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would have a better frustration of the tilts and a stronger ferroelectric tendency. 10 

The density functional (DFT) calculations were done using two methods. We used the general potential lin- earized augmented planewave (LAPW) method 35 for the structure relaxations. This is a full potential all electron method. We did the calculations using the local density approximation at a scalar relativistic level. Well con- verged basis sets were employed with LAPW sphere radii of 2.3 Bohr for Pb, Sr and Bi, 1.90 Bohr for Zn and Nb and 1.55 Bohr for O. We included local orbitals to relax linearization and to accurately treat semicore states. 36 The polarization and Born effective charges were calcu- lated with an ultrasoft pseudopotential method as im- plemented in the quantum espresso package, 37 using the relaxed structures from the LAPW calculations. The re- sults were converged with respect to the Brillouin zone sampling, which was tested. The polarization was ob- tained using the Berry’s phase method. 

We started by assuming a pseudocubic structure, i.e. setting the lattice parameters of our 40 atom super- cells to be orthogonal and equal. We then fully relaxed the internal coordinates of all atoms in the cell, with no symmetry constraints. This was done as a function of the lattice parameter to obtain the equilibrium cell volumes for BiPbZnNbO$_{6}$ and BiSrZnNbO$_{6}$. This was done in the LDA using the LAPW method. The ef- fective perovskite lattice parameters for the minimum energy were 3.99 ˚ A, for BiPbZnNbO$_{6}$ and 3.97 ˚ A, for BiSrZnNbO$_{6}$. As mentioned, (Bi$_{0}$$_{.}$$_{5}$Sr$_{0}$$_{.}$$_{5}$)(Zn$_{0}$$_{.}$$_{5}$Nb$_{0}$$_{.}$$_{5}$)O$_{3}$ was synthesized by Kosyachenko and co-workers. 29 They reported a weakly tetragonal structure with lattice pa- rameters a = b =4.002, and c =4.015 ˚ A ( c/a =1.003) at 293 K. These values are ∼ 0.9 % larger than our calculated LDA lattice parameter. This size of underestimation is typical of LDA errors. 

The structure of the 40 atom pseudocubic BiPbZnNbO$_{6}$ supercell at the LDA equilibrium volume is depicted in Fig. 1. As may be seen, there are large off-centerings of the Bi ions and also some octahedral tilting. Fig. 2 shows the cation off-centerings in this cell for the different sites with respect to the centers of their O cages (the 12 nearest O ions for the A -site and the 6 nearest for the B -site). The largest off-centerings are of the Bi ions, consistent with Fig. 1, and what might be expected based on the small size and stereochemical activity of Bi $^{3+}$. These displacements are large (0.64 ˚ A) and collinear along a [111] direction. Importantly, there are also large collinear displacements of the other cations. The Pb ions displace by ∼ 0.39 ˚ A, the Zn by ∼ 0.27 ˚ A, and the Nb by ∼ 0.27 ˚ A, all very close to [111]. This cooperative nature of the ferroelectricity, where all cations off-center significantly, is a characteristic of good perovskite ferroelectrics such as KNbO$_{3}$, BaTiO$_{3}$ 

## III. RESULTS 

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