π-ACID shows predominating 15 role of the local (atomic and diatomic) ring currents, especially 16 over the metal center (see Figure 5c). However, the results of 17 the EDDB orbital-decomposition presented in Figure 5b seem 18 to rationally reconcile this incompatibility between structural, 19 topological, and magnetic characteristics unambiguously identi- 20 fying the cupraheterobicycle as quasi-aromatic.[28] Indeed, since 21 neither 3d nor 3d contribute to electron delocalization 22 xz yz (acting as doubly-occupied lone-pair-like orbitals) there is no 23 cyclic delocalization in each heterocyclic unit and consequently 24 no diatropic ring current can be observed; this is in full 25 agreement with the previous findings by Krygowski et al.[28a] On 26 the other hand, electrostatic interactions with the copper cation 27 (the natural charge on Cu is +1.308) and noticeable σ- 28 delocalization involving 3d orbital and the lone-pairs from 29 x2-y2 heteroatoms (which seem to hold the structure in a plane) 30 support delocalization of π-electrons in both quasi-rings. It is Figure 5. a) Contour plots of the occupied π-MOs in cupraheterobicycle. b) 31 Results of the σ- and π-EDDB analyses with the corresponding electron worth noting that, according to the π-EDDB and LCAO π-MO- 32 R populations based on the LCAO MO coefficients (blue numbers). c) The σ/π- based electron populations, only a half of the total number of ACID plots with the corresponding averaged σ/π-NICS(1) value. 33 ZZ π-electrons in the heterocyclic quasi-rings (6.076jej) is delocal- 34 ized, which gives on average 0.608jej per each quasi-ring 35 member; for comparison, the corresponding values for furan 36 and pyrrole are 0.503jej and 0.724jej, respectively.[12a] Interest- 4. Conclusions 37 ingly, both quasi-rings contain in total ten atoms and therefore, 38 in accordance with Figure 1b, the maximum of number of The results presented in this study clearly show that the 39 electrons that can be delocalized through the π-system of reported by Mauksch and Tsogoeva correlation of aromaticity 40 conjugated 2p orbitals is 10. Thus, since the π-system contains and the topology of the π-HOMO (provided it contains neither 41 z more electrons (12.023jej) than can actually be delocalized, the p nor d metal orbitals)[14] strictly holds only for the d- 42 z z2 balance-equivalence rule[15] is not satisfied and, consequently, metallacycles with the total population of electrons (based on 43 the direct link between the number of “4n+2” (“4n”) π- the LCAO-MO coefficients) approximately equal to the number 44 electrons and aromaticity (antiaromaticity) no longer holds. In of members in the ring. And so, 6-membered 6π-electron 45 fact, the overlapping of the metal 3d and 3d orbitals with 2p metallacycles are predominated by the π-conjugation topology 46 xz yz z orbitals of the heteroatoms makes it very difficult to predict and hence they are mainly Hückel aromatics. In contrast, in the 47 quasi-aromatic character of cupraheterobicycle basing exclu- 8-membered 8π-electron systems δ-type topology (involving 48 sively on the frontier π-orbital topology rule.[14] Admittedly, the d metal orbitals) prevails over the π-type one, thus 49 xz negligible d-orbital valencies can be deduced straightforwardly indicating Möbius aromaticity (usually with small dope of the 50 from the LCAO π-MO coefficients, but still, the effect of π- d orbital contributing to the cyclic delocalization of electrons). 51 yz conjugation of all the 2p orbitals is assessable only within the But, for the 7-membered 8π-electron d-metallacycles as well as 52 z framework of the electronic criterion of aromaticity, i.e. by the studied d-metallabicycles, there is more electrons in the π- 53 means of the EDDB method. system than can actually be delocalized, and thus the direct link 54 between aromaticity and the π-HOMO topology no longer 55 holds. For such systems it is very difficult to predict the effect of 56 the π-MO interference on the orbital valency and overlapping, 57

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and their aromaticity is hardly assessable by means of the 797335 “MulArEffect” and INFRAIA-2016-1 Grant Agreement No. 1 magnetic, structural, and topological criteria; good examples 730897 “HPC-Europa3” (Contract No: HPC17158J2). It was also 2 here are 7-membered metallacycles and the cupraheterobicycle partially supported by the National Science Centre, Poland under 3 molecule revealing a hybrid Hückel-Möbius aromatic and quasi- the Sonata IX Grant No. 2015/17/D/ST4/00558, as well as Faculty 4 aromatic nature, respectively. of Chemistry at Jagiellonian University in Krakow and Institute of 5 In this work we have demonstrated that the EDDB method Computational Chemistry and Catalysis at University of Girona. 6 can successfully be used not only to quantify and visualize We also acknowledge the financial support from the Spanish 7 aromaticity in such difficult cases, but also – in contrast to ACID MINECO (project CTQ2017-85341-P), the Catalan DIUE 8 plots or NICS values – to provide a great deal of information on (2017SGR39, XRQTC, and ICREA Academia 2014 Award to M.S.), 9 the real role of the metal d-orbitals in metallacycles. Moreover, and the FEDER fund (UNGI10-4E-801). Calculations were carried 10 the proposed methodology does not rely on bookkeeping of out within the PL-Grid Infrastructure of the Academic Computer 11 electrons within the π-system of the metallacyclic ring (no Centre (CYFRONET) and the Supercomputer center of the Consorci 12 dissection into σ- and π-MOs is needed); in other words, no de Serveis Universitaris de Catalunya (CSUC). 13 assumption on the formal number of π-electrons (“4n+2”/“4n”) 14 is required to assess aromaticity and the d-conjugation top- 15 ology in metallacycles. Conflict of Interest 16 17 The authors declare no conflict of interest. 18 Computational Details 19 20 All the DFT calculations with full geometry optimizations were Keywords: Metallacycle · Möbius aromaticity · Molecular orbital [29] [30] 21 performed using Gaussian 09; the standard B3LYP exchange- topology · Electron delocalization · Quasi-aromaticity correlation functional was used in all cases and the stationary 22 points showed minima within the frequency calculations. All 23 species studied, except the cupraheterobicycle one, have singlet 24 closed-shell ground states. Optimized geometries are available on [1] For recent reviews see the following works (and references herein): 25 request. For the 15 test-set d-metallacycles we employed two basis a) J. R. Bleeke, Chem. Rev. 2001, 101, 1205–1228; b) G. He, H. Xia, G. Jia, 26 sets: Stuttgart-Dresden (SDD)[31] with effective core potentials (4d Chin. Sci. Bull. 2004, 49, 1543–1553; c) L. J. Wright, Dalton Trans. 2006, 15, 1821–1827; d) W. C. Landorf, M. M. Haley, Angew. Chem. Int. Ed. 27 and 5d metals) and 6-31G* (other atoms). For osmapentalenes we used SDD (Os) and 6-31G** (C,H), while the restricted open-shell 2006, 45, 3914–3936; e) A. F. Dalebrook, L. J. Wright, Adv. Organomet. 28 Chem. 2012, 60, 93–177; f) I. Fernández, G. Frenking, G. Merino, Chem. calculation of the cupraheterobicycle in its doublet ground state 29 Soc. Rev. 2015, 44, 6452–6463; g) R. Islas, J. Poater, M. Solà, Organo- was performed using 6-31+ +G** (C,H,N,O) and the Watcher’s metallics 2014, 33, 1762–1773; h) Z. N. Chen, G. Fu, I. Y. Zhang, X. Xu, 30 primitive 14s9p5d set supplemented with s, p, d (diffuse), and f Inorg. Chem. 2018, 57, 9205–9214. 31 (polarization) functions (Cu).[32] The axial (zz) components of the [2] L. J. Wright, Metallabenzenes: An Expert View, Wiley-VCH, Hoboken, 2017 32 nucleus-independent chemical shifts calculated at 1 Å above/below (and references herein). [3] a) D. 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Krygowski, H. Szatylowicz, O. A. Stasyuk, J. Dominikowska, 52 M. Palusiak, Chem. Rev. 2014, 114, 6383–6422. The authors are very grateful to Dr. Michael Mauksch for several 53 [7] a) E. Matito, M. Duran, M. Solà, J. Chem. Phys. 2005, 122, 014109; b) J. stimulating discussions and critical analysis of the results. This Poater, M. Duran, M. Solà, B. Silvi, Chem. Rev. 2005, 105, 3911–3947. 54 research was supported by the European Union’s Framework [8] M. K. Cyrański, Chem. Rev. 2005, 105, 3773–3811. 55 [9] a) M. Giambiagi, M. S. de Giambiagi, C. D. dos Santos, A. P. de Figueir- Programme for Research and Innovation Horizon 2020 (2014– 56 edo, Phys. Chem. Chem. Phys. 2000, 2, 3381–3392; b) P. Bultinck, R. 2020) under the Marie Skłodowska-Curie Grant Agreement No. Ponec, S. V. Damme, J. Phys. Org. Chem. 2005, 18, 706–718; c) J. 57

