Matta, Boyd. The quantum theory of atoms in molecules

392 14 Applications of the Quantum Theory of Atoms in Molecules in Organic Chemistry

Fig. 14.7 Isodensity maps of rðrÞ for (a) benzene, (b) tetralin, (c) naphthalene and (d) phenanthrene obtained at the HF/6-31G(d,p) approximation. Dark envelopes are for rðrÞ ¼ 0:3274 au, the value for the benzene CaC BCP. A value of rðrÞ ¼ 0:001 au was used for the outer envelopes. Adapted from Ref. [61].

maps shown in Fig. 14.7. Notice, for example, how rðrÞ is more localized in the inner ring of phenanthrene, as indicated by the greater volume contained in the envelope encompassing the C9aC10 bond. In addition, because more electron delocalization between bonded C atoms is associated with a more negative ‘2rbðrÞ and HbðrÞ, greater bond stabilization is associated with a larger bond order quantified by d(C, C0).

All of the values reported in Table 14.5 are intermediate between those of single and double bonds. For example, the RHF/6-31G(d,p) value of rbðrÞ at the CaC BCP for cyclohexane is 0.255 and the corresponding d(C, C0) is 0.961, whereas for the CaC double bond of cyclohexene they are rbðrÞ ¼ 0:363 and d(C, C0) ¼ 1.811. These values are to be compared with those of benzene, the reference aromatic molecule, or with the C9aC10 bond in phenanthrene, the latter of which have more double-bond character. The values reported in Table 14.5 are consistent with the envelopes of Fig. 14.7.

Aromatic monoradicals and diradicals are also of interest. For example, m- benzyne-derived compounds are stable toward rearrangement to their ortho and para isomers, and the experimental information suggests this molecule is in the

14.4 Aromatic Molecules 393

Table 14.5 rbðrÞ, ‘2rbðrÞ, and HbðrÞ at nonequivalent CaC BCPs and the corresponding d(C, C0) values of several PBHs.[a]

Molecule	Bond	r	(r)	‘2r	(r)	H (r)	d(C, CO)
		b		b		b
Benzene	CaC	0.3274		1.0163		0.3506	1.388
Naphthalene	C1aC2	0.3432		1.0912		0.3855	1.533
	C1aC9	0.3095		0.9320		0.3128	1.222
	C2aC3	0.3106		0.9386		0.3161	1.245
	C9aC10	0.3170		0.9598		0.3273	1.278
Phenanthrene	C1aC2	0.3391		1.0744		0.3761	1.479
	C1aC11	0.3160		0.9608		0.3257	1.270
	C2aC3	0.3188		0.9791		0.3325	1.296
	C3aC4	0.3376		1.0640		0.3729	1.476
	C4aC12	0.3137		0.9448		0.3217	1.276
	C9aC10	0.3549		1.1520		0.4129	1.630
	C10aC11	0.2993		0.8884		0.2934	1.143
	C11aC12	0.3191		0.9656		0.3316	1.308
	C12aC13	0.2880		0.8252		0.2720	1.145
m-Benzyne	C1aC2	0.3061		0.6579		0.3247	1.436
	C2aC3	0.3022		0.7956		0.3200	1.393
	C3aC4	0.3027		0.7955		0.2943	1.347

a HF/6-31G(d,p) wavefunctions used for benzene, tetralin, naphthalene, and phenanthrene [61]. In the case of m-benzyne in the singletdiradical electronic state, a B3LYP/6-31þþG(d,p) wavefunction within the broken symmetry approach was used [68]. The carbon atomic numbering is given in Scheme 14.4.

