Matta, Boyd. The quantum theory of atoms in molecules

412 15 Aromaticity Analysis by Means of the Quantum Theory of Atoms in Molecules

Scheme 15.3 Reprinted, with permission, from Ref. [40]; copyright 2003, Wiley–VCH.

3He@C70 and 3He@C60 follow the same trend as calculated NICS for C70 and C60 [44].

15.5.2

E ect of Substituents on Aromaticity

Benzene is regarded as the archetype of aromaticity, fulfilling all the criteria attributed to this property [2]. This molecule has been used as the reference for the proposal of quantitative descriptors of substituent e ects, that is, the Hammett substituent constants [45]. We have tried to establish a relationship between the substituent e ect and the aromaticity of a series of monosubstituted derivatives of benzene [46]. Table 15.4 contains the HF/6-31þG(d)//B3LYP/6- 311þG(d,p) NICS, the B3LYP/6-311þG(d,p) HOMA, and the B3LYP/6- 311G(d,p)//B3LYP/6-311þG(d,p) PDI aromaticity measures, with di erent substituent constants (explained elsewhere [46, 47]). It is apparent that, although the nature of the substituents varies substantially along the series (sp varying from 0.66 for a strongly electron-donating NH2 substituent to 1.91 for a strongly electron-accepting NNþ substituent), no significantly large changes of aromaticity are observed. This proves the high resistance of the p-electron structure of benzene, in agreement with its preference for substitution rather than addition reactions. In addition, PDI is the only index that gives a direct correlation between aromaticity and the substituent constants (Fig. 15.1), thus proving to be a good descriptor of changes of p-electron delocalization in substituted benzenes [46].

In another study the substituent e ect was analyzed for a series of carbazole derivatives (Scheme 15.4) and an attempt was made to predict the reactivity of these systems quantitatively as a function of the substituent by measuring di er-

15.5 Applications of QTAIM to Aromaticity Analysis 413

Scheme 15.4 Reprinted, with permission, from Ref. [48]; copyright 2004, Royal Society of Chemistry.

ent local aromaticity criteria [48]. As is apparent from Fig. 15.2 for the substituted ring, the results for the three aromaticity criteria are scattered over a narrow range of values. As in the previous example, the p-electron structure of the aromatic ring is slightly a ected by substituents. There is also clear divergence in

Table 15.4 GIAO/HF/6-31þG(d) NICS, B3LYP/6-311þG(d,p) HOMA, and B3LYP/6-311G(d,p) PDI aromaticity indices, calculated at B3LYP/ 6-311þG(d,p) geometry, for di erently substituted benzene (C6H5X) structures. Also listed are substituent constants sþ, s , sm, and sp and resonance constants Rþ and R (m and p refer to meta and para substitution and þ and indicate the ability of the substituent to e ectively delocalize either a positive or negative charge).[a]

xX	NICS	HOMA	PDI	sB/sC	sm	sp	RB/RC
	(ppm)		(electrons)
aNNþ		0.96	0.080	3.43	1.76	1.91	1.85
	10.6
aNO	9.8	0.98	0.091	1.63	0.62	0.91	1.14
aNO2	10.9	0.99	0.096	1.27	0.71	0.78	0.62
aCN	10.3	0.98	0.096	1	0.56	0.66	0.49
aCOCl	9.9	0.98	0.095	1.24	0.51	0.61	0.78
aCOCH3	9.7	0.98	0.097	0.84	0.38	0.5	0.51
aCOOCH3	9.8	0.98	0.097	0.75	0.37	0.45	0.14
aCOOH	9.7	0.98	0.097	0.77	0.37	0.45	0.43
aCHO	9.6	0.97	0.095	1.03	0.35	0.42	0.70
aCONH2	9.9	0.98	0.098	0.61	0.28	0.36	0.35
aCCH	10.1	0.97	0.096	0.53	0.21	0.23	0.31
aCl	10.7	0.99	0.099	0.19	0.37	0.23	0.31
aF	11.7	0.99	0.098	0.03	0.34	0.06	0.52
aH	9.7	0.99	0.103	0	0	0	0
aPh	9.3	0.98	0.098	0.18	0.06	0.01	0.30
aCH3	9.7	0.98	0.100	0.31	0.07	0.17	0.32
aOCH3	10.8	0.98	0.094	0.78	0.12	0.27	1.07
aNH2	9.8	0.98	0.093	1.3	0.16	0.66	1.38
aOH	10.8	0.99	0.095	0.92	0.12	0.37	1.25

a Reprinted, with permission, from Ref. [46]; copyright 2004, American Chemical Society.

