  We provide for the first time accurate assessments of the consequences of
violations of self-consistency in the Hartree-Fock based random phase
approximation (RPA) as commonly used to calculate the energy $E_c$ of the
nuclear breathing mode. Using several Skyrme interactions we find that the
self-consistency violated by ignoring the spin-orbit interaction in the RPA
calculation causes a spurious enhancement of the breathing mode energy for spin
unsaturated systems. Contrarily, neglecting the Coulomb interaction in the RPA
or performing the RPA calculations in the TJ scheme underestimates the
breathing mode energy. Surprisingly, our results for the $^{90}$Zr and
$^{208}$Pb nuclei for several Skyrme type effective nucleon-nucleon
interactions having a wide range of nuclear matter incompressibility ($K_{nm}
\sim 215 - 275$ MeV) and symmetry energy ($J \sim 27 - 37$ MeV) indicate that
the net uncertainty ($\delta E_c \sim 0.3$ MeV) is comparable to the
experimental one.

Nuclear Theory