Vol. 3, No. 2  The Chemical Educator  © 1998 Springer-Verlag New York, Inc.

ISSN 1430-4171  http://journals.springer-ny.com/chedr  S 1430-4171(98)02190-6

A semicircular mnemonic is described which permits one to anticipate the Hückel eigenvalue patterns for symmetrical contiguous nonfused bicycles (e.g., biphenyl). The mnemonic constitutes a new technique that is complementary to the existing Frost circle mnemonic for hydrocarbon monocycles. Pedagogically, the development of this semicircular mnemonic in class will serve to establish that the widespread practice of applying Hückel's rule to polycycles is unsound. 

Introduction

For some time now, students of Hückel theory have been able to anticipate the relative energies of the molecular orbitals of the annulenes by employing the Frost circle mnemonic.[1]. To anticipate the relative energies of the molecular orbitals (i.e., their eigenvalues) for an annulene: (1) draw a circle whose center represents the nonbonding (or ) level and (2) inscribe the appropriate polygon in the circle with a corner down. Each point at which a corner of the polygon touches the circumference of the circle represents the relative eigenvalue of a molecular orbital of the annulene. Figure 1 depicts the application of the "Frost-circle trick" to cyclobutadiene.
 

Quite apart from introductory Hückel theory, the Frost circle trick is often given in lectures which focus on aromaticity in hydrocarbons [2].

The Frost circle mnemonic provides a convenient basis for classroom generalizations about 4n + 2 (aromatic) hydrocarbon monocycles and 4n (antiaromatic) hydrocarbon monocycles which are collectively covered under the rubric of Hückel's rule (notwithstanding the fact that Breslow coined the term antiaromatic [3]). The prototypical 4n system, cyclobutadiene (Figure 1), has two nonbonding electrons while the prototypical 4n + 2 system, benzene (Figure 2), has bonding electrons exclusively.
 

It is seductively convenient to apply Hückel's rule (monocyclic hydrocarbons which have 4n + 2 electrons will be aromatic) and Breslow's extension (monocyclic hydrocarbons which have 4n electrons will be antiaromatic) to any molecule which has a cyclic system. Nonetheless, it is instructive to examine the validity of conclusions reached by those who apply Hückel's rule to non-monocycles. In this report I consider a subset of hydrocarbon bicycles (at the Hückel level) to see whether it is possible to enunciate meaningful generalizations about aromatic/antiaromatic bicycles.

Results and Discussion

Underlying qualitative analyses, like the Frost-circle trick, is the quantitative side of Hückel theory. Quantitative treatments [4] of hydrocarbon systems begin with the topological information implicit in their Lewis structures and proceed using simple linear algebra to produce a polynomial called the secular equation; for example, the secular equation for butadiene is x⁴ - 3x² + 1 = 0. The roots of the secular equation (in this case -1.62, -0.62, 0.62, 1.62) lead to the eigenvalues (energies) for the molecular orbitals (see Figure 3 for butadiene orbital energies).

Let's consider relationships between an annulene, A, the corresponding acyclic polyene with one less carbon atom, A-c and the symmetrical contiguous nonfused bicycle, A-A. As an example, for A representing benzene, A-c would be pentadienyl and A-A would be biphenyl. Some time ago, Heilbronner [5] published a theorem which relates the secular equation of A (i.e., S^A(x)) and the secular equation of A-c (i.e. S^A-c(x)) to the secular equation of the corresponding symmetrical nonfused bicycle A-A (see equation 1).

                                     (1)

As would be expected from a simple perturbational argument [6], Hückel calculations on the bicycles consistently give degenerate pairs of eigenvalues "sandwiched" between a pair of nondegenerate orbital energies, producing a quartet of eigenvalues as shown in Figure 4. Unique eigenvalues for A lead to a unique pair of eigenvalues for A-A.

One can now use the foregoing generalizations to deduce the relative orbital energies for any symmetrical contiguous nonfused bicycle. Figure 5 presents the procedure as it applies to biscyclobutadiene.

Application of this new semi-circular mnemonic to the smallest bicycles A-A (i.e., C₆, C₈, C₁₀, C₁₂) produces four different occupancy patterns for the orbitals. Biscyclobutadiene 1 results are shown in Figure 5 and the other bicycles 2, 3 and 4 are shown in Figure 6.

The basic patterns repeat as the bicycles grow so that the C₁₄ bicycle shows the same pattern as the C₆ bicycle, C₁₆ shows the same pattern as C₈, and so on. In general, there is one pattern for all 8n bicycles, one for all 8n + 2 bicycles, one for all 8n + 4 bicycles and one for all 8n + 6 bicycles.
 

