Clar's rule

In organic and physical organic chemistry, Clar's rule is an empirical rule that relates the chemical stability of a molecule with its aromaticity. It was introduced in 1972 by the Austrian organic chemist Erich Clar in his book The Aromatic Sextet. The rule states that given a polycyclic aromatic hydrocarbon, the resonance structure most important to characterize its properties is that with the largest number of aromatic π-sextets i.e. benzene-like moieties. ^[1]

The rule

In general, the chemical structure of a given polycyclic aromatic hydrocarbon admits more than one resonance structure: these are sometimes referred to as Kekulé resonance structures. Some of such structures may contain aromatic π-sextets, namely groups of six π-electrons localized in a benzene-like moiety and separated by adjacent rings by formal C–C bonds. An aromatic π-sextet can be represented by a circle, as in the case of the anthracene molecule. Clar's rule states that for a benzenoid polycyclic aromatic hydrocarbon (i.e. with only hexagonal rings), the resonance structure with the largest number of disjoint aromatic π-sextets is the most important to characterize its chemical and physical properties. Such resonance structure is called the Clar structure. In other words, a polycyclic aromatic hydrocarbon with a given number of π-sextets is more stable than its isomers with less π-sextets.^[1][2] In 1984, Glidewell and Lloyd provided an extension of Clar's rule to polycyclic aromatic hydrocarbons containing rings of any size.^[3] More recently, Clar's rule was further extended to biradicaloids in their singlet state.^[4]

 

Two representations of the same resonance structure of anthracene. Above, each covalent bond between carbon atoms is represented by one or two segments. Below, the aromatic π-sextet is put in evidence by means of a circle.

Writing a Clar structure

When writing a Clar structure, the following rules must be satisfied:^[5]

1. each vertex of the molecular graph representing the polycyclic aromatic hydrocarbon either belongs to a double bond or a circle;
2. such double bonds and circles never join;
3. there are no rings with three double bonds, since they are always represented by circles; moreover, the number of circles in the graph is maximized;
4. when a ring with a circle is adjacent to a ring with two double bonds, an arrow is drawn from the former to the latter ring.

Some observations about these rules are worth to be put into evidence. Following Clar,^[1] rules at points 1 and 2 imply that circles can never be in adjacent rings; rule at point 3 means that only four options are viable for rings, namely (i) having only one double bond, (ii) having two double bonds, (iii) having a circle, or (iv) being empty, i.e. having no double bonds; finally, the arrow mentioned in the rule at point 4 can be interpreted in terms of mobility of π-sextets (in this case we speak of migrating π-sextets) or, equivalently, of a quantum-mechanical resonance between different Clar structures.^[5]

Examples

In the following, Clar's rule is applied to three different cases.

The resonance structures of phenanthrene

 

Two resonance structures of phenanthrene: above, one with only one circle; below, one with two circles, which is also a Clar's structure. Clar's rule states that the latter structure contributes the most to the properties of phenanthrene.

According to the rules exposed above, the phenanthrene molecule admits two different resonance structures: one of them presents a single circle in the center of the molecule, with each of the two adjacent rings having two double bonds; the other one has the two peripheral rings each with one circle, and the central ring with one double bond. According to Clar's rule, this last resonance structure gives the most important contribution to the determination of the properties of phenanthrene.^[2][6]

The migrating π-sextet of anthracene

 

Representation of the anthracene molecule: above, three equivalent resonance structures; below, its Clar structure, with the arrow denoting a migrating π-sextet.

The anthracene molecule admits three resonance structures, each with a circle in one ring and two sets of double bonds in the other two. Following the rule at point 4 exposed above, anthracene is better described by a superposition of these three equivalent structures, and an arrow is drawn to indicate the presence of a migrating π-sextet. Following the same line of reasoning, one can find migrating π-sextets in other molecules of the acene series, such as tetracene, pentacene, and hexacene.^[2]

The role of angular rings

Fusing angular rings around a benzene moiety leads to an increase in stability. The Clar structure of anthracene, for instance, has only one π-sextet, but moving one ring into the angular position phenanthrene is obtained, the Clar structure of which carries two circles instead of one – notice that this molecule can be thought of as a benzene moiety with two fused rings; a third ring can be fused to obtain triphenylene, with three aromatic π-sextets in its Clar structure. The chemical stability of these molecules is greatly influenced by the degree of aromaticity of their Clar structures. As a result, while anthracene reacts with maleic acid, phenanthrene does not, and triphenylene is the most stable species of these three.^[1]

 

Three Clar structures with an increasing number of fused rings around a benzene moiety: anthracene (on the left), phenanthrene (in the middle), and triphenylene (on the right). The chemical stability of these compounds increases from left to right due to the increase in the number of π-sextets.

Experimental evidence and applications

Since its formal statement in 1972, Clar's rule has received a vast amount of experimental evidence. The dependence of the color and reactivity of some small polycyclic aromatic hydrocarbons on the number of π-sextets in their structures is reported by Clar himself in his seminal contribution.^[1] Similarly, it was shown that the HOMO-LUMO gap, and therefore the color, of a series of heptacatafusenes depends on the number of π-sextets.^[5] Clar's rule has also been supported by experimental results about the distribution of π-electrons in polycyclic aromatic hydrocarbons,^[7] valence bond calculations,^[8] and nucleus independent chemical shift studies.^[9]

Clar's rule is widely applied in the fields of chemistry and materials science. For instance, Clar's rule can be used to predict several properties of graphene nanoribbons.^[10] Aromatic π-sextets play an important part in the determination of the ground state of open shell biradical-type structures.^[11] , Clar's rule can rationalize the observed a decrease of the bandgap of holey graphenes with increasing size.^[12]

