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Validation of the Recently Developed Aromaticity Index D3BIA for Benzenoid Systems

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Validation of the Recently Developed Aromaticity Index D3BIA for Benzenoid Systems J Mol Model (2015) 21: 248 DOI 10.1007/s00894-015-2791-1 ORIGINAL PAPER Validation of the recently developed aromaticity index D3BIA for benzenoid systems. Case study: acenes Diógenes Mendes Araújo1 & Tamires Ferreira da Costa2 & Caio Lima Firme2 Received: 27 April 2015 /Accepted: 13 August 2015 /Published online: 2 September 2015 # Springer-Verlag Berlin Heidelberg 2015 Abstract There are four types of aromaticity criteria: energet- Introduction ic, electronic, magnetic and geometric. The delocalization, density and degeneracy-based index of aromaticity, D3BIA, Aromaticity is a key concept in chemistry due to its ap- is an electronic aromaticity index from QTAIM that is not plication to organic and inorganic molecules [1, 2], de- reference dependent and can be used for aromatic, scribing so-called aromatic, homoaromatic [3] and all- homoaromatic, sigma aromatic and other aromatic systems metal aromatic molecules [1, 4]. These molecules have with varying ring size containing hetereoatoms or not. We distinctive geometrical [5], energetic [6] and magnetic used B3LYP, MP2 and MP3 methods to search for linear [7–11] features as well as a unique reactivity behavior relations between well-known aromaticity indices and [5, 12]. Nonetheless, the definition of aromaticity lacks D3BIA for a series of acenes. We found that the D3BIAversus consensus [13] and some new attempts to account for this FLU correlation exceeded 91 % and reasonably good correla- phenomenon have arisen, for instance, the ELF bifurcation tions exist between D3BIA and HOMA and between D3BIA value [14], the concept of π distorcivity from Shaik and and PDI. Previous works have shown that D3BIA can be used collaborators [15, 16] and Nascimento’s multicenter bond for homoaromatic systems and tetrahedrane derivatives (sig- [17], in which π distorcivity has been proved experimen- ma aromaticity), but no previous work has validated D3BIA tally [18]. Moreover, pure σ-aromaticity has been discov- for benzenoid systems. This is the first time we have shown ered recently in molecular systems such as Li clusters [19] that D3BIA can be used successfully for benzenoid systems, and the tetrahedrane cage [20, 21], and the concepts of for example, acenes. This work supports and validates the use multifold aromaticity and conflicting aromaticity have of D3BIA in classical aromatic systems. emerged for metal clusters [22, 23]. An aromaticity index represents an attempt to quantify aro- maticity based on a specific criterion (energetic, geometric, Keywords D3BIA . Aromaticity index . HOMA . FLU . magnetic, reactivity or electronic). However, all the aromatic- PDI . Acenes ity indices proposed to date are somewhat arbitrary, which leads to a lack of agreement on the best aromaticity index. As a consequence, it is advisable to use two or more aroma- ticity indices to evaluate a set of aromatic molecules. The harmonic oscillator model of aromaticity (HOMA) is a * Caio Lima Firme [email protected]; [email protected] well-known geometric aromaticity index that quantifies aro- maticity in comparison with a reference molecule (e.g., ben- 1 zene) with highest bond length equalization [24]. HOMA Departamento de Química, Faculdade de Filosofia, Ciências e Letras values are derived from Eq. 1. de Ribeirão Preto, Universidade de São Paulo, 14040-901 Ribeirao Preto, SP, Brazil 2 Instituto de Química, Universidade Federal do Rio Grande do Norte, α XI ÀÁ Campus Lagoa Nova, Natal, Rio Grande do Norte, Brazil CEP 2 HOMA ¼ 1− Ropt−RI ð1Þ 59078-970 n 248 Page 2 of 10 J Mol Model (2015) 21: 248 Where n is the number of chemical bonds; α is an empirical In this work, we used a set of acenes in order to find constant where HOMA disappears for hypothetical Kekulé correlations between D3BIA and some well-known aro- structures and HOMA reaches its maximum value for struc- maticity indices [37] (HOMA, NICS, FLU and PDI). tures with bond lengths similar to Ropt (α=257.7); RI is the Calculations in acenes were performed with different individual bond length, and Ropt is 1.388 Å. methods (B3LYP, MP2 and MP3) in order to investigate Nucleus independent chemical shift (NICS) is one of the how these correlations change with the method used. most used magnetic aromaticity indices [25] and exists in There was a very good correlation between D3BIA several variations in an attempt to improve its correlation with and FLU and a reasonably good correlation between other aromaticity indices, e.g., NICS(1), NICSzz, and so on D3BIA and HOMA and between D3BIA and PDI for [26]. the studied acenes. For the first time, we have shown More