If the words aromatic and antiaromatic are to imply similar things for both monocycles and bicycles, then the 8n bicycle pattern (triplet ground state), which best matches the 4N monocycle pattern (triplet ground state - see Figures 1 and 5), should be appropriate for antiaromatic symmetric contiguous nonfused bicycles. Like the 4n + 2 monocycles, the 8n + 2 bicycle pattern exhibits degenerate, fully filled, highest-occupied bonding molecular orbitals (see Figure 2 and compound 3, Figure 6). Thus the 8n + 2 bicycles should be appropriate for aromatic, symmetric, contiguous nonfused bicycles. Unlike the monocycles, the bicycles produce two other patterns (one for 8n + 4 and one for 8n + 6 systems), which can simply be classed as nonaromatic.

A simple, established measure of aromaticity (at the Hückel level) for the annulenes is the Dewar resonance energy [6, 7]. A plot of annulene carbon content versus Dewar resonance energy (DRE) looks like a picket fence with large DREs for the 4n + 2 annulenes and small DREs for the 4n annulenes (see Figure 7). The corresponding plot for 8n and 8n + 2 contiguous symmetric nonfused bicycles gives a similar picture (see Figure 8).

While the generalizations regarding aromatic (8n + 2) and antiaromatic (8n) contiguous nonfused bicycles are nicely supported at the Hückel level, there is a surprising feature. Planar biphenyl 4 (Figure 6) is classified as nonaromatic (8n + 4 system) by these rules. Of course biphenyl 4 is widely discussed as an aromatic molecule. In the gas phase, however, biphenyl 4 is not flat. The two phenyl rings occupy separate planes which are tilted about 45 with respect to each other [8] (see Figure 9). 
 
 

Furthermore, the C-C bond linking the biphenyl rings is about 1.48 Å long [8] indicating that it has a weaker resonance interaction than styrene, which has a corresponding bond length of 1.44 Å [8]. Nonplanarity for biphenyl is usually explained in terms of crowding between the ortho hydrogens in the flat structure [8, 9].

Nonetheless, it may well be, in accord with the Hückel picture, that planar biphenyl is also destabilized by the relatively high-lying electrons in its HOMO, which would enjoy substantial stabilization in approximately perpendicular conformations. This
 

would certainly be the conclusion if a kinetic based criterion for aromaticity were to be employed [10] (E_FMO for planar biphenyl: 1.4; for perpendicular biphenyl: 2).

It is now clear that Hückel's rule, along with Breslow's extension of it, which were designed to describe annulenes, should not be applied to polycycles, even those as simple as bicycles. For example, a monocyclic structure which is antiaromatic (4n  electrons) can easily have a planar nonaromatic bicyclic counterpart (e.g., 12  electrons). It is for this reason that the usual approach to aromaticity in transition states [10] requires analysis in terms of monocyclic orbital arrays.

References

1. Frost, A. A.; Musulin, B. J. Chem. Phys. 1953, 21, 572.
2. For example: Carey, F. A.; Sundberg, R. A. Advanced Organic Chemistry, 2nd Ed., Part A.; Plenum Press: New York; p 43.
3. (a) Breslow, R.; Mohacsi, E. J. Am. Chem. Soc. 1963, 85, 431; (b) Breslow, R. Chem. Eng. News 1965, 43, 90; (c) Breslow, R. Acc. Chem. Res. 1973, 6, 393.
4. (a) Liberles, A. Introduction to Molecular Orbital Theory; Holt, Rinehart and Winston: New York, 1966; (b) Yates, K. Hückel Molecular Orbital Theory; Academic Press: New York, 1978; (c) Higasi, K.; Baba, H.; Rembaum, A. Quantum Organic Chemistry, Interscience: New York, 1965; (d) Carroll, F.A. Perspectives on Structure and Mechanism in Organic Chemistry; Brooks/Cole: Pacific Grove, 1998; pp 178-224.
5. Heilbronner, E. Helv. Chim. Acta 1953, 36, 170.
6. Dewar, M. J. S.; Doughertry, R. C. The PMO Theory of Organic Chemistry; Plenum: New York, 1975.
7. Durkin, K. A.; Langler, R. F. J. Phys. Chem. 1987, 91, 2422.
8. Pauling, L. The Nature of The Chemical Bond, 3rd ed.; Cornell University Press: Ithaca, NY, 1960. pp 291, 294-295.
9. Streitwieser, A. Jr.; Heathcock, C. H. Introduction to Organic Chemistry, 3rd. ed.; Macmillan: New York, 1985; p 971.
10. Langler, R. F. J. Chem. Educ. 1996, 73, 899; and references therein.