Using $textit{ab-initio}$ crystal structure prediction we study the high-pressure phase diagram of $textit{A}BiO_3$ bismuthates ($A$=Ba, Sr, Ca) in a pressure range up to 100$~$GPa. All compounds show a transition from the low-pressure perovskite structure to highly distorted, low-symmetry phases at high pressures (PD transition), and remain charge disproportionated and insulating up to the highest pressure studied. The PD transition at high pressures in bismuthates can be understood as a combined effect of steric arguments and of the strong tendency of bismuth to charge-disproportionation. In fact, distorted structures permit to achieve a very efficient atomic packing, and at the same time, to have Bi-O bonds of different lengths. The shift of the PD transition to higher pressures with increasing cation size within the $textit{A}BiO_3$ series can be explained in terms of chemical pressure.
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