CROATICA CHEMICA ACTA CCACAA 79 (3) 363¿371 (2006) ISSN-0011-1643 CCA-3101 Original Scientific Paper Generalized Polansky Index as an Aromaticity Measure in Polycyclic Aromatic Hydrocarbons Patrick Bultinck,a,* Robert Ponec,b Ana Gallegos,c Stijn Fias,1 Sofie Van Damme,a and Ramon Carbó-Dorcaa,d aDepartment of Inorganic and Physical Chemistry, Ghent University Krijgslaan 281, B-9000 Gent, Belgium bInstitute of Chemical Process Fundamentals, Czech Academy of Sciences, Suchdol, 165 02 Prague 6, Czech Republic cQSAR – European Chemicals Bureau, TP 582, Institute for Health & Consumer Protection, European Commission – Joint Research Centre, Via Enrico Fermi 1, 21020 Ispra (Varese), Italy dInstitute of Computational Chemistry, University of Girona, Campus de Montilivi, 17005 Girona, Spain RECEIVED DECEMBER 6, 2005; REVISED DECEMBER 27, 2005; ACCEPTED FEBRUARY 28, 2006 In this work, the ideas of molecular quantum similarity are used to generalize the Polansky similarity index. The newly developed index gauges the aromaticity of individual benzenoid rings in polyaromatic hydrocarbons by its similarity to benzene beyond the scope of simple Keywords Hückel theory on which it was originally based. The reported generalization allows the new in- Polansky dex to be calculated at a realistic contemporary ab initio level of theory, opening the possibility NOEL of its use as a new measure of aromaticity. As will be shown, the new index correlates very quantum similarity well not only with the original Polansky index but also with the Generalized Population Analy- aromaticity sis based multicenter index. INTRODUCTION Many different measures of aromaticity have alrea- dy been introduced through literature. Many aspects and Aromaticity, despite being quite an old concept,1–2 re- aromaticity related phenomena have been reviewed by mains a lustrous and debated issue since there is no clear Schleyer et al.3 A frequently used classification of these unique definition with a sound quantum chemical basis. measures divides them into structural, energetic, magne- As a result, many different quantities have been derived tic and reactivity based indices.4 Structural indices are to express the degree of aromaticity in various mole- mainly based on equalization of bond lengths and on cules. A striking feature of these different measures is planarity of molecules.5 Energetic indices are mainly ba- that some of them clearly contradict the conclusions sed on the extra energetic stabilization of aromatic com- drawn from other measures. Such divergence is naturally pounds, including various types of resonance energies, hard to reconcile with the unique perception of aromatic aromatic stabilization energies gauged from isodesmic character. reactions, etc. Magnetic indices are based on the special * Author to whom correspondence should be addressed. (E-mail: [email protected]) 364 P. BULTINCK et al. magnetic properties of aromatic compounds, such as proach.16 For this purpose, let us consider a polycyclic chemical shifts and ring currents. In the final group, dif- aromatic hydrocarbon consisting of K fused benzenoid ferent reactivity descriptors are used to assess the degree rings and let us characterize the p electron structure of of aromaticity,6 using e.g. conceptual DFT quantities.7 this hydrocarbon by the set of Hückel molecular orbitals As mentioned above, different (classes of) indices can ji expressed as a linear combination of atomic pp orbi- be contradictory.8–10 An illustration is the orthogonality tals cm. between aromatic stabilization energy and the NICS jcii= ∑ cmm (1) (Nucleus Independent Chemical Shift, a magnetic crite- m 11 rion) values, although there may be a some correlation where the summation runs over all N atoms in the mole- 12–14 within limited sets of molecules. cule. The contemporary situation relating to the definition Based on these orbitals, it is straightforward to in- of aromaticity was recently reviewed in a special issue troduce the charge density-bond order matrix (Eq. (2)) of Chemical Reviews. The introduction to this issue, 3 occ written by P. v. R. Schleyer, clearly stressed that further * p mn= ∑ c mc n (2) efforts to characterize aromaticity and to propose new i i i aromaticity measures and indices are still worth ´ pursuing. In the present work, the electron density itself This matrix, whose dimension is N N, characterizes is used as a natural way to investigate aromaticity, rather the distribution of electron density in the whole molecu- than using structural features or quantities derived from le. In addition to this global information, the matrix also it. To that end, molecular quantum similarity theory is allows one to get information on the electron structure of used as a technique to investigate how different benze- any particular benzenoid ring within the molecule. Such noid rings are compared to benzene itself in different po- information about the particular ring L is inherently con- lycyclic aromatic hydrocarbons. The