This chapter has described the calculation of local atomic properties
from plane-wave electronic structure calculations. Section
described the theory underlying these calculations
and details of the implementation of this theory may be found in Appendix
. Section gave some brief
examples of the application of these techniques to simple molecules
and Section demonstrated how they may be used
in a system of practical interest. The application of population
analysis to simple bulk crystals was described in Section
and a relationship was found for overlap population and effective
valence charge with covalency and ionicity respectively. The utility
of overlap population as a measure of bond modulus was discussed in
Section . The relationship between these quantities was
found to be statistically significant. However, it was found that the
overlap population can only be used as an approximate guide to bond
modulus. Finally, Section showed that population analysis
could be used to investigate the distribution of spin density in a
spin-dependent calculation.
Section described how both Mulliken and Löwdin
populations may be calculated from the overlap and density
matrices. The calculation of these populations was demonstrated for
simple molecules. However, in the remaining analyses presented in this
chapter only the Mulliken populations have been calculated. Mulliken's
definitions of these quantities were chosen as these are used
more widely within the scientific community. Many other population
analysis schemes have been proposed, such as those due to Roby
[57] and Mayer [58, 59]. These analyses can
also be performed using the overlap and density matrices calculated by
the projection techniques described herein.
Provided a consistent choice of basis sets is made, population analysis offers an objective, reliable measure of differences in charge and spin distribution which can greatly aid in the detailed microscopic understanding of a system. Previously, analysis of plane wave calculations generally relied on subjective descriptions based on plots of charge density distributions. The overlap population offers a useful guide to bonding within a system. At the simplest level, it provides an quantitative criterion for the presence of a bond, in contrast to previous methods which relied on empirical measures such as interatomic distance or subjective examination of charge density distributions. However, the overlap population also provides additional information on the nature of the atomic interactions.
The work described in Section through
was undertaken in collaboration with Dr R. Shah and C.J. Pickard and
resulted in the publication of [60] and
[61]. The techniques developed have been used in the
analysis of calculations of practical interest, examples of which may
be found in [50], [62] and the remainder of
this thesis.