Matta, Boyd. The quantum theory of atoms in molecules

402 15 Aromaticity Analysis by Means of the Quantum Theory of Atoms in Molecules

through all space yields the negative of the total number of electrons in this molecule.

The Fermi and Coulomb hole density is another two-electron function directly connected to the exchange-correlation density. It is defined as:

r	~r ;~r	Þ ¼	Gxcð~r1;~r2Þ		4	Þ
	xcð 1 2		r ~r	Þ	ð
			ð 1

and represents the decrease/increase, relative to the uncorrelated probability density, in the probability of finding an electron in position ~r2 when a reference electron is fixed at position ~r1. In two landmark works [18, 19], Bader and Stephens using a Hartree–Fock (HF) wavefunction, which includes only the Fermi hole, showed that the extent of localization or delocalization of an electron at ~r1 is determined by the corresponding localization/delocalization of its Fermi hole. A localized Fermi hole indicates the presence of localized electronic charge in the position of the electron of reference and vice versa. For this reason, Fermi density hole maps have been widely used to analyze the electronic localization/ delocalization [20].

Equation (4) shows that the exchange-correlation density is nothing but the Fermi and Coulomb hole density weighted by the density of the reference electron. Bader and coworkers [19, 21] used this quantity to define the localization and delocalization indexes (LIs and DIs) from the double integration of the exchange-correlation density over the atomic basins defined within the QTAIM theory:

lðAÞ ¼ ðA ðA Gxcð~r1;~r2Þ d~r1 d~r2		ð5Þ
ð	ð
dðA; BÞ ¼ 2	Gxc ð~r1;~r2Þ d~r1 d~r2	ð6Þ

BA

The term dðA; BÞ is a quantitative measure of the number of electrons delocalized or shared between atomic basins A and B, and lðAÞ is a measure of the average number of electrons localized on basin A. The following sum rule can easily be demonstrated:

		1			X
		NðAÞ ¼ lðAÞ þ		B0A dðA; BÞ			ð7Þ
			2
	Equation (7) proves that the total number of electrons belonging to a given
basin can be exactly partitioned into its localized ðlðAÞÞ and							delocalized
1	P
2		B0A dðA; BÞ parts. In addition, for closed-shell wavefunctions, one can de-
fine		the global delocalization				or valence of atomic basin A as:

	15.3	Electron Delocalization in Aromatic Systems	403
VðAÞ ¼	X	ð8Þ
	dðA; BÞ ¼ 2½NðAÞ lðAÞ&
	B0A

As shown in the next sections of this chapter, we have used these quantities to define new electronically based aromaticity descriptors.

15.3

Electron Delocalization in Aromatic Systems

In 1996, Bader and coworkers [22], in a seminal work, analyzed electron delocalization in benzene, the archetypal aromatic molecule. They observed by means of contour maps of the Fermi-hole density in conjugated species that the interatomic delocalization of the p-electrons on a given atom, in general, decreases with the distance of the second atom from the one in question. For benzene, however, there is significantly more delocalization of the p density into the basins of the para-related carbon atoms than into those of the meta-related carbon atoms, despite the shorter distance to the latter. These data are in accordance with the energy ordering of the principal resonance structures of benzene – the two Kekule´ structures are the most important, then the Dewar structure connecting para-related carbon atoms, which is, in turn, more relevant than that connecting meta-related atoms. To corroborate what happens in benzene, Bader and coworkers [22] studied the e ect of geometrical distortion on delocalization of the p electrons in benzene by considering a symmetrical distortion (S) in which each equilibrium CaC bond length of 1.42 A˚ was increased by 0.06 A˚ in an a1g stretching mode, and a b2u unsymmetrical one (U) obtained by alternately increasing and reducing the bond lengths to 1.54 and 1.34 A˚ , respectively. The results indicated there is no significant change in the delocalization of the p electrons for S; for U, however, it is seen how delocalization between para carbon atoms largely decreases. The values of the DIs for benzene also confirm the larger delocalization in the para form than in the meta form. In particular, Bader and coworkers

obtained the following DIs at the HF/6-31G(d) level of theory: dðC; C0Þpara ¼ 0:101 e and dðC; C0Þmeta ¼ 0:070 e. At the same level of theory, it is important to recognize that Fulton and Mixon reported, some years previously, almost identical

values for dðC; C0Þpara and dðC; C0Þmeta [23]. Interestingly, these authors showed

that the dðC; C0Þpara has a large p component (0.09 e), at variance with dðC; C0Þmeta. The larger DI found between para-related carbon atoms than between meta-related carbons has been corroborated by use of larger basis sets

and higher levels of calculation. For example, the CISD/6-311G(d,p) values of

dðC; C0Þpara and dðC; C0Þmeta in benzene are 0.071 and 0.054 e, respectively [24]. This result is, therefore, not an artifact of the method used and, consequently, it

has sound physical foundation for being the basis of the definition of our para-DI (PDI) index of aromaticity (vide infra).