414 15 Aromaticity Analysis by Means of the Quantum Theory of Atoms in Molecules

Fig. 15.1 Plot of PDI (electrons) against sm (a) and sp (b). The correlation coe cients are 0.83 and 0.91, respectively. Reprinted, with permission, from Ref. [46]; copyright 2004, American Chemical Society.

Fig. 15.2 Comparative plot of HOMA, NICS (ppm), and PDI (electrons) for the substituted ring (Subs) for the series of carbazole derivatives studied. The x-plot enumeration can be found in Scheme 15.4. Adapted, with permission, from Ref. [48]; copyright 2004, Royal Society of Chemistry.

								15.5	Applications of QTAIM to Aromaticity Analysis									415
Table 15.5	NICS (ppm), PDI (electrons), HOMA, and FLU measures of
the aromaticity of the five and six-membered rings of guanine (G) and
six-membered ring of cytosine (C) computed by the B3LYP method.[a]
System		NICS					PDI
		(ppm)					(electrons)			HOMA			FLU
		G-5		G-6	C-6	G-6		C-6	G-5		G-6	C-6	G-5		G-6	C-6
GC	11.94			4.10	1.86	0.036		0.040	0.848		0.795	0.703	0.025		0.033	0.035
[GC]þ			5.41			0.023		0.042	0.829		0.550	0.773	0.028		0.048	0.031
				0.31	2.49
Ca2þ-GC						0.044		0.045	0.843		0.886	0.797	0.023		0.024	0.029
	10.67			4.76	2.53
Cuþ-GC						0.040		0.043	0.869		0.898	0.761	0.021		0.027	0.032
	10.64			4.59	2.25
Cu2þ-GC			7.37			0.022		0.040	0.915		0.760	0.822	0.033		0.058	0.039
				2.00	3.07

a Adapted, with permission, from Ref. [49]; copyright 2005, Taylor and Francis. Details of the basis set used can be found in Ref. [49].

the ordering of the di erent systems by local aromaticity values given by the B3LYP/6-31þþG(d,p) values of NICS, HOMA, and PDI, at variance with previous analyses of aromaticity which resulted in relatively good agreement among the di erent aromaticity criteria. One must be very cautious with the results, because it is not being possible to give a definite answer about the relative aromaticity of these rings. Finally, it is worth saying that the di erent trends found by use of these criteria are not completely unexpected, because we are considering descriptors based on di erent physical properties.

The last study on this subject has consisted in the analysis of how metal cations and ionization a ect the aromaticity of the Watson–Crick guanine–cytosine base pair (GC) [49]. Table 15.5 contains the B3LYP aromaticity results (HOMA, PDI, NICS, and FLU) for the 6-MRs and 5-MRs studied (Fig. 15.3). H-bond formation in GC implies some loss of p-charge in N1 and a gain in O6, thus increasing the relevance of the resonance structure 2 (Fig. 15.3), which favors intensification of

Fig. 15.3 Schematic representation of charge transfer in the GC base pair. G is guanine and C is cytosine. Reprinted, with permission, from Ref. [49]; copyright 2005, Taylor and Francis.

416 15 Aromaticity Analysis by Means of the Quantum Theory of Atoms in Molecules

the aromatic character of the guanine 6-MR. The increased aromaticity of the guanine and cytosine 6-MRs which results from interactions with Cuþ and Ca2þ is also attributed to the strengthening of hydrogen-bonding in the GC pair, which stabilizes the charge-separation resonance structure 2 [50]. This e ect is stronger for the divalent Ca2þ metal cation than for the monovalent Cuþ. The observed reduction of the aromaticity of the 5-MRs and 6-MRs of guanine because of ionization or interaction with Cu2þ is caused by the oxidation process that removes one p electron, disrupting the p-electron distribution.

