The competitions between ferroelectric and rotational instabilities in rhombohedral PbZrxTi1-xO3 near x = 0.5 are investigated using first-principles density-functional supercell calculations. As expected, we find a strong ferroelectric instability. However, we also find a substantial R-point rotational instability, close to but nor as deep as the ferroelectric one. This is similar to the situation in pure PbZrO3. These two instabilities are both strongly pressure dependent, but in opposite directions so that lattice compression of less than 1% is sufficient to change their ordering. Because of this, and local stress fields due to B-site cation disorder may lead to coexistence of both types of instability are likely present in the alloy near the morphotropic phase boundary.
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