Hückel method
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. 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. A more dramatic extension of the method to include σ-electrons, known as the extended Hückel method, was developed by Roald Hoffmann. The extended Hückel method gives some degree of quantitative accuracy for organic molecules in general and was used to provide computational justification for the Woodward–Hoffmann rules. To distinguish the original approach from Hoffmann's extension, the Hückel method is also known as the simple Hückel method.
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 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: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, general solutions exist:
The energy levels for cyclic systems can be predicted using the mnemonic. 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 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 and comparing the energy of a doubly-occupied π orbital with the energy of electrons in two isolated p orbitals, a value of |β| = 32.5 kcal/mole can be inferred. On the other hand, using the resonance energy of benzene and comparing benzene with a hypothetical "non-aromatic 1,3,5-cyclohexatriene", a much smaller value of |β| = 18 kcal/mole emerges. These differences are not surprising, given the substantially shorter bond length of ethylene compared to benzene. 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 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| as high as 3 eV 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:
- The predicted molecular orbital energies as stipulated by Koopmans' theorem correlate with photoelectron spectroscopy.
- The Hückel delocalization energy correlates with the experimental heat of combustion. This energy is defined as the difference between the total predicted π energy and a hypothetical π energy in which all ethylene units are assumed isolated, each contributing 2β.
- Molecules with molecular orbitals paired up such that only the sign differs are called alternant hydrocarbons and have in common small molecular dipole moments. This is in contrast to non-alternant hydrocarbons, such as azulene and fulvene that have large dipole moments. The Hückel theory is more accurate for alternant hydrocarbons.
- For cyclobutadiene the theory predicts that the two high-energy electrons occupy a degenerate pair of molecular orbitals that are neither stabilized nor destabilized. Hence the square molecule would be a very reactive triplet diradical. In fact, all cyclic conjugated hydrocarbons with a total of 4n π-electrons share this molecular orbital pattern, and this forms the basis of Hückel's rule.
- Dewar reactivity numbers deriving from the Hückel approach correctly predict the reactivity of aromatic systems with nucleophiles and electrophiles.
- The benzyl cation and anion serve as simple models for arenes with electron-withdrawing and electron-donating groups, respectively. The π-electron population correctly implies the meta- and ortho-/''para-''selectivity for electrophilic aromatic substitution of π electron-poor and π electron-rich arenes, respectively.
Application in optical activity analysis
Instead of traditional chiroptical solution measurements, Hückel theory helps focus on oriented π systems by separating from σ electrons especially in the planar, -symmetric cases. Transition dipole moments derived by multiplying each wavefunction of individual planar molecule one by one, contribute to the directions of the most optical activity, where sit at the bisectors of two orthogonal ones. Despite the zero value for the trace of the tensor, cis-butadiene shows considerable off diagonal component which was computed as the first optical activity evaluation of achiral molecule.
Consider 3,5-dimethylene-1-cyclopentene as an example. Transition electric dipole, magnetic dipole and electric quadrupole moments interactions result in optical rotation, which can be described by both tensor components and chemical geometries. The in phase overlap of two molecular orbitals yield negative charge while depleting charge out of phase. The movement can be interpreted quantitatively by corresponding π and π* orbitals coefficients.