Computational Chemistry

necessarily conventional atomic orbitals: they can be any set of mathematical
functions that are convenient to manipulate and which in linear combination give
useful representations of MO’s. With this reservation, LCAO is a useful acronym.
Physically, several (usually) basis functions describe the electron distribution
around an atom and combining atomic basis functions yields the electron distribu-
tion in the molecule as a whole. Basis functions not centered on atoms (occasionally
used) can be considered to lie on “ghost atoms”; see basis set superposition error,
Section 5.4.3.3.
The simplest basis sets are those used in the simple H€uckel and the extended
H€uckel methods (SHM and EHM,Chapter 4). As applied to conjugated organic
compounds (its usual domain), the simple H€uckel basis set consists of justpatomic
orbitals (or “geometricallyp-type” atomic orbitals, like a lone-pair orbital which
can be considered not to interact with thesframework). The extended H€uckel basis
set consists of only the atomicvalenceorbitals. In the SHM we don’t worry about
the mathematical form of the basis functions, reducing the interactions between
them to 0 or#1 in the SHM Fock matrix (e.g. Eqs. 4.64 and 4.65). In the EHM
the valence atomic orbitals are represented as Slater functions (Sections 4.4.1.2
and 4.4.2).

5.3.2 Gaussian Functions; Basis Set Preliminaries; Direct SCF.......

The electron distribution around an atom can be represented in several ways.
Hydrogenlike functions based on solutions of the Schr€odinger equation for the
hydrogen atom, polynomial functions with adjustable parameters, Slater functions
(Eq.5.95), and Gaussian functions (Eq.5.96) have all been used [ 34 ]. Of these,
Slater and Gaussian functions are mathematically the simplest, and it is these that
are currently used as the basis functions in molecular calculations. Slater functions
are used in semiempirical calculations, like the extended H€uckel method (Sec-
tion 4.4) and other semiempirical methods (Chapter 6). Modern molecular ab initio
programs employ Gaussian functions.
Slater functions are good approximations to atomic wavefunctions and would be
the natural choice for ab initio basis functions, were it not for the fact that the
evaluation of certain two-electron integrals requires excessive computer time if
Slater functions are used. The two-electron integrals (Sections 5.2.3.6.3, 5.2.3.6.5
of theGmatrix (Eq.5.104) involve four functions, which may be on from one to
four centers (normally atomic nuclei). Those two-electron integrals with three or
four different functions ((rs|tt), (rs|rt) and (rs|tu)) and three or four nuclei (three-
center or four-center integrals) are extremely difficult to calculate with Slater
functions, but are readily evaluated with Gaussian basis functions. The reason is
that the product of two Gaussians on two centers is a Gaussian on a third center.
Consider ans-type Gaussian centered on nucleusAand one on nucleusB; we are
considering real functions, which is what basis functions normally are:

gA¼aAe#aAjr#rAj

2

; gB¼aBe#aBjr#rBj

2

ð 5 : 153 Þ

5.3 Basis Sets 233