Essentials of Computational Chemistry

5.2 EXTENDED HÜCKEL THEORY 135
The only terms remaining to be defined in Eq. (5.1), then, are the resonance integrals
H . For diagonal elements, the same convention is used in EHT as was used for simple
Hückel theory. That is, the value for Hµµ is taken as the negative of the average ionization
potential for an electron in the appropriate valence orbital. Thus, for instance, when
µ is a hydrogen 1s function, Hµµ =−13.6 eV. Of course in many-electron atoms, the
valence-shell ionization potential (VSIP) for the ground-state atomic term may not necessarily
be the best choice for the atom in a molecule, so this term is best regarded as an
adjustable parameter, although one with a clear, physical basis. VSIPs have been tabulated
for most of the atoms in the periodic table (Pilcher and Skinner 1962; Hinze and Jaffé 1962;
Hoffmann 1963; Cusachs, Reynolds and Barnard 1966). Because atoms in molecular environments
may develop fairly large partial charges depending on the nature of the atoms to
which they are connected, schemes for adjusting the neutral atomic VSIP as a function of
partial atomic charge have been proposed (Rein et al. 1966; Zerner and Gouterman 1966).
Such an adjustment scheme characterizes so-called Fenske–Hall effective Hamiltonian calculations,
which still find considerable use for inorganic and organometallic systems composed
of atoms having widely different electronegativities (Hall and Fenske 1972).
The more difficult resonance integrals to approximate are the off-diagonal ones. Wolfsberg
and Helmholtz (1952) suggested the following convention
Hµν = 1
2 Cµν(Hµµ + Hνν)Sµν
(5.3)
where C is an empirical constant and S is the overlap integral. Thus, the energy associated
with the matrix element is proportional to the average of the VSIPs for the two orbitals µ
and ν times the extent to which the two orbitals overlap in space (note that, by symmetry,
the overlap between different STOs on the same atom is zero). Originally, the constant C
was given a different value for matrix elements corresponding to σ -andπ-type bonding
interactions. In modern EHT calculations, it is typically taken as 1.75 for all matrix elements,
although it can still be viewed as an adjustable parameter when such adjustment is warranted.
All of the above conventions together permit the complete construction of the secular
determinant. Using standard linear algebra methods, the MO energies and wave functions
can be found from solution of the secular equation. Because the matrix elements do not
depend on the final MOs in any way (unlike HF theory), the process is not iterative, so
it is very fast, even for very large molecules (however, the process does become iterative
if VSIPs are adjusted as a function of partial atomic charge as described above, since the
partial atomic charge depends on the occupied orbitals, as described in Chapter 9).
The very approximate nature of the resonance integrals in EHT makes it insufficiently
accurate for the generation of PESs since the locations of stationary points are in general
very poorly predicted. Use of EHT is thus best restricted to systems for which experimental
geometries are available. For such cases, EHT tends to be used today to generate qualitatively
correct MOs, in much the same fashion as it was used by Wolfsberg and Helmholz
50 years ago. Wolfsberg and Helmholz used their model to explain differences in the UV
spectroscopies of MnO4 − ,CrO4 2− ,andClO4 − by showing how the different VSIPs of the
central atom and differing bond lengths gave rise to different energy separations between