Limitations

Despite the experimental support mentioned above, Clar's rule suffers from some limitations. In the first place, Clar's rule is formulated only for species with hexagonal rings,^[13] and thus it cannot be applied to species having rings different from the benzene moiety, even though an extension of the rule to molecules with rings of any dimension has been provided by Glidewell and Lloyd.^[13] Secondly, if more than one Clar structure exist for a given species, Clar's rule does not allow to determine the relative importance of each of them in the determination of the physicochemical properties.^[6] Finally, it is important to mention that exceptions to the Clar's rule exist, such as in the case of triangulenes.^[14]

See also

• Hückel's rule
• Baird's rule

References

1. Erich Clar (1972). "The Aromatic Sextet". In D. Rondia; M. Cooke; R. K. Haroz (eds.). Mobile Source Emissions Including Policyclic Organic Species. John Wiley & Sons. doi:10.1007/978-94-009-7197-4_4.
2. Miquel Solà i Puig (17 October 2013). "Forty years of Clar's aromatic π-sextet rule". Frontiers in Chemistry. 1: 22. doi:10.3389/FCHEM.2013.00022. ISSN 2296-2646. PMC 3982536. PMID 24790950. Wikidata Q38208843.
3. Christopher Glidewell; Douglas Lloyd (1984). "MNDO study of bond orders in some conjugated BI- and tri-cyclic hydrocarbons". Tetrahedron. 40 (21). doi:10.1016/S0040-4020(01)98821-0. ISSN 0040-4020. Wikidata Q112830674.
4. Zhe Sun; Sangsu Lee; Kyu Hyung Park; et al. (20 November 2013). "Dibenzoheptazethrene isomers with different biradical characters: an exercise of Clar's aromatic sextet rule in singlet biradicaloids". Journal of the American Chemical Society. 135 (48): 18229–18236. doi:10.1021/JA410279J. ISSN 0002-7863. PMID 24206273. Wikidata Q44732390.
5. Alexandru Balaban; Douglas J. Klein (2009). "Claromatic Carbon Nanostructures". The Journal of Physical Chemistry C (113): 19123–19133. doi:10.1021/JP9082618. ISSN 1932-7447. Wikidata Q112828750.
6. Guillem Portella; Jordi Poater; Miquel Solà (5 May 2005). "Assessment of Clar's aromatic π-sextet rule by means of PDI, NICS and HOMA indicators of local aromaticity". Journal of Physical Organic Chemistry. 18 (8): 785–791. doi:10.1002/POC.938. ISSN 0894-3230. Wikidata Q56387336.
7. Ivan Gutman; ŽeljkoTomović; Klaus Müllen; Jürgen P. Rabe (12 October 2004). "On the distribution of π-electrons in large polycyclic aromatic hydrocarbons". Chemical Physics Letters. 397 (4–6): 412–416. doi:10.1016/J.CPLETT.2004.08.138. ISSN 0009-2614. Wikidata Q112830992.
8. Remco W A Havenith; Haijun Jiao; Leonardus W Jenneskens; et al. (1 March 2002). "Stability and aromaticity of the cyclopenta-fused pyrene congeners". Journal of the American Chemical Society. 124 (10): 2363–2370. doi:10.1021/JA011538N. ISSN 0002-7863. PMID 11878993. Wikidata Q43905733.
9. Yosadara Ruiz-Morales (10 October 2009). "Aromaticity in pericondensed cyclopenta-fused polycyclic aromatic hydrocarbons determined by density functional theory nucleus-independent chemical shifts and the Y-rule — Implications in oil asphaltene stability". Canadian Journal of Chemistry. 87. doi:10.1139/V09-052. ISSN 0008-4042. Wikidata Q112831105.
10. Tobias Wassmann; Ari P. Seitsonen; A. Marco Saitta; Michele Lazzeri; Francesco Mauri (1 March 2010). "Clar's theory, pi-electron distribution, and geometry of graphene nanoribbons". Journal of the American Chemical Society. 132 (10): 3440–3451. arXiv:1003.3572. doi:10.1021/JA909234Y. ISSN 0002-7863. PMID 20178362. Wikidata Q83008058.
11. Zhe Sun; Sangsu Lee; Kyu Hyung Park; et al. (20 November 2013). "Dibenzoheptazethrene isomers with different biradical characters: an exercise of Clar's aromatic sextet rule in singlet biradicaloids". Journal of the American Chemical Society. 135 (48): 18229–18236. doi:10.1021/JA410279J. ISSN 0002-7863. PMID 24206273. Wikidata Q44732390.
12. Karol Strutyński; Aurelio Mateo-Alonso; Manuel Melle-Franco (16 January 2020). "Clar Rules the Electronic Properties of 2D π-Conjugated Frameworks: Mind the Gap". Chemistry: A European Journal. doi:10.1002/CHEM.201905087. ISSN 0947-6539. PMID 31944437. Wikidata Q92685111.
13. Ouissam El Bakouri; Jordi Poater; Ferran Feixas; Miquel Solà (August 2016). "Exploring the validity of the Glidewell–Lloyd extension of Clar's π-sextet rule: assessment from polycyclic conjugated hydrocarbons". Theoretical Chemistry Accounts. 135 (8). doi:10.1007/S00214-016-1970-1. ISSN 1432-2234. Wikidata Q61857678.
14. Eduardo Martín Rico-Sotomayor; José E Barquera-Lozada (26 October 2020). "Triangulenes and theirs ions: reaching the limits of Clar's rule". Physical Chemistry Chemical Physics. doi:10.1039/D0CP03305G. ISSN 1463-9076. PMID 33104146. Wikidata Q100996684.