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Wiley VCH Mittwoch, 20.02.2019 1902 / 130576 [S. 227/227] 1 The electron density of delocalized bonds (EDDBs) as a 8 measure of local and global aromaticity

Dariusz W. Szczepanik a,b, Miquel Solà a a Institute of Computational Chemistry and Catalysis and Department of Chemistry, University of Girona, C/ M. Aurèlia Capmany, Girona, Spain. b Department of Theoretical Chemistry, Faculty of Chemistry, Jagiellonian University, Gronostajowa, Kraków, Poland

Introduction

Aromatic rings are among the most important chemical entities in the world: no aerobic life on this planet can survive without the characteristic carbon- and nitrogen-based aromatic macrocycles which carry oxygen in the bloodstream (as a part of hemoglobin) and allow plants to capture sunlight’s energy with their chloroplasts. Over the last decades, the exceptional electron-transport and energy-harnessing capabilities of the macro- and polycyclic aromatic species have been utilized in cancer therapy, drug delivery, bioimaging, molecular electronics, solar cells, lighter converters, biosensors, quantum computing, photoluminescent materials, photodetectors, and many, many others, making aromaticity one of the most commonly exploited theoretical concepts in chemistry—according to the ISI Web of Science, in 2020 there were about 42 papers published every day that contained the word “aromatic” (or its antithesis) in the title, keywords, or abstract. On the other hand, the lack of a rigorous definition and the resulting superfluous diversity (dozens of types and rules of aromaticity) and numerous examples of the discrepancies between different aromaticity criteria proposed in the literature have become the main reasons for this concept being perceived by some members of the chemical community as an elusive, questionable, and suspicious concept [1]. But, if rightly [2]? In fact, aromaticity shares the same lack of strict definition as the most fruitful concepts in chemistry [3]; for instance, chemical bond is not strictly related to any quantum-mechanical observable either, but we can measure its strength experimentally and even observe it using the noncontact atomic force microscopy (nc-AFM) [4]. Aromaticity is very often linked with thermodynamic stability by means of the π- electron-bookkeeping rules, like the “4n + 2” and “4n” rules known from chemistry textbooks [5,6]. Although these qualitative criteria of aromaticity (and antiaromaticity) adequately relate topology, symmetry, and degeneracy of molecular orbitals (MOs) in

Aromaticity: Modern Computational Methods and Applications. DOI: 10.1016/C2019-0-04193-3 Copyright © 2021 Elsevier Inc. All rights reserved. 260 Aromaticity: Modern Computational Methods and Applications the annulene-like systems predominated by covalent resonance forms at their singlet or lowest-lying triplet states, their use in a more general context regarding topologically diversified poly- and macrocyclic species (e.g., expanded porphyrins predominated by ionic forms), non-Kekulé molecules (e.g., radicals), etc., is not well-founded. Over the last decades an overwhelming number of quantitative “measures” of aromaticity has been proposed in the literature, based on energetic, structural, magnetic, and electronic properties of molecules, thus providing a far more accurate account of aromatic stabilization than the electron-bookkeeping criteria [7]. The most commonly used quantitative criteria of aromaticity within each of these groups are: (1) the aromatic stabilization energy (ASE) [8], which is an energetic measure of π-aromaticity that emanates from the theory of valence bonds, and it can be evaluated by means of thermodynamic data for isodesmic, homodesmotic, or isomerization reactions (also, a multitude of schemes can be found in the literature that allows one to efficiently estimate the ASEs for a specific class of aromatic species); (2) the harmonic-oscillator model of aromaticity (HOMA) [9], which is a π-aromaticity index based on structural properties of molecules (being a normalized measure of deviations of bond lengths in aromatic molecule from the corresponding optimum bond lengths in an idealized nonaromatic molecule as a reference)—HOMA is close to 0 for nonaromatic species, approaches 1 for highly aromatic ones, while for potentially antiaromatic rings it usually assumes negative values; (3) the nucleus-independent chemical shift (NICS) [10], which quantifies the effective magnetic shielding at the centroid (or above) of the aromatic ring in external magnetic field—the more negative (positive) value of NICS, the more aromatic (antiaromatic) is the molecular ring in question; (4) the multicenter index (MCI) [11–13], which is a nonreference index of aromaticity that can be calculated from both the ab initio molecular wave function as well as the electron density (ED); MCI has been shown to be superior to other aromaticity criteria as regards to its performance and reliability (it is the only descriptor that passed a set of rigorous tests for aromaticity quantifiers) [14]. Unfortunately, each of the above-mentioned aromaticity “measures” has shortcomings and limitations that sometimes may lead to wrong predictions [2].TheASE seems to outwardly be the most adequate measure of global aromaticity since it can be evaluated by means of thermodynamical data and it emerges for the direct relationship between structural consequences of electron delocalization and the stability [8]. However, designing of isodesmic and homodesmotic reaction scenarios is very difficult in practice and opens the door to a lot of arbitrariness. In contrast, an unquestionable advantage of HOMA is its computational and interpretative simplicity—it allows one to straightforwardly classify any molecular ring as aromatic, nonaromatic or (potentially) antiaromatic [9]. Unfortunately, the principal problem with HOMA is the necessity of parameterization of bond lengths for an idealized reference molecule, which obviously cannot be chosen unambiguously. Consequently, the practical use of HOMA is limited to aromatic and heteroaromatic systems since the parameters for chemical bonds with metal atoms are not available. Furthermore, parameterization of HOMA should be performed using exactly the same quantum-chemical method as used in calculations of equilibrium geometries of the molecule under study, since routinely computed HOMA with the experimentally determined parameters is bound to suffer The electron density of delocalized bonds (EDDBs) as a measure of local and global aromaticity 261 from large unsystematic errors and strong sensitivity to the choice of the basis set and the exchange-correlation functional in the density functional theory (DFT)-based calculations [15]. Aromaticity descriptors based on magnetic properties of molecules, especially NICS(0), NICS(1), and its axial component, NICS(1)zz, dissected NICS, etc., are one of the most preferable measures of local aromaticity due to their relation to experiment; diatropic (aromatic) and paratropic (antiaromatic) ring currents indirectly manifest itself in the NMR spectra [10]. However, the magnetic-based measures of aromaticity have also come under bitter criticism due to their complexity (NICS relies on the condensation of potentially complicated patterns of induced currents to a single number), and methodological shortcomings [16–18]. Finally, the main disadvantage connected with the calculation of MCI is its numerical instability and computational cost (at the DFT level employing a moderate basis set one can use MCI to evaluate aromaticity of molecular fragments with no more than 10–12 atoms) [13], but also the method dependence (in particular, the exchange-correlation functional selection at the DFT level) [18]. To summarize, most of the aromaticity descriptors suffer from serious issues such as the arbitrariness of choice of a reference system, lack of parameterization, ring-size extensivity issue, limited applicability, computational inefficiency, as well as methodological shortcomings and interpretative mistiness. We refer the reader to the previous chapters in this book for an extensive review of the main shortcomings of the above-commented aromaticity descriptors. However, introducing new aromaticity measures makes sense nowadays only if their performance provides an advantage over the already existing descriptors or they offer similar quality but at a significantly lower computational cost [2]. In this chapter, the recently proposed electron density of delocalized bonds (EDDBs) [19] method will be described, which may fulfill these conditions by a wide margin providing a uniform approach notonly to quantify but also visualize electrons delocalized through the system of all (global) or selected (local) conjugated bonds in a wide range of aromatic species regardless of their size and topology.

