THE FORMATION AND ABSENCE OF ASYMMETRIC ELECTRON
DENSITIES IN PbO AND PbS: THE END OF THE Pb 6s2 LONE PAIR MYTH
Aron WALSH and Graeme. W. WATSON
Department of Chemistry, Trinity College, Dublin 2, Ireland
The concept of a chemically inert but stereochemically active 6s2 lone pair is
commonly associated with Pb(II). We have performed density functional theory
calculations on PbO and PbS in both the rock salt and litharge structures which
confirm that this lone pair does not exist. Analysis of the electronic structure shows
that the asymmetric density formed by Pb(II) is a direct result of anion-cation
interactions. The formation has a strong dependence on the electronic states of the
anion and while oxygen has the required states, sulphur does not, which can explain
for the first time why PbO forms distorted structures and possesses an asymmetric
density and PbS does not. This analysis finally ends the myth that the distorted Pb(II)
structures are the result of chemically inert, sterically active lone pairs.
1. INTRODUCTION
Lead oxides have been the subject of great interest due their unusual structural and
electrical properties, and their range of applications. Of particular interest has been
the so-called 6s2 sterically active lone pair which Pb displays in its +2 oxidation state.
Lone pairs have also been associated with other materials showing a valence state
two lower their groups including Tl(I), Sn(II) and Bi(III). A full understanding of
their electronic structure would contribute to understanding the properties of the
materials which contain these ions.
The concept of an inert electron pair is commonly associated with Pb(II) in solid
state materials1. It is considered to form from the hybridization of the 6s and 6p
atomic orbitals, with one of the resulting orbitals being filled by the 6s2 electrons. It is
deemed to be a chemically inactive feature, but sterically active in distorting the
crystal structure. PbO adopts a highly asymmetric crystal structure, litharge2, which
can be seen as a distortion of the symmetric CsCl structure. In litharge each Pb has
four oxygen nearest neighbours, all of which are on the same side of Pb with a lone
pair considered to be projected in the opposite direction (Fig.1). This sterically active
electron pair has always been directly associated with the Pb(II) species however not
all Pb(II) compounds display the same distortion in crystal structure. PbS adopts the
rock salt structure in which the Pb sites are six coordinate and symmetric3. If the
asymmetric electron density of PbO was a lone pair arising purely from the Pb(II) ion
such a feature would be expected to form in all Pb(II) compounds yet this is not the
case.
FIGURE 1
Representation of litharge PbO (left) and rock salt PbS (right), Pb atoms are coloured
light with the anions coloured dark.
In previous work4 we have shown that PbO is not a purely ionic material, as there
are significant interactions between the Pb 6s states and the O 2p states that result in
filled antibonding orbitals near the Fermi level which contain Pb 6s states. This study
and similar work5, 6 indicate that it is these near Fermi level states, and not the main
6s states at lower energy, which give rise to lone pair formation in PbO. In this work
we investigate the electronic structure of PbO and PbS in both the rock salt and
litharge structures with an aim of fully explaining the origin of asymmetry in the
Pb(II) electron density.
2. COMPUTATIONAL METHODS AND OPTIMISATIONS
The calculations were performed using gradient corrected, periodic density
functional theory (DFT) using the parameterization of Perdew, Burke and Ernzerhof7
(PBE) as implemented in the code VASP8. A plane wave basis set was used to
expand the valence electrons with the core electrons (Pb: [Xe], O: [He]) treated using
the projector augmented wave approach9. The calculations were checked for
convergence with respect to both plane wave cut off (400 eV and 300 eV were used
for PbO and PbS) and k-point sampling (6×6×4 and 6×6×6 for litharge and rock salt).
Relaxations at a series of volumes was performed with the resulting energy volume
curve fitted to the Murnaghan equation of state to obtain the equilibrium cell volumes.
The equilibrium lattice vectors, binding energies and nearest Pb-anion distances
are shown in Table 1. The calculated lattice vectors are in good agreement with
experiment with the exception of the c vector of litharge PbO, which represents
interlayer interactions. This vector is overestimated to a greater extent due to the
inability of DFT to accurately describe the weak non bonding forces in this direction.
Both PbO and PbS are stable in the rock salt structure, however while the litharge
structure is more stable for PbO as expected for the thermodynamically favoured
phase, PbS is not. Optimisation of PbS in the litharge structure resulted in an
expansion of the a and b vectors and contraction of the c vector. The structure
attempts to create a symmetric Pb site by relaxing toward the undistorted CsCl
structure. To aid proper comparison of litharge PbO and PbS an additional
optimization was performed with constant a:c ratio taken from PbO.
TABLE 1
Calculated data for rock salt and litharge PbO and PbS and error with respect to
experimental data2,3 where available.
