ORIGINAL PAPER
D2BIA
—flexible, not (explicitly) arbitrary
and reference/structurally invariant
—a very effective
and improved version of the D3BIA aromaticity index
Caio Lima Firme1&Diógenes Mendes Araújo1Received: 23 May 2017 / Accepted: 26 July 2017 / Published online: 7 August 2017 # Springer-Verlag GmbH Germany 2017
Abstract Although there are a multitude of aromaticity in-dexes, only a few have a widespread usage. All famous aro-maticity indexes are limited: HOMA and FLU are reference-dependent; ELF isπ-bond-dependent; PDI is structurally de-pendent and NICS is ring size dede-pendent. These limitations stimulate the continuous search for better (i.e., having no de-pendency), more flexible (i.e., applied to any aromatic system) and more effective (i.e., with excellent correlations with other indexes) aromaticity indexes. The D3BIA was our first topo-logical aromaticity index. It is flexible, reference-independent and effective for planar and caged aromatic molecules. However, one of its terms, the degree of degeneracy (δ), is arbitrary and difficult to carry out for new users. Thus, in this work, we show that D2BIA—an improved version of D3BIA—is a good candidate to be used widely, since it retains the strong points of D3BIA while avoiding its weak point. In particular cases where all studied systems haveδ = 1 (e.g., for acenes), then D2BIA equals D3BIA. For our recent study with acenes, D3BIA (and, as a consequence, D2BIA) has (have) an excellent correlation with FLU according to the MP3 method. In this work, by using DFT calculations for a series involving several six-membered and five-membered heteroaromatic rings, only D2BIA and NICS have very good correlation. All other well known aromaticity indexes used in this work
(FLU, HOMA and ELF) gave poor correlations. As to homoaromatic systems, only D2BIA vs NICS and D2BIA vs FLU plots have excellent correlations. HOMA has the worst results in this series. Thus, D2BIA proved to be flexible and effective for the analysis of heteroaromatic rings of different sizes and for caged homoaromatic systems. Moreover, D2BIA has better correlations than D3BIA for planar aromatic sys-tems, and same correlations for caged-homoaromatic systems.
Keywords D3BIA . D2BIA . Aromaticity index . QTAIM . Heteroaromatic rings . Homoaromaticity . NICS
Introduction
Few concepts in Chemistry are theoretically invariant and physically well-defined. Aromaticity is an iconic example of a poorly defined concept having several definitions derived from different quantum chemical approaches from molecular orbital (MO) and valence bond (VB) to electron density and topology [1, 2]. Extended Lewis description of chemical bonds can be used to characterize aromatic and anti-aromatic systems according to adaptive natural density partitioning (AdNDP) [3] or localized ELF or-bitals (ELF-LOC) [4].
The study of aromaticity seems not to have produced a consistent theory [5], even after more than a century and a half of several theories developed from Kekule’s benzene hexag-onal structure [6], MO Huckel method [7], Pauling’s VB
res-onance theory [8] going towards most modern aromaticity theories such as the VB spin-coupled study of Cooper et al. [9], the interference-VB energy portioning approach proposed by Cardozo et al. [10,11], and the multicenter bonding con-cept for benzene [12].
Electronic supplementary material The online version of this article (doi:10.1007/s00894-017-3433-6) contains supplementary material, which is available to authorized users.
* Caio Lima Firme [email protected]
1 Instituto de Química, Universidade Federal do Rio Grande do Norte,
Campus Lagoa Nova, Natal, Rio Grande do Norte CEP 59058-970, Brazil
Apart from the consistency dilemma, aromaticity indexes (and/or concepts) have been used to understand an intermina-ble list of intricate chemical structures [13], such as the fam-ilies of bisnoradamantenyl dications [14], tetrahedranes [15], cyclophanes [16], all-metal aromatic clusters [17], caged structures with spherical aromaticity [18], structures with su-pramolecular aromaticity [19], carbo-aromatic compounds [20], etc.