40415 Aromaticity Analysis by Means of the Quantum Theory of Atoms in Molecules

15.4

Aromaticity Electronic Criteria Based on QTAIM

Sondheimer defined as aromatic those molecules with a ‘‘measurable degree of cyclic delocalization of a p-electron system’’ [25]. Likewise, Schleyer and coworkers [26] considered the aromaticity ‘‘associated with cyclic arrays of mobile electrons with favorable symmetries’’ and ‘‘the unfavorable symmetry properties of antiaromatic systems lead to localized, rather than to delocalized electronic structures’’. These definitions stress the existence of a direct connection between aromaticity and electron delocalization. As said in the introduction, unfortunately, aromaticity does not have a universally accepted quantitative descriptor. This is the reason for the continuous search for new aromaticity indexes, and the exhaustive revision of the existing ones in this quest for a less ambiguous index which, at the same time, agrees with the most elementary chemical basis of aromaticity. Two aromaticity measures based on DIs have recently been proposed, the PDI and the fluctuation aromatic index (FLU) measures. In this context it is worth mentioning, first, the work of Matta and Herna´ndez-Trujillo [27] who attempted to construct an HOMA-like index from the QTAIM by substituting the bond length by the total electron delocalization, and, second, the use by Bultinck et al. [28] of the n-center electron DIs as descriptors of aromaticity.

15.4.1

The para-Delocalization Index (PDI)

As already said, Bader and coworkers reported that delocalization in benzene is greater for para-related (para-DI) than for meta-related carbon atoms (meta-DI). With this idea in mind, Poater et al. [29] decided to undertake a study to validate the averaged para-DI in six-membered rings (6-MRs) as a measure of local aromaticity. The PDI is a specific measure of aromaticity for 6-MR, in which there are three para-related positions, namely ð1; 4Þ, ð2; 5Þ and ð3; 6Þ:

PDI	¼	dð1; 4Þ þ dð2;	5Þ þ dð3; 6Þ	9	Þ
		3		ð

where dðA; BÞ, the DI, is defined as in Eq. (6).

The correlation between PDI and other measures of aromaticity (HOMA, magnetic susceptibility, and NICS) for the series of M1 to M10 polycyclic aromatic hydrocarbons (PAHs) given in Scheme 15.1 served to validate this measure as a reliable index of aromaticity [29]. The above cited work by Matta and Herna´ndezTrujillo [27] appeared shortly after the introduction of the PDI with the proposal of another local aromaticity index, y, similar to the geometric HOMA, but using the DI as a measure of electron sharing alternation within a ring. This index y was calculated for a series of PAHs revealing perfect correlation with HOMA, and a relatively good correlation with NICS. It has been also found that PDI and the index y result in the same trends for a series of common aromatic molecules [7].

15.4 Aromaticity Electronic Criteria Based on QTAIM 405

Scheme 15.1 Reprinted, with permission, from Ref. [31]; copyright 2005, American Institute of Physics.

The main shortcoming of PDI is that it can only be applied to 6-MR. This disadvantage was somehow overcome by the introduction of DDI [29]. DDI is based on the intuitive idea of comparing the electron delocalization between formal single and double bonds. For compounds with a formal Lewis structure, the di erence between the DI for double and single bonds in a given ring was suggested as a measure of aromaticity. The smaller the di erence (the lower the value of DDI) the closer is electron delocalization in double and single bounds, an indication that both single and double bonds have delocalization typical of aromatic com-

406 15 Aromaticity Analysis by Means of the Quantum Theory of Atoms in Molecules

pounds. The greater the di erence, on the other hand, the closer the structure to the Lewis structure, indicative of quite localized electrons, which is known to prevent aromaticity. This relationship was especially useful for five-membered ring (5-MR) species, for example the series C4H4 aX (X ¼ CH , NH, S, O, SiH , PH, CH2, AlH, SiHþ, BH, CHþ), for which a reasonable agreement between DDI and NICS values was obtained. Nonetheless, DDI needs a clear Lewis structure to compare double and single bonds, and its application to rings of di erent sizes is less clear. In addition, nonaromatic and antiaromatic systems are not well differentiated by DDI values. In this sense, the quest for a new aromaticity index based on DIs, powerful enough to deal with rings of any size and able to distinguish between nonaromatic and antiaromatic species, led to the aromatic fluctuation index (FLU).

