Hückel method

The proposed by Erich Hückel Hückel approximation, also: Hückel-Molekülorbital-Modell/Methode ( HMO-Modell/Methode ), is a method of semi-empirical quantum chemistry, which provides with little computational effort surprisingly good results. It allows to approximate molecular orbitals in conjugated systems. The two main conclusions are the Hückel rule and the Woodward -Hoffmann rules.

Application

The approximation is derived from the Ritz'schen method. In this case, the eigenvalue problem is formally reduced to. In the first case is a natural function of the molecule Hamiltonian. In order to accurately determine in an n -atom molecule has a 6N -dimensional partial differential equation would have to be solved, which is not analytically possible. In the second case, the n-dimensional vector containing the coefficients for the linear combination.

Once calculated, the molecular orbital can be specified as corresponding linear combination of the individual pz orbitals. The value of E is the energy of the orbital again.

In Ritz'schen process is iteratively found by the Hartree- Fock method the best possible solution. For a large number of integrals to be solved in each step what high computational complexity. The simplification of Hückelnäherung is that all integrals are parameterized.

The n conjugated atoms in the molecule are numbered. The matrix is ​​an n x n matrix. It sets:

Hii =? i

Hij = β, if the two atoms adjacent to ( and linked through conjugation)

Hij = 0 otherwise

? i is the Coulomb integral of the atom i in the molecule (? i <0) (* Is the complex conjugation, V is the whole volume )

β the resonance integral between atoms i, j ( is assumed to be equal for all pairs of atoms ) ( β <0)

The meaning of the approximation is that the two integrals are not calculated. They can be estimated, for example, on the basis of empirical data. For two atoms of the same kind the Coulombintegrale be equated. Particularly simple one can therefore calculate pure hydrocarbons. There are only two constants α and β left. The eigenvectors are independent of its value.

Derivation of the Hückel rule

The frost - circle provides an easy way to estimate the stability of cyclic conjugated planar molecules. It is based on the surprising fact that the energy level scheme of a cyclic conjugated planar n- atomic molecule can be represented as a regular n-gon.

The corresponding to an n- ring nxn matrix is ​​formed by α writes in the main diagonal, left and right next to β and β in the left lower and right upper corner:

The eigenvalues ​​of this matrix yield to him. It now transmits energy along the y- axis that noted that β is negative and maintains an appropriate distance in the x- direction, one obtains a standing on the top of n-gon.

• Examples of aromatic compounds

Cyclopentadienanion

Cycloheptatrienkation

This n-gon can now be set with an odd number of pairs of electrons (4n 2 π - electrons) with high energy gain. These molecules are referred to as " aromatics ." When an even number of pairs of electrons (4n π -electron ), the two upper levels would be half full, the molecule is paramagnetic and the total energy gain low. These compounds are unstable and are called " antiaromatics ".

• Examples of anti -aromatic compounds

Cyclopentadiene cation

Cycloheptatrienanion

Example

Benzene

The Hückelmatrix is:

The eigenvalues ​​found to be:

( α 2 β, α β, α β, α - β, α - β, α - 2 β )

The eigenvectors of the respective eigenvalues ​​:

( (1, 1, 1, 1, 1, 1 ), (1, 0, -1, -1, 0, 1), (-1, -1, 0, 1, 1, 0), (-1, 1, 0, -1, 1, 0), (-1, 0, 1, -1, 0, 1), (-1, 1, -1, 1, -1, 1))

A section (parallel to the xy plane at 75 PM removal) with the corresponding orbitals is drawn on the right. The coordinates of the vectors are each the sign of pz - wave functions. These are started from the right by counting counterclockwise.

According to one set at a point 6 -gon, there are 3 stabilized and destabilized three orbitals. 6 electrons can occupy the lower three orbitals ie at a high energy gain. The total binding energy is 2.2 β 4 β = β 8. This value is much higher than 6 for three isolated β π bonds.

Swell

• Peter W. Atkins: Physical Chemistry. Weinheim [ inter alia ]: VCH- Verl -Ges.
• Dietmar Dorninger: Basic Mathematics for Chemists. Eisenstadt: Prugg

• Theoretical Chemistry