A multitude of aromaticity indexes have been created [21] even nowadays [22]. Several aromaticity indexes are based on geometric (such as HOMA [23] and HOMHED [24]), magnetic (e.g., NICS [25], free of in-plane compo-nent NICS, FiPC-NICS [26], diamagnetic susceptibility exaltation [27] and ring current [28, 29]), energetic (e.g., aromatic stabilization energy [30] and bond resonance en-ergy [31–33]) and topological (for instance, D3BIA [34,
35], PDI, FLU and MCI [36]) criteria. More recently, an aromaticity index based on vibrational stretches [37] has been developed, proving that an open avenue for devel-oping new aromaticity indexes still exists. However, only a few are popular, namely HOMA, FLU, NICS, ELF and PDI. Even these popular aromaticity indexes are limited: NICS is not a reliable indicator of aromaticity for systems (e.g., Al2X6) with large anisotropy in the center of the
ring [38], HOMA and FLU are reference dependent, while ELF and PDI are electronically and structurally depen-dent, respectively, which prevents them from being ap-plied successfully to all types of aromatic systems. These limitations in the well known aromaticity indexes trigger the search for better aromaticity indexes with im-proved characteristics: having no dependency; being ap-plicable to any aromatic system (flexible aromaticity in-dex); and having excellent correlations with other aroma-ticity indexes (effective aromaaroma-ticity index).
Our previously developed aromaticity index, D3BIA, has no type of dependency. However, one of its terms, the degree of degeneracy (δ), has a couple of arbitrary rules that are somewhat difficult to carry out for new users. In order to overcome this weak point, we improved D3BIA by removing δ, yielding the so-called D2BIA, which has four reasons to become successful: (1) it is easy to calculate; (2) it has no type of dependency; (3) it can be used for any sort of aromatic, all-metal aromatic, sigma aromatic or homoaromatic systems; and (4) it has very good correlations with some famous aromaticity indexes. Besides, D2BIA can be easily implemented in open source programs like MultiWFN [39]. Since only rarely are all famous aromaticity indexes coincidently unsuc-cessful (i.e., no good correlation between aromaticity in-dexes can be found) for a specific series of aromatic sys-tems, our aim was to test the quality of D2BIA in a search for correlation with one or more of well known indexes in our studied series.
D2BIA—density and delocalization-based index of aromaticity
Being inspired by the VB concept of multicenter bonding for aromaticity of benzene [12], the aromaticity index D3BIA was developed [34] in an attempt to associate this VB concept with topological data from the quantum theory of atoms in mole-cules (QTAIM). By assuming that aromaticity is directly de-pendent on multicenter bonding of the corresponding system, and that the latter could be related with the overlap of VB singly occupied p orbitals (from modern VB theory [9]) of the corresponding system, then two sets of topological data were associated with the VB concept of multicenter bonding: (1) the charge density in the center of the ring, and (2) the degree of degeneracy of the atoms in the ring (i.e., arbitrary rules based upon the virial partitioning scheme of atomic en-ergy in order to determine the uniformity of the enen-ergy of atoms in the aromatic center). In addition to these two topo-logical factors, we observed the importance of the uniformity of delocalization index from all bonded atomic pairs in the ring, the so-called DIU (delocalization index uniformity). Thus, D3BIA [34] is a simple product of three terms: DIU, RDF (ring density factor, whose formula varies for planar and caged structures [35]) andδ (degree of degeneracy). D3BIA proved to be an effective index (i.e., it has very good correla-tions with well-known aromaticity indexes) for homoramatic and sigma-bonded caged structures [14,15] as well as for acenes [35]. Unlike HOMA [40], FLU [41] and PDI [42], D3BIA is a non-reference aromatic index, and it is also not size dependent such as NICS [43,44]. However, some re-searchers inclined towards using D3BIA found that the rules of degree of degeneracy were difficult to implement. Moreover, a good aromaticity index should have very low level of arbitrariness. Thus, we decided to improve this aro-maticity index by retaining its advantages and removing its deficiencies.