recently, electronic criteria have been developed that D3BIA can be used for benzenoid systems. This for aromaticity quantification [27, 28], and several elec- work supports and validates the use of D3BIA for ben- tronic aromaticity indices based on quantum theory of zenoid systems. atoms in molecules (QTAIM) now exist [29, 30]. Most of these are dependent on the delocalization index (DI), Delocalization, density and degeneracy-based index which relates to the number of shared electrons between of aromaticity each atomic pair derived from the Fermi hole density [31, 32]. The D3BIA is based on three electronic factors: the electronic Based on the assumption that the DI between para-related density within the aromatic ring; the uniformity of electronic carbon atoms is higher than that between meta-related carbon delocalization in the aromatic ring; and the degree of degen- atoms in benzene, the para-DI (PDI) was developed as an eracy of the atoms in the aromatic ring [35]. The general aromaticity index. The PDI is an arithmetic average of all formula of D3BIA is shown in Eq. (3). para-DI in six-membered-rings [33]; however, PDI can be ðÞ¼½RDF ⋅½⋅δ ð Þ applied only to six-membered rings. D3BIA general formula DIU 3 In an attempt to find an aromaticity index that in- Where RDF stands for ring density formula; DIU stands for volved all DIs in the target ring, the aromatic fluctua- DI uniformity; and δ is related to the degeneracy of atoms in tion index (FLU) was developed [34]. Like HOMA, the aromatic site. FLU depends on an aromatic molecule as reference. The D3BIA was inspired by Nascimento’s statement that Its formula is shown in Eq. 2. Bthe stabilization of benzene is brought about by a six-electron ^ 20X 1α 32 bond based on benzene SCVB calculation and its quantum- RINGX δðÞB; A 1 A≠B δðÞA; B −δref ðÞA; B FLU ¼ 4@X A 5 ∴ ð Þ dynamical type analysis for chemical bond [17], from which n δ ðÞA; B 2 A−B δðÞA; B ref we associated the degree of degeneracy of atoms in the aro- B≠A matic ring, δ, and the electronic density inside the aromatic X X 1 if δðÞB; A > δðÞA; B ring, RDF. The third factor of D3BIA, DIU, on the other hand, α ¼ A≠XB XB≠A was obtained empirically from the analysis of several aromatic −1 if δðÞB; A ≤ δðÞA; B molecules, including benzenoid, hetero and ionic aromatic A≠B B≠A systems, with varying ring size [35]. The DIU formula is shown in Eq. (4). Where the summation involves all atomic pairs in the aro- 100σ DIU ¼ 100− ð4Þ matic site and δ(A,B) is the DI value between atoms A and B. DI Similarly, the delocalization, density and degeneracy-based index of aromaticity (D3BIA) is a QTAIM-based aromaticity Where DI is the average of all DIs from the atoms of the index having the DI in its formula [35]. Like all other aroma- aromatic ring and σ is the mean deviation of all DIs from the ticity indices, the D3BIA is an arbitrary aromaticity index, but atoms in the aromatic ring. has the advantage that it can be applied to all aromatic and The DIU formula was not inspired by HOMA and FLU; homoaromatic systems with or without heteroatoms and it however, all have nearly the same grounding, i.e., the unifor- has no reference parameter. Early works have shown that mity of bond length or DI, though their formulas are not math- D3BIA has very good correlation with NICS for a set of ematically equivalent. This close conceptual relation between homoaromatic bisnoradamantenyl derivatives [36]and HOMA, FLU and DIU corroborates the use of DIU in the tetrahedranes derivatives [21]. However, no previous work D3BIA formula. has shown that D3BIA can be used to quantify the aromaticity The D3BIA can be used for the analysis of aromatic and of benzenoid systems. homoaromatic molecules. However, the RDF is different in J Mol Model (2015) 21: 248 Page 3 of 10 248 the two cases. The RDF formulas are shown bellow: symmetric) are 0.83, 0.67, 0.50 and 0.33, respectively [35], hi according to Eq. 9. RDF ¼ 1 þ λ2 ⋅ρRCP ∴For aromatic molecules ð5Þ n δ ¼ ð9Þ hi N RDF ¼ ρRCP=CCP ∴For homoaromatic molecules ð6Þ Where n is the number of atoms in the aromatic circuit that follow the rules for maximum degree of degeneracy shown λ where 2 is the average of all second eigenvalues from the below, and N is the total number of atoms in the aromatic λ Hessian matrix of the charge density, 2, of all bond critical circuit. !u points (BCP) in the aromatic ring, whose eigenvector, 2, The rules for maximum degree of degeneracy for points from the BCP belonging to the aromatic ring toward aromatic systems are shown below. Although these rules ρ the center of the ring; RCP is the charge density of the ring are arbitrary, they were obtained empirically from a ρ critical point (RCP); and CCP is the charge density of the cage large range of aromatic (neutral and ionic aromatic mol- critical point (CCP).
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