Hückel approximation and ESR spectra.

According to the McConnell relation, the hyperhine constants on hydrogens, a_H, in aromatic ion radicals are proportional to the unpaired p electron density, r_p, of the carbon that this hydrogen is bonded with:

The latter can be 'relatively' easy calculated using the Hückel approximation - the simplest realization of the molecular orbital (MO) method in application to solving Schrodinger Equation by variational method.

In the Hückel approximation only p_z orbitals are used to calculate energies of p molecular orbitals, while all s orbitals are taken for granted as holding the molecular skeleton and are ignored in calculations. Moreover, molecular orbitals are expressed as linear combinations of the atomic orbitals (LCAO), namely p_z - orbitals that are perpedicular to the molecular skeleton. Since s -bonding in planar molecules has different symmetry from the p -bonding, corresponding molecular orbitals, y_MO, are separate in the Hamiltonian - do not have cross off-diagonal elements and thus can be solved separately. Besides, s -bonding is much stronger than p -bonding: p - and p* molecular orbitals lie within the s - s* gap. Thus HOMO and LUMO orbitals are either of p, p* -type or nonbonding, n-type. All that allows considering only p_z -orbitals, p_i, in conjugated molecules for treating their spectroscopic features, while s and p_x, and p_y orbitals will be responsible primarily for s - bonding, i.e. molecule’s shape.

Thus, we are solving the Schrodinger equation: H|y_MO> = E|y_MO> , where molecular orbital, |y_MO> = S_i c_i|p_i>, is sought as a linear combination of atomic (p_z) orbitals, |p_i>, and we are trying to find the coefficients c_i that minimize the energy of the system. In a matrix form, this results in a secular equation for the energy:

Huckel approximation, though quite crude, provides very useful results. It can be summarized in following statements:

• all overlap integrals are zero: <p_i|p_j> = d_ij {0, when i = j, and 1, when i = j}
• diagonal elements equal atomic energies of p_z electrons: <p_i|H|p_i> = a
• <p_i|H|p_j> = b (neighbors) - the interaction energy is nonzero only for neighboring carbons. Both, a and b, are negative and can be parameterized for a typical conjugated carbon and carbon-carbon bond, respectively. Such a parameterization can be also introduced for hetero atoms, S, O, N, in different oxidation states. By convention, all these parameters are expressed through a of carbon and b of carbon-carbon bond. If we write the secular equation for benzene

Let's consider a benzene molecule as an example. We can solve the Schodinger equation for this molecule by considering only p-orbitals of six carbons under the Huckel approximation. Thus the secular equation will be obtained from setting the secular determinant on the right to zero. Each carbon has two neigbors, which makes the matrix appear with only three nonzero elements on each row. The secular equaiton is of a sixth order:

and can be solved straight forward. Its solutions in the order of increasing energy (remeber that both a, b< 0) include: E = a + 2b; E = a + b; E = a - b; E = a - 2b. It helps a great deal in constrcuting the orbitals and solving the secular equation if one uses symmetry restricted functions/orbitals. But even without that, the eigen functions can be obtained as shown (not normalized):

The graph on the left shows the amplitudes of these eigenfuctions and their symmetries.

For an arbitrary cyclic molecule with identical conjugated bonds, the solutions can be written as E = a + 2bcos(2pk/n) and graphically represented as the energies using horizontally leveled lines placed at the corners of the appropriate polygon pointing down by one of its corners.

When considering a neutral molecule, like benzene, all six electrons from six p_z carbons should be placed in accordance with the Pauli exclusion principle (see graph to left). For its anion or cation, 7 and 5 electrons should be distributed, correspondingly.

Similarly, the problem with semiquinone radical ions can be resolved. Now the coulombic integrals for oxygens, a_O, are different from those of carbon, and are, according to the table: a_O = a + b. But the off-diagonal elements, b_CO, are the same as for C-C bonds: b_CO = b.

• Enumerate all carbons and oxygens, and write the secular equationobtain
• Identify the orbital where unpaired electron should be placed, y_LUMO = S_i c_i|p_i>, and calculate the electron densities, r_i = c_i², from this orbital on each carbon
• Use Eq. 1 to calculate a_H.

Last updated 04/18/05.