Hückel method

Hückel characteristics

The method has several characteristics:

It limits itself to conjugated hydrocarbons.

Only pi electron molecular orbitals (MO's) are included because these determine the general properties of these molecules and the sigma electrons are ignored. This is referred to as sigma-pi separability. It is justified by the orthogonality of sigma and pi orbitals in planar molecules. For this reason, the Hückel method is limited to planar systems.

The method takes as inputs the LCAO MO Method, the Schrödinger equation and simplifications based on orbital symmetry considerations. Interestingly the method does not take in any physical constants.

The method predicts how many energy levels exist for a given molecule, which levels are degenerate and it expresses the MO energies as the sum of two other energy terms, called alpha, the energy of an electron in a 2p-orbital, and beta, an interaction energy between two p orbitals which are still unknown but importantly have become independent of the molecule. In addition it enables calculation of charge density for each atom in the pi framework, the bond order between any two atoms, and the overall molecular dipole moment.

Hückel results

The results for a few simple molecules are tabulated below:

The theory predicts two energy levels for ethylene with its two pi electrons filling the low-energy HOMO and the high energy LUMO remaining empty. In butadiene the 4 pi electrons occupy 2 low energy MO's, 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:

Linear: E k = α + 2 β cos ⁡ k π ( n + 1 )

Cyclic: E k = α + 2 β cos ⁡ 2 k π n

The energy levels for cyclic systems can be predicted using the Frost circle mnemonic. A circle centered at α with radius 2β is inscribed with a polygon with one vertex pointing down; the vertices represent energy levels with the appropriate energies. A related mnemonic exists for linear systems.

Many predictions 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:

Δ E = − 4 β sin ⁡ π 2 ( n + 1 ) from which a value for β can be obtained between −60 and −70 kcal/mol (−250 to −290 kJ/mol).

The predicted MO 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 pi energy (in benzene 8β) and a hypothetical pi energy in which all ethylene units are assumed isolated, each contributing 2β (making benzene 3 × 2β = 6β).

Molecules with MO's paired up such that only the sign differs (for example α ± β) 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 MO's that are neither stabilized or destabilized. Hence the square molecule would be a very reactive triplet diradical (the ground state is actually rectangular without degenerate orbitals). In fact, all cyclic conjugated hydrocarbons with a total of 4nπ electrons share this MO 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.

Mathematics behind the Hückel method

The Hückel method can be derived from the Ritz method, with a few further assumptions concerning the overlap matrix S and the Hamiltonian matrix H.

It is assumed that the overlap matrix S  is the identity matrix. This means that overlap between the orbitals is neglected and the orbitals are considered orthogonal. Then the generalised eigenvalue problem of the Ritz method turns into an eigenvalue problem.

The Hamiltonian matrix H = (H_ij) is parametrised in the following way:

H_ii = α for C atoms and α + h_Aβ for other atoms A. H_ij = β if the two atoms are next to each other and both C, and k_AB β for other neighbouring atoms A and B. H_ij = 0 in any other case.

The orbitals are the eigenvectors, and the energies are the eigenvalues of the Hamiltonian matrix. If the substance is a pure hydrocarbon, the problem can be solved without any knowledge about the parameters. For heteroatom systems, such as pyridine, values of h_A and k_AB have to be specified.

Hückel solution for ethylene

In the Hückel treatment for ethylene, the molecular orbital Ψ is a linear combination of the 2p atomic orbitals ϕ at carbon with their ratios c :

  Ψ = c 1 ϕ 1 + c 2 ϕ 2

This equation is substituted in the Schrödinger equation:

  H Ψ = E Ψ with H the Hamiltonian and E the energy corresponding to the molecular orbital

to give:

H c 1 ϕ 1 + H c 2 ϕ 2 = E c 1 ϕ 1 + E c 2 ϕ 2

This equation is multiplied by ϕ 1 and integrated to give the equation:

c 1 ( H 11 − E S 11 ) + c 2 ( H 12 − E S 12 ) = 0

The same equation is multiplied by ϕ 2 and integrated to give the equation:

c 1 ( H 21 − E S 21 ) + c 2 ( H 22 − E S 22 ) = 0

This really can be represented as a matrix. After converting this set to matrix notation,

[ c 1 ( H 11 − E S 11 ) + c 2 ( H 12 − E S 12 ) c 1 ( H 21 − E S 21 ) + c 2 ( H 22 − E S 22 ) ] = 0

Or more simply as a product of matrices.

[ H 11 − E S 11 H 12 − E S 12 H 21 − E S 21 H 22 − E S 22 ] × [ c 1 c 2 ] = 0

where:

H i j = ∫ ϕ i H ϕ j d v S i j = ∫ ϕ i ϕ j d v

All diagonal Hamiltonian integrals H i i are called coulomb integrals and those of type H i j , where atoms i and j are connected, are called resonance integrals. The Hückel method assumes that all overlap integrals equal the Kronecker delta, S i j = δ i j , and all nonzero resonance integrals are equal. Resonance integral H i j is nonzero when the atoms i and j are bonded.

H 11 = H 22 = α H 12 = H 21 = β

Other assumptions are that the overlap integral between the two atomic orbitals is 0

S 11 = S 22 = 1 S 12 = S 21 = 0

leading to these two homogeneous equations:

[ α − E β β α − E ] × [ c 1 c 2 ] = 0

dividing by β :

[ α − E β 1 1 α − E β ] × [ c 1 c 2 ] = 0

Substituting x for α − E β :

[ x 1 1 x ] × [ c 1 c 2 ] = 0

This is convenient for computation, but it is also convenient as the energy and coefficients can be easily found:

x = α − E β x β = α − E E = α − x β c 2 = − x c 1 c 1 = − x c 2

The trivial solution gives both wavefunction coefficients c  equal to zero which is not useful so the other (non-trivial) solution is:

| x 1 1 x | = 0

which can be solved by expanding its determinant:

x 2 − 1 = 0 x 2 = 1 x = ± 1

Knowing that E = α − x β , the energy levels can be found to be:

E = α − ± 1 × β E = α ∓ β

The coefficients can be found by using the previous relationship determined:

c 2 = − x c 1 c 1 = − x c 2

Only one equation is necessary however:

c 2 = − ± 1 × c 1 c 2 = ∓ c 1

The second constant can be replaced giving the following wave equation.

Ψ = c 1 ( ϕ 1 ∓ ϕ 2 )

After normalization, the coefficient is obtained:

c 1 = 1 2 ,

Leaving

Ψ = 1 2 ( ϕ 1 ∓ ϕ 2 ) = ϕ 1 ∓ ϕ 2 2

The constant β in the energy term is negative; therefore, α + β with Ψ = 1 2 ( ϕ 1 + ϕ 2 ) is the lower energy corresponding to the HOMO energy and α − β with Ψ = 1 2 ( ϕ 1 − ϕ 2 ) is the LUMO energy.

Hückel solution for butadiene

In the Hückel treatment for butadiene, the MO Ψ is a linear combination of the 4p ϕ AO's at carbon with their ratios c :

  Ψ = c 1 ϕ 1 + c 2 ϕ 2 + c 3 ϕ 3 + c 4 ϕ 4

The secular equation is:

[ α − E β 0 0 β α − E β 0 0 β α − E β 0 0 β α − E ] × [ c 1 c 2 c 3 c 4 ] = 0

which leads to

( α − E ) ( α + β − E ) − β 2 = 0

and:

E ± = α + 1 ± 5 2 β