approaches like the PPP method (Section 6.2.2) are available, but mainly because of
the phenomenal success of all-valence-electron semiempirical methods (Chapter 6),
which are applicable to quite large molecules, and of the increasing power of
all-electron ab initio (Chapter 5) and DFT (Chapter 7) methods.
4.3.6.2 Weaknesses
The defects of the SHM arise from the fact that it treats onlypelectrons, and these
only very approximately. The basic H€uckel method described here has been
augmented in an attempt to handle non-psubstituents, e.g. alkyl groups, halogen
groups, etc., and heteroatoms instead of carbon. This has been done by treating the
substituents aspcenters and embodying empirically altered values ofaandb, so
that in the Fock matrix values other than"1 and 0 appear. However, the values of
these modified parameters that have been employed vary considerably [ 54 ], which
tends to diminish one’s confidence in their reliability.
The approximations in the SHM are its peremptory treatment of the overlap
integralsS(Section 4.3.4, discussion in connection with Eqs.4.55), its drastic
truncation of the possible values of the Fock matrix elements into justa,band
0(Section 4.3.4, discussion in connection with Eqs. 4.61), its complete neglect of
electron spin, and its glossing over (although not exactly ignoring) interelectronic
repulsion by incorporating this into theaandbparameters.
Theoverlap integrals Sare divided into just two classes:
Z
fifjdv¼Sij¼1 or 0
depending on whether the orbitals on the atomsiandjare on the same or
different atoms. This approximation, as explained earlier, reduces the matrix
form of the secular equations to standard eigenvalue formHC¼C«(Eq.4.59),
so that the Fock matrix can (after giving its elements numerical values) be
diagonalized without further ado (the ado is explained in Section 4.4.1, in
connection with the extended H€uckel method). In the older determinant, as
opposed to matrix, treatment (Section 4.3.7), the approximation greatly simplifies
the determinants. In fact, however, the overlap integral between adjacent carbon
porbitals is ca. 0.24 [ 55 ].
Setting the Fock matrix elements equal to justa,band 0: Setting
Z
fiH^fjdv¼Hij¼a;bor 0
depending on whether the orbitals on the atomsiandjare on the same, adjacent or
further-removed atoms is an approximation, because all theHiiterms are not the
same, and all the adjacent-atomHijterms are not the same either; these energies
depend on the environment of the atom in the molecule; for example, atoms in the
4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 145