Lattice instabilities, isotope effect, and high-${\mathit{T}}_{\mathit{c}}$ superconductivity in ${\mathrm{La}}_{2\mathrm{-}\mathit{x}}$${\mathrm{Ba}}_{\mathit{x}}$${\mathrm{CuO}}_4$

Unlike ${\mathrm{La}}_{2\mathrm{-}\mathit{x}}$ ${\mathrm{Sr}}_{\mathit{x}}$ ${\mathrm{CuO}}_4$, ${\mathrm{La}}_{2\mathrm{-}\mathit{x}}$ ${\mathrm{Ba}}_{\mathit{x}}$ ${\mathrm{CuO}}_4$, x∼${\mathit{x}}_{\mathrm{cr}}$=0.12, undergoes a structural change to a low-temperature tetragonal (LTT) phase below 60 K, and this transformation is known to suppress ${\mathit{T}}_{\mathit{c}}$ drastically. We present self-consistent local-density-functional calculations for this system that indicate the ${\mathrm{CuO}}_6$ octahedrons are unstable to an X-point tilt in any direction, thereby leading to strongly anharmonic X-point vibrations. The LTT phase is most favored energetically, consistent with experiment. The effect of the LTT distortion on the electronic structure accounts for the suppression of ${\mathit{T}}_{\mathit{c}}$ in the LTT phase, and for the enhanced isotope effect at x=${\mathit{x}}_{\mathrm{cr}}$.