  Typical density-functional theory calculations that wrongly predict undoped
cuprates to be metallic also predict Cu-O half- and full-breathing phonon
energies that are significantly softer than observed, presumably because of
weak on-site Coulomb repulsion on the Cu 3d orbitals. We used DFT+U
calculations with antiferromagnetic supercells of La$_2$CuO$_4$ to establish
correlation between the on-site repulsion strength, tuned via adjusting the
value of U, and phonon dispersions. We find that breathing and half-breathing
phonons reach experimental values when U is tuned to obtain the correct optical
gap and magnetic moments. We demonstrate that using distorted supercells within
DFT+U is a promising framework to model phonons in undoped cuprates and other
perovskite oxides with complex, interrelated structural and electronic degrees
of freedom.