Electron density of delocalized bonds

EDDBs are a part of the original method of one-ED decomposition into “layers” representing different levels of electron delocalization [20]: (1) density of electrons localized on atoms representing inner shells, lone pairs, etc.; (2) electron density of localized bonds (EDLB) representing typical (two-center two-electron) Lewis-like bonds; (3) EDDB, which represents ED that cannot be assigned to atoms or bonds due to its (multicenter) delocalized nature. On the basis of natural atomic orbitals (NAOs) [21],{χ μ,ν(r)}, or any other representation of well-localized orthonormalized atomic orbitals [22], the spinless global EDDBs function, EDDBG(r), for a single-determinant molecular wavefunction is defined as follows [23,24]: † G EDDBG(r) = χμ(r)Dμ,ν χν (r), (8.1) μ,ν 262 Aromaticity: Modern Computational Methods and Applications where G σ σ ,σ σ 2 σ σ D G = C ε G λ C † , 2 P a,b a,b a,b a,b P (8.2) σ =α,β a,b

In the above equation, Pσ (σ = α,β) stands for the σ spin-resolved charge and Cσ bond-order (CBO) matrix, a,b is the matrix of linear coefficients of the appropriately orthogonalized [25,26] σ spin-resolved two-center bond-order orbitals (2cBO) of the chemical bond Xa–Xb (obtained by diagonalization of the appropriate off-diagonal λσ blocks of the CBO matrix) [27], a,b represents the diagonal matrix collecting the ,σ ε G corresponding 2cBO eigenvalues (occupation numbers [ONs]), a,b is a diagonal matrix of the σ -spin bond-conjugation factors [28], and for an n-atomic molecular system, G represents the set of all n(n − 1)/2 possible atomic pairs (regardless of whether the atoms are formally bonded or not) [19]. The definition of the key ,σ ε G matrix a,b is based on the bond-orbital projection (BOP) criterion developed by one of the authors [23,29], which relies on sophisticated orbital projection cascades involving 2cBOs, their three-center counterparts (3cBO), and canonical MOs [28]. According to BOP, for a typical well-localized (Lewis-like) bond Xa–Xb, all diagonal ,σ ε G elements of the a,b matrix are close to zero, which means that the 2cBOs associated with this bond do not form effectively linear combinations with 2cBOs of all other bonds in a molecule [23]. On the other hand, when the Xa–Xb bond is effectively ,σ ε G conjugated with any other adjacent bond in the system, the a,b matrix has at least one element on its diagonal that approaches 1 (for systems with double and higher multifaceted aromaticity, the number of nonzero diagonal elements is equal to the number of delocalization “channels”) [28]. The trace of such defined D G matrix can be straightforwardly interpreted as the population of electrons delocalized through the system of all conjugated bonds in a molecule, and as such, it can be used as a “measure” of global aromaticity [24]. One of the most distinctive features of the BOP technique is that one can easily restrict the set bonds/atomic pairs in G giving rise to a series of different variants of global and local EDDB functions, as depicted in Fig. 8.1. In the case of the EDDBH(r) function, H contains all possible atomic pairs in the molecule excluding hydrogen

Fig. 8.1 Isosurface contours and the corresponding populations of delocalized electrons in naphthalene by different global and local EDDB(r) functions. Method: CAM-B3LYP/ 6-311G(d,p), equilibrium geometry. The electron density of delocalized bonds (EDDBs) as a measure of local and global aromaticity 263

atoms from consideration. Thus, EDDBH(r)isalsoaglobal function that can be especially useful in the analysis of global π-aromaticity in organic molecules since the 2cBO involving hydrogen atoms tend to conjugate with the adjacent σ -bond orbitals noticeably increasing the delocalization in the σ -subsystem, which may sometimes lead to less precise conclusions. Moreover, the difference between EDDBG(r) and EDDBH(r) can be very useful in the identification of multivalent hydrogen interactions [30]. For instance, as shown in Fig. 8.1, the net effect of eight C–H bonds in naphthalene on electron delocalization is 0.8054e, which means that only about 0.1e shared between carbon and hydrogen atoms is delocalized due to conjugation with other σ -bonds in the system; a much higher values (usually greater than 0.3e) are expected for bonds containing hydrogen and dihydrogen bonds [30]. In contrast to EDDBG(r) and EDDBH(r), the next two EDDB functions can be regarded as local aromaticity measures. In the case of the EDDBF(r) function, F contains all possible atomic pairs in the selected molecular fragment (usually cyclic unit without H atoms), which in the case of a single six-membered ring (6-MR) in naphthalene gives rise to 6×5/2 = 15 atomic pairs (the same number of linear combinations of 2cBO has to be considered within the BOP procedure). In turn, six chemical bonds are considered within the definition of the congeneric local EDDBP(r) function, which “measures” electron delocalization along the selected pathway of adjacent bonds. Thus, in contrast to EDDBF(r), the EDDBP(r) function does not take into account the cross-ring delocalization effects, which for 6-MR are associated with the resonance involving Dewar structures; in a sense, the relation between EDDBP and EDDBF corresponds to that between the MCI originally proposed by Giambiagi et al. (IRing) [11] and its averaged-over-permutations variant by Bultinck et al. (MCI) [12]. The EDDBP(r) function and the corresponding electron population (denoted simply by EDDBP) can be used to visualize and quantify local aromaticity in a wide range of molecular rings regardless of their size and topology, but also to assess macrocyclic aromaticity associated with particular delocalization pathways in larger structures like expanded porphyrins. Each of the EDDB(r) functions can easily be dissected into σ -, π- and higher- symmetry components by diagonalization of the corresponding density matrix defined in Eq. (8.2) [31]. The resulting eigenfunctions, called the natural orbitals for bond delocalization (NOBD), can be particularly useful in the identification of nonplanar multifaceted aromatic compounds because they do not tend toward mixing of different symmetry types like the canonical MOs, even for twisted Möbius-type hetero- and metallacyclic aromatics [32,33]. Within the current implementation of the EDDB method [34], the NOBDs are ordered from the high occupied to the low occupied ones, and, since the NOBDs responsible for aromaticity usually have the ONs by order of magnitude greater than the rest of orbitals, the symmetry dissection of EDDB is straightforward. For instance, diagonalization of the EDDBH density matrix in naphthalene (Fig. 8.1) gives rise to five NOBDs (no. 1–5) with ONs frome 1.5381 to 1.1907e, 29 NOBDs (no. 6–34) with ONs from 0.0600e to 0.0011e, and the rest of NOBDs (no. 35–228) have strictly zero ONs. By summation of the corresponding NOBD ONs, one gets the following: π-EDDBH = 6.5100e and σ -EDDBH = 0.5721e; thus, even though the σ -delocalization effects are still noticeable, naphthalene is clearly π-aromatic with about 65% (6.5e/10.0e) of electrons effectively delocalized in the 264 Aromaticity: Modern Computational Methods and Applications

