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2 p states as may be seen from the Bi and Pb s character at the top of the O 2 p bands. However, this is a mixing of occupied states and as such does not signify chemi- cal bonding. Instead, as has been noted previously for ferroelectric perovskites, 4,5 the important hybridization is between the O 2 p states and unoccupied metal states, which favors ferroelectricity due to chemical bonding and leads to enhanced Born effective charges. The energy position of the Nb 4 d states, which are dominant in the lower conduction bands from the conduction band mini- mum (CBM) to ∼ 1.2 eV above it is important because the low energy favors hybridization with O 2 p states and ferroelectric off-centering, as was discussed in the case of PMN and KNbO$_{3}$ in relation to KTaO$_{3}$. 39,40 In the present case, there is a strong mixing of O 2 p and Nb 4 d states. This is evident from the Nb d character in the O 2 p bands. The Bi 6 p and Pb 6 p states, which oc- cur in the conduction bands from the CBM to ∼ 4 eV above, also mix strongly with the O 2 p bands. There is also cross-gap hybridization evident involving the Zn sp states, although this mixing is weaker perhaps because the Zn states occur further above the CBM. The values of the Born effective charges are enhanced by this cross gap hybridization. The average values for Bi, Pb, Zn and Nb are 4.4, 3.6, 2.9 and 5.8, respectively. The calculated polarization for the supercell is 85 µ C/cm $^{2}$. This num- ber was obtained using a Berry’s phase calculation. The estimate of the polarization that is obtained by multiply- ing the Born charges and cation off-centerings in their O cages is 83 µ C/cm 2 in close agreement with the Berry’s phase calculation. For comparison the polarization of BiFeO$_{3}$ from DFT calculations is 90 – 100 µ C/cm $^{2}$.16 

The above discussion in terms of cross gap hybridiza- 

tion due to interaction between O 2 p states and extended, low lying metal orbitals making up the conduction bands is different from a discussion in terms of a polarizable atomic like s orbital. However, it should be emphasized that atomic character is not a uniquely defined quantity since it is basis set dependent (note that an expansion in terms of radial functions times spherical harmonics about a single site is a complete basis, and so expan- sion about the different sites on a lattice is necessarily overcomplete). Furthermore, polarization of an s orbital implies a mixing in of some p character from unoccu- pied orbitals, and in these perovskites this is 6 p charac- ter that arises from conduction bands. The distinction between the two views is that in the lone pair description the discussion is in terms of response to crystal potential as in a discussion of crystal field, while in the cross-gap hybridization it is structure dependent hybridization as in ligand field. This is analogous to the origin of crys- tal field splittings in transition metal oxide compounds, which arise primarily from hybridization with O 2 p states and not the non-spherical crystal potential around the metal site. 

As mentioned, high piezoelectric coupling in ferroelec- tric perovskites is generally found near an MPB, most commonly between a pseudocubic phase and a tetragonal phase, as in PZT. We did calculations as a function of im- posed tetragonal strain for our supercell at a fixed volume equal to the equilibrium volume for the pseudocubic cell ( a =3.99 ˚ A). Over the range considered (0.992 < c/a < 1.076), the cation off-centerings stayed collinear and pri- marily along a [111] direction, and in particular, at least up to the maximum strain considered, did not switch to an [001] average direction. As an example, cation off- centerings for c/a = 1 . 0606 are shown in Fig. 4. The lowest energy is at c/a ∼ 1 . 015. However, near this min- imum the cation displacements remain very close to a [111] direction. Instead it seems that the deviation of c/a from unity is due to the octahedral tilts and not strain coupling for the tetragonal ferroelectric state. Since the tilts are expected to be coupled to the A -site cation or- dering, this may very well be an artifact of the specific ordering used in the necessarily small supercell studied here (note that the A -site cations would most likely be disordered in this material). In any case, BiPbZnNbO$_{6}$ is most likely pseudocubic with a rhombohedral ferroelec- tric state or weakly tetragonal. As such it may be of interest to investigate the BiPbZnNbO$_{6}$ – PbTiO$_{3}$ solid solution to determine whether an MPB is present. We note that the tetragonality may be particularly sensitive to volume, as is the case in PbTiO$_{3}$, meaning that the LDA volume error may be particularly important for this, and also that errors due to the use of a small ordered su- percell may be particularly important for this. 

BiPbZnNbO$_{6}$ has two A -site ions with stereochemi- cal activity. Bi shows large off-centerings while the Pb off-centerings are smaller, and furthermore Bi carries a higher charge. This raises the question of the role of Pb in the ferroelectricity. This question is also of interest 