The aromaticity index D2BIA is a product of two terms: DIU and IDF (inner density factor). Both terms have the same formulas as those from D3BIA.
The IDF formula varies according to the molecular system: whether it is planar (IDFplanar) or topologically caged
(IDFcaged). It is important to emphasize that RDF (from
D3BIA) and IDF (from D2BIA) have the same formulas, but IDF represents the most suitable acronym since the word ‘ring’ is not appropriate for caged systems.
IDFplanar¼ 1 þ λ2
⋅ρRCP ð1Þ
IDFcaged¼ ρCCP ð2Þ
Whereλ2 is the averaged eigenvalue of Hessian of charge
density from u!2pointing towards the center of ring;ρRCPis
the charge density of the cage critical point (CCP). This value is taken from each bond critical point (BCP) in the ring circuit of the aromatic molecule. The charge density from each atom-ic pair in the ring decreases towards the center of the ring. Then, theλ2is used in IDFplanarto take into account the
steep-ness of the slope of the charge density decrease from each bonded atomic pair in the ring towards the center of the ring. In our earlier work with D3BIA, it was noted that some aro-matic systems have a steep descent of charge density from each bond in the ring towards the center of the ring (e.g., Si6H6) while others have a smooth decrease of charge density
[34].
The IDF term is an attempt to associate the VB concept of multicenter bond in aromatic systems [12] with topology. Coincidently, Mohajeri and Ashrafi [45] developed an aroma-ticity index based solely on charge density of RCP. There are plenty of situations where one may observe that aromaticity and the charge density in the ring are directly related. For example, in our recent study of acenes [35], we observed that the aromaticity of the terminal ring in linear acenes decreases as the number of fused rings in the corresponding acene in-creases. The MP3 calculations from this work clearly indicate that by increasing the number of fused benzenoid rings in the acene, theρRCPfrom the terminal ring decreases and both
corresponding local D3BIA and FLU also decrease. From the standpoint of multicenter bonding [12] (in association with electron density) for benzenoid rings, one may assume that by decreasing the ring charge density the multicenter bonding also decreases, which means the aromaticity decreases.
From empirical observation of several aromatic systems [34] it was noticed that our aromaticity index could not have topological term(s) associated only with multicenter bond in aromatic systems. Thus, by taking for granted the importance of bond length uniformity in the aromatic ring as a geometric criterion, DIU was incorporated in D3BIA and D2BIA formu-las (Eq.3). The DIU is an attempt to associate bond length uniformity of the aromatic ring with topology. It is important to make clear that DIU was not inspired in either HOMA or FLU.
The DIU formula is the following: DIU ¼ 100− 100σ
DI
ð3Þ Where DI is the averaged delocalization index value (DI), involving all bonded atomic pairs from caged (in a mathemat-ical combination) or planar aromatic structures; andσ is the mean deviation of these DIs. The DIs from the bonded atomic pairs in the ring of benzene are: DI (C1–C2), DI (C2–C3), DI (C3–C4), DI (C4–C5), DI (C5–C6) and DI (C6–C1). For a caged structure such as tetrahedrane, all DIs involving the C1– C4 cage are used: DI (C1–C2), DI (C2–C3), DI (C3–C4), DI (C4–C1), DI (C2–C4), DI (C1–C3). It is important to
emphasize that DI is the (rational) number of electrons shared between any atomic pair [46].
D2BIA has two distinct formulas depending on the molec-ular structure (planar or cage): D2BIA(p) and D2BIA(c), where p means planar structure and c means caged structure.
D2BIA pð Þ ¼ 1þ λ2 ⋅ρRCP h i ⋅DIU ð4Þ D2BIA cð Þ ¼ ρCCP⋅DIU ð5Þ
It is also important to note that when the set of studied systems has a unity of degree of degeneracy, then D3BIA automatically equals D2BIA (Eq. 6). By using D3BIA for acenes [35], all studied systems haveδ = 1, and, since there is an excellent correlation between FLU and D3BIA from MP3 method, we can assume that also there is an excellent correlation between FLU and D2BIA for acenes. This means that D2BIA has already been proved to be effective even before being formally formulated.
