Hückel method
The theory predicts the molecular orbitals for π-electrons in π-delocalized molecules, such as ethylene, benzene, butadiene, and pyridine.
It was later extended to conjugated molecules such as pyridine, pyrrole and furan that contain atoms other than carbon and hydrogen (heteroatoms).
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.
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.
[6] 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.
The method has several characteristics: The results for a few simple molecules are tabulated below: HOMO/LUMO/SOMO = Highest occupied/lowest unoccupied/singly-occupied molecular orbitals.
For linear and cyclic systems (with N atoms), general solutions exist:[9] The energy levels for cyclic systems can be predicted using the Frost circle [de] mnemonic (named after the American chemist Arthur Atwater Frost [de]).
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.
This quantity is negative, since the electron is stabilized by being electrostatically bound to the positively charged nucleus.
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.
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.
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.
[13] 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.
The analysis of the optical activity of a molecule depends to a certain extent on the study of its chiral characteristics.
However, for achiral molecules applying pesudoscalars to simplify the calculations of optical activity cannot be achieved due to the lack of spatial average.
[16] Instead of traditional chiroptical solution measurements, Hückel theory helps focus on oriented π systems by separating from σ electrons especially in the planar,
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.
The Hückel definition of bond order attempts to quantify any additional stabilization that the system enjoys resulting from delocalization.
In a sense, the Hückel bond order suggests that there are four π-bonds in benzene instead of the three that are implied by the Kekulé-type Lewis structures.
The "extra" bond is attributed to the additional stabilization that results from the aromaticity of the benzene molecule.
The π-electron population is calculated in a very similar way to the bond order using the orbital coefficients of the Hückel MOs.
Then, we substitute the ansatz into the expression for E and expand, yielding In the remainder of the derivation, we will assume that the atomic orbitals are real.
with nontrivial solution vectors represent reasonable estimates of the energies of the remaining π orbitals.
For typical bond distances (1.40 Å) as might be found in benzene, for example, the true value of the overlap for C(2pz) orbitals on adjacent atoms i and j is about
If the substance is a planar, unsaturated hydrocarbon, the coefficients of the MOs can be found without appeal to empirical parameters, while orbital energies are given in terms of only
, the energy levels are The coefficients can then be found by expanding (***): Since the matrix is singular, the two equations are linearly dependent, and the solution set is not uniquely determined until we apply the normalization condition.
can be found: Finally, the Hückel molecular orbitals are The constant β in the energy term is negative; therefore,
