Biomedical Engineering Reference
In-Depth Information
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
Distance/a0
Figure 16.6
GTO versus STO for hydrogen atom
GTOs show the wrong behaviour at the nucleus, where they should have a cusp because
the mutual potential energy of the electron and the nucleus becomes infinite as the distance
becomes zero. GTOs also fall off far too quickly with the distance from the nucleus.
16.9.1
STO/
n
G
The next step was to address the long-distance behaviour, and Hehre, Stewart and Pople
proposed the idea of fitting a fixed linear combination of
n
GTOs to a given STO. The GTOs
are not explicitly included in a HF-LCAO calculation, they are just used to give a good fit
to an STO for integral evaluation. The resulting HF-LCAO orbitals can be thought of as
minimal basis STOs. The GTOs are called
primitive
GTOs, and we say that the resulting
atomic (STO) orbital is
contracted
. So, for example, we would use least squares fitting
techniques to find the best three primitive GTO exponents α
i
and contraction coefficients
d
i
in the STO/3G fit to a 1s STO orbital with exponent 1:
STO
(
ζ
=
1
)
=
d
1
GTO
(
α
1
)
+
d
2
GTO
(
α
2
)
+
d
3
GTO
(
α
3
)
(16.29)
The next keynote paper is therefore that of Hehre
et al.
(1969). As usual I will let the
authors explain their ideas.
Least Squares representations of Slater-type atomic orbitals as a sum of Gaussian-type orbitals
are presented. These have the special feature that common Gaussian exponents are shared
between Slater-type 2s and 2p functions. Use of these atomic orbitals in self-consistent
molecular-orbital calculations is shown to lead to values of atomization energies, atomic pop-
ulations, and electric dipole moments which converge rapidly (with increasing size of the
Gaussian expansion) to the values appropriate for pure Slater-type orbitals. The ζ exponents
(or scale factors) for the atomic orbitals which are optimized for a number of molecules are