singlet electronic state [62], as can be inferred from its reactivity as an electrophile rather than as a free radical [63]. The importance of these compounds is illustrated, for example, by the use of o-benzyne nickel(0) complexes in the synthesis of PBHs [64]. Benzyne diradicals have also been investigated by theoretical methods. Nicolaides et al. [65] reported CI calculations on didehydrobenzenes and compared their results with thermochemical data [66], and Clark et al. [67] analyzed the relative stability of dehydrobenzenes to H-abstraction processes by means of CASSCF and GVB calculations. The electronic structure of aromatic radicals in the doublet state and of diradicals in both their singlet open shell and triplet states have also been investigated with QTAIM [68]. In these cases, in addition to d(C, C0) and the properties of rbðrÞ at the CaC BCPs, the critical points of ‘2rðrÞ have provided insight into bonding pattern for benzene and pyrenederived radicals and diradicals – the charge concentrations revealed by the minima of ‘2rðrÞ clearly indicate the radical character of phenyl and dehydropyrenes in the doublet state, and the properties of rbðrÞ at the CaC BCPs and the CaC0 delocalization indices are similar to the corresponding values for benzene.

More pronounced electronic redistribution occurs for singlet diradicals. For example:

39414 Applications of the Quantum Theory of Atoms in Molecules in Organic Chemistry

1.the CaC bond properties of the dehydrogenated carbon atoms of o-benzyne are similar to those of a triple bond;

2.the open-shell nature of p-benzyne in the singlet diradical state is revealed by the existence of critical points of ‘2rðrÞ on the dehydrogenated C atoms; and

3.the lack of a bond path between C1 and C5 in m-benzyne (Table 14.5 and Scheme 14.4) prevents one from assigning a chemical bond between these two atoms, despite the large electron delocalization between them, thus explaining the large distortion of the molecular geometry.

Scheme 14.4 Carbon-atom labeling for naphthalene, phenanthrene, and m-benzyne.

Similar properties have been reported for aromatic triradicals [69] by means of the QTAIM.

Although the properties of rðrÞ at the BCP have not been regarded as useful as those at RCPs for discussion of aromaticity [60], this analysis has shown how the electronic structure of PBHs is recovered well from them and from their relationship with d(C, C0). A useful aromaticity index can be defined [61]:

y		1	n sn d0 di 2			12
	¼			c	Xi 1 ð Þ	ð Þ
					¼

where n is the number of C atoms on a ring of a PBH, and do and di are the total delocalization of a carbon atom with all other carbon atoms in benzene and on the given ring of the PBH, respectively. In addition, c is a constant such that y is zero for cyclohexane and unity for benzene and its value depends on the level of approximation used. y does not rely on the s–p separability (for example, the hydrogenated ring of tetralin is not planar) and can be regarded as a reformulation of the geometric HOMA [70] index taking advantage of the high correlation between CaC bond distances and d(C, C0). y recovers bonding characteristics of PBHs – aromatic dilution, greater electron localization and di erential reactivity. This is illustrated, for example, by the following RHF/6-31G(d,p) values [61]:

14.5 Conclusions 395

y ¼ 1 and 0.807 for benzene and naphthalene, respectively; 0.960 and 0.278 for the aromatic and hydrogenated rings of tetralin, respectively; and 0.863 and 0.650 for the outer and inner rings of phenanthrene, respectively. The bonding pattern obtained from rðrÞ, the individual CaC delocalization indices, and the y aromaticity index, agree with the description provided by Clar’s aromatic sextet model for these molecules [71]. For example:

1.the rings in naphthalene are less aromatic than in benzene; and

2.the aromaticity of the inner ring of phenanthrene is low, with the C9aC10 bond having the characteristics of a double bond, in agreement with the known addition reactivity of the molecule.

It must also be considered that further improvement of delocalization indices should account for the multidimensional nature of aromaticity with regard to the structural, electronic, magnetic and energetic aspects involved in this notion [72].

14.5 Conclusions

This chapter has shown how the QTAIM enables quantification of chemical concepts that are elusive in other approaches, for example those of the chemical bond and of aromaticity. The formality of a theory based on an observable property – the electron density – provides the experimenter with useful tools to construct models that enable explanation of their observations on key chemical concepts, for example reactivity and conformation.

