Biomedical Engineering Reference
In-Depth Information
depend on regions of the valence shells that are far from the nucleus, it is necessary
to include
diffuse
basis functions (primitives with very small exponents), and they are
denoted
+
++
and
.
16.9.4 Extended Basis Sets
Huzinaga's set of uncontracted GTOs provides the classic example for our next topic. These
large sets of primitive (uncontracted) GTOs comprise ten primitive s-type and six primitive
p-type, for a first-row atom. The orbital exponents were carefully optimized for every first-
row atom and for hydrogen. There is no reason in principle why we should not use them
as they stand for molecular calculations but the contraction process is found to give great
computational efficiency with little cost in energy. What is needed is a way of effectively
grouping them together for a molecular calculation, i.e. a contraction scheme.
Many authors performed HF-LCAO calculations on atoms and small molecules, and
looked for groupings of the primitive GTOs that did not change their relative weight-
ings from orbital to orbital and from molecule to molecule. I can illustrate these ideas by
mentioning Dunning's (1975) work, with the by-now inevitable synopsis.
Contracted [5s3p] and [5s4p] Gaussian basis sets for the first-row atoms are derived from
the (10s6p) primitive basis sets of Huzinaga. Contracted [2s] and [3s] sets for the hydrogen
atom obtained from primitive sets ranging from (4s) to (6s) are also examined. Calculations on
the water and nitrogen molecules indicate that such basis sets when augmented with suitable
polarization functions should yield wavefunctions near the Hartree-Fock limit.
Dunning's notation and ideas can be explained with the example in Table 16.4, for the
oxygen atom.
Table 16.4
Dunning's [5s3p] contraction scheme for Huzinaga's
(10s6p) GTO set
GTO type
Exponent
Contraction
coefficient
s
1805.0
0.000757
2660.0
0.006066
585.7
0.032782
160.9
0.132609
51.16
0.396839
17.90
0.542572
s
17.90
0.262490
6.639
0.769828
s
2.077
1
s
0.7736
1
s
0.2558
1
p
49.83
0.016358
11.49
0.106453
3.609
0.349302
1.321
0.657183
p
0.4821
1
p
0.1651
1
