# Hückel method

The Hückel method or Hückel molecular orbital method (HMO), proposed by Erich Hückel in 1930, is a very simple linear combination of atomic orbitals molecular orbitals (LCAO MO) method for the determination of energies of molecular orbitals of pi electrons in conjugated hydrocarbon systems, such as ethene, benzene and butadiene.[1][2] It is the theoretical basis for the Hückel's rule. It was later extended to conjugated molecules such as pyridine, pyrrole and furan that contain atoms other than carbon, known in this context as heteroatoms.[3] The extended Hückel method developed by Roald Hoffmann is computational and three-dimensional and was used to test the Woodward–Hoffmann rules.[4]

It is a very powerful educational tool, and details appear in many chemistry textbooks.

## 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:

 Molecule Energy Frontier orbital HOMO–LUMO energy gap Ethylene E1 = α - β LUMO −2β E2 = α + β HOMO Butadiene E1 = α + 1.62β E2 = α + 0.62β HOMO −1.24β E3 = α − 0.62β LUMO E4 = α − 1.62β Benzene E1 = α + 2β E2 = α + β E3 = α + β HOMO −2β E4 = α − β LUMO E5 = α − β E6 = α − 2β Cyclobutadiene E1 = α + 2β E2 = α SOMO 0 E3 = α SOMO E4 = α − 2β Table 1. Hückel method results.  Lowest energies of top α and β are both negative values.[5] HOMO/LUMO/SOMO = Highest occupied/lowest unoccupied/singly-occupied molecular orbitals.

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:[6]

Frost circle mnemonic for 1,3-cyclopenta-5-dienyl anion
• Linear: ${\displaystyle E_{k}=\alpha +2\beta \cos {\frac {k\pi }{(n+1)}}}$
• Cyclic: ${\displaystyle E_{k}=\alpha +2\beta \cos {\frac {2k\pi }{n}}}$

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.[7] A related mnemonic exists for linear systems.[8]

Many predictions have been experimentally verified:

${\displaystyle \Delta E=-4\beta \sin {\frac {\pi }{2(n+1)}}}$
from which a value for β can be obtained between −60 and −70 kcal/mol (−250 to −290 kJ/mol).[9]
• The predicted MO energies as stipulated by Koopmans' theorem correlate with photoelectron spectroscopy.[10]
• 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 = (Hij) is parametrised in the following way:

Hii = α for C atoms and α + hAβ for other atoms A.
Hij = β if the two atoms are next to each other and both C, and kAB β for other neighbouring atoms A and B.
Hij = 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 hA and kAB have to be specified.

## Hückel solution for ethylene

Molecular orbitals ethylene ${\displaystyle E=\alpha -\beta }$
Molecular orbitals ethylene ${\displaystyle E=\alpha +\beta }$

In the Hückel treatment for ethylene,[11] the molecular orbital ${\displaystyle \Psi \,}$ is a linear combination of the 2p atomic orbitals ${\displaystyle \phi \,}$ at carbon with their ratios ${\displaystyle c\,}$:

${\displaystyle \ \Psi =c_{1}\phi _{1}+c_{2}\phi _{2}}$

This equation is substituted in the Schrödinger equation:

${\displaystyle \ H\Psi =E\Psi }$
with ${\displaystyle H\,}$ the Hamiltonian and ${\displaystyle E\,}$ the energy corresponding to the molecular orbital

to give:

${\displaystyle Hc_{1}\phi _{1}+Hc_{2}\phi _{2}=Ec_{1}\phi _{1}+Ec_{2}\phi _{2}\,}$

This equation is multiplied by ${\displaystyle \phi _{1}\,}$ and integrated to give the equation:

${\displaystyle c_{1}(H_{11}-ES_{11})+c_{2}(H_{12}-ES_{12})=0\,}$

The same equation is multiplied by ${\displaystyle \phi _{2}\,}$ and integrated to give the equation:

${\displaystyle c_{1}(H_{21}-ES_{21})+c_{2}(H_{22}-ES_{22})=0\,}$

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

${\displaystyle {\begin{bmatrix}c_{1}(H_{11}-ES_{11})+c_{2}(H_{12}-ES_{12})\\c_{1}(H_{21}-ES_{21})+c_{2}(H_{22}-ES_{22})\\\end{bmatrix}}=0}$

Or more simply as a product of matrices.

${\displaystyle {\begin{bmatrix}H_{11}-ES_{11}&H_{12}-ES_{12}\\H_{21}-ES_{21}&H_{22}-ES_{22}\\\end{bmatrix}}\times {\begin{bmatrix}c_{1}\\c_{2}\\\end{bmatrix}}=0}$

where:

${\displaystyle H_{ij}=\int \phi _{i}H\phi _{j}\mathrm {d} v\,}$
${\displaystyle S_{ij}=\int \phi _{i}\phi _{j}\mathrm {d} v\,}$

All diagonal Hamiltonian integrals ${\displaystyle H_{ii}\,}$ are called coulomb integrals and those of type ${\displaystyle H_{ij}\,}$, where atoms i and j are connected, are called resonance integrals. The Hückel method assumes that all overlap integrals equal the Kronecker delta, ${\displaystyle S_{ij}=\delta _{ij}\,}$, and all nonzero resonance integrals are equal. Resonance integral ${\displaystyle H_{ij}\,}$ is nonzero when the atoms i and j are bonded.