Rock salt
E (eV/Pb) -10.53
a(Å)
5.27
b(Å)
c(Å)
Pb-O (Å) 2.63
PbO
Litharge
-10.52
4.06 (+2.4%)
4.06 (+2.4%)
5.66 (+13%)
2.35 (+1.3%)
PbS
Litharge Litharge
Fixed a:c
-8.91
-8.61
-8.51
6.01 (+1.3%) 5.13
5.08
5.13
5.08
4.21
7.08
3.01 (+1.3%) 2.86
2.82
Rock salt
3. RESULTS AND DISCUSSION
Analysis of the electron densities reveals that both materials show symmetric
electron distributions in the rock salt structure as expected. However, litharge PbO
shows a marked asymmetric electron density directed away from the Pb atom, while
PbS displays similar but much weaker asymmetry, Fig.2. To simplify the
interpretation of the electronic structure we have calculated the partial (ion and l and
m-quantum number decomposed) electronic density of states (PEDOS). These were
obtained by projecting the wave functions onto spherical harmonics centred on each
atom with a radius of 1.55Å for both Pb and O atoms and 1.85Å for S atoms. These
radii give rise to reasonable space filling, but the results (at least the qualitative
aspects) are insensitive to a change of the radii.
FIGURE 2
Partial electron densities from -10eV to the fermi level for litharge PbO (left) and
litharge structured PbS (right), shown from 0 to 0.5 e/Å3.
Fig.3 shows the PEDOS curves from -10 eV to +5 eV (relative to the top of the
valence band) for Pb and O in both rock salt and litharge structures, while Fig.4
shows the PEDOS for PbS. The basic structure of the PEDOS are very similar. The
first peak at around -8 eV is mainly Pb 6s with O 2p (or S 3p) also present. The
second is mainly O 2p (S 3p) with a small amount of Pb 6p. The peak at the top of the
valence band contains mainly O 2p but with some Pb 6s character and can be shown
through crystal orbital overlap population analysis to be the filled antibonding
combination of the -8 eV peak. In this way Pb 6s states are found at a substantially
higher energy than expected and close to the Fermi level.
FIGURE 3
Density of states for (a) Pb and (b) O in rock salt PbO and (c) Pb and (d) O in litharge
PbO. The light grey lines correspond to s states, black to p states and dark grey to pz.
FIGURE 4
Density of states for (a) Pb and (b) S in rock salt PbS and (c) Pb and (d) S in litharge
PbS. The light grey lines correspond to s states, black to p states and dark grey to pz.
However, for litharge PbO an additional interaction occurs. The antibonding
combination close to the Fermi level is able to interact with the unoccupied Pb 6p
states resulting in a stabilization of the structure and a corresponding shift of the
peaks close to the Fermi level. This can be seen in the PEDOS (Fig.3c) in which Pb
6pz interacts with the antibonding combination partially made up of Pb 6s. Additional
evidence for this interaction is found in the unoccupied states in which a new Pb 6s
peak is observed for litharge PbO. The partial electron density plot for this region of
the EDOS (Fig.5a) confirms that these states are responsible for the asymmetry in the
Pb electron density.
For PbS the antibonding states in the PEDOS, and thus the Pb 6s states near the
Fermi level, are significantly smaller, resulting in a much weaker interaction with Pb
6p and subsequent weaker asymmetry. The origin of the asymmetry in the electron
density comes from a combination of the bonding Pb 6p and antibonding states
arising from a Pb 6s–anion interaction. For PbO the antibonding density is pushed out
further from Pb through the interaction of Pb 6p. For PbS, although the antibonding
density is directed away from the S layers, it is much less active than PbO through a
combination of reduced antibonding states and a reduced Pb 6p interaction (Fig.5b).
FIGURE 5
Partial electron densities from -3 eV to the fermi level for (a) litharge structured PbO
and (b) litharge structured PbS, shown from 0 to 0.3 e/Å3.
4. CONCLUSIONS
This analysis confirms that it is the coupling of the Pb 6p and the antibonding Pb
6s states that gives rise to the net asymmetry in the electron density. However this
coupling can only take place when there is an appropriate anion that can generate
significant Pb 6s states close to the Fermi level. The Pb 6s and 6p states are too
distant in energy to couple directly. Oxygen clearly has the required energy levels to
achieve this while sulphur does not. The formation of the asymmetric electron density
in PbO is therefore dependent on both the cation and anion and hence it is clearly not
a lone pair at all.
We have demonstrated that the asymmetric electron density formed by Pb(II), in
contrast to traditional lone pair theory, is a result of the interaction of the antibonding
Pb 6s and O 2p states with unfilled Pb 6p. This causes, in the case of litharge PbO, a
shift of the states at the Fermi level to lower energy and the appearance of
unoccupied Pb 6s states in the conduction band. The filled antibonding states are
much weaker in PbS and the unoccupied Pb 6s states are not seen, indicating that
strong Pb 6p coupling does not occur. The asymmetry of PbS is therefore weak and
cannot stabilize the distorted structure relative to a symmetric structure of higher
coordination, explaining why rock salt is the thermodynamically stable phase of PbS.
These results prove conclusively that the so-called Pb(II) inert lone pair is a direct
result of the cation–anion interactions, having implications for understanding the
properties of materials that display such asymmetry.
ACKNOWLEDGEMENT
We would like to acknowledge Trinity College for a postgraduate studentship, the
HEA for a PRTLI grant and Compaq and JREI (JR99BAPAEQ) for funding to
purchase and support a 20 processor Compaq SC cluster at the Rutherford Appleton
Laboratory.
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