Research: Science and Education
The Mechanism of Covalent Bonding: Analysis within the Hückel Model of Electronic Structure
W
Sture Nordholm* and Andreas Bäck Department of Chemistry, Göteborg University, SE-412 96 Göteborg, Sweden; *
[email protected] George B. Bacskay School of Chemistry, University of Sydney, NSW 2006, Australia
Covalent bonding undoubtedly constitutes the very heart of chemistry. The unique properties of a molecule are largely determined by the nature of the bonds between its constituent atoms. Unfortunately, it appears to be far easier to observe the consequences of covalent bonding than to understand its physical origin, since the latter requires a careful analysis and interpretation of results obtained by the application of quantum mechanics. The prevailing view expressed in many textbooks is that covalent bonding is primarily an electrostatic phenomenon, whereby the critical contribution to the bond energy is due to increased electron density in the interatomic “bonding” region that is attracted to both adjoining nuclei (1–3), acting as an “electronic glue”. This view appears to be consistent with the requirements of the virial theorem (4), which states that the change in potential energy as atoms are brought from infinite separation to the equilibrium molecular geometry is binding, that is, negative and twice as large in magnitude as the corresponding increase in kinetic energy. Further, molecular orbital (MO) theory does predict increased electron density in the bonding regions when bonding MOs are occupied. However, as noted by Ruedenberg (5, 6), the electrostatic view has overlooked the phenomenon of orbital contraction, which is also associated with covalent bond formation. Thus, the drop in potential energy that occurs on bonding is mostly due to electron density contracting towards the nuclei. This is an atomic effect as it produces atoms in “promoted” states. The crucial molecular component of bonding however is the constructive interference of the (contracted) AOs as electron delocalization occurs, which is a quantum effect. Maximum molecular stability, that is, lowest total energy, can thus be accompanied by a net increase and decrease in the kinetic and potential energies, respectively. A similar explanation of bonding, although without the contractive promotion step, was proposed by Hellman in the early 1930s (7). In our earlier work, building on Ruedenberg’s analysis, we argued that the essence of covalent bonding is the delocalization of electronic motion in a molecule, that is, the bonding mechanism is fundamentally dynamical in origin (8–10). The concomitant changes in potential and kinetic energies, being sensitive to the details of orbital contraction, are not sufficient in themselves to explain covalent bonding. Given the importance and subtlety of these issues one is led to search for the simplest physically correct and pedagogically satisfying approach to describe covalent bonding. We propose that the Hückel π-MO model of conjugated hydrocarbons, being a quantum mechanical theory, is ideal for such a purpose. In this short article we briefly discuss and analyze the Hückel model and describe the advantages it offers in terms of a clear view of bonding. A more comprehenwww.JCE.DivCHED.org
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sive and detailed presentation of our ideas can be found in the Supplemental Material.W The Hückel Model of Covalent Bonding The Hückel model of π MOs in planar conjugated hydrocarbons (11) is standard material in many physical and organic chemistry textbooks (12, 13). Briefly, the MOs are expressed in terms of carbon 2pz AOs (assuming the molecule lies in the xy plane), where the expansion coefficients are obtained as eigenvectors Cπ of the Hückel Hamiltonian (Fock) matrix Fπ, defined as (1) Fπ Cπ = (α1 + β M) C π = (1 + γ M) C π ε where M is the topological or connectivity matrix. The parameters α, β, and γ represent the energy of an electron in a 2pz AO in the molecule and the interaction and overlap between AOs on neighboring, that is, bonded, atoms, respectively, and ε is the energy of the molecular orbital. The structure of M reflects the tight-binding approximation, whereby interaction and overlap between non-neighbor atoms are neglected. Standard Hückel theory formally neglects differential overlap (γ = 0), hence the energies of the resulting MOs, εi, are εi = α + λi β, i = 1, 2, ...
