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The phonon dynamics of the low-temperature superconductor Sr$_{2}$RuO$_{4}$is calculated quantitatively in linear response theory and compared with thestructurally isomorphic high-temperature superconductor La$_{2}$CuO$_{4}$. Ourcalculation corrects for a typical deficit of LDA-based calculations whichalways predict a too large electronic $k_{z}$-dispersion insufficient todescribe the c-axis response in the real materials. With a more realisticcomputation of the electronic band structure the frequency and wavevectordependent irreducible polarization part of the density response function isdetermined and used for adiabatic and nonadiabatic phonon calculations. Ouranalysis for Sr$_{2}$RuO$_{4}$ reveals important differences from the latticedynamics of $p$- and $n$-doped cuprates. Consistent with experimental evidencefrom inelastic neutron scattering the anomalous doping related softening of thestrongly coupling high-frequency oxygen bond-stretching modes (OBSM) which isgeneric for the cuprate superconductors is largely suppressed or completelyabsent, respectively, depending on the actual value of the on-site Coulombrepulsion of the Ru4d orbitals. Also the presence of a characteristic$\Lambda_{1}$-mode with a very steep dispersion coupling strongly with theelectrons is missing in Sr$_{2}$RuO$_{4}$. Moreover, we evaluate thepossibility of a phonon-plasmon scenario for Sr$_{2}$RuO$_{4}$ which has beenshown recently to be realistic for La$_{2}$CuO$_{4}$. In contrast toLa$_{2}$CuO$_{4}$ in Sr$_{2}$RuO$_{4}$ the very low lying plasmons areoverdamped along the c-axis.

Microscopic calculation of the phonon dynamics of Sr$_{2}$RuO$_{4}$ compared with La$_{2}$CuO$_{4}$