D2BIA¼ D3BIA∴δ ¼ 1 ð6Þ
Computational details
All geometry optimizations, frequency and NICS calculations were done in Gaussian 09 [47]. For geometry optimizations and frequency calculations, ωB97XD [48]/6–311++G (d,p)
[49,50] level of theory was used. All optimized geometries are minima in the potential energy surface since they present-ed no imaginary frequencies. NICS calculations were done using B3LYP [51–53]/6–311++G (d,p) level of theory from
single point calculations of the optimized structure. By using a gauge-independent atomic orbital method [54], all NICS cor-respond to the negative values of absolute magnetic shielding calculated in the center of both rings and cages (for derivatives of 1,3-dehydro-5,7-adamantanediyl dication). Data for the cal-culation of the topological aromaticity indexes were obtained from the ωB97XD/6–311++G (d,p) wave function. Topological data related to FLU, D3BIA and D2BIA were generated from the AIMALL package [55] in order to be calculated in a spreadsheet program according to their corre-sponding formulas (see details of D2BIA calculation in
Supplementary Material). FLU (π) and ELF (π) were directly
obtained from the MultiWFN package [39]. From MO analy-sis of the studied systems, one can obtain the π-MOs, and hence obtain ELF (π) from the ELF value of (3, −1) critical point of the bifurcation position of ELF domains [56].
Results and discussion
In order to test the flexibility of D2BIA for planar aromatic rings and caged homoaromatic systems, we analyzed two dif-ferent cases (or series): (1) five-membered heteroaromatic
rings and NxCy-benzenoid systems; and (2) derivatives of
1,3-dehydro-5,7-adamantanediyl dication. Aromatic systems of the first series are iconic examples of planar aromaticity be-sides acenes (already used in our former work in which D3BIA equals D2BIA), and 1,3-dehydro-5,7-adamantanediyl dication and its derivatives are other iconic example of
caged-homoaromatic molecules, which makes them a good choice for studying D2BIA.
To evaluate the effectiveness of D2BIA, its coefficients of determination with well-known aromaticity indexes [HOMA, FLU, ELF (π) and NICS] were obtained. All values of aro-maticity indexes for both series are depicted in Table 1and Table 1 Values of D3BIA, D2BIA, FLU, NICS, HOMA and ELF for each studied molecule within its series
D3BIA D2BIAa FLUa,b,c NICSd HOMAb,e ELF(π)f First series 1 Benzene 1.0187 1.0187 0.0000 −8.05 1.0000 0.9418 2 Pyridine 0.8310 0.9972 0.0022 −6.83 0.5619 0.8852 3 Pyrazine 0.6631 0.9946 0.0050 −5.33 0.1495 0.8182 4 Pyridazine 0.5640 0.8461 0.0038 −5.33 0.2837 0.8790 5 Pyrimidine 0.6350 0.9525 0.0070 −5.52 0.1755 0.8381 6 1,3,5-triazine 0.4478 0.8956 0.0133 −4.05 −0.2627 0.8003 7 1,2,4-triazine 0.3952 0.7904 0.0422 −3.75 −0.1296 0.8221 8 1,2,4,5-tetrazine 0.4025 0.6038 0.1116 −1.72 −0.5376 0.8238 9 1,2,3,4-tetrazine 0.3582 0.5373 0.1116 −2.62 −0.3229 0.8965 10 Dichloro-1,3,5-triazine 0.3102 0.9219 0.0166 −4.88 −0.4598 0.8181 11 Dichloro-1,2,4,5-tetrazine 0.4217 0.6325 0.0156 −1.91 −0.5642 0.8481 12 Furan 1.7628 2.2035 0.1504 −11.85 0.2910 0.7278 13 