π-subsystem (for comparison, the corresponding effectiveness of delocalization of π- electrons in benzene is close to 90%) [19]. It has to be stressed that the BOP technique underlying the EDDB method provides a strict criterion for bonding orbitals to form linear combinations with each other, which can be directly related to the effectiveness of conjugation of double bonds resulting from different resonance forms contributing to the wavefunction. For instance, naphthalene can be represented by three different resonance forms as presented in Fig. 8.2. Assuming equal contribution of each resonance form to the wavefunction, each of 11 bond positions in naphthalene can be thus represented either by two single and one double bond or one single and two double bonds; e.g., the central bond (Fig. 8.2A) is single in two forms (red circles) and double in one form (blue circle). One should realize that the superposition of the resonance form containing a double bond in the center (blue circle) with one of the resonance forms containing a single bond in the same position (red circle) gives rise to the Clar structure with a single π-sextet, while the superposition of two resonance forms containing a single bond in the center (red circles) results in a single π-dectet circuit (Fig. 8.2B). In the former, the central bond can be regarded as fully delocalized (one-and-a-half bond), whereas in the latter, it does not take part in delocalization. Since there are two possible combinations of the resonance structures (two different Clar π-sextets) responsible for delocalization of the central bond and only one combination (π-dectet) responsible for its localization, the effectiveness of delocalization of this bond is about 67%; alternatively, one can say that the central bond in naphthalene has a 67% “aromatic” and a 33% “aliphatic” character (the bond is indeed noticeably longer than that in benzene, RCC = 1.43 Å compared to RCC = 1.40 Å, respectively). The same degree of delocalization features all other bond positions in the system; e.g., the four “rim” bonds (connecting carbon atoms in positions α and β) can be represented by one double and two one-and-a-half (delocalized) bonds, which means they have a 67% “aromatic” and a 33% “olefinic” character (RCC = 1.37 Å, so the bonds are noticeably shorter than the one in benzene). From the energetic point of view, the second-order perturbation theory involving the natural bond orbitals (NBOs) [35] predicts almost the same average stabilization effects (ca. 25 kcal/mol) associated with the delocalization of each double bond in all three resonance structures in naphthalene, which means that, e.g., the delocalization of each “rim” bond and the central bond contributes similarly to the ASE. Therefore, it is clear that all chemical bonds in naphthalene have partially “aromatic” character as they equally participate in the resonance, and the effectiveness of electron delocalization in naphthalene according to the EDDB predictions based on ab initio calculations (65%) is in full agreement with the classical resonance theory. In this context, it has to be emphasized that the representation of aromaticity in naphthalene exclusively by a single π-dectet circuit or the migrating Clar’s π-sextet is incorrect as it always discriminates delocalization effects associated with certain bond positions; thus, a reliable description of π-aromaticity in naphthalene requires both π-dectet and π- sextets circuits (Fig. 8.2B) [36]. Moreover, in the presence of external magnetic field, local diatropic ring currents associated with Clar’s π-sextets in naphthalene to a large extent cancel each other (Fig. 8.2C), thus favoring the envelope π-dectet diatropic current (which is depicted in Fig. 8.2D by utilizing the π-component of the anisotropy The electron density of delocalized bonds (EDDBs) as a measure of local and global aromaticity 265

Fig. 8.2 (A) The effectiveness of electron delocalization in naphthalene based on the classical resonance theory; (B) Alternative representations of the resonance forms involving Clar’s π-sextet and π-dectet notation; (C) Schematic representation of the π-circuit cancellation effect in naphthalene under the external magnetic field; (D) 3D-plots of π-EDDBH and π-ACID for naphthalene. Method: CAM-B3LYP/6-311G(d,p), equilibrium geometry. of current-induced density, π-ACID) [37]. The case of naphthalene clearly shows that the correspondence between aromaticity and the ring-current diatropicity may be not strict, and, in this context, the results of the ring-current analysis in polycyclic and topologically diversified aromatics should be always interpreted with caution [17]. To summarize, the EDDBs allow one to quantify and visualize the population of electrons delocalized through the system of all (global) or selected (local) conjugated 266 Aromaticity: Modern Computational Methods and Applications bonds in a wide range of aromatic species, and, in contrast to the induced ring-current methods, EDDB is derived from unperturbed one-ED and, as such, it can be directly related to chemical resonance and its structural and energetic consequences.

The RunEDDB program

The current implementation of the EDDB method, called RunEDDB, has been coded in the cross-platform R-script language, and the detailed installation and usage instruc- tions are available online [34]. The program requires two input files: the formatted checkpoint file (.fchk) and the file generated by the NBO program and containing the density matrix within the representation of NAOs (.49 file). The former can be generated by the popular quantum-chemical packages (e.g., Gaussian, Q-Chem, PSI4, etc.) and tools (e.g., MultiWFN); a full list of the software supporting the .fchk format can be found here [34]. To generate the .49 file, the following line has to be specified within the NBO calculations: $NBO AONAO=W49 DMNAO=W49 $END. It is strongly rec- ommended to install and use the newest version of the NBO program, especially if the diffuse functions are used or the calculations involve the correlated wavefunctions. RunEDDB works well with different types of the wavefunction (i.e., RHF, ROHF, UHF, CUHF, etc.) from the calculations at different levels of the theory (including DFT, CI, CC, SACCI, CASSCF, MP2, etc.) and spin states [34]. For illustration, let us consider the calculations of the EDDB populations for the naphthalene molecule (Fig. 8.1)by the RunEDDB script. The Gaussian input file used to generate .fchk and .49 files reads:

