The strengths of the SHM lie in the qualitative insights it gives into the effect
of molecular structure onporbitals. Its main triumph in this regard was its
spectacularly successful prediction of the requirements for aromaticity (the H€uckel
4 nþ2 rule).
The weaknesses of the SHM arise from the fact that it treats onlypelectrons
(limiting its applicability largely to planarsp^2 arrays), its all-or-nothing treatment
of overlap integrals, the use of just two values for the Fock integrals, and its neglect
of electron spin and interelectronic repulsion. Because of these approximations it is
not used for geometry optimizations and its quantitative predictions are sometimes
viewed with suspicion. For obtaining eigenvectors and eigenvalues from the secular
equations an older and inelegant alternative to matrix diagonalization is the use of
determinants.
The extended H€uckel method (EHM, extended H€uckel theory, EHT) follows
from the SHM by using a basis set that consists not just ofporbitals, but rather of all
the valence AOs (a minimal valence basis set), by calculating (albeit very empiri-
cally) the Fock matrix integrals, and by explicitly calculating the overlap matrixS
(whose elements are also used in calculating the Fock integrals). BecauseSis not
taken as a unit matrix, the equationHC¼SC«must be transformed to one without
Sbefore matrix diagonalization can be applied. This is done by a matrix multipli-
cation process called orthogonalization, involvingS"1/2, which converts the origi-
nal Fock matrixH, based on nonorthogonal atom-centered basis functions, into a
Fock matrixH^0 , based on orthogonal linear combinations of the original basis
functions. With these new basis functions,H^0 C^0 ¼C^0 «, i.eH^0 ¼C^0 «C^0 "^1 , so
that diagonalization ofH^0 gives the eigenvectors (of the new basis functions, which
are transformed back to those corresponding to the original set:C^0 !C) and
eigenvalues ofH.
Because the overlap integrals needed by the EHM depend on molecular geome-
try, the method can in principle be used for geometry optimization, although for the
conventional EHM the results are generally poor, so known geometries are used as
input. Applications of the EHM involve largely the study of big molecules and
polymeric systems, often containing heavy metals.
The strengths of the EHM derive from its simplicity: it is very fast and so can be
applied to large systems; the only empirical parameters needed are (valence-state)
ionization energies, which are available for a wide range of elements; the results of
calculations lend themselves to intuitive interpretation since they depend only on
geometry and ionization energies, and on occasion the proper treatment of overlap
integrals even gives better results than those from more elaborate semiempirical
methods. The fact that the EHM is conceptually simple yet incorporates several
features of more sophisticated methods enables it to serve as an excellent introduc-
tion to quantum mechanical computational methods.
The weaknesses of the EHM are due largely to its neglect of electron spin and
electron-electron repulsion and the fact that it bases the energy of a molecule simply
on the sum of the one-electron energies of the occupied orbitals, which ignores
electron–electron repulsion and internuclear repulsion; this is at least partly the
reason it usually gives poor geometries.
4.5 Summary 167