Pyrrole 1.8730 2.3413 0.0489 −13.59 0.8665 0.8161 14 Thiophene 1.6888 2.1110 0.0947 −12.94 0.7799 0.9248 15 Pirazole 1.3120 2.1866 0.0337 −13.62 0.9326 0.8144 16 Imidazole 1.3840 2.3066 0.0589 −13.09 0.8887 0.7670 17 Isoxazole 1.1639 1.9398 0.1477 −12.29 0.5280 0.7601 18 Oxozole 1.2943 2.1572 0.1670 −11.44 0.3374 0.6788 19 Izothiazole 1.3672 2.2787 0.0756 −13.32 0.8806 0.8119 20 Thiazole 1.2636 2.1060 0.1069 −12.91 0.8071 0.8583 Second series 21 1,3-Dehydro-5,7-adamantanediyl +2 5.6252 5.6252 0.0000 −47.53 0.9999 -22 6-Azo-substituted analog 5.3513 5.3513 0.0084 −45.42 0.9493 -23 6-Bora-substituted analog 4.6230 4.6230 0.0467 −45.01 0.8685 -24 6-Oxo-substituted analog 5.3604 5.3604 0.0169 −44.7 0.8373 -25 6-Sylil-substituted analog 4.4082 4.4082 0.0518 −46.64 0.6944 -26 6-Tio-substituted analog 4.9721 4.9721 0.0179 −47.56 0.9928 -27 8-PH-substituted analog 4.4758 4.4758 0.0733 −46.91 0.9070 -28 1,3-Diazo-substituted analog −0.0070 −0.0130 2.8707 −7.88 0.4720 -29 1,3-Dioxo-substituted analog 0.0864 0.1728 1.8210 −5.79 0.4358 -30 1,3-Ditio-substituted analog −0.0080 −0.0160 2.7457 −2.88 −0.5656 -31 1,3,5,7-Sylil-substituted analog 3.1945 3.1945 0.2371 −28.1 0.8754 -a
D2BIA(c) for second series and FLU (π) for the first series
bReference for FLU and HOMA in second series was 21. FLU (π) was not used since there is no electron π for this series cFor the subset of five-membered heteroaromatic rings (12–20), FLU (π) was used instead of FLU
d
In ppm
e
Regarding HOMA, the reference molecule for the subset of five-membered heteroaromatic rings (12–20) was benzene since no neutral heteroaromatic five-membered ring can be chosen as reference
f
will be analyzed below in conjunction with other data in each series. For the second series, 1,3-dehydro-5,7-adamantanediyl dication was chosen as the reference for HOMA and FLU. As with ELF for the second series, it was not possible to separate σ- and π-MOs and, as a consequence, the corresponding cal-culations of ELF (σ) and ELF (π) were not successful. The reference of HOMA for 1 to 20 was benzene. Although FLU(π) is reference-invariant [57], it was not used for the second series since there are noπ electrons for these caged homoaromatic molecules. TheSupplementary Materialshows all topological data used to calculate D2BIA for both series. First series
The first series involves benzene (as a reference), NxCy
-ben-zenoid systems (2–11) and five-membered heteroaromatic rings (12–20). Figure1 depicts their optimized geometries. Table2shows the correlation coefficients involving all aro-maticity indexes: D3BIA, D2BIA HOMA, FLU (π), NICS and ELF. It is important to emphasize that only D2BIA and NICS have an excellent correlation (R2= 0.9527). The second
best correlation derives from D2BIA and ELF (π) (R2= 0.7767) (see their plots in Fig.2). Other than the corre-lation between HOMA and NICS (R2= 0.7512), all other coefficient of determination are very poor. FLU (π) does not have any reasonable correlation with any other studied aroma-ticity index in this series. These results prove that D2BIA is a very effective aromaticity index.