# CAM-B3LYP/6-311G(d,p) Pop(NBORead) FormCheck Naphthalene 01 C 0.00000 1.24008 1.39429 C 0.00000 2.41940 0.70603 C 0.00000 2.41940 -0.70603 C 0.00000 1.24008 -1.39429 C 0.00000 0.00000 -0.70953 C 0.00000 0.00000 0.70953 C 0.00000 -1.24008 1.39429 C 0.00000 -2.41940 0.70603 C 0.00000 -2.41940 -0.70603 C 0.00000 -1.24008 -1.39429 H 0.00000 1.23660 2.47868 H 0.00000 3.36162 1.24086 H 0.00000 3.36162 -1.24086 H 0.00000 1.23660 -2.47868 H 0.00000 -1.23660 2.47868 H 0.00000 -3.36162 1.24086 H 0.00000 -3.36162 -1.24086 H 0.00000 -1.23660 -2.47868 $NBO DMNAO=W49 AONAO=W49 $END The electron density of delocalized bonds (EDDBs) as a measure of local and global aromaticity 267

The execution of Gaussian generates, among others, two files required by the RunEDDB program: FILE.49 and Test.FChk. The easiest way to perform the EDDB calculations at this point is to simply execute the RunEDDB script (by double click or in terminal—depending on the operating system) [34] inside the directory containing those files. After successful parsing the input files, the program asks for the selection of the EDDB function type: EDDBG (1), EDDBH (2), EDDBF (3), EDDBE (4), EDDBP (5). The EDDBE function is similar to EDDBF, but in this case the density of electrons delocalized in selected fragment is extracted from the EDDBG function, and thus it includes also the “external” (nonlocal) orbital conjugation effects; the comparison of EDDBF and EDDBE allows us to assess, e.g., to what extent the local aromaticity in particular molecular ring is affected by the surrounding rings in polycyclic aromatic hydrocarbons (PAHs). Selection of 3–5 requires user to define molecular fragment/pathway by providing the list of atoms/bonds. Next, the program performs a series of BOPs to get the corresponding EDDB density matrix, and when completed, it prints the nonzero NOBD ONs, the results of the EDDB-based population analysis in atomic resolution (supplemented by the natural atomic populations of electrons, NPA), and, at the end, the desired total population of delocalized electrons. For instance, selection of the EDDBG function (1) in the example above gives the following output: NOBD ___1__ ___2__ ___3__ ___4__ ___5__ ___6__ ___7__ ___8__ ___9__ __10__ 1 1.5385 1.3366 1.2412 1.2055 1.1914 0.0972 0.0958 0.0947 0.0937 0.0928 11 0.0832 0.0792 0.0765 0.0764 0.0762 0.0759 0.0549 0.0541 0.0537 0.0536 21 0.0509 0.0509 0.0475 0.0471 0.0022 0.0022 0.0020 0.0020 0.0020 0.0020 31 0.0019 0.0019 0.0018 0.0018 ______o Results of the NPA and EDDB_G population analyses in the NAO basis: ______|||| | Atom NPA EDDB_G | Atom NPA EDDB_G | Atom NPA EDDB_G | | ””” ””” ””” | ””” ””” ””” | ””” ””” ””” | | 1 C 6.1789 0.7595 | 2 C 6.2007 0.7268 | 3 C 6.2007 0.7268 | | 4 C 6.1789 0.7595 | 5 C 6.0563 0.8854 | 6 C 6.0563 0.8854 | | 7 C 6.1789 0.7595 | 8 C 6.2007 0.7268 | 9 C 6.2007 0.7268 | | 10 C 6.1789 0.7595 | 11 H 0.7968 0.0223 | 12 H 0.7954 0.0204 | | 13 H 0.7954 0.0204 | 14 H 0.7968 0.0223 | 15 H 0.7968 0.0223 | | 16 H 0.7954 0.0204 | 17 H 0.7954 0.0204 | 18 H 0.7968 0.0223 | |______|______|______| | | | Total population of electrons (from NPA): 68.0000e ( 3.7778e / atom ) | | Total population of delocalized electrons: 7.8873e ( 0.4382e / atom ) | |______|

To enable visualization of the EDDB functions, the SAVE_EDDB_TO_GAUSSIAN_FCHK variable defined at the beginning of the RunEDDB script code has tobeset TRUE.The generated new checkpoint file (named EDDB_X.FChk) is a copy of the original one (Test.FChk) in which the Alpha MO energies and coefficients as well as the Total SCF density are replaced by the NOBD ONs and coefficients, and the corresponding EDDB density, respectively. The new formatted checkpoint file can be used to generate cube file by the Cubegen utility from the Gaussian suite, or it can directly be visualized by different molecular editor programs (e.g., Avogadro, GaussView, MultiWFN, etc.) [34]. By default, the RunEDDB runs in interactive mode, which allows user to select 268 Aromaticity: Modern Computational Methods and Applications different options during the program execution. Working in the system terminal, however, it is much more practical to disable this mode (-q option) and use the command-line options and redirect the output into the file. For example, to calculate all the EDDB(r) functions and the corresponding delocalized-electron populations presented in Fig. 8.1, export the results into .FChk and .out files, and finally to generate the .cube files for visualization, the following commands have to be executed inthe Linux/Unix terminal:

./RunEDDB.R -q -i Test.FChk FILE.49 -g -o EDDB_G.FChk >& EDDB_G.out cubegen 1 FDensity=SCF EDDB_G.FChk EDDB_G.cube 100 h ./RunEDDB.R -q -i Test.FChk FILE.49 -h -o EDDB_H.FChk >& EDDB_H.out cubegen 1 FDensity=SCF EDDB_H.FChk EDDB_H.cube 100 h ./RunEDDB.R -q -i Test.FChk FILE.49 -f 1:6 -o EDDB_F.FChk >& EDDB_F.out cubegen 1 FDensity=SCF EDDB_F.FChk EDDB_F.cube 100 h ./RunEDDB.R -q -i Test.FChk FILE.49 -p 1-2-3-4-5-6-1 -o EDDB_P.FChk >& EDDB_G.out cubegen 1 FDensity=SCF EDDB_P.FChk EDDB_P.cube 100 h The complete tutorial on the command-line based use of the RunEDDB program is available online [34]. The EDDB method is under active development and new functionalities are being developed and implemented, such as the energetic analysis of bond-orbital conjugation effects, the spin-resolved EDDB analysis, etc.