It was discussed how electron delocalization is responsible for the 3JHH coupling constants for several systems – the Karplus-type behavior of 3JHH of ethane during internal rotation and trends of 3JHH values for cyclic aliphatic and aromatic compounds. The rotational barrier of ethane and substituted ethanes was explained on energetic grounds that are a consequence of the charge redistribution along the barrier. For example, as the Ehrenfest forces in both eclipsed and staggered conformations of ethane are attractive, but of smaller magnitude in the former arrangement, there is no need to use a repulsive model to explain the barrier. Analysis of the anomeric e ect on heterocyclic molecules provides an illustrative application of how the structural criteria obtained from the QTAIM can guide the design of compounds useful for demonstrating molecular e ects of interest. The electronic structure of polybenzenoid hydrocarbons obtained from the topology of the electron density and two-electron properties such as the electronic energy density and the electron delocalization index provide a clear bonding pattern and a definition of aromaticity. This broad range of examples provides a coherent description of chemical bonding and interactions in organic chemistry characterized by the QTAIM.

396 14 Applications of the Quantum Theory of Atoms in Molecules in Organic Chemistry

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399

15

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

Eduard Matito, Jordi Poater, and Miquel Sola`

15.1 Introduction

Aromaticity is a concept formulated to account for the unusual properties of an important class of organic and inorganic molecules – the aromatic compounds [1, 2]. Aromaticity is currently enjoying a resurgence of interest as its scope of application has expanded from the organic to the inorganic realm of chemistry [3]. The new class of all-metal and inorganic aromatic compounds which has been synthesized has prompted new definitions of aromaticity applicable to both classic and novel aromatic molecules. According to the most recent definition of Schleyer and coworkers [4], aromaticity is a manifestation of electron delocalization in closed circuits, in either two or three dimensions, which results in energy reduction, often quite substantial, and a variety of unusual chemical and physical properties. These include a tendency toward bond length equalization, unusual reactivity, characteristic spectroscopic features, and distinctive magnetic properties related to the strong induced ring currents in aromatic systems [5].

Although aromaticity is not a directly observable quantity, its importance as a central concept in chemistry has not diminished since the discovery of benzene by Michael Faraday in 1825 [6]. Yet its quantification has proved remarkably elusive. Nearly everyone agrees there is not yet a well-established method for quantifying the aromatic character of molecules. Aromaticity is usually evaluated indirectly by measuring a physicochemical property that reflects the aromatic character of molecules. Thus, most aromaticity indicators are based on the classical aromaticity criteria, namely, structural, magnetic, energetic, and reactivitybased measures [2], although, more recently, new ways of quantifying aromaticity based on the electronic properties of the molecules have been devised (a review has recently been published [7]). Although the existence of many aromaticity descriptors complicates matters, it is also true that the use of di erently-based aromaticity criteria is recommended for aromaticity analysis, because of its multidimensional character [8, 9].

400 15 Aromaticity Analysis by Means of the Quantum Theory of Atoms in Molecules

Structure-based measures of aromaticity rely on the idea that important manifestations of aromaticity are equalization of bond lengths and symmetry. Among the most common structure-based indices of aromaticity, one of the most e ective is the harmonic oscillator model of aromaticity (HOMA) index [10], defined by Kruszewski and Krygowski as:

	a	n
HOMA ¼ 1		Xi 1	ðRopt RiÞ2;	ð1Þ
	n
		¼

where n is the number of bonds considered and a is an empirical constant fixed to give HOMA ¼ 0 for a model nonaromatic system and HOMA ¼ 1 for a system with all bonds equal to an optimum value Ropt, assumed to be achieved for fully aromatic systems. Ri stands for a running bond length. The HOMA value can be split into the energetic (EN) and geometric (GEO) contributions according to the relationship [11]:

				a
HOMA ¼ 1 EN GEO ¼ 1 aðRopt RÞ2					Xi	ðR RiÞ2:			ð2Þ
				n

The GEO contribution measures the decrease/increase in bond length alternation and the EN term takes into account the lengthening/shortening of the mean bond lengths ðRÞ of the ring. The higher the HOMA value the more aromatic the system.

