The Hückel method or Hückel molecular orbital theory, proposed by Erich Hückel in 1930, is a simple method for calculating molecular orbitals as linear combinations of atomic orbitals. The theory predicts the molecular orbitals for π-electrons in π-delocalized molecules, such as ethylene, benzene, butadiene, and pyridine.E. Hückel, Zeitschrift für Physik, 70, 204 (1931); 72, 310 (1931); 76, 628 (1932); 83, 632 (1933).Hückel Theory for Organic Chemists, C. A. Coulson, B. O'Leary and R. B. Mallion, Academic Press, 1978.P. R. Bunker and P. Jensen (2005),Fundamentals of Molecular Symmetry (CRC Press) Hückel method for benzene: Sections 3.4.3 and 10.2 It provides the theoretical basis for Hückel's rule that cyclic, planar molecules or ions with π-electrons are aromatic. It was later extended to conjugated molecules such as pyridine, pyrrole and furan that contain atoms other than carbon and hydrogen (heteroatoms).Andrew Streitwieser, Molecular Orbital Theory for Organic Chemists, Wiley, New York (1961). A more dramatic extension of the method to include σ-electrons, known as the extended Hückel method (EHM), was developed by Roald Hoffmann. The extended Hückel method gives some degree of quantitative accuracy for organic molecules in general (not just planar systems) and was used to provide computational justification for the Woodward–Hoffmann rules."Stereochemistry of Electrocyclic Reactions", R. B. Woodward, Roald Hoffmann, J. Am. Chem. Soc., 1965; 87(2); 395–397. . To distinguish the original approach from Hoffmann's extension, the Hückel method is also known as the simple Hückel method (SHM).
Although undeniably a cornerstone of organic chemistry, Hückel's concepts were undeservedly unrecognized for two decades. Pauling and Wheland characterized his approach as "cumbersome" at the time, and their competing resonance theory was relatively easier to understand for chemists without fundamental physics background, even if they couldn't grasp the concept of quantum superposition and confused it with tautomerism. His lack of communication skills contributed: when Robert Robinson sent him a friendly request, he responded arrogantly that he is not interested in organic chemistry.
In spite of its simplicity, the Hückel method in its original form makes qualitatively accurate and chemically useful predictions for many common molecules and is therefore a powerful and widely taught educational tool. It is described in many introductory quantum chemistry and physical organic chemistry textbooks, and organic chemists in particular still routinely apply Hückel theory to obtain a very approximate, back-of-the-envelope understanding of π-bonding.
Hückel characteristics
The method has several characteristics: It limits itself to conjugated molecules. Only π electron molecular orbitals are included because these determine much of the chemical and spectral properties of these molecules. The σ electrons are assumed to form the framework of the molecule and σ connectivity is used to determine whether two π orbitals interact. However, the orbitals formed by σ electrons are ignored and assumed not to interact with π electrons. This is referred to as σ-π separability. It is justified by the orthogonality of σ and π orbitals in planar molecules. For this reason, the Hückel method is limited to systems that are planar or nearly so. The method is based on applying the variational method to linear combination of atomic orbitals and making simplifying assumptions regarding the overlap, resonance and Coulomb integrals of these atomic orbitals. It does not attempt to solve the Schrödinger equation, and neither the functional form of the basis atomic orbitals nor details of the Hamiltonian are involved. For hydrocarbons, the method takes atomic connectivity as the only input; empirical parameters are only needed when heteroatoms are introduced. The method predicts how many energy levels exist for a given molecule, which levels are degenerate and it expresses the molecular orbital energies in terms of two parameters, called α, the energy of an electron in a 2p orbital, and β, the interaction energy between two 2p orbitals (the extent to which an electron is stabilized by allowing it to delocalize between two orbitals). The usual sign convention is to let both α and β be negative numbers. To understand and compare systems in a qualitative or even semi-quantitative sense, explicit numerical values for these parameters are typically not required. In addition the method also enables calculation of charge density for each atom in the π framework, the fractional bond order between any two atoms, and the overall molecular dipole moment.
Hückel results
Results for simple molecules and general results for cyclic and linear systems
The results for a few simple molecules are tabulated below: Molecule EnergyFrontier orbitalHOMO–LUMO gapNotesE1 = α + β HOMO 2βE2 = α – β LUMOE1 = α + 1.618...β 1.236...β1.618... and 0.618... = E2 = α + 0.618...β HOMO E3 = α – 0.618...β LUMO E4 = α – 1.618...β E1 = α + 1.802...β0.890...β1.802..., 1.247..., and 0.445... = 2cos(nπ/7) for n = 1, 2, and 3E2 = α + 1.247...βE3 = α + 0.445...βHOMOE4 = α – 0.445...βLUMOE5 = α – 1.247...βE6 = α – 1.802...βE1 = α + 2β 0(E2, E3) are degenerate, both are singly occupied for D4h (square) cyclobutadiene, to comply with Hund's ruleThe actual ground state of cyclobutadiene is D2h (rectangular) with non-degenerate E2 (HOMO) and E3 (LUMO). E2 = α SOMO E3 = α SOMO E4 = α − 2β E1 = α + 2β 2β(E2, E3) and (E4, E5) are degenerateE2 = α + β HOMOE3 = α + β HOMOE4 = α − β LUMOE5 = α − β LUMOE6 = α − 2β Table 1. Hückel method results. Because α and β are negative,The chemical bond, 2nd ed., J.N. Murrel, S.F.A. Kettle, J.M. Tedder, orbitals are in order of increasing energy. HOMO/LUMO/SOMO = Highest occupied/lowest unoccupied/singly-occupied molecular orbitals.