When analyzing the subset of six-membered rings (1– 11), only the plots D2BIA versus NICS, NICS versus HOMA and ELF (π) versus HOMA have reasonable cor-relations, i.e., R2 > 0.7500 (see footnote in Table 2). All other relations (for the studied aromaticity indexes) in-volving 1 to 11 are very poor. No studied aromaticity index [D2BIA, FLU (π), NICS, HOMA and ELF (π)] has a very good correlation (R2 > 0.9000) for the subset of six-membered heteroaromatic rings. As to the subset of five-membered rings (12–20), only the relations NICS vs FLU (π), NICS vs HOMA, NICS vs ELF (π) have excel-lent coefficients of determinations (see footnote in Table2). Thus, when analyzing the whole series and both subsets, NICS is the most effective. Taking for granted that the whole series is more important than each subset, since it encompasses a greater number of molecules, D2BIA is the second most effective aromaticity index because it has excellent and reasonable correlations in the whole series and in the six-membered subset, respec-tively. Other aromaticity indexes have neither excellent nor reasonable correlations for the whole series: some
(12)
(13)
(14)
(15)
(17)
(16)
(18)
(19)
(20)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
Fig. 1 Optimized geometries of benzene (1), NxCy-benzenoid systems
(2–11) and five-membered heteroaromatic rings (12–20)
Table 2 Coefficients of determination (R2) involving all studied aromaticity indexes (D3BIA, D2BIA, FLU, NICS, HOMA and ELF) for the first series of six-membered (6-M) and five-membered (5-M) heteroaromatic rings (1–20)a. Values >0.9000 are highlighted in bold
D3BIA D2BIA FLU(π) NICS HOMAb ELF(π) D3BIA 1 0.8696 0.1889 0.8699 0.6427 0.5666 D2BIA 1 0.2415 0.9527 0.5782 0.7767 FLU 1 0.1627 0.0044 0.5640 NICS 1 0.7512 0.6381 HOMA 1 0.1773 ELF(π) 1
aBy considering each six-membered (6-M) and five-membered (5-M)
subsets separately, the best R2values were 0.8012 (D2BIA vs NICS for
6-M subset), 0.8289 (HOMA vs ELF for 6-M subset), 0.8288 (HOMA vs ELF for 6-M subset), 0.9150 (FLU vs NICS for 5-M subset), 0.8537 (FLU vs HOMA for 5-M subset), 0.8875 (FLU vs ELF for 5-M subset), 0.8400 (NICS vs HOMA for 6-M subset), 0.9198 (NICS vs HOMA for 5-M subset), 0.9187 (NICS vs ELF for 5-5-M subset), and 0.8728 (HO5-MA vs ELF for 5-M subset). All other R2 for 6-M and 5-M subsets were very poor (< 0.7000). As to D3BIA, no subset was evaluated since our main focus is D2BIA
b
Regarding HOMA, the reference molecule for the subset of 5-M heteroaromatic rings (12–20) was benzene since no neutral heteroaromatic 5-M ring can be chosen as reference
[ELF (π) and HOMA] have reasonable correlations in the six-membered subset, along with NICS and D2BIA, and all of them (FLU (π), ELF (π) and HOMA) have excellent correlations in the five-membered subset.
The influence on the aromaticity of the number of nitro-gen atoms in the benzenoid ring and the symmetry was analyzed in the NxCy-benzenoid systems where x varied
from 0 to 4 and y from 6 to 2 concomitantly. In Fig.1, they are arranged according to increasing number of nitrogen atoms in the ring. As to the number of nitrogen atoms in the ring, except for ELF (π), all other aromaticity indexes agreed that, when going from x = 0 to x = 4, the aromaticity decreased accordingly. Regarding the symmetry analysis of NxCy-benzenoid systems with x = 2, and considering only
Csand C2symmetry elements, pyrazine (3) has 3 Csand 3
C2, pyrimidine (5) has 2 Csand 1 C2and pyridazine (4) has 1
Csand 1 C2. Only D2BIA indicates a correlation between
symmetry and aromaticity, i.e., as the symmetry of the mo-lecular system decreases, the aromaticity decreases. In that case, pyrazine has D2BIA = 0.9946, pyrimidine has D2BIA = 0.9525, and pyridazine has D2BIA = 0.8461, which accompanies the symmetry trend. In the symmetry analysis of NxCy-benzenoid systems with x = 3 and x = 4,
1,3,5-triazine and 1,2,4,5-tetrazine are more symmetric than their isomers 1,2,4-triazine and 1,2,3,4-tetrazine. D2BIA also indicates that the more symmetric isomers (6 and 8) are more aromatic. It seems reasonable to rationalize that more symmetric systems are more aromatic. Benzene, for instance, is the most aromatic and most symmetric of the six-membered subset. However, no aromaticity index other than D2BIA shows this linear trend between aromaticity a n d s y m m e t r y f o r t h i s s u b s e t o f s i x - m e m b e r e d heteroaromatic systems.
