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Bonding in Bulk Crystals.

  Electronic structure calculations of several simple bulk crystals were analysed using the techniques described in Section gif. In each case the LCAO basis set used was the atomic pseudo-orbitals corresponding to the shell containing the valence electrons. The spilling parameters and atomic charges resulting from these calculations are presented in Table gif. The spilling parameters for these systems are very low, indicating a good representation of the electronic bands using the LCAO basis set. A spilling parameter in the region of tex2html_wrap_inline2786 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 ommission of the Si d-orbitals from the LCAO basis set used in the analysis of SiC suggests a charge transfer of 1.25e rather than 0.66e. The spilling parameter when the Si d-orbitals are absent is only tex2html_wrap_inline2792 , indicating that this change is not due to an under-representation of the electronic bands. The discrepancy in the Mulliken charges is caused by the change in the number of basis states associated with the Si atoms used in the representation of the charge distribution and the consequent effect on all of the other orbitals due to the nonorthogonality of the basis functions. Table gif also lists the effective ionic valences for each of the crystals. This is defined as the difference between the formal ionic charge and the Mulliken charge on the anion species in the crystal. The effective valence charge 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.

 

Material Spilling Anion Cation Effective
Parameter Charge (|e|) Charge (|e|) Valence (|e|)
NaF tex2html_wrap_inline2800 -0.59 0.59 0.41
NaCl tex2html_wrap_inline2802 -0.42 0.42 0.58
TiO2 tex2html_wrap_inline2800 -0.73 1.45 2.55
NaI tex2html_wrap_inline2800 -0.42 0.42 0.58
MgO tex2html_wrap_inline2800 -0.76 0.76 1.24
TiC tex2html_wrap_inline2810 -0.23 0.23 1.77
MgS tex2html_wrap_inline2812 -0.50 0.50 1.50
GaAs tex2html_wrap_inline2810 -0.29 0.29 2.71
SiC tex2html_wrap_inline2816 -0.66 0.66 3.34
Si tex2html_wrap_inline2816 N/A N/A 4.00
Table: Spilling parameters, atomic Mulliken charges and effective valence charges calculated from PW electronic structure calculations.

 

Table gif 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 is -0.11e while in NaCl this population is -0.03e. 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 gif is the difference between Mulliken and Pauling electronegativities of the species in each crystal. The Mulliken electronegativity of a species is defined as

equation854

where A is the electron affinity of an atom of the species and I is the ionisation energy of the atom. The Pauling electronegativity, tex2html_wrap_inline2828 , is defined empirically from the bond energies of diatomic molecules containing the species [51]. 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 indicates high ionicity. Pauling suggests that the degree of ionicity is given by tex2html_wrap_inline2830 , 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.

 

Material Structure Overlap tex2html_wrap_inline2834 (ref [52]) tex2html_wrap_inline2836 (ref [53])
Population (|e|) (eV)
NaF NaCl 0.18 7.56 3.1
NaCl NaCl 0.22 5.45 2.1
TiO2 Rutile 0.35,0.43 4.09 1.9
NaI NaCl 0.19 3.93 1.5
MgO NaCl 0.34 3.82 2.3
TiC NaCl 0.52 2.84 0.9
MgS NaCl 0.40 2.47 1.3
GaAs Zincblende 0.65 2.1 0.4
SiC Zincblende 0.83 1.5 0.7
Si Diamond 0.87 0 0
Table: Mulliken overlap populations calculated from PW pseudopotential calculations, Mulliken and Pauling electronegativity differences.

 

The calculations provide overlap population and effective valence charge as measures of covalency and ionicity. These may be compared with those derived from electronegativities. Figures gif and gif show graphs of the overlap populations against the Mulliken and Pauling electronegativity differences. Figure gif 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 gif is a fit of the data to a function of the form

  equation880

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, there is a constant offset indicating that a completely ionic bond is not possible within our definition. The agreement between the overlap populations and Pauling electronegativities, shown in Figure gif, is not as good. This may be due to the fact that the Pauling electronegativity scale is derived from the energies of diatomic molecules and may not be suitable for application to bulk materials. A graph of the effective valence charge against the difference in Mulliken electronegativities, Figure gif, again shows a correlation between these values. The notable exception is TiO2 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. The correlation between effective valence charge and Mulliken electronegativity difference indicates that the effective valence charge is also a good measure of ionicity, although it must be used with care. A fit of the data in Figure gif has also been performed to a function of the form shown in Equation gif. The standard error of the fit is 0.11 which demonstrates that this measure is similar to that provided by the overlap population and electronegativities.

   figure889
Figure: Graph of overlap population against Mulliken electronegativity difference. The best fit function tex2html_wrap_inline2846 is plotted for comparison.

   figure896
Figure: Graph of overlap population against Pauling electronegativity difference.

   figure903
Figure: Graph of effective valence charge against Mulliken electronegativity difference. Note that TiO2 is a special case (see text). The best fit function tex2html_wrap_inline2848 is plotted for comparison.


next up previous contents
Next: Overlap Population as a Up: Population Analysis Previous: Practical Example

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