EDDBP as a local aromaticity descriptor

In this section, we will provide a comprehensive view on the performance of the EDDBP index, i.e., the population of delocalized electrons associated with the EDDBP(r) function, in quantifying local aromaticity. The EDDBP predictions were compared with those derived from well-known descriptors based on structural, magnetic, and electronic criteria of aromaticity [9–13]. Two test sets of molecules were used. The firsts one, T1 was designed by Andrzejak et al. [15] to include mainly Hückel’s aromatic and antiaromatic mono- and polycyclic hydrocarbons, as for such a homogeneous set of carbon rings it is reasonable to expect the differences reflected by different aromaticity criteria to be more or less proportional to each other (Fig. 8.3). The following aromaticity indices were used in the benchmark study involving the test set T1: EDDBP, HOMA [9], the axial component of NICS calculated at 1 Å above the ring centroid, denoted as NICS(1)zz, [10] IRing [11], the para-delocalization index, PDI (which measures the average electron delocalization between three para- related atoms in a benzenoid unit) [38], Shannon aromaticity, SA (which measures the Kullback–Leibler distance of the bonding ED distribution in aromatic ring from uniformity) [39], and the fluctuation index of aromaticity, FLU (which has a similar interpretation as the SA index but depends upon parameters determined for an idealized aromatic system) [40]. Results of the correlation analysis based on T1 are collected in Fig. 8.4. The test set T2, in turn, contains various types of aromatic systems and has been used to assess if and to what extent the newly proposed local aromaticity index EDDBP reproduces the aromaticity trends predicted by IRing;T2 is based on the collection of benchmarks originally designed by one of the authors to evaluate aromaticity indicators proposed in the literature, and it accumulates chemical experience The electron density of delocalized bonds (EDDBs) as a measure of local and global aromaticity 269

Fig. 8.3 The T1 test set of aromatic and antiaromatic species used in the comparative study between EDDBP and different aromaticity descriptors. 270 Aromaticity: Modern Computational Methods and Applications

Fig. 8.4 The T1-based correlation analysis between EDDBP and different aromaticity indices for the entire test set (black), and the subsets of 5- (blue), 6- (green), and 7-MRs (red). Method: CAM-B3LYP/def2-TZVPP, equilibrium geometries. about expected trends in aromaticity changes in such compounds as distorted benzene, substituted benzene, metal complexes, penta- and heptafulvenes, PAHs, heteroaromatic systems, as well as the aromatically stabilized transition states (TSs) in selected chemical reactions [16]. The results of the T2-based comparative study are collected in Figs. 8.5–8.7. Formal definitions of all the above-listed indices as well as computational details can be found in the source paper by Szczepanik et al. [19].

The correlation analyses (T1)

Fig. 8.4 illustrates in detail the correlation analysis results based on the test set T1. It is worth noticing that, by definition, PDI was used in the evaluation of aromaticity of 6-MRs only, while the utilization of NICS(1)zz index was restricted to monocyclic The electron density of delocalized bonds (EDDBs) as a measure of local and global aromaticity 271

Fig. 8.5 Different benzene distortions together with the corresponding IRing and EDDBP values 3 as well as the EDDBP(r) contour maps. IRing values have been multiplied by 10 . Method: B3LYP/6-311++G(d,p). species due to its nonlocal character [36]. Even a cursory glance at Fig. 8.4A and F clearly shows that the ring-size extensivity issue [13] makes IRing and NICS(1)zz incomparable with the rest of indices when size diversified molecular rings are considered. Admittedly, the renormalized MCI (defined as the nth root of IRing) [13] was shown to be superior to the original one as regards the comparison of T1 rings of different sizes, but, at the same time, renormalization of IRing turned out to give only a slight adjustment to the r-squared coefficients when we consider 5-, 6-, and 7-MRs separately [19]. In contrast, reasonably tight nonlinear correlation is observed within the subset of SA, FLU, and HOMA indices, regardless of whether we consider rings

Fig. 8.6 Plots of the EDDBP and IRing relative changes along (A) the Diels–Alder cycloaddition of butadiene and ethylene and (B) the acetylene trimerization reaction; for TSs and the products (P) the EDDBP(r) isosurfaces are displayed. Method: B3LYP/6-311++G(d,p). 272 Aromaticity: Modern Computational Methods and Applications

Fig. 8.7 Summary of the T2-based performance comparison between EDDBP and IRing. Method: B3LYP/6-311++G(d,p), equilibrium geometries. of different sizes separately or collectively. The fact that aromaticity changes predicted by different aromaticity measures may not necessarily be linearly proportional to each other is commonly known [41–44]. One should bear in mind that the relation between EDDBP (a quantity based on the bond-order orbitals) and HOMA (an index involving bond lengths) explicitly refers to the bond-distance–bond-order exponential relationship established by Pauling [45]. Additionally, in the case of SA (Fig. 8.4C), excluding the central ring (27) in triphenylene from the 6-MR subset of T1 significantly improves The electron density of delocalized bonds (EDDBs) as a measure of local and global aromaticity 273

the correlation with EDDBP (as well as with other indices like HOMA and FLU) rising the r-squared coefficient up to R2 = 0.950. On the other hand, as follows from Fig. 8.4A and E, in the case of PAHs the correlation between EDDBP and IRing as well as PDI is significantly weaker; this effect, however, is associated with a more general problem of the definition of local aromaticity in polycyclic systems [36]. Nevertheless, it should be emphasized that a tight correlation between EDDBP and HOMA once again demonstrates that the effectiveness of electron delocalization as defined within the BOP criterion underlying the EDDB method is closely related to the chemical resonance and its structural consequences in the molecular systems.

The benzene distortions test Benzene, an archetypical aromatic molecule, can exclusively be used to assess the performance of aromaticity descriptors in a series of in-plane deformation modes such as bond length alternation (BLA) and clamping, as well as the out-of-plane distortions such as pyramidalization, boat-like aromatic, and chair-like aromatic deformations [14]. The distortions are often observed in large and strained molecular systems like fullerenes, graphene, nanotubes, and they usually lead to a reduction of the local aromatic character of particular units. As shown in Fig. 8.5, both EDDBP and IRing in full compliance reproduce the expected drop of aromaticity when each of these distortions is applied. Both delocalization indices consistently identify BLA as the most dearomatizing deformation, while pyramidalization and boat-like aromatic are found to affect delocalization to a very limited extent (less than 5%). Interestingly, in the case of BLA EDDBP is clearly much more sensitive than IRing as it predicts a reduction of aromaticity of about 85% vs. ca. 45% reduction predicted by the latter. Since for the bond-alternation parameter R = 0.25 Å one actually gets the hypothetical resonance- free structure of 1,3,5-cyclohexatriene (the NBO analysis confirms that all three double bonds are localized in 90%), EDDBP can be regarded as an even more reliable and adequate descriptor for electron delocalization and aromaticity in BLA-distorted benzene than IRing (a more comprehensive study of differences between the EDDBP and IRing predictions of aromaticity in distorted benzene and other topologically diversified molecules is under preparation).

Aromaticity changes along chemical reactions The TS aromaticity plays a key role in determining the mechanisms of countless chemical reactions in organic chemistry [46]. For this benchmark, let us consider two model reactions with aromatic TS: the Diels–Alder (DA) cycloaddition of butadiene and ethylene and the acetylene trimerization. In the former, it is known that the reaction takes place through a boat-like aromatic TS [46], while in the latter, an increase of aromaticity is expected when going from reactants to TS and after this point, a significant reduction of the aromatic character is observed until a final peak of aromaticity due to the formation of the product—benzene. Fig. 8.6 presents plots of the EDDBP and IRing changes along the reaction coordinates. Despite some minor differences in the shape of the curves, it is clear that both aromaticity indices precisely identify the σ -aromatic TS and the π-aromatic product of the acetylene trimerization as 274 Aromaticity: Modern Computational Methods and Applications

well as the aromatic TS in the DA cycloaddition. Moreover, the visualized EDDBP(r) isosurfaces clearly show that the TS in the DA-reaction has a boat-like structure with the characteristic cyclic delocalization pattern in the plane between butadiene and ethylene fragments.

