We have analysed electronic structure calculations of several simple bulk
crystals using the techniques described in
refs Sanchez-Portal95,Segall96. In each case the LCAO
basis set used was the atomic pseudo-orbitals corresponding to the
shell containing the valence electrons. The
spilling parameter and atomic charges resulting from these
calculations are presented in Table i. It was found that
the spilling parameters for these systems were very low, indicating a
good representation of the electronic bands using the LCAO basis set.
A spilling parameter in the region of 
indicates that only
approximately 0.1% of the valence charge has been missed in the
projection. As an example of the sensitivity to basis set, the
ommision of the Si d-orbitals from the LCAO basis set used in the
analysis of SiC gives rise to a charge transfer of 1.25 rather than
0.66. The spilling parameter when the Si d-orbitals are absent is only
, indicating that this change is not due to an
underrepresentation of the electronic bands. The discrepancy in the
Mulliken charges is explained by the change in the number of basis
states associated with the Si atoms used in the representation of the
charge distribution. Table i also lists the effective
ionic valences for each of the crystals. This is defined to be the
difference between the formal ionic charge and the Mulliken charge on
the anion species in the crystal. This is also used as a measure of
ionicity; a value of zero implies an ideal ionic bond while values
greater than zero indicate increasing levels of covalency.
Table i: Spilling parameters, atomic Mulliken charges and Valence
charges calculated from PW electronic structure calculations.
Table ii shows the overlap populations for nearest neighbours in the crystal. Positive and negative values indicate bonding and anti-bonding states respectively. A value for the overlap population close to zero indicates that there is no significant interaction between the electronic populations of the two atoms. For example, in GaAs the overlap population between next-nearest neighbours was found to be -0.11 while in NaCl this population is -0.03. This indicates that the anti-bonding interaction between atoms in the second coordination shell is stronger in GaAs than in NaCl. A high overlap indicates a high degree of covalency in the bond. Also shown in Table ii is the difference in Mulliken and Pauling electronegativities of the species in each crystal. The Mulliken electronegativity of a species is defined as
where A is the electron affinity of an atom of the species and I
is the ionization energy of the atom. The Pauling electronegativity,
, is defined empirically from the bond energies of diatomic
molecules containing the species.[14] The difference in
electronegativities between two species is used as a guide to the
ionicity of the interaction between two such atoms, a high value
indicating high ionicity. Pauling suggests that the degree of ionicity
is given by 
, where a is a constant. It is
notable that using this method the two electronegativity scales
disagree even on the ordering of the ionicity of the crystals studied.
Table ii: Mulliken overlap populations calculated from PW
pseudopotential calculations, Mulliken and Pauling electronegativity
differences and bulk moduli.
Our calculations provide us with overlap population and effective valence charge as measures of ionicity. These may be compared with those derived from electronegativities. Figures 1 and 2 show graphs of the overlap populations against the Mulliken and Pauling electronegativity differences. Figure 1 indicates that there is a correlation between the overlap population of nearest neighbours and the covalency of the bonds within the crystal as measured by the Mulliken electronegativity. Also shown in Figure 1 is a fit of the data to a function of the form
where a, b and c are constants. The standard error in this fit
is 0.08.This demonstrates that our measure of covalency in terms of
overlap population is proportional to that of Pauling. However, we
find a constant offset indicating that a completely ionic bond is not
possible within our definition. The agreement between the overlap
populations and Pauling electronegativities as shown in Figure
2 is not as good. This may be due to the fact that the
Pauling electronegativity scale is derived from the energetics of
diatomic molecules and therefore may not be suitable for application
to bulk materials. A graph of the effective valence charge against
the difference in Mulliken electronegativities, Figure 3,
again shows a correlation between these values.
The notable exception is 
which has a higher effective valence
charge than predicted by the electronegativity difference between Ti
and O. However, this is due to the fact that there are two O atoms for
every Ti atom. This result indicates that the effective valence
charge is also a good measure of ionicity although it must be used
with care. A fit has also been performed to a function of the form
shown in Equation 2. The standard error of such a fit is 0.11.
This demonstrates that this measure is similar to that provided by the
overlap population and electronegativities.
Figure 1: Graph of overlap population against Mulliken
electronegativity difference. The best fit function 
is plotted for comparison.
Figure 2: Graph of overlap population against Pauling
electronegativity difference.
Figure 3: Graph of effective valence charge against Mulliken
electronegativity difference. Note that 
is a special case (see
text). The best fit function 
is
plotted for comparison.
Finally, Figure 4 shows a graph of bulk modulus against overlap population for the crystals in Table ii with a NaCl structure. This suggests a correlation between the overlap populations of the bonds within the crystal and the bulk modulus of the crystal. If we take the bulk modulus as a measure of the strength of the inter-atomic bonds, this result indicates that the bond strength increases with overlap population.
Figure 4: Graph of bulk modulus against Mulliken
overlap population for crystals with the NaCl structure.