15.4.2

The Aromatic Fluctuation Index (FLU)

Sondheimer’s definition [25] suggests using cyclic electron delocalization as a measure of aromaticity. Although PDI focused on para-related carbon atoms and DDI on a couple of bonds in the ring, there was no attempt to construct an aromaticity index by examining the DI of all bonded pairs in a given ring. It is worth mentioning, however, that Bird [30] has compared Gordy bond orders of all bonded pairs in a given ring to define a measure of aromaticity that has some resemblance to our FLU index. These two indexes are similar in the sense that both compare the values of the bond orders for all pairs of adjacent atoms in the ring with a value of reference to give a measure of aromaticity. The FLU index was constructed by following the HOMA philosophy, i.e. measuring divergences (DI di erences for each single pair bonded) from aromatic molecules chosen as a reference; cf. Eq. (1). The formula below was given for FLU [31]:

FLU ¼ n

A B

V

ðAÞ

ð

dref A; Bð

Þ

ð10Þ

1

RING

V B

a

d

A; BÞ

dref

A; B

2

X

ð

Þ

ð

Þ

where the summation runs over all adjacent pairs of atoms around the ring, n is equal to the number of atoms of the ring, VðAÞ is the global delocalization of atom A given in Eq. (8), dðA; BÞ and dref ðA; BÞ are the DI values for the atomic pairs A and B and its reference value, respectively, and

1 VðBÞ > VðAÞ

a ¼

ð11Þ

1 VðBÞ aVðAÞ

The second factor in Eq. (10) measures the relative divergence with respect to a typical aromatic system, and the first factor in Eq. (10) penalizes those with highly localized electrons. The reference DI values for CaC and CaN bonds were obtained from benzene (1.4 e) and pyridine (1.2 e) at the HF/6-31G(d) level of

15.4 Aromaticity Electronic Criteria Based on QTAIM 407

theory. In a forthcoming work we will also give the reference data for BaN bonds, which will be taken from (0.77 e) [32].

As is readily apparent, unlike PDI, FLU can cope with rings of di erent sizes and furnishes global and local measures of aromaticity. The weakness of FLU, however, as it is also for HOMA, is the need for typical aromatic systems as references. One must also be aware that FLU is actually measuring the relative electronic divergence of a given ring in a molecule relative to a molecule chosen as a reference. It is, therefore, worth noting that FLU (the same is true for HOMA) must be applied with care when studying the change of aromaticity along a reaction coordinate, because it fails to recognize instances when aromaticity is enhanced on deviation from the equilibrium geometry (vide infra) [33].

15.4.3

The p-Fluctuation Aromatic Index (FLUp)

To overcome the need for reference data which prevents the FLU to be applied in a straightforward manner to any molecule, another index based on the QTAIM was designed – the FLUp; this measures the divergence of p-delocalization from its average [31]:

FLU

p ¼ n

A B

Vp

A

ð

dav

ð Þ

1

RING

VpðBÞ

a

dp

A; BÞ dav

2

12

X

ð

Þ

where dav is the average value of the p-DI for the bonded pairs in the ring, and the other symbols denote the aforementioned quantities calculated using p-orbitals only. FLUp can only be exactly calculated for planar molecules, where DI can be (exactly) decomposed into its s and p contributions. Nevertheless, orbital localization schemes can always be used to obtain approximate p-DI for nonplanar molecules.

The correlation between FLUp and FLU is shown to be excellent for a series of organic compounds (cf. Table 15.1 and Scheme 15.1). There is, however, no reason to expect this always to be true. Indeed, FLUp is measuring the amount of homogeneous delocalization in a p-system, whereas FLU is measuring the extent of similarity with reference aromatic molecules. Hence, for organic species for which aromaticity comes from delocalization of the p-system, one expects both indexes to reproduce the same trends; di erences will, however, arise for inorganic species, the aromaticity of which is not exclusively driven by p-electron delocalization.

Correlation of PDI, FLU, and FLUp with other aromaticity indexes is usually reasonably good (cf. Table 15.2) for the series of compounds given in Scheme 15.1, perhaps with the exception of NICS values, as one would expect. One can, therefore, see how QTAIM provides a formidable scheme for measurement of aromaticity. It is also worth noticing that some other molecular decompositions, for example fuzzy-atom [34], can furnish measures of aromaticity in excellent agreement with those derived from QTAIM DIs [35].