The performance comparison of EDDBP and IRing (T2)

Fig. 8.7 displays a summary of the EDDBP and IRing performance comparison involving distorted benzene, the above-mentioned chemical reactions, and completing the T2 test set by 16 substituted benzenes, 1 metalloorganic species, 2 charged monocyclic hydrocarbons, 7 six-membered heterocycles, 6 five-membered (hetero)cycles, 5 PAHs containing in total 13 Clar’s π-sextets, 5 penta- and 5 heptafulvenes. The results leave no doubt that the EDDBP index predicts exactly the same trends of local aromaticity changes as the IRing index. Furthermore, in the vast majority of cases the predictions agree with general expectations; [14,19] only three systems can be identified for which the discrepancies between expectations and the predictions based on both indices are observed. Nevertheless, EDDBP perfectly reproduces the performance of IRing in this particular test, which clearly shows that the BOP formalism provides a widely applicable and reliable criterion for quantitative evaluation of the multicenter electron sharing and aromatic stabilization effects (at least in the case of the ground-state aromatic organic species) [19]. It should be pointed out, however, that the calculation of EDDBP is far more efficient than IRing; for instance, it takes from a dozen of seconds (5- MRs) to several hours depending on the size of the ring (up to 12-MRs for the physical- space partitioning and slightly less for the atomic-orbital based ones, especially when large basis sets are used) and the computational method used to calculate IRing, while the calculation of EDDBP takes less than 1s regardless of the method and size of the ring [19]. Moreover, in contrast to IRing, EDDBP does not suffer from ring-size extensivity and numerical instability issues.

Hückel’s vs. Craig–Möbius aromaticity in metallacycles

It should be noticed that EDDBP can also be used to quantify aromaticity in topologically diversified d-block metallacyclic species [47–49], in which chemical bonding between transition metal and the adjacent atoms allows two different topologies of d-orbital conjugation: the π-type (Hückel’s) topology involving the dyz metal orbital (without orbital phase inversion), and the δ-type (Craig–Möbius) topology, in which dxz metal orbital acts as a “phase switch” allowing electron delocalization to fall into the opposite phase side over the cyclic unit [50]. We recently demonstrated that, in contrast to any other aromaticity descriptor, especially those based on the magnetic- response criteria, the EDDBP(r) function provides a great deal of information on the role of the transition metal orbitals in metallacycles [31]. The additive decomposition of the EDDBP density into carbon ring and metal fragments followed by the diagonalization of the corresponding atomic blocks of the EDDBP density matrix allows us to quantitatively assess the contribution of each type of metal d-orbital to the cyclic delocalization of electrons within the metallacyclic ring (Fig. 8.8). Moreover, the The electron density of delocalized bonds (EDDBs) as a measure of local and global aromaticity 275

Fig. 8.8 Relative contributions of three different d-orbital-conjugation topologies to electron delocalization in selected 15 metallacycles based on the atomic-block orbital decomposition of the EDDBP density matrix. Method: B3LYP/6-31G(d)∼SDD.

proposed EDDBP-based orbital decomposition scheme does not rely on bookkeeping of electrons in the molecular ring (which in nonplanar rings might be problematic), and thus no assumption on the formal number of π-electrons (“4n + 2”, “4n”, etc.) is required to assess whether the metallacyclic ring can be classified as Hückel’s, Craig– Möbius, or the Hückel–Möbius hybrid aromatic [31].

Visualization of global aromaticity using the EDDBG/H(r) function

The EDDB method provides a detailed description of local aromaticity in a selected molecular ring, but it can also be utilized as a bird’s-eye view of the nonlocal resonance effects and global aromaticity in nanoscopic-size systems like fullerenes, graphene petals, carbon nanotubes, or even DNA fragments. The EDDBG/H(r) function can be especially useful to study topologically diversified macrocyclic aromatic systems like expanded porphyrins, since, in contrast to EDDBP or any other local aromaticity descriptor, it does not rely on the arbitrary selection of the particular electron delocalization pathway. As the default definition of G in Eq. (8.2) calls for the BOP procedure to consider all possible orbital conjugations between n(n − 1)/2 atomic pairs in a molecule, the resulting layer of the ED visualized by EDDBG/H(r) represents only the electrons that cannot be assigned to any atom or bond due to their (multicenter) delocalized character (Fig. 8.9). A multitude of global aromaticity visualization techniques has been proposed in the literature over the last two decades, among which the most commonly used 276 Aromaticity: Modern Computational Methods and Applications

10+ Fig. 8.9 Electron delocalization in exemplary organometallic complex, fullerene C60 , + expanded hexaphyrin complex with the d-block metal, Möbius aromatic C9H9 cation, and the DNA fragment (∼1000 atoms), visualized by the EDDBH(r) function. based on the ground-state ED are the electron localization function (ELF) [51], localized-orbital locator (LOL) [52], electron delocalization range (EDR) function [53], and the density-overlap region indicator (DORI) [54]. These functions of the one- ED are easily available from the DFT and wavefunction calculations, however, they usually require additional processing, orbital-symmetry dissection, or/and bifurcation analysis to give a look on the electron delocalization and identify the most aromatic pathways/fragments. On the other hand, the magnetic-response methods such as the ACID [37] and the gauge-including magnetically induced current (GIMIC) [55] have become standard tools for global aromaticity evaluation in poly- and macrocyclic systems due to their clear and distinct pictorial character that allows one to straightforward get a definitive classification of the compound as globally aromatic (when a diatropic induced ring current is observed) or antiaromatic (when a paratropic ring current is induced in the external magnetic field). The main advantage of ACID is its orientation-independence while a unique feature of the GIMIC method is that it enables calculations of strengths of the induced currents by numerical integration of the current flows. Unfortunately, the correspondence between global aromaticity (antiaromaticity) and diatropicity (paratropicity) may be not strict as there are examples of thermodynamically stable molecules revealing a high degree of electron delocalization and aromaticity by different ground-state criteria, but featuring global paratropic ring current [56]. Moreover, in poly- and macrocyclic systems the interference of different local ring currents with the global ones sometimes results in incidental enhancements or cancelations, which may lead to unrealistic electron delocalization patterns that do not correspond to the actual resonance structure of a molecule (cf. Fig. 8.2). The EDDBG/H(r) function is directly related to the chemical resonance and its structural and energetic consequences, and hence it does not share the drawbacks of ACID and GIMIC. Fig. 8.10 presents the performance of EDDBG, GIMIC, ACID, ELF, LOL, EDR and DORI in the context of global aromaticity visualization in a model The electron density of delocalized bonds (EDDBs) as a measure of local and global aromaticity 277

Fig. 8.10 Global aromaticity in 24π-electron carboporphine visualized by different functions. Method: CAM-B3LYP/6-311G(d,p), equilibrium geometry.

