  We measure the dispersion of the Cu-O bond-stretching phonon mode in the
high-temperature superconducting parent compound Ca$_2$CuO$_2$Cl$_2$. Our
density functional theory calculations predict a cosine-shaped bending of the
dispersion along both the ($\xi$00) and ($\xi\xi$0) directions, while
comparison with previous results on Ca$_{1.84}$CuO$_2$Cl$_2$ show it only along
($\xi$00), suggesting an anisotropic effect which is not reproduced in
calculation at optimal doping. Comparison with isostructural
La$_{2-x}$Sr$_x$CuO$_4$ suggests that these calculations reproduce well the
overdoped regime, however they overestimate the doping effect on the Cu-O
bond-stretching mode at optimal doping.
