BaBiO$_3$ is a mixed-valence perovskite which escapes the metallic state through a Bi valence (and Bi-O bond) disproportionation or CDW distortion, resulting in a semiconductor with a gap of 0.8 eV at zero pressure. The evolution of structural and electronic properties at high pressure is, however, largely unknown. Pressure, one might have hoped, could reduce the disproportionation, making the two Bi ions equivalent and bringing the system closer to metallicity or even to superconductivity, such as is attained at ambient pressure upon metal doping. We address the high-pressure phase diagram of pristine BaBiO$_3$ by ab initio DFT calculations based on GGA and hybrid functionals in combination with crystal structure prediction methods based on evolutionary algorithms, molecular dynamics and metadynamics. The calculated phase diagram from 0 to 50 GPa indicates that pristine BaBiO$_3$ resists metallization under pressure, undergoing instead at room temperature structural phase transitions from monoclinic textit{I2/m} to nearly tetragonal textit{P-1} at 7 GPa, possibly to monoclinic textit{C2/m} at 27 GPa, and to triclinic textit{P1} at 43 GPa. Remarkably, all these phases sustain and in fact increase the inequivalence of two Bi neighboring sites and of their Bi-O bonds and, in all cases except semimetallic textit{C2/m}, the associated insulating character. We then present high-pressure resistivity data which generally corroborate these results, and show that the insulating character persists at least up to 80 GPa, suggesting that the textit{C2/m} phase is probably an artifact of the small computational cell.
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