${\displaystyle H_{11}=H_{22}=\alpha \,}$
${\displaystyle H_{12}=H_{21}=\beta \,}$

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

${\displaystyle S_{11}=S_{22}=1\,}$
${\displaystyle S_{12}=S_{21}=0\,}$

leading to these two homogeneous equations:

${\displaystyle {\begin{bmatrix}\alpha -E&\beta \\\beta &\alpha -E\\\end{bmatrix}}\times {\begin{bmatrix}c_{1}\\c_{2}\\\end{bmatrix}}=0}$

dividing by ${\displaystyle \beta }$:

${\displaystyle {\begin{bmatrix}{\frac {\alpha -E}{\beta }}&1\\1&{\frac {\alpha -E}{\beta }}\\\end{bmatrix}}\times {\begin{bmatrix}c_{1}\\c_{2}\\\end{bmatrix}}=0}$

Substituting ${\displaystyle x}$ for ${\displaystyle {\frac {\alpha -E}{\beta }}}$:

${\displaystyle {\begin{bmatrix}x&1\\1&x\\\end{bmatrix}}\times {\begin{bmatrix}c_{1}\\c_{2}\\\end{bmatrix}}=0}$

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

${\displaystyle x={\frac {\alpha -E}{\beta }}\,}$
${\displaystyle x\beta =\alpha -E\,}$
${\displaystyle E=\alpha -x\beta \,}$
${\displaystyle c_{2}=-xc_{1}\,}$
${\displaystyle c_{1}=-xc_{2}\,}$

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

${\displaystyle {\begin{vmatrix}x&1\\1&x\\\end{vmatrix}}=0}$

which can be solved by expanding its determinant:

${\displaystyle x^{2}-1=0\,}$
${\displaystyle x^{2}=1\,}$
${\displaystyle x=\pm 1\,}$

Knowing that ${\displaystyle E=\alpha -x\beta }$, the energy levels can be found to be:

${\displaystyle E=\alpha -\pm 1\times \beta }$
${\displaystyle E=\alpha \mp \beta }$

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

${\displaystyle c_{2}=-xc_{1}\,}$
${\displaystyle c_{1}=-xc_{2}\,}$

Only one equation is necessary however:

${\displaystyle c_{2}=-\pm 1\times c_{1}\,}$
${\displaystyle c_{2}=\mp c_{1}\,}$

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

${\displaystyle \Psi =c_{1}(\phi _{1}\mp \phi _{2})\,}$

After normalization, the coefficient is obtained:

${\displaystyle c_{1}={\frac {1}{\sqrt {2}}},}$

Leaving

${\displaystyle \Psi ={\frac {1}{\sqrt {2}}}(\phi _{1}\mp \phi _{2})={\frac {\phi _{1}\mp \phi _{2}}{\sqrt {2}}}\,}$

The constant β in the energy term is negative; therefore, ${\displaystyle \alpha +\beta }$ with ${\displaystyle \Psi ={\frac {1}{\sqrt {2}}}(\phi _{1}+\phi _{2})\,}$ is the lower energy corresponding to the HOMO energy and ${\displaystyle \alpha -\beta }$ with ${\displaystyle \Psi ={\frac {1}{\sqrt {2}}}(\phi _{1}-\phi _{2})\,}$ is the LUMO energy.

## Hückel solution for butadiene

In the Hückel treatment for butadiene, the MO ${\displaystyle \Psi \,}$ is a linear combination of the 4p ${\displaystyle \phi \,}$ AO's at carbon with their ratios ${\displaystyle c\,}$:

${\displaystyle \ \Psi =c_{1}\phi _{1}+c_{2}\phi _{2}+c_{3}\phi _{3}+c_{4}\phi _{4}}$

The secular equation is:

${\displaystyle {\begin{bmatrix}\alpha -E&\beta &0&0\\\beta &\alpha -E&\beta &0\\0&\beta &\alpha -E&\beta \\0&0&\beta &\alpha -E\\\end{bmatrix}}\times {\begin{bmatrix}c_{1}\\c_{2}\\c_{3}\\c_{4}\\\end{bmatrix}}=0}$

${\displaystyle (\alpha -E)(\alpha +\beta -E)-\beta ^{2}=0\,}$
${\displaystyle E\pm =\alpha +{\frac {1\pm {\sqrt {5}}}{2}}\beta }$