(2)
where λi are the eigenvalues of M. The Hückel matrices, orbital energies, occupancies, and total π energies (Eπ) for ethene, butadiene, and benzene are shown in Figure 1, along with diagrams depicting the composition of the MOs in terms of 2pz AOs. Comparison of the total π energies of butadiene and benzene with that of ethene immediately shows a delocalization (resonance) enhancement, that is, an excess beyond what could be expected on the basis of localized ethene-like π bonds. Such resonance effects have been the principal focal point of the Hückel model. Now we focus on the basic mechanism of bonding as demonstrated in its simplest form in ethene. Orbital interaction, as quantified by the coupling parameter β in the Hückel Hamiltonian, is responsible for the formation of two delocalized MOs at energies α ± β. Since β is negative, the lowest (bonding) MO energy is ε1 = α + β. When the MO is doubly occupied, an energetic stabilization by 2β occurs, corresponding to a covalent π bond. The antibonding MO with energy ε2 = α − β is unoccupied in the ground state. Excitation of a bonding electron to the antibonding MO yields an excited state with zero π-energy stabilization, that is, a cancellation of the π bond. Clearly, the coupling between the carbon AOs results in orbital splitting, hence stabilization, and
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The Hückel parametrization yields a model of molecular electronic structure that is independent of the precise nature of the interactions of the electrons and nuclei as well as of the AOs. For example, the eigenvalue equations, which yield the MOs of H2+ in a minimal basis of 1s AOs and their energies, are isomorphic with the Hückel equations (eq 1) of ethene. Moreover, a generalized form of Hückel theory, with multiple sets of parameters, can be readily developed and applied to more complex molecules including ionic systems as well as solids, indicating that the bonding principles, as developed above with the aid of Hückel theory, have universal applicability (see the Supplemental MaterialW). In summary, we conclude that covalent bonding results from a delocalization of electronic motion over at least two, but often more than two, atoms in a molecule. The stabilization follows from a splitting of degenerate or nearly degenerate atomic orbital energies in the formation of delocalized molecular orbitals. Figure 1. Hückel π MOs, energy levels, total π energies (Eπ) and Hückel matrices (showing nonzero elements only) of ethene, butadiene, and benzene. The relative signs and magnitudes of the MO expansion coefficients are indicated by the shading and sizes of the circles representing the AOs.
the delocalization of electronic motion. The global form of the π MOs determines the extent of the motion, while the splitting 2β provides a measure of its speed, that is, the rate of transfer between the bonded atoms. Evidently, the Hückel model describes in a simple quantum mechanical way the onset of π-electron motion between the bonded carbon atoms, emphasizing that such motion of shared electrons is the fundamental feature of the covalent bond. In the case of butadiene and benzene the same coupling of neighboring carbon atoms applies but there is significant further delocalization with the electrons moving along the entire chain or ring of bonded carbon atoms. Note that the extent of delocalization of the electronic motion is revealed by the corresponding delocalization of the molecular orbitals. As shown in Figure 1, the original local atomic carbon 2pz orbitals are transformed into delocalized MOs by the coupling represented by β in the Hückel Hamiltonian. The fact that the MOs are delocalized over the entire chain or ring, not just over C⫺C atom pairs, demonstrates the presence of an additional degree of delocalization that is often referred to as a resonance effect. Benzene represents the best known example of this, where the equivalence of the C⫺C π bonds may be explained in terms of resonance, that is, by superposition of the two Kekulé structures with localized π bonds (1). The most valuable feature of the Hückel model is that it captures the quantum mechanical nature of covalent bonding and displays the underlying physical mechanism in the form of electron delocalization directly without the need to resolve the stabilization in terms of kinetic and potential energy components. Moreover, the model clearly demonstrates that resonance stabilization, a familiar property of conjugated hydrocarbons such as butadiene and benzene, is simply an extension of the two-center bonding mechanism of ethene. 1202