The theory predicts two energy levels for ethylene with its two π electrons filling the low-energy HOMO and the high energy LUMO remaining empty. In butadiene the 4 π-electrons occupy 2 low energy molecular orbitals, out of a total of 4, and for benzene 6 energy levels are predicted, two of them degenerate.
For linear and cyclic systems (with N atoms), general solutions exist:Quantum Mechanics for Organic Chemists. Zimmerman, H., Academic Press, New York, 1975.
Linear system (polyene/polyenyl): . Energy levels are all distinct.
Cyclic system, Hückel topology (annulene/annulenyl): . Energy levels are each doubly degenerate. Cyclic system, Möbius topology (hypothetical for N < 8Due to the twisted geometry required for a molecule to take on Möbius aromaticity, the idealized Möbius versions of the annulenes (or annulenyl ions) are hypothetical species for small ring sizes. Three- to seven-membered Möbius annulene/annulenyl systems are too twisted to be reasonably considered as stable species. Computations have considered whether certain Möbius topology isomers and conformers of eight-membered and larger annulene/annulenyl systems are aromatic. While Möbius aromatic configurations have indeed been found, they are still generally less stable than their nonaromatic counterparts. The sole exception appears to be penta-trans-annulenyl cation, whose ground state is believed to be Möbius aromatic (Herges and coworkers, Org. Lett. 2010, 12, 1708). In fact, there are only a handful of ground state species that are thought to be Möbius aromatic. However, Möbius aromaticity is conceptually important, as many pericyclic transition states take on Möbius aromatic character, and the distinction between Hückel and Möbius topologies forms the basis of the Dewar-Zimmerman approach to the generalized pericyclic selection rules (Woodward-Hoffmann rules).): . Energy levels are each doubly degenerate.
The energy levels for cyclic systems can be predicted using the mnemonic (named after the American chemist ). A circle centered at α with radius 2β is inscribed with a regular N-gon with one vertex pointing down; the y-coordinate of the vertices of the polygon then represent the orbital energies of the [N]annulene/annulenyl system. Related mnemonics exists for linear and Möbius systems.
The values of α and β The value of α is the energy of an electron in a 2p orbital, relative to an unbound electron at infinity. This quantity is negative, since the electron is stabilized by being electrostatically bound to the positively charged nucleus. For carbon this value is known to be approximately –11.4 eV. Since Hückel theory is generally only interested in energies relative to a reference localized system, the value of α is often immaterial and can be set to zero without affecting any conclusions.
Roughly speaking, β physically represents the energy of stabilization experienced by an electron allowed to delocalize in a π molecular orbital formed from the 2p orbitals of adjacent atoms, compared to being localized in an isolated 2p atomic orbital. As such, it is also a negative number, although it is often spoken of in terms of its absolute value. The value for |β| in Hückel theory is roughly constant for structurally similar compounds, but not surprisingly, structurally dissimilar compounds will give very different values for |β|. For example, using the π bond energy of ethylene (65 kcal/mole) and comparing the energy of a doubly-occupied π orbital (2α + 2β) with the energy of electrons in two isolated p orbitals (2α), a value of |β| = 32.5 kcal/mole can be inferred. On the other hand, using the resonance energy of benzene (36 kcal/mole, derived from heats of hydrogenation) and comparing benzene (6α + 8β) with a hypothetical "non-aromatic 1,3,5-cyclohexatriene" (6α + 6β), a much smaller value of |β| = 18 kcal/mole emerges. These differences are not surprising, given the substantially shorter bond length of ethylene (1.33 Å) compared to benzene (1.40 Å). The shorter distance between the interacting p orbitals accounts for the greater energy of interaction, which is reflected by a higher value of |β|. Nevertheless, heat of hydrogenation measurements of various polycyclic aromatic hydrocarbons like naphthalene and anthracene all imply values of |β| between 17 and 20 kcal/mol.
However, even for the same compound, the correct assignment of |β| can be controversial. For instance, it is argued that the resonance energy measured experimentally via heats of hydrogenation is diminished by the distortions in bond lengths that must take place going from the single and double bonds of "non-aromatic 1,3,5-cyclohexatriene" to the delocalized bonds of benzene. Taking this distortion energy into account, the value of |β| for delocalization without geometric change (called the "vertical resonance energy") for benzene is found to be around 37 kcal/mole. On the other hand, experimental measurements of electronic spectra have given a value of |β| (called the "spectroscopic resonance energy") as high as 3 eV (~70 kcal/mole) for benzene. Given these subtleties, qualifications, and ambiguities, Hückel theory should not be called upon to provide accurate quantitative predictions – only semi-quantitative or qualitative trends and comparisons are reliable and robust.
Other successful predictions With this caveat in mind, many predictions of the theory have been experimentally verified: The HOMO–LUMO gap, in terms of the β constant, correlates directly with the respective molecular electronic transitions observed with UV/VIS spectroscopy. For linear polyenes, the energy gap is given as:
from which a value for β can be obtained between −60 and −70 kcal/mol (−250 to −290 kJ/mol)."Use of Huckel Molecular Orbital Theory in Interpreting the Visible Spectra of Polymethine Dyes: An Undergraduate Physical Chemistry Experiment". Bahnick, Donald A., J. Chem. Educ. 1994, 71, 171.