24π-carboporphine from the original study by Sundholm et al. [57].Attheveryfirst glance it becomes clear that the EDDBG(r) function provides exceptionally precise and unambiguous picture of electrons delocalized through the system of π-conjugated bonds. The lack of ED in the “rim” saturated carbon–carbon bond is well reflected by only two methods, EDDBG and GIMIC. In the latter, however, the delocalization pattern changes completely with the reorientation of the external magnetic field and with different selection of the plane on which the vector field is projected. It should be emphasized that at certain isovalue almost all other functions including ACID reveal significant contributions of the electrons from the nonaromatic σ -subsystem, while EDDBG shows a clear and distinct π-delocalization pattern regardless of the isovalue. In fact, the orbital-symmetry dissection of the EDDBG(r) function indicates that ca. 85% of delocalized electrons comes from the π-subsystem, and the effectiveness of delocalization of all the π-electrons approaches 72%. The EDDBG/H(r) function together with the complement EDLBG/H(r) function (representing the density of electrons localized in the Lewis-type bonds) [20] can be used as a unique lens for controlling and predicting site-specific reactivity of large PAHs; the former allows us to identify “aromatic” regions of a system (e.g., Clar’s π- sextets), while the latter can be used to locate the “olefinic” fragments prone to addition reactions. Fig. 8.11 demonstrates the performance of both functions in the analysis of the electron delocalization pattern in the 168π-electron [6]coronoid system, C168H60. [6]Coronoid is a promising model structure of the planar porous graphene, which has recently been synthesized and characterized by Giovannantonio et al. [58].Thelow- temperature scanning tunneling microscopy/spectroscopy and the nc-AFM revealed that the [6]coronoid molecule entails a large cavity (>1 nm) with inner zig-zag edges 278 Aromaticity: Modern Computational Methods and Applications

Fig. 8.11 (A) Electron density (ED), global aromaticity (EDDBG), and olefinicity (EDLBG) maps calculated at 2.0 Å above the molecular plane of the 168π-electron [6]coronoid system; (B) 3D-plots of the π-ACID (bottom) and π-EDDBG (top) functions with the corresponding total population of delocalized electrons, percentages of benzene aromaticity of individual (circled) benzenoid units, and percentage of the “aromatic” character of (boxed) double bonds. Method: UCAM-B3LYP/6-311G(d,p), equilibrium geometry.

distinct from their outer armchair edges, and the delocalization pattern suggests the Clar structure with 24 π-sextets and 12 “double” bonds. The results of the aromaticity and olefinicity analyses involving the EDDBG(r) and EDLBG(r) functions, as presented in Fig. 8.11A, fully support this hypothesis. Although the total ED map calculated at 2.0 Å above the molecular plane does not allow to differentiate between fragments containing localized and delocalized electrons, its EDLBG and EDDBG components (visualized using appropriately renormalized scales) reveal with details the characteristic benzenoid circles representing 24 Clar π-sextets and 12 highly olefinic bond positions in the inner edge, respectively. Each of the double bonds identified by EDLBG has a bond length of 1.36 Å, which is exactly in the middle between the perfectly olefinic double bond in ethylene (1.33 Å) and the ideally aromatic (one-and-a-half) bond in benzene (1.39 Å). Thus, assuming a linear correlation between bond length and electron delocalization (for carbon–carbon bonds within the given range) one should expect more or less the 50% olefinic–50% aromatic character of these 12 distinct bonds in the inner zig-zag edge. The quantitative analysis of the orbital-symmetry dissected π-EDDBG(r) functions clearly shows that ∼81% of the electrons in the π-subsystem is delocalized (i.e., 135.7e from 168.0e), but (despite visual overlapping of different The electron density of delocalized bonds (EDDBs) as a measure of local and global aromaticity 279

6-MRs units) the π-delocalization effect has a local character since all 24 Clar π- sextets preserve from 95% to 97% of the benzene aromaticity (Fig. 8.11A). Moreover, the π-EDDBG-based population analysis reveals that each of 12 “double” bonds in the inner edge is populated by 1.9942e and a half of it (exactly 1.008e) contributes to π-delocalization, which fully supports the prediction based on the structural (bond length) considerations. In contrast, the analysis of the dissected π-ACID does not lead to such definitive conclusions due to the complex delocalization pattern resulting from the coexistence of different local (6π-Hückel’s aromatic benzenoid units), macrocyclic (28π-Hückel’s antiaromatic phenanthroperylenoid units), and global (78π-Hückel’s aromatic circuit) diatropic currents together with the global (42π-Hückel’s aromatic circuit) paratropic current induced in the inner edge. It should be mentioned that the EDLBG(r) function has recently been utilized to decipher and interpret the experimental nc-AFM image of pentacene revealing a striking resemblance between the AFM image and the corresponding EDLBG(r)map calculated above the molecular plane (i.e., within the noncontact region on the border of the sample-carbon atoms and the tip-oxygen atom van der Waals radii) [36].This observation suggests that the localized (olefinic) π-electrons might be responsible for distinct repulsive interactions between the sample and CO-tip giving rise to enhanced frequency shifts and, consequently, the exceptionally bright regions observed in the AFM image. If so, the complementarity of the EDLBG(r) and EDDBG(r) functions could be used to assess electron delocalization patterns “hidden” in the AFM images, and hence, in a sense, to indirectly observe the ground-state aromaticity experimentally for the very first time—a comprehensive study in this regard is in progress.

Summary

Over the last two decades, a great number of aromaticity types have been discovered and, as far as the physicochemical and electronic-structure properties are concerned, many of them differ dramatically from the archetypical π-aromaticity in benzene [6]. This has stimulated research progress toward the quantification of aromatic stabilization, and every now and then new criteria of aromaticity are proposed in the literature. Unfortunately, numerous examples of the discrepancies between different aromaticity criteria have contributed toward criticism lowering the reputation of this concept. With that said, introducing new aromaticity measures makes sense nowadays only if their performance exhibits a clear advantage over the already existing descriptors or they enable the study of molecular systems whose size and complex structure challenge the use of current tools. The results presented in this chapter clearly show that the EDDB method safely fulfills the above conditions. For a wide range of organic compounds, the EDDBP index predicts the same trends of local aromaticity as most of the aromaticity indices from different criteria at a small fraction of the computational cost (especially as regards the MCI). The EDDBG/H function, in turn, provides a clear a distinct picture of global aromaticity in large and topologically diversified systems, and, in contrast to the induced ring-current methods, it is directly related to chemical resonance and its structural and energetic consequences. Universality 280 Aromaticity: Modern Computational Methods and Applications

(EDDB can be used to study almost all known types of the ground- and excited-state aromaticity) [24,49,59,60], low computational costs (enabling aromaticity assessment in molecules containing hundreds of atoms), intuitiveness and interpretative simplicity (quantification and visualization of delocalized electrons within the same paradigm), nonreferential character (no parameterization needed), ability to strict separation of different symmetry components and orbital-conjugation topologies even for nonplanar aromatic species [31], etc.—all these features clearly show that the newly proposed EDDB method offers a profound paradigmatic change of the aromaticity quantification, and therefore, it has a great chance to become a tool of reference in the realm of aromatic compounds.

Acknowledgments

D.W.S. acknowledges the financial support by the European Union’s Framework Programme for Research and Innovation Horizon 2020 (2014–2020) under the Marie Skłodowska-Curie Grant Agreement no. 797335 “MulArEffect.” M.S. is grateful to the Ministerio de Economía y Competitividad of Spain (project CTQ2017-85341-P) and the Generalitat de Catalunya (project 2017SGR39) for financial support.

References

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