use of electron den- tained in the fragment of the whole density matrix (2), sity, and a fortiori molecular quantum similarity theory, involving only the atoms contributing to this ring. The to assess the degree of aromaticity was previously pro- basic idea of the Polansky approach to the classification posed by Giambiagi et al., who suggested its use to »open of aromaticity of such a ring is based on the ingenious up new insights into the concept of aromaticity, with solid comparison of the fragment density matrices character- chemical and mathematical foundations«.15 Although izing the benzenoid ring L in the polycyclic molecule A suggested several years ago, no in-depth report has yet with the density matrix of benzene, B. Such a compari- been published on the application of the molecular quan- son is quantitatively expressed by the index (3): tum similarity theory in the context of aromaticity. This = 1 ppA B was the motivation for the present study. PL,B ∑∑ mn mn (3) 2NL m∈>Lnm n∈L Theoretical Development where NL is the number of atoms involved in the ring The natural starting point for the present study is the im- considered (6 in the case of benzenoid ring). The reason portant work by Derflinger and Polansky published in for including this parameter is to ensure proper normali- 1967.16 Based on the Clar postulate17 that individual zation of the index so as to provide maximum similarity benzenoid rings in polycyclic aromatic hydrocarbons (identity) for the comparison of benzene with itself. In (PAH) can be regarded as local benzene-like regions, all other cases, the values will be smaller than 1 and the Polansky and Derflinger proposed to characterize the more the index deviates from its idealized value 1, the aromaticity of these rings in PAH by the »similarity« to less similar is a given ring L to benzene and, conse- benzene itself. This similarity was characterized by the quently, the smaller will be its aromaticity. In this way, a value of a certain index derived from the charge-density simple Hückel Molecular Orbital program can be used bond order matrix. This approach was, however, formu- to compute very quickly the necessary similarity mea- lated only at the level of the nowadays sometimes con- sures for all benzenoid rings that will be considered in sidered outdated Hückel Molecular Orbital theory the present work. (HMO) and despite the attractiveness of this approach, After being reminded of the basic idea of the origi- no attempt has so far been reported to incorporate this nal Polansky approach, let us now address the problem aromaticity measure as such into the framework of more of its generalization beyond the scope of the HMO the- sophisticated contemporary computational tools. Our ory. As already said above, the basic idea of the aim in this study is to fill this gap and to attempt a gen- Polansky approach was to gauge the aromaticity of a eralization of the Polansky approach so as to be applica- given ring in PAH by its similarity to benzene itself. ble at the ab initio level of theory. While in the original HMO-like approach this similarity Prior to describing the basic idea of our generaliza- is straightforwardly given by the index (3), the same ap- tion, it is worthwhile to describe briefly the original ap- proach cannot be straightforwardly extended to more so- Croat. Chem. Acta 79 (3) 363¿371 (2006) GENERALIZED POLANSKY INDEX AS AN AROMATICITY MEASURE 365 phisticated levels of theory. To overcome the drawback NOEL indices as in equation (8) is naturally very quick, of the original approach, we found it useful to benefit since one needs only the MO overlap matrix between the from our experience with quantum molecular similarity molecules involved. This gives the NOEL index an and to attempt a generalization of the index (3) in a way important computational advantage over the MQSM in that would resemble as much as possible the original ap- equation (6). Up to now, the NOEL index has been proach by Polansky. mainly used to study the similarity between benzene and For a detailed account of molecular quantum simi- a small number of substituted benzene molecules, such larity, the reader is referred to recent reviews.18–19 For as aniline, nitrobenzene, etc. and to study the Gamma the present goals, it suffices that the similarity between AminoButyric Acid (GABA) agonists.20,22 Cioslowski et two molecules, A and B, is expressed via the Molecular al. also noted the apparent similarity between the NOEL Quantum Similarity Measure (MQSM) as in equation index and the Polansky approach, but no in-depth (4): analysis of the performance of the NOEL index for aromaticity has been performed thus far. = []rWrrrrrr r ZdA,B∫ A()(,112212 ) B ()d (4) Our aim in this study is to explore the above close parallel of both approaches and to demonstrate that the W r r r r where ( 1, 2) is a positive definite operator, and A( 1) appropriately defined NOEL index can indeed be used r is the one electron density for molecule A at 1.