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Application to Simple Molecules

  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 LCAOgif 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**
tex2html_wrap_inline2704 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
tex2html_wrap_inline2744 H 0.007 0.49 0.40 0.16 0.25
tex2html_wrap_inline2746 C 0.009 -0.74 -0.54 -0.20 -0.27
H 0.36 0.25 0.10 0.15
tex2html_wrap_inline2748 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
Table: Gross atomic charges calculated by Mulliken and Löwdin analysis of PW calculations, and Mulliken analysis of LCAO calculations performed with GAUSSIAN94.


The results of these calculations are presented in Table gif. 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 gif. It is also generally recognised that the molecular dipole is not simply related to the sum of the products tex2html_wrap_inline2750 as can be seen from the analysis of the tex2html_wrap_inline2744 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
tex2html_wrap_inline2744 O-H 0.54 0.05 0.23 0.31
H-H -0.01 0.00 -0.04 -0.04
Table: Mulliken and Löwdin atomic overlap populations.


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, tex2html_wrap_inline2744 and SiO are given in Table gif. 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 tex2html_wrap_inline2746 are shown in Figure gif. Again, a sensitivity to the basis set is demonstrated by the results in Table gif, 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.

next up previous contents
Next: Practical Example Up: Population Analysis Previous: Methods

Matthew Segall
Wed Sep 24 12:24:18 BST 1997