Computational Chemistry

To initiate the process we need aninitial guessof the coefficients, to calculate the
density matrix valuesPtu. The guess can come from a simple H€uckel calculation (for
apelectron theory like the PPP method) or from an extended H€uckel calculation (for
an all-valence-electron theory, like CNDO and its descendants). The Fock matrix of
Frselements is diagonalized repeatedly to refine energy levels and coefficients.
The semiempirical methods we consider here diverge from ab initio calculations
through the use of several approximations. These are discussed in detail by Dewar
[ 7 ]. An excellent yet compact survey of the principles behind all the major
semiempirical methods is given by Levine [ 8 ], and semiempirical methods have
also been reviewed by Thiel [ 9 ]; a detailed exposition of the basic (pre-1970) theory
behind these methods can be found in the book by Pople and Beveridge [ 10 ]. Clark
has written a very thoughtful review of the “philosophy” of the semiempirical
approach, its strengths and weaknesses, its past and future [ 11 ]. The divergence
from the ab initio method lies in (1) treating only valence orpelectrons, i.e. in the
meaning of the “core”, (2) the mathematical functions used to expand the MOs (the
nature of the basis set functions), (3) how the core and two-electron repulsion
integrals are evaluated, and (4) the treatment of the overlap matrix.
Expanding on points (1)–(4):

1.Treating only valence orpelectrons, i.e. the meaning of the “core”.In an ab
initio calculationHrscoreis the kinetic energy of an electron moving in the force-
field of the atomic nuclei, plus the potential energy of attraction of the electron to
these atomic nuclei: the electron is moving under the influence of a positive core
composed of atomic nuclei. Semiempirical calculations handle at most valence
electrons (the PPP method handles onlypelectrons), so each element of the core
becomes an atomic nucleusplus its core electrons(for the PPP method, a
nucleus with the core electrons plus alls valence electrons). Instead of a
cloud of all the electrons moving in a framework of nuclei, we have a cloud of
valenceelectrons (for the PPP method,pelectrons) moving in a framework of
atomic cores (atomic core¼nucleusþnon-valence electrons, or for PPP,
nucleusþall electrons that don’t contribute to thepsystem). The SCF semiem-
pirical energy is calculated in a manner analogous to that of an ab initio
calculation of the Hartree-Fock energy (cf. Eq. 5.149), butnin Eq.6.2is not
half the total number of electrons, but rather half the number of valence electrons
(half the number ofpelectrons for a PPP calculation), i.e.nis the number of
MOs formed from the those electrons being included in the basis set.ESEis the
valence electronic (p electronic for the PPP method) energy, rather than
the total electronic energy, andVCCis the core–core repulsion, rather than the
nucleus–nucleus repulsion:

EtotalSE ¼ESEþVCC¼

Xn

i¼ 1

eiþ

1

2

Xm

r¼ 1

Xm

s¼ 1

PrsHcorers þVCC (6.2)

Treating the core electrons in effect as part of the atomic nuclei means that we

need basis functions only for the valence electrons. With a minimal basis set

394 6 Semiempirical Calculations