Ring	EN	GEO	HOMA	NICS(0)	NICS(0)zz	NICS(1)	NICS(1)zz	PDI	FLU	FLUp

Benzene		0.001	0.000	1.001	11.5
Naphthalene		0.022	0.199	0.779	10.9
Anthracene	A	0.067	0.416	0.517	8.7
	B	0.045	0.072	0.884	14.2
Naphthacene	A	0.110	0.565	0.325	6.7
	B	0.073	0.153	0.774	13.8
Chrysene	A	0.011	0.130	0.859	11.1
	B	0.122	0.325	0.553	8.2
Triphenylene	A	0.003	0.068	0.930	10.6
	B	0.609	0.324	0.067	2.6
Pyracylene	A	0.000	0.329	0.671	4.9
	B	0.406	0.923	0.328	10.1
Phenanthrene	A	0.007	0.091	0.902	11.4
	B	0.189	0.409	0.402	6.8
Acenaphthylene	A	0.011	0.192	0.797	9.6
	B	0.337	0.703	0.039	2.2
Biphenylene	A	0.000	0.193	0.807	6.7
	B	1.360	0.571	0.930	17.4
Benzocyclobutadiene	A	0.001	0.501	0.497	4.0
	B	0.910	1.526	1.437	20.2
Pyridine		0.006	0.001	1.005	9.5
Pyrimidine		0.015	0.000	0.985	7.5
Triazine		0.023	0.000	0.977	5.3
Quinoline	A	0.018	0.190	0.792	11.0
	B	0.008	0.161	0.830	9.1
Cyclohexane		5.340	0.000	4.340	2.1
Cyclohexene		2.955	1.647	3.601	1.6
Cyclohexa-1,4-diene		0.779	1.984	1.763	1.5
Cyclohexa-1,3-diene		0.931	2.207	2.138	3.2

17.2	12.9	32.5	0.105	0.000	0.000
13.8	12.6	30.6	0.073	0.012	0.116
6.9	10.7	24.8	0.059	0.024	0.254
21.9	15.5	38.4	0.070	0.007	0.024
0.9	9.0	19.6	0.051	0.031	0.355
20.4	15.2	37.2	0.063	0.011	0.073
13.7	12.8	30.7	0.079	0.008	0.068
3.6	10.6	23.3	0.052	0.019	0.185
11.8	12.3	29.3	0.086	0.003	0.026
13.9	6.2	9.4	0.025	0.027	0.181
6.9	7.2	13.3	0.067	0.014	0.132
56.4	4.8	24.2	[b]	0.050	0.686
14.7	13.0	31.5	0.082	0.005	0.045
0.1	9.4	20.0	0.053	0.025	0.257
8.7	11.4	26.4	0.070	0.013	0.117
33.7	1.8	4.6	[b]	0.045	0.587
0.7	8.1	17.3	0.088	0.008	0.068
87.0	7.5	35.7	[b]	0.048	0.297
8.4	5.4	9.5	0.080	0.022	0.196
96.2	10.7	44.1	[b]	0.071	1.072
15.2	12.5	31.6	0.097	0.001	0.001
11.8	11.7	29.8	0.089	0.005	0.003
6.9	10.8	27.0	0.075	0.013	0.000
14.7	12.6	30.7	0.072	0.015	0.126
11.5	12.1	29.5	0.071	0.017	0.129
23.5	2.0	3.1	0.007	0.091	[c]
18.1	3.6	1.8	0.019	0.089	[c]
25.4	0.8	2.9	0.014	0.084	[c]
30.0	0.8	7.3	0.031	0.078	[c]

a Reprinted, with permission, from Ref. [31]; Copyright 2005, American Institute of Physics. b PDI cannot be computed for non 6-MRs.

c Nonplanar molecules that prevent easy and exact s–p separation.

Molecules in Atoms of Theory Quantum the of Means by Analysis Aromaticity 15 408

				15.5 Applications of QTAIM to Aromaticity Analysis						409
Table 15.2	Pearson coe cient (r 2) for correlation between di erent
aromaticity indexes for the series of molecules in Table 15.1 at the HF/
6-31G(d) level of theory.
	HOMA	NICS(0)	NICS(0)zz	NICS(1)	NICS(1)zz	PDI	FLU	FLUp
HOMA	1.00	0.30	0.33	0.43	0.41	0.68	0.90	0.78
NICS(0)		1.00	0.98	0.96	0.97	0.50	0.47	0.67
NICS(0)zz			1.00	0.96	0.98	0.64	0.49	0.70
NICS(1)				1.00	0.99	0.55	0.60	0.72
NICS(1)zz					1.00	0.60	0.58	0.72
PDI						1.00	0.79	0.53
FLU							1.00	0.89
FLUp								1.00

15.5

Applications of QTAIM to Aromaticity Analysis

In this section we present some applications of the QTAIM to the analysis of aromaticity in buckybowls, fullerenes, substituted and unsubstituted PAHs, and in the simplest Diels–Alder reaction.

