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Typical density-functional theory calculations that wrongly predict undopedcuprates to be metallic also predict Cu-O half- and full-breathing phononenergies that are significantly softer than observed, presumably because ofweak on-site Coulomb repulsion on the Cu 3d orbitals. We used DFT+Ucalculations with antiferromagnetic supercells of La$_2$CuO$_4$ to establishcorrelation between the on-site repulsion strength, tuned via adjusting thevalue of U, and phonon dispersions. We find that breathing and half-breathingphonons reach experimental values when U is tuned to obtain the correct opticalgap and magnetic moments. We demonstrate that using distorted supercells withinDFT+U is a promising framework to model phonons in undoped cuprates and otherperovskite oxides with complex, interrelated structural and electronic degreesof freedom.

Effect of the electronic charge gap on LO bond-stretching phonons in undoped La$_2$CuO$_4$ calculated using LDA+U