As a graduate student, I was once asked as part of an oral examination to explain the semiempirical Hu¨ckel method. The examiner, a professor of theoretical physics, was not convinced by my attempts to justify the various approximations that were required for its derivation. I finally said: ‘‘I may not be able to justify the Hu¨ckel method, but it works extremely well and it has been successful in interpreting and predicting many complex and sophisticated phenomena in organic chemistry.’’ In retrospect, this does not seen a bad description of Hu¨ckel theory. Incidentally, I passed the examination.
During the 1930s, the German physicist Erich Hu¨ckel proposed a semiempirical theory to describe the electronic structure of aromatic and conjugated organic mole- cules. The latter are reactive compounds that had become important in a number of practical applications. For example, some aromatic compounds were starting pro- ducts in the early industrial production of dyestuffs. Chemists proposed that some of the characteristic properties of aromatic and conjugated molecules could be attributed to the presence of delocalized electron orbitals. Such orbitals were not confined to a single chemical bond but could extend over a number of bonds or even the entire molecule.
Hu¨ckel’s theory offered a theoretical description of these delocalized orbitals based on a number of rather drastic approximations. It is best illustrated for the ben- zene molecule C6H6, which is both the smallest aromatic and the prototype of the aromatics. The benzene molecule is planar, all 12 atoms are located in the XY plane, and its geometry is that of a regular hexagon (see Figure 12-9) It is important to note that all carbon-carbon bonds are equivalent; they all have the same length and energy. The six carbon-hydrogen bonds are also equivalent.
THE HU¨ CKEL MOLECULAR ORBITAL METHOD 183
Thes bonding orbitals in the plane of the molecules may all be expressed in terms of sp2 hybridized orbitals that are linear combinations of the carbon (2s), (2px), and (2py) orbitals and of the hydrogen (1s) orbitals. This bonding skeleton involves three of the four valence electrons of each carbon atom, which leaves six electrons unaccounted for. At the same time, there are six carbon (2pz) or p orbitals available. It therefore seems logical to assume that these six electrons may be distributed over the six availableporbitals and that their molecular orbitals fmay be represented as linear combination of the atomic orbitalspi:
fẳX
i
aipi ð12-80ị
The specific form of the expansion coefficientsaimay be derived by means of the variational principle described in Sections 9.II and 9.III and by means of Eq. (9-18) in particular. In our present situation, we have a finite basis set so that we may write the variational equations as
XN
kẳ1
ðHjkeSjkịakẳ0 jẳ1;2;. . .N ð12-81ị
The matrix elementsHjk andSjkare defined as
Hjkẳ hpjjHjpri Sjkẳ hpjjpki ð12-82ị HereHrepresents the effective Hamiltonian acting on an electron in a delocalizedp orbital. It is the sum of the kinetic energy and the electrostatic interactions between a particularp electron and the nuclei, the selectrons, and the other p electrons.
C
C C
C C C H
H
H
H
H H
Figure 12-9 Geometry of the benzene molecule.
It will appear that its particular form does not really matter because of the semi- empirical nature of the theory.
Hu¨ckel proposed that the matrix elementsHij andSij may be approximated as semiempirical parameters as follows:
Hjkẳa if jẳk
Hjkẳb if j andkare separated by one bond
Hjkẳ0 if jandkare separated by more than one bond Sjkẳ1 if jẳk
Sjkẳ0 if j6ẳk
ð12-83ị
The Hu¨ckel equations may then be obtained by substituting the approximate equa- tions (12-83) into the variational equation (12-81). In general, these areN homo- geneous linear equations with N unknowns. These equations were discussed in Section 2.VIII, where we showed that they have nonzero solutions only if the deter- minant of the coefficients is zero. The standard procedure for solving the Hu¨ckel equations consists of first evaluating the values of the parametereof Eq. (12-81) in order to determine the energy eigenvalues and then solving the linear equations in order to determine the corresponding eigenvectors and molecular orbitals.
In a few specific situations, the Hu¨ckel equations may be solved by a much sim- pler procedure in which there is no need to evaluate the determinant. The first case is a ring system containingN carbon atoms, and the second case is a conjugated hydrocarbon chain ofN atoms with alternate single and double bonds.