In the five-membered subset, D2BIA also indicates the decrease of aromaticity as the number of heteroatoms in the ring increases. For example, pyrrole (13) is more aromatic than pirazole (15) and imidazole (16); furan (12) and pyrrole are more aromatic than isoxazole (17) and oxozole (18). No other aromaticity index indicates this trend.
When comparing D2BIA and D3BIA with other aromatic-ity indexes, D2BIA has the best correlations with all of them except for HOMA. Thus, for the first series, D2BIA is more effective than D3BIA.
Second series—derivatives
of 1,3-dehydro-5,7-adamantanediyl dication
The topological features of some derivatives of 1,3-dehydro-5,7-adamantanediyl dication (21), along with dozens of differ-ent types of neutral and cation analogs, were investigated by QTAIM [58]. No correlation was found between aromaticity indexes due to the highly distinguished electronic nature of the studied molecules in that very large series (which was made up of neutral, cation, dication, neutral dehydro, neutral bisdehydro, double structure of adamantane moiety, bisadamantyl, and all of their analogs). More straightforward-ly, the electronic structure of the adamantane moiety cannot change drastically when attempting to obtain correlations be-tween aromaticity indexes. Thus, our strategy in this work was to generate derivatives of 1,3-dehydro-5,7-adamantanediyl dication, but leaving apart neutral, cation and bisdehydro an-alogs. As a consequence, one carbon vicinal to the 4c-2e bonding was changed into trivalent nitrogen (22), trivalent boron (23), divalent oxygen (24), tetravalent silicon (25), di-valent sulfur (26) and tridi-valent phosphorous (27). In addition, R² = 0.9527 -16 -14 -12 -10 -8 -6 -4 -2 0 0.00 0.50 1.00 1.50 2.00 2.50 D2BIA D2BIA vs NICS R² = 0.7767 0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.50 1.00 1.50 2.00 2.50 ELF D2BIA D2BIA vs ELF (A) (B) NICS / ppm
Fig. 2 a D2BIA vs NICS and b D2BIA vs ELF plots for the first series (1–20)
Table 3 Coefficients of determination (R2) involving all studied
aromaticity indexes (D2BIA, FLU, NICS and HOMA) for the series of 1,3-dehydro-5,7-adamantanediyl dication (21) and its derivatives (22– 30)a. Values >0.900 are highlighted in bold
D2BIA(a) FLU(b) NICS HOMA(b)
D2BIA 1 0.909 0.9633 0.6231
FLU 1 0.893 0.6806
NICS 1 0.6374
HOMA 1
a
The correlation between D3BIA and D2BIA is 99.99%; the reference aromatic system in second series for FLU and HOMA was 21
two carbon atoms of the 4c–2e bonding were changed into two cationic (tetracoordinated) nitrogen atoms (28), two cat-ionic (tricoordinated) oxygen atoms (29) and two catcat-ionic (tricoordinated) sulfur atoms (30). Finally, four carbon atoms of the 4c–2e bonding were changed into silicon atoms (31). Their optmized geometries are depicted in Fig.3.