15.5.3

Assessment of Clar’s Aromatic p-Sextet Rule

The introduction of Hu¨ckel’s 4n þ 2 rule enabled better comprehension of aromaticity [51], although strictly it could only be applied to monocyclic conjugated systems. This was solved later by Clar’s model of the extra stability of 6n p-electron benzenoid species [42, 52]. According to Clar’s rule, the Kekule´ resonance structure with the largest number of disjoint aromatic p-sextets, i.e. benzene-like moieties, is the most important for the characterization of the properties of polycyclic aromatic hydrocarbons (PAHs). Aromatic p-sextets are defined as six p-electrons localized in a single benzene-like ring separated from adjacent rings by formal CaC single bonds. For example, application of this rule to phenanthrene reveals that resonance structure 2 (Scheme 15.5) is more important than resonance structure 1; which is translated into major local aromaticity of the outer rather than inner rings. The Clar structure of a given PAH is the resonance structure with the maximum number of isolated and localized aromatic p-sextets, with a minimum number of localized double bonds. A PAH with more p-sextets is kinetically more stable than its isomer with fewer. In addition, p-sextets are regarded as the most aromatic rings of PAHs. Some PAHs (e.g. phenanthrene) have a single Clar structure whereas others have several. For these latter, Clar’s rule cannot di erentiate which of the resonance structures is mainly responsible

Scheme 15.5 Reprinted, with permission, from Ref. [53]; copyright 2005, Wiley–VCH.

15.5 Applications of QTAIM to Aromaticity Analysis 417

Table 15.6 PDI (in electrons), HOMA, and NICS (in ppm) values for the

PAHs studied. The numbering is given in Scheme 15.6.[a]

Ring Molecule

		1	2	3	4	5	6	7	8	9	10
PDI	A	0.080	0.069	0.086	0.084	0.083	0.076	0.066	0.080	0.069	0.079
	B	0.047	0.043	0.026	0.034	0.041		0.066	0.053	0.066	0.057
	C			0.044		0.068				0.038	0.031
	D			0.073						0.084	0.085
	E										0.085
HOMA	A	0.856	0.834	0.889	0.811	0.872	0.769	0.619	0.829	0.697	0.749
	B	0.435	0.553	0.030	0.383	0.356		0.696	0.542	0.730	0.305
	C			0.518		0.788				0.266	0.097
	D			0.838						0.883	0.820
	E	10.06	12.74	8.63	9.44	9.93	9.98	8.84	9.94	9.30	0.883
NICS	A										10.19
	B	6.82	5.07	1.18	4.13	5.38		12.60	7.69	11.69	7.68
	C			5.47		11.27				4.58	3.91
	D			11.58						9.81	9.55
	E										8.99

a Adapted, with permission, from Ref. [53]; copyright 2005, Wiley–VCH.

for the aromaticity of the system. In this study we investigate whether three local aromaticity criteria, PDI, HOMA, and NICS, give results consistent with Clar’s original qualitative p-sextet rule [53].

The PAHs studied are depicted in Scheme 15.6. PAHs 1–5 have a single Clar structure and 6–10 have several Clar valence structures, also represented. The corresponding local aromaticity values, calculated at the B3LYP/6-31G(d) level, can be found in Table 15.6. First, for systems with a single Clar structure (1–5), p-sextet rings have higher PDI values, higher HOMA, and more negative NICS than non-p-sextet rings. Hence, all three aromaticity criteria used agree perfectly with the qualitative description given by Clar’s rule. Second, for systems with several Clar structures, it is apparent that the overall aromaticity of the system given by PDI and HOMA agrees with the superimposition of all possible Clar structures. For example, for 7, PDI and HOMA attribute very similar aromaticity to rings A and B, which proves the non-localizability of the p-sextet. NICS, however, attributes much more aromatic character to ring B than ring A, although it is claimed this is because of overestimation by NICS of the local aromaticity of the inner rings of PAHs [54].

418 15 Aromaticity Analysis by Means of the Quantum Theory of Atoms in Molecules

Scheme 15.6 Adapted, with permission, from Ref. [53]; copyright 2005, Wiley–VCH.