The Hu¨ckel equations of the ring system and of the chain are similar, but there are minor differences. The equations for the ring are
baNỵ ðaeịa1ỵba2ẳ0
bak1ỵ ðaeịakỵbakỵ1ẳ0 kẳ2;3;. . .;N1 baN1ỵ ðaeịaNỵba1ẳ0
ð12-84ị
and those for the chain are
ðaeịa1ỵba2ẳ0
bak1ỵ ðaaeịakỵbakỵ1ẳ0 kẳ2;3;. . .;N1 baN1ỵ ðaeịaN ẳ0
ð12-85ị
The set of equations (12-84) are solved by substituting
akẳeikl ð12-86ị
The middle equations are then
beilỵ ðaeị ỵbeilẳ0 ð12-87ị
THE HU¨ CKEL MOLECULAR ORBITAL METHOD 185
and the expression (12-87) is a solution if
eẳaỵbðeilỵeilị ẳaỵ2b cosl ð12-88ị Substitution of Eqs. (12-86) and (12-88) into the first and last equation (12-84) gives
eiNl1ẳ0 ð12-89ị
and these equations are solved for
lẳ2pn
N nẳ0;1;2; etc: ð12-90ị The eigenvalues of Eq. (12-84) are therefore given as
enẳaỵ2bcosð2pn=Nị nẳ0;1;2; etc: ð12-91ị In order to solve the Hu¨ckel equations (12-85) for the chain system we must sub- stitute
akẳAeiklỵBeikl ð12-92ị This provides a solution of all equations except the first and the last one if we take en ẳaỵ2bcosl ð12-93ị We again substitute the solutions (12-92) and (12-93) into the first and last Eq. (12-85) and we obtain
AỵBẳ0
AeðNỵ1ịilỵBeðNỵ1ịilẳ0 ð12-94ị or
Bẳ A
sinẵðNỵ1ịl ẳ0 ð12-95ị
The eigenvalues are now given by lẳ np
Nỵ1 nẳ1;2;3;. . .;N ð12-96ị
or
enẳaỵ2b cos np
Nỵ1 ð12-97ị
The corresponding eigenvectors are
akẳsin nkp
Nỵ1 ð12-98ị
It may be instructive to present a few specific examples of the above results. The benzene molecule is an aromatic ring systems of six carbon atoms, and its eigen- values are described by Eq. (12-90) by substitutingNẳ6. The results are
e0ẳaỵ2bcos 0ẳaỵ2b e1ẳe1ẳaỵ2bcos 60ẳaỵb e2ẳe2ẳaỵ2bcos 120ẳab e3ẳaỵ2bcos 180ẳa2b
ð12-99ị
We have sketched the energy level diagram in Figure 12-10. It should be realized thatbis negative and thate0is the lowest energy eigenvalue. The next eigenvalue, e1, is twofold degenerate. The molecular ground state has a pair if electrons in the eigenstatese0,e1, and e1, and its energy is therefore
Eðbenzeneị ẳ2ðaỵ2bị ỵ4ðaỵbị ẳ6aỵ8b ð12-100ị
– –
– –
– –
α–2β
α–β
α+β
α+2β Figure 12-10 Energy level diagram of the benzene molecule.
THE HU¨ CKEL MOLECULAR ORBITAL METHOD 187
It is easily shown that a localizedporbital has an energyaþb, so that one of the two benzene structures of Figure 12-9 with fixedpbonds has an energy
EIẳ6aỵ6b ð12-101ị
It follows that the energy of the delocalized bond model is lower by an amount 2b than the energy of the structure with localizedp bonds. This energy difference is called the resonance energy of benzene.
In early theoretical work on the benzene structure it was assumed that the mole- cule ‘‘resonated’’ between the two structures I and II of Figure 12-11 and that this resonance effect resulted in lowering the energy by an amount that was defined as the resonance energy. However, the molecular orbital model that we have used is better suited for numerical predictions of the resonance energies of aromatic molecules than the corresponding VB model.
As a second example, we calculate the energy eigenvalues of the hexatriene molecule. We have sketched its structure containing localized bonds in Figure 12-12;
it has six carbon atoms, and the energy of the three localized p bonds is again 6aþ6b. The three lowest energy eigenvalues, according to the Hu¨ckel theory, may be derived from Eq. (12-97); they are
e1ẳaỵ2bcos 25:71ẳaỵ1:8019b e2ẳaỵ2bcos 51:71ẳaỵ1:2470b e3ẳaỵ2bcos 77:14ẳaỵ0:4450b
ð12-102ị
The total molecular energy, according to the Hu¨ckel theory, is therefore 6aþ 6:9879band the resonance energy is 0.9879b.
It turned out that the Hu¨ckel MO theory could be successfully applied to the pre- diction of molecular geometries, electronic charge densities, chemical reactivities,
II I
Figure 12-11 Resonance structures of the benzene molecule.
C H
C H
C H
C H
CH2 H2C
Figure 12-12 Structure of the 1,3,5 hexatriene molecule.