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Covalent Bonding as a Result of Relaxation of Nonergodic Constraints The analysis above indicates that the covalent bonding mechanism is inherently dynamical. The delocalization of MOs associated with bond formation in a molecule implies delocalized motion of valence electrons encompassing the bonded atoms. The rapidity of this motion, determined by the degree of splitting of the MO energies, directly determines the strength of the covalent bond. Such dynamics, as determined by Hückel MOs, is illustrated in our Supplemental Material.W The change of dynamical character of the electronic motion in the covalent bonding mechanism turns out to be closely related to the classical concept of ergodicity (14). A classical system is ergodic at an energy E if the dynamics at that energy covers the entire energy surface, that is, if the dynamics is fully delocalized and constrained only by energy conservation. Any localization of the dynamics is then referred to as a nonergodic effect. We find that in the electronic structure of a molecule covalent bonding corresponds to the removal of the dynamical constraints of localization to the constituent atoms, allowing the system to approach the ergodic limit of full delocalization (8–10). The Hückel model is ideally suited to illustrate the application of these concepts of ergodicity and nonergodic effects to quantized electron dynamics in molecules (15, 16). In particular, it is readily shown that the Hückel MOs in polyenes, for example, butadiene, predict ergodic translational dynamics (see Supplemental MaterialW). Thus we also conclude that covalent bonding is associated with relaxation of nonergodic constraints on valence electron dynamics. Discussion and Conclusion The familiar Hückel model provides a simple but also profound and far-reaching approach to covalent bonding. A key advantage is that this model captures the quantum mechanical character of the mechanism in its simplest form and focuses on its dynamical nature rather than on its expression in terms of kinetic or potential energy variations. Thereby the subtleties of the virial theorem, orbital contraction, and choice of basis set in the context of covalent bonding mecha-
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nism are avoided. The Hückel approach, by virtue of the essential elements of quantum theory, has the capacity to clearly demonstrate the relationship between covalent bonding and the dynamical concept of ergodicity that plays a fundamental role in classical dynamics. We conclude that the Hückel model provides us not just with an understanding of conjugation and aromaticity of planar hydrocarbon molecules but with a valuable pedagogical model for the understanding of the basic mechanism of covalent bonding, which is a quantum effect related to electron dynamics. This mechanism is one and the same in H2+ and in the π bonding in ethene as well as in butadiene and benzene. The key concept is electron delocalization, rather than any interplay of kinetic and potential energies. W
Supplemental Material
A more comprehensive and detailed presentation is available in this issue of JCE Online. Literature Cited 1. Coulson, C. A. Valence, 2nd ed.; Clarendon: Oxford, 1961. 2. Gray, H. B. Chemical Bonds: An Introduction to Atomic and Molecular Structure; University Science Books: Mill Valley, 1994.
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3. Burdett, J. K. Chemical Bonds—A Dialog; Wiley: Chichester, U.K., 1997. 4. Slater, J. C. Quantum Theory of Matter, 2nd ed.; McGrawHill: New York, 1968; p 400, and references therein. 5. Ruedenberg, K. Rev. Mod. Phys. 1962, 34, 326. 6. Ruedenberg, K. In Localization and Delocalization in Quantum Chemistry; Chalvet, O., Daudel, R., Diner, S., Malrieu, J. P., Eds.; Reidel: Dordrecht, 1975; Vol. 1, p 223. 7. Hellman, H. Z. Phys. 1933, 85, 180. 8. Nordholm, S. J. Chem. Phys. 1987, 86, 363. 9. Nordholm, S. J. Chem. Educ. 1988, 65, 581. 10. Bacskay, G. B.; Reimers, J. R.; Nordholm, S. J. Chem. Educ. 1997, 74, 1494. 11. Hückel, E. Z. Phys. 1930, 60, 423. Hückel, E. Z. Phys. 1931, 70, 204. Hückel, E. Z. Phys. 1931, 72, 310. Hückel, E. Z. Phys. 1932, 76, 628. 12. Engel, T.; Reid, P. Physical Chemistry; Pearson Education: San Francisco, 2005; Chapter 25. 13. Yates, K. Hückel Molecular Orbital Energy; Academic Press: New York, 1978. 14. Jancel, R. Foundations of Classical and Quantum Statistical Mechanics; Pergamon Press: London, 1969. 15. Nordholm, S.; Rice, S. A. J. Chem. Phys. 1974, 61, 203. 16. Bäck, A.; Nordholm, S.; Nyman, G. J. Phys. Chem. A 2004, 108, 8782.
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