By analyzing the geometries of 28–30 (Fig.3), the C–C
bond in the 4c-2e bonding is much smaller than that from 21
(2.050 Å), indicating that there is less charge transfer from this bond to the cationic heteroatoms (nitrogen atoms in 28, oxy-gen atoms in 29 and sulfur atoms in 30) as compared with all other dications from this series. Except for HOMA, the values of D2BIA, FLU and NICS for 28–30 establish their non-aromatic character (Table1). HOMA values for 28 and 29 were higher than those from the most aromatic systems in the first series, i.e., HOMA results for 28 and 29 show that they are homoaromatic, which is in disaccord with either the geometric data (small C–C bond lengths in the 4c–2e bond in Fig. 3) or all other aromaticity indexes for both dications. Moreover, HOMA has neither excellent nor reasonable corre-lations with any other aromaticity index (Table 3). Then, HOMA was not effective for the study of the second series.
Table 3depicts all coefficients of determination between the studied aromaticity indexes (D2BIA, NICS, FLU and HOMA). These data indicate that only D2BIA has two very good correlations (R2> 0.9000) with NICS and FLU (see also Fig.4). The results in Table3suggest that D2BIA can be used successfully for the study of caged homoaromatic systems.
For the second series, the correlation between D3BIA and D2BIA was 99.99%. Thus, for caged-homoaromatic series, D2BIA is as effective as D3BIA.
Conclusions
In our previous work on aromaticity of acenes [35], D3BIA and FLU had excellent correlation using the MP3 method. This result is also valid for D2BIA since all studied acenes haveδ = 1, then D3BIA = D2BIA for that series.
For the series of six-membered and five-membered heteroaromatic systems (1–20), only D2BIA and NICS have an excellent correlation (R2= 0.9527). None of other aroma-ticity indexes [HOMA, FLU and ELF (π)] gave a very good correlation for this series.
For the subset of six-membered rings (1–11), only the plots D2BIA versus NICS, NICS versus HOMA and ELF (π) ver-sus HOMA had reasonable correlations (R2> 0.7500). All other relations were very poor. For the subset of
five-R² = 0.9633 -60 -50 -40 -30 -20 -10 0 10 -1.00 1.00 3.00 5.00 7.00 mp p/ S CI N D2BIA D2BIA vs NICS R² = 0.909 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 -2.0 0.0 2.0 4.0 6.0 FLU D2BIA D2BIA vs FLU (A) (B)
Figure 4 a D2BIA vs NICS and b D2BIA vs FLU plots for the second series (21–31) (21) (22) (23) (24) (25) (26) (28) (29) (30) (27) (31)
Fig. 3 Optimized geometries of 1,3-dehydro-5,7-adamantanediyl dication (21) and their analogs (22–31)
membered rings (12–20), only the plots NICS vs FLU, NICS vs HOMA, NICS vs ELF (π) had correlations > 90%. Thus, regarding correlations with other aromaticity indexes, NICS is the most effective for this series. The second most effective is D2BIA because it has excellent and reasonable correlations with NICS for the whole series and the 6-membered subset, respectively. Likewise, HOMA and ELF (π) had reasonable correlations only in the 6-membered subset; however, they do not have very good correlations for the whole series, although they have excellent correlations for the 5-membered subset. FLU (π) has poor correlations for the whole series as well as the 6-membered subset.
Only D2BIA and NICS indicate a decreasing trend of aro-maticity as the number of heteroatoms increase in both 6-membered and 5-6-membered subsets. More importantly, only D2BIA indicates a clear and direct relation between aromatic-ity and symmetry for this subset.
For the second series of derivatives of 1,3-dehydro-5,7-adamantanediyl dication (21), HOMA was not effective since it does not have any reasonable correlation with other aroma-ticity indexes, and also it fails in not indicating two non-homoaromatic systems (28 and 29). On the other hand, D2BIA was the most effective aromaticity index because it has two excellent correlations with NICS and FLU. Thus, D2BIA can be used successfully for caged homoaromatic systems.
When comparing with D3BIA, D2BIA has an improved quality since, for the first series, D2BIA is more effective than D3BIA, and, for the second series, D2BIA is as effective as D3BIA.
Therefore, D2BIA is a flexible, non-arbitrary, not structur-ally or reference dependent, and very effective aromaticity index. D2BIA is a good candidate to be widely used for the study of a variety of aromatic or homoaromatic systems.
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