Supplementary information on the projection radii:
Electronic structures of rock salt, litharge and herzenbergite SnO by
density functional theory
Aron Walsh and Graeme W. Watson*
Department of Chemistry, Trinity College, Dublin 2, Ireland
Tel: +353 1 6081357; E-mail: [email protected]
In our plane wave calculations the wave function character is calculated by projecting
the wavefunctions onto spherical harmonics that are non zero within spheres of the
specified radius around each ion, the same way that an A.O. basis set represents the
wavefunction/charge density in a region of space. The radius defines the atoms the
electron density is assigned to which is then projected onto the spherical harmonics,
determining the orbital type. This assumes that the electron density localised around
an atom belongs to it, which is the same as the assumptions of a localised basis set.
By using a value of 1.55Å for both Sn and O we are performing a treatment based on
ionic radii. The covalent radii are 1.41Å for Sn and 0.73Å for O, giving a ratio of 2:1.
The Sn-O bond lengths in the three structures are 2.6Å, 2.2Å and 2.2/2.7Å. With an
ionic radius of 1.24Å for oxygen, Sn is approximately the same (ratio 1:1). This is
also consistent with charge density plots, Fig.1. With spheres it is impossible to
account for all the charge density without using radii that overlap (i.e. 1.55Å). In
addition using a smaller radius for Sn would exclude the outer charge density relating
to the lone pair.
Figure 1. Electron density map for litharge SnO. The pink and blue spheres represent
projection radii of 1.55Å for Sn an O respectively.
Although overlap occurs between the Sn and O spheres, using radii of 1.55 Å gives
rise to the correct total number of electrons for the system. The DOS is shown below
using radii of 1.35Å for both Sn and O (maintaining a ratio of 1:1), Fig.2. While it
shows the general features are relatively insensitive to change in the radii, integration
results in three too few electrons being accounted for.
The ratios of the radii have also been tested. The DOS are shown below with a Sn:O
ratios of 1.25:1 and 1:1.25 (while maintaining the total number of electrons).
Although the relative magnitude of the peaks changes (as expected since we exclude
some of the lone pair density with a small Sn radius) the general features remain the
same showing that the results are also insensitive to the ratio of the radii.
The radii we employ (1.55 Å) are consistent with the charge density plots and this is
also reflected in the DOS. The Sn(5s)-O(2p) interaction at -8eV does not show
significant artifacts. For example there is little Sn(5p) due to the assignment of density
from the O(2p) to Sn, as expected if the radii on the Sn were significantly too large.
There is a small trace of Sn(5p) shown in the test case with a large (1.78 Å) Sn radius.
Figure 2. Projected electron density of states for litharge SnO for various projection
radii.
In summary, the size and ratio of the radii used in our projection onto spherical
harmonics was examined and the qualitative results found to be insensitive to the
exact values. By using radii consistent with the charge density plots, scaled to
reproduce the total number of electrons analysis of the electronic structure was carried
out.