Biomedical Engineering Reference
In-Depth Information
and fluorine, where they give poor bond lengths and just about every other property you
can think of. Eventually it was realized that whilst most of the energy comes from the inner
shell regions, some flexibility ought to be given to the valence regions. The valence orbitals
are therefore split into (
n
−
1) primitives and one primitive so we represent a hydrogen
atom as two basis functions as shown in Table 16.3. We think of an inner (3 GTO) and an
outer (1 GTO) basis function.
Table 16.3
STO/4-31G hydrogen atom basis functions
Orbital exponent
Contraction coefficient
13.00773
0.0334960
1.962079
0.22472720
0.4445290
0.8137573
0.1219492
1
For other atoms, the inner shell basis functions are left in an STO/
n
G contraction. Again,
you might like to read the synopsis of the keynote paper by Ditchfield
et al.
(1971).
An extended basis set of atomic functions expressed as fixed linear combinations of Gaussian
functions is presented for hydrogen and the first row atoms carbon to fluorine. In this set,
described as 4-31G, each inner shell is represented by a single basis function taken as a sum
over four Gaussians and each valence orbital is split into inner and outer parts described by three
and oneGaussian function respectively. The expansion coefficients andGaussian exponents are
determined by minimizing the total calculated energy of the electronic ground state. This basis
set is then used in single-determinant molecular-orbital studies of a group of small polyatomic
molecules. Optimization of valence-shell scaling factors shows that considerable rescaling of
atomic functions occurs in molecules, the largest effects being observed for hydrogen and
carbon. However, the range of optimum scale factors for each atom is small enough to allow
the selection of a standard molecular set. The use of this standard basis gives theoretical
equilibrium geometries in reasonable agreement with experiment.
16.9.3 Gaussian Polarization and Diffuse Functions
I mentioned polarization functions briefly in Section 16.8. The best thing is for me to quote
the synopsis of a keynote paper by Collins
et al.
(1976) at this point.
Three basis sets (minimal s-p, extended s-p and minimal s-p with d functions on the second
row atoms) are used to calculate geometries and binding energies of 24 molecules containing
second row atoms. d functions are found to be essential in the description of both properties
for hypervalent molecules and to be important in the calculations of two-heavy-atom bond
lengths even for molecules of normal valence.
The addition of a single set of polarization functions to a heavy atomSTO/4-31G basis set
gives the so-called STO/4-31G*, and further addition of (p-type) polarization functions to
hydrogen gives STO/4-31G** in an obvious notation. There are more explicit conventions
when using more than one set of polarization functions per atom.
Polarization functions essentially allow spherical atomic charge distributions to dis-
tort on molecule formation or in the presence of external electric fields. In order to
treat species that carry formal negative charges or deal with molecular properties that
