The Hückel method is a semi-empirical method that makes additional assumptions to the -electron approximation in order to determine the energies of molecular orbitals in planar molecules that are conjugated.

###### Question

What is a semi-empirical method?

###### Answer

A semi-empirical method combines theoretical approximations with experimental data to derive useful parameters like the energy of a molecular orbital (MO).

Like the -electron approximation, the Hückel method assumes that valence electrons in planar conjugated organic compounds are relatively unreactive for reactions that do not involve in breaking these bonds. Furthermore, they are assumed not to interact with the valence electrons, which participate in many reactions. Consequently, the valence electrons are ignored in deriving the energies of the MOs.

To solve for the energies of -electron MOs, we refer to eq157 of the Hartree-Fock-Roothaan method. The total wavefunction is , where is the antisymmetriser and . The index refers to the number of carbon atoms forming the conjugated framework of the planar molecule, while is a real and normalised orbital of the -th carbon atom. Non-trivial solutions are given by the secular determinant:

is known as the * Coulomb integral* if .

is known as the * resonance integral* if .

is the normalisation expression of if .

is known as the ** overlap integral** if .

is the eigenvalue associated with .

Unlike *ab initio* methods, the exact form of the effective Hamiltonian in is not important (read on for explanation). The additional assumptions used in the Hückel method are:

In other words, all Coulomb integrals (interaction energy) have the same value of , which is reasonable if all the atoms forming the conjugated framework are the same. It follows that for adjacent atoms is similarly a good approximation. From *ab initio* calculations, both and are negative. As non-adjacent carbons atoms are well separated in space, the conjecture that for non-adjacent carbons atoms and is also reasonable. The integral is a result of the normalisation of orbitals. However, for adjacent carbon atoms is a poor approximation because orbitals of adjacent carbon atoms are not orthogonal to one another. In reality, the overlap of orbitals of adjacent carbon atoms leads to a stabilisation (lower energy) of bonding MOs and a destabilisation (increase in energy) of antibonding MOs. Nevertheless, the relative order of the MOs along the energy axis remains unchanged.

Let’s look at a simple example: ethene. The wavefunction is and the secular determinant is

Since the orbitals are real and is Hermitian, and . Furthermore, and we have

This implies that and the ground state electron configuration of the electrons in ethene is

Consequently, the Hückel method provides a quick way of predicting the number and order of a conjugated molecule’s valence MOs that likely to participate in a reaction, without explicitly specifying the Hamiltonian. The parameters and are adjusted to give the best fit to experimental data, which is why the Hückel method is considered a semi-empirical method. The total ground state electronic energy of ethene is .

###### Question

Show that i) when and when , and that ii) the normalisation constant of is .

###### Answer

i) Multiplying the eigenvalue equation by , integrating over all space and using the Hückel assumptions, we get

For and , we have and , respectively.

Therefore, the occupied MO wavefunction and unoccupied MO are and respectively. Since the two electrons occupying lead to a stable molecule, the MO is known as the * bonding MO*. The higher energy MO, which obviously results in a relatively less state when occupied, is known as the

*.*

**antibonding MO**ii)

Since , we have

Let’s consider another molecule: 1,3-Butadiene. The secular determinant is:

The Hückel assumptions give

Using the determinant identity , we multiply both sides of the above equation by to give

where .

Expanding the determinant, we have . The roots to the polynomial are and the energies of the MOs are

If we regard two electrons in 1,3-Butadiene as localised between carbons atoms and , and another two between carbon atoms and , we can compare the total ground state electronic energy of 1,3-Butadiene with that of two molecules of ethene. The difference of implies an energy stabilisation for 1,3-Butadiene that is due to the delocalisation of electrons over the length of the molecule, rather than being localised.

The general form of the wavefunction for the four MOs is . To determine each individual wavefunction, we apply the Hückel assumptions to eq157, giving

From eq162 and eq165, we have

Substituting eq166 in eq163 gives . When , we have .

Substituting in eq166 and comparing with eq167, . It follows that . Therefore, , which becomes after normalisation. Using the same logic, we have

For more complicated molecules, such as benzene, the Hückel method is combined with group theory to derive the wavefunctions and solve for their associated eigenvalues.