Magnetic indices of aromaticity are based on the p-electron ring current that is induced when the system is exposed to external magnetic fields. Probably the most widely used magnetic-based indicator of aromaticity is the nucleusindependent chemical shift (NICS) proposed by Schleyer and coworkers [4, 12]. It is defined as the negative value of the absolute shielding computed at a ring center or at some other interesting point of the system. Rings with large negative NICS values are regarded as aromatic. The more negative the NICS values, the more aromatic the rings. Antiaromatic rings, in contrast, are characterized by positive values of NICS.

Finally, energy-based indices of aromaticity make use of the fact that conjugated cyclic p-electron compounds are more stable than their chain analogues. The most commonly used energetic measure of aromaticity is the aromatic stabilization energy (ASE), calculated as the reaction energy of a homodesmotic reaction [13].

Although aromaticity plays a prominent role in current chemistry, aromaticity measures based on electronic descriptors are rare. Among these, we can mention the HOMO–LUMO gap, absolute and relative hardness, the electrostatic potential, and the polarizability [14]. None refers directly to the electronic delocalization in aromatic species. If we take into account that electron delocalization is the

15.2 The Fermi Hole and the Delocalization Index 401

main factor responsible for manifestation of aromaticity, this fact is surprising. The electron structure of the molecules is obviously strongly connected with aromaticity, and QTAIM provides a wide set of concepts to characterize electronic structure – atomic populations and charges, electron localization and delocalization (through the so-called LIs and DIs, vide infra), BCP, RCP (one should bear in mind that aromaticity manifests in molecules with ring structures), among others [15]. In this sense, the QTAIM provides a formidable scheme for the construction of measures of aromaticity.

In the last five years we have concentrated our research e orts on quantifying aromaticity by measuring the extent of electron delocalization in molecules using the set of tools that the QTAIM theory provides for this task [7]. In this chapter we briefly review the new aromaticity criteria defined by us that make use of the QTAIM concepts. The chapter is organized as follows. Section 15.2 covers the definition of electron delocalization in the QTAIM theory; Section 15.3 analyzes electron delocalization in benzene, the quintessential aromatic molecule; Section 15.4 gives the definition of the new aromatic indexes based on analysis of electron delocalization; Section 15.5 discusses several applications of the newly defined indexes to the analysis of aromaticity; and, finally, in Section 15.6, we summarize the main conclusions.

15.2

The Fermi Hole and the Delocalization Index

Most methods used to determine the extent of electronic localization/ delocalization in molecules employ the two-electron density, Gð~r1s1;~r2s2Þ, also named second-order density or pair density [16]. This is the simplest quantity that describes the pair behavior and is usually interpreted as the probability density of finding two electrons with spins s1 and s2 simultaneously at positions ~r1 and ~r2, respectively, irrespective the position and spin of the other N 2 electrons. Integration of this function over the spin variables yields the spin-less pair density, Gð~r1;~r2Þ, which can be split into an uncorrelated pair density and a part that gathers all exchange and correlation e ects:

		sX1; 2		;~r2s2Þ ¼ Gð~r1;~r2Þ ¼ rð~r1Þrð~r2Þ þ Gxc ð~r1;~r2Þ:					ð3Þ
			Gð~r1s1
			s
The		uncorrelated			component of	the pair	density, given	by	the product
~		~	Þ, provides		the probability	of finding	simultaneously	two	independent
rðr1	Þrðr2

electrons at positions ~r1 and ~r2. The di erence between Gð~r1;~r2Þ and rð~r1Þrð~r2Þ is known as the exchange-correlation density [17], Gxcð~r1;~r2Þ, which is a measure of the extent to which density is excluded at ~r2 because of the presence of an electron at ~r1. Integration of the exchange-correlation density of a given molecule