The techniques described in the previous section were used to perform
calculations of Mulliken and Löwdin atomic charges in four simple
molecules. These were compared with Mulliken charges calculated from
DFT calculations using LCAO
basis sets with GAUSSIAN94 [49].
The basis set used for projection of the PW states in each case was that of the atomic pseudo-orbitals corresponding to the shell occupied by the valence electrons. The LCAO calculations were performed using the STO-3G and 6-311G** basis sets provided with GAUSSIAN94.
| Molecule | Atom | Plane Wave | STO-3G | 6-311G** | ||
 | Mulliken | Löwdin | Mulliken | Mulliken | ||
| Charge (|e|) | Charge (|e|) | Charge (|e|) | Charge (|e|) | |||
| CO | C | 0.016 | 0.42 | 0.18 | 0.08 | 0.04 |
| SiO | Si | 0.009 | 0.71 | 0.28 | 0.32 | 0.46 |
 | H | 0.007 | 0.49 | 0.40 | 0.16 | 0.25 |
 | C | 0.009 | -0.74 | -0.54 | -0.20 | -0.27 |
| H | 0.36 | 0.25 | 0.10 | 0.15 | ||
 | 0.35 | 0.27 | 0.08 | 0.12 | ||
| O | -0.78 | -0.62 | -0.24 | -0.38 | ||
| H | 0.47 | 0.39 | 0.18 | 0.24 | ||
The results of these calculations are presented in Table
. It should be noted that the absolute values of
these charges are thought to have little physical meaning as they
display an extreme sensitivity to the atomic basis set with which they
are calculated [45]. This is demonstrated by the results
in Table . It is also generally recognised that
the molecular dipole is not simply related to the sum of the products
as can be seen from the analysis of the molecule, for which the dipole naıvely calculated from the
Mulliken charges is 2.83 D, in poor agreement with the experimental
value of 1.87 D. However, the dipole calculated for this molecule from
the full charge density obtained in the plane wave calculation, 1.86
D, is much closer to the experimental value. In contrast to the
absolute values, redistribution of Mulliken charges in response to
system changes should have more significance.
| Molecule | Overlap | Plane Wave | STO-3G | 6-311G** | |
| Mulliken | Löwdin | Mulliken | Mulliken | ||
| Population (|e|) | Population (|e|) | Population (|e|) | Population (|e|) | ||
| CO | C-O | 0.91 | 0.77 | 0.46 | 0.52 |
| SiO | Si-O | 0.97 | 1.04 | 0.32 | 0.50 |
|
| O-H | 0.54 | 0.05 | 0.23 | 0.31 |
| H-H | -0.01 | 0.00 | -0.04 | -0.04 | |
Figure: The structure of methanol
showing the overlap populations in electronic units between bonded
atoms as calculated by Mulliken analysis of a PW calculation.
The overlap populations between the atoms in CO, and SiO
are given in Table . A large positive value for this
population indicates that the atoms in question are bonded, a large
negative value indicates the atoms are in an antibonded state. The
Mulliken overlap populations for bonded atoms in are
shown in Figure . Again, a sensitivity to the basis set
is demonstrated by the results in Table , e.g. the
Mulliken overlap population for Si-O is markedly smaller than that
for C-O using an STO-3G basis, but not in the case of the 6-311G**
basis. In the PW calculation the overlap population is actually
slightly larger for Si-O. It is notable that Löwdin analysis of
the O-H overlap population gives a significantly smaller result than
expected, although this is still much larger than the
overlap of unbonded atoms.