Essentials of Computational Chemistry

134 5 SEMIEMPIRICAL IMPLEMENTATIONS OF MO THEORY
5.2 Extended Hückel Theory
Prior to considering semiempirical methods designed on the basis of HF theory, it is instructive
to revisit one-electron effective Hamiltonian methods like the Hückel model described
in Section 4.4. Such models tend to involve the most drastic approximations, but as a result
their rationale is tied closely to experimental concepts and they tend to be intuitive. One
such model that continues to see extensive use today is the so-called extended Hückel
theory (EHT). Recall that the key step in finding the MOs for an effective Hamiltonian is
the formation of the secular determinant for the secular equation
H11
− ES11 H12 − ES12 ... H1N − ES1N
H21
− ES21 H22 − ES22 ... H2N − ES2N
. = 0 (5.1)
.
. .. .
HN1 − ESN1 HN2 − ESN2 ... HNN − ESNN
The dimension of the secular determinant for a given molecule depends on the choice
of basis set. EHT adopts two critical conventions. First, all core electrons are ignored. It
is assumed that core electrons are sufficiently invariant to differing chemical environments
that changes in their orbitals as a function of environment are of no chemical consequence,
energetic or otherwise. All modern semiempirical methodologies make this approximation. In
EHT calculations, if an atom has occupied d orbitals, typically the highest occupied level of
d orbitals is considered to contribute to the set of valence orbitals.
Each remaining valence orbital is represented by a so-called Slater-type orbital (STO). The
mathematical form of a normalized STO used in EHT (in atom-centered polar coordinates) is
ϕ(r,θ,φ; ζ,n,l,m) = (2ζ)n+1/2
[(2n)!] 1/2 rn−1e −ζr Y m
l
(θ, φ) (5.2)
where ζ is an exponent that can be chosen according to a simple set of rules developed by
Slater that depend, inter alia, on the atomic number (Slater 1930), n is the principal quantum
number for the valence orbital, and the spherical harmonic functions Y m
l (θ,φ), depending
on the angular momentum quantum numbers l and m, are those familiar from solution of
the Schrödinger equation for the hydrogen atom and can be found in any standard quantum
mechanics text. Thus, the size of the secular determinant in Eq. (5.2) is dictated by the total
number of valence orbitals in the molecule. For instance, the basis set for the MnO4 − anion
would include a total of 25 STO basis functions: one 2s and three 2p functions for each
oxygen (for a subtotal of 16) and one 4s, three 4p, and five 3d functions for manganese.
STOs have a number of features that make them attractive. The orbital has the correct
exponential decay with increasing r, the angular component is hydrogenic, and the 1s orbital
has, as it should, a cusp at the nucleus (i.e., it is not smooth). More importantly, from a
practical point of view, overlap integrals between two STOs as a function of interatomic
distance are readily computed (Mulliken Rieke and Orloff 1949; Bishop 1966). Thus, in
contrast to simple Hückel theory, overlap matrix elements in EHT are not assumed to be
equal to the Kronecker delta, but are directly computed in every instance.