15.5.4

Aromaticity Along the Diels–Alder Reaction. The Failure of Some Aromaticity Indexes

This work [33] analyzes the aromaticity along the Diels–Alder reaction between 1,3-butadiene and ethane to yield cyclohexene (Scheme 15.7) [55, 56], which is often taken as a prototypical pericyclic concerted reaction. This reaction is characterized by an aromatic transition state (TS) [56, 57], thus along the reaction path a

15.6 Conclusions 419

Scheme 15.7 Reprinted, with permission, from Ref. [33]; copyright 2005, Elsevier.

peak of aromaticity around the TS is expected. The trends of aromaticity along the path for the di erent criteria applied, at the B3LYP/6-31G(d) level, can be found in Fig. 15.4. Only the magnetic NICS(1) and the electronic PDI criteria find the most aromatic point along the reaction path around the TS of the reaction. In contrast, both geometric HOMA and electronic FLU regard cyclohexene as the most aromatic species in this reaction. This latter trend is also given by the RSS (root summed squares) of the best fitted plane for atoms in the ring, which is an unambiguous measure of molecular planarity [33]. It shows the product as the most planar species, thus in principle implying greater p-electron delocalization. The failure of RSS proves that the flatter structure is not necessarily the more aromatic. On the other hand, HOMA and FLU measure variances of structural and electronic patterns around the ring, and might fail if they are not applied to stable species, for example in a reaction with major structural and electronic changes. The failure of some indices to detect the aromaticity of the TS in the simplest Diels–Alder cycloaddition thus reinforces the idea of the multidimensional character of aromaticity [8] and the need to use several criteria to quantify it.

15.6 Conclusions

A key aspect of aromatic compounds is the p-electron delocalization (and s and even d-electron delocalization in all-metal and inorganic aromatic species) present in these molecules. In this chapter we have defined three new aromaticity indexes founded on evaluation of electron delocalization in the framework of the QTAIM, i.e. the para-delocalization (PDI), aromatic fluctuation (FLU), and FLUp indexes. We have shown that theoretical studies of electron delocalization using QTAIMbased tools have significantly improved our understanding of aromaticity in fullerenes, substituted benzene derivatives, polycyclic aromatic hydrocarbons, and chemical reactivity. The lack of a universally accepted measure of aromaticity, its multidimensional character, and the limitations of almost all descriptors of aromaticity stress the need for new aromaticity criteria in addition to those defined

420 15 Aromaticity Analysis by Means of the Quantum Theory of Atoms in Molecules

Fig. 15.4 Plot of NICS(1) (ppm), PDI (electrons), FLU, HOMA (values divided by 10), and RSS (values divided by 2) against the reaction coordinate (IRP in amu1/2 bohr). Negative values of the IRP correspond to the reactants side of the reaction path, positive values to the product side, and IRP ¼ 0.000 corresponds to the TS of the DA cycloaddition. Reprinted, with permission, from Ref. [33]; copyright 2005, Elsevier.

in this chapter. In a given study of a series of compounds, one can safely reach a definite conclusion about their aromaticity only when di erently based indicators of aromaticity lead to the same results. For this reason, in our opinion, careful analysis of the aromaticity of a set of molecules must be performed using electronically based descriptors, for example the PDI or FLU indexes, and geometrybased indicators such as the HOMA index, magnetically based measures, for example NICS, and energetically based descriptors, for example ASEs.

Acknowledgments

The authors are grateful to Professor Dr Miquel Duran and Dr Xavier Fradera for helpful discussions. Financial help was obtained from the Spanish Ministerio de Educacio´n y Ciencia (MEC) project No. CTQ2005-08797-C02-01/BQU and from

References 421

the Catalan Departament d’Universitats, Recerca i Societat de la Informacio´ (DURSI) project No. 2005SGR-00238. E.M. and J.P. thank the MEC for doctoral fellowship no. AP2002-0581 and the DURSI for the postdoctoral fellowship 2005BE00282, respectively. Excellent service by the Centre de Supercomputacio´ de Catalunya (CESCA) is gratefully acknowledged.

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