15.5.1

Aromaticity of Buckybowls and Fullerenes

Di erent studies have attributed ambiguous aromatic character to fullerenes [36], which are regarded as aromatic by some criteria and as nonaromatic by others [37–39]. For example, their magnetic properties are indicative of extensive cyclic p-electron delocalization and substantial ring currents; evidence against their aromaticity, however, is that they are very reactive. We have analyzed the local aromaticity of fullerenes and buckybowls [40, 41], which also have fullerene-like physicochemical properties, to clarify their aromatic character. Table 15.3 lists the HF/ 6-31G(d)//AM1 values of the NICS, PDI, and HOMA measures of aromaticity, with the average pyramidalization angles, for a series of planar and bowl-shaped PAHs and fullerenes, i.e. from benzene to buckminsterfullerene (C60), represented in Scheme 15.2.

The three local aromaticity criteria almost give the same trend for the di erent rings of the PAHs studied. Clear aromatic character is assigned to the hexagonal rings of benzene, naphthalene, and C20H10 and to the outer 6-MRs of C26H12 (C) and C30H12 (C and D), whereas the inner 6-MRs of C26H12 (A), C30H12 (A), and C60 are found to be moderately aromatic. In contrast with the significant local aromaticity of 6-MRs, the 5-MRs have antiaromatic character. It is also worth noticing some convergence in the local aromaticity of the inner 6-MRs when go-

41015 Aromaticity Analysis by Means of the Quantum Theory of Atoms in Molecules

Table 15.3 HF/6-31G(d)//AM1 calculated values of NICS (ppm), HOMA, para-delocalization (PDI) (electrons) indices, and average pyramidalization angles for the carbon atoms present in a given ring (Pyr, in degrees) for a series of aromatic compounds.[a]

Molecule	Ring	NICS	HOMA	PDI	Pyr
C6H6	6A	11.7	0.987	0.101	0.0
C10H8	6A	11.3	0.807	0.074	0.0
C14H8	6A	2.7	0.603	0.067	0.0
	5B	13.1	0.205		0.0
C20H10	6A	8.6	0.652	0.058	4.6
	5B	7.6	0.357		9.1
C26H12	6A	5.6	0.474	0.037	6.9
	5B	3.9	0.142		6.3
	6C	10.0	0.746	0.078	2.8
C30H12	6A	6.5	0.390	0.043	9.2
	5B	6.8	0.113		10.1
	6C	9.4	0.652	0.061	5.1
	6D	8.1	0.614	0.057	4.6
C60	6A	6.8	0.256	0.046	11.6
	5B	6.3	0.485		11.6
C70	5A	2.8	0.481		11.9
	6B	11.5	0.294	0.046	11.8
	5C	1.3	0.301		11.0
	6D	8.8	0.141	0.028	10.1
	6E	17.3	0.697	0.059	9.6

a Adapted, with permission, from Ref. [40]; copyright 2003, Wiley– VCH.

ing from the most aromatic benzene to the partially aromatic C60. Unexpectedly for the bowl-shaped PAHs, the most pyramidalized outer rings have the largest local aromaticities, even though this trend agrees with the Clar model of aromaticity, which attributes more aromaticity to outer rings [42].

Another interesting fullerene is C70 [40], the structure of which arises from the insertion of an equatorial belt of five 6-MRs into C60 (Scheme 15.3), causing it to have di erent reactivity and local aromaticity [43]. Experimentally, the pole is less planar than the equatorial belt, which should imply lower aromaticity of the former. The HF/6-31G(d)//AM1 aromaticity criteria in Table 15.3 confirm that ring E, as expected, has the largest aromaticity, followed by ring B, located at the pole and, unexpectedly, by ring D, even though this is located in the equatorial belt. Compared with C60, rings B and E of C70 are more aromatic than 6-MRs of C60, in line with the accepted greater aromaticity of C70 [37]; despite this is more reactive [38]. NICS values of C70 are, in contrast, surprisingly high compared with those for C60, even though they have similar geometric environments and pyramidalization angles. It must be noticed that experimental chemical shifts in

15.5 Applications of QTAIM to Aromaticity Analysis 411

Scheme 15.2 Reprinted, with permission, from Ref. [40]; copyright 2003, Wiley–VCH.