(Section5.3.3)anabinitiocalculationonethene,C 2 H 4 ,needsfivebasis
functions (1s,2s,2px,2py,2pz)foreachcarbonandonebasisfunction(1s)
for each hydrogen, a total of 14 basis functions, while a semiempirical cal-
culation needs four functions for eachcarbon and one for each hydrogen, for
atotalof12;forcholesterol,C 27 H 46 O, the numbers of basis functions are
186 and 158 for ab initio and semiempirical, respectively. For both molecules
the semiempirical calculation needs about 85% as many basis functions as the
ab initio calculation. The semiempirical basis set advantage is small compared
to a minimal basis set ab initio calculation, a kind not much used nowadays,
but it is large compared to ab initio calculations with split valence and split
valence plus polarization (Section5.3.3)basissets.Forethene,comparinga
6-31G* ab initio calculation with a minimal basis semiempirical calculation,
the numbers of basis functions are 38 and 12, for cholesterol, 522 and 158;
the semiempirical calculation needs only about 30% as many basis functions
for both molecules. Semiempirical calculations use only a minimal basis set
and hope to compensate for this by parameterization of the two-electron
integrals (below).
2.The basis set functions.In semiempirical methods the basis functions correspond
to atomic orbitals (valence AOs orp-pAOs), while in ab initio calculations this
is strictly true only for a minimal basis set, since an ab initio calculation can use
many more basis functions than there are conventional AOs. The SCF-type
semiempirical methods we are considering in this chapter use Slater functions,
rather than approximating Slater functions as sums of Gaussian functions
(Section5.3.2). Recall that the only reason ab initio calculations use Gaussian,
rather than the more accurate Slater, functions, is because calculation of the
electron–electron repulsion two-electron integrals is far faster with Gaussian
functions (Section5.3.2). In semiempirical calculations these integrals have
been parameterized into the calculation (see below). Mathematical forms of
the basis functionsfare still needed, to calculate overlap integralshfr|fsi,
for although these methods treat the overlap matrix as a unit matrix, some
overlap integrals are evaluated rather than simply being taken as zero or one.
Approximate MO theory has some apparent logical contradictions [ 7 ]. The
calculated overlap integrals are used to help calculate core integrals and
electron-repulsion integrals. As in ab initio calculations linear combinations of
the basis functions are used to construct MOs, which in turn are multiplied by
spin functions and used to represent the total molecular wavefunction as a Slater
determinant as in ab initio theory (Section5.2.3.1).
3.The integrals.The core integrals and the two-electron repulsion integrals
(electron-repulsion integrals), Eq.6.1¼5.82, are not calculated from first prin-
ciples (i.e. not from an explicit Hamiltonian and basis functions, as illustrated in
Section5.2.3.6.5), but rather many integrals are taken as zero, and those thatare
used are evaluated in an empirical way from the kinds of atoms involved and
their distances apart. Recall that calculation of the two-electron integrals, par-
ticularly the three- and four-center ones (those involving three or four different
atoms) takes up most of the time in an ab initio calculation. The integrals to be
6.2 The Basic Principles of SCF Semiempirical Methods 395