Biomedical Engineering Reference
In-Depth Information
1
0.8
0.6
0.4
0.2
-8
-6
-4
-2
0
2
4
6
Figure 16.4
Product of two one-dimensional Gaussians
Credit for the introduction of Gaussian basis functions is usually given to S.F. Boys for
his 1950 paper, and here is the famous synopsis (Boys 1950).
This communication deals with the general theory of obtaining numerical electronic wave-
functions for the stationary states of atoms and molecules. It is shown that by taking Gaussian
functions, and functions derived from these by differentiation with respect to the parameters,
complete systems of functions can be constructed appropriate to any molecular problem, and
that all the necessary integrals can be explicitly evaluated. These can be used in connection
with the molecular orbital treatment, or localized bond method, or the general method of linear
combinations of many Slater determinants by the variation procedure. This general method of
obtaining a sequence of solutions converging to the accurate solution is examined. It is shown
that the only obstacle to the evaluation of wavefunctions of any required degree of accuracy
is the labour of computation. A modification of the general method applicable to atoms is
discussed and considered to be extremely practicable
GTOs have one great advantage over STOs; the nasty integrals (especially the two-
electron integrals) we need for molecular quantummechanics are relatively straightforward
because they can be reduced from at most a four-centre integral to a one-centre integral
by repeated application of the principle above. Consider for example an overlap integral
between the two s-type GTOs shown in Figure 16.5. This is a three-dimensional extension
to the one-dimensional problem discussed above.
Gaussian A has exponent α
A
and is centred at
r
A
; Gaussian B has exponent α
B
and is
centred at
r
B
. If the position vector of the electron is
r
then its position vector relative to
the centre of
G
A
is
r
-
r
A
, with a similar expression for
G
B
. The overlap integral (apart from
the normalizing constants
N
and
N
)is
