  We present an extensive study of nuclear matrix elements (NME) for the
neutrinoless double beta decay of the nuclei $^{48}$Ca, $^{76}$Ge, $^{82}$Se,
$^{96}$Zr, $^{100}$Mo, $^{116}$Cd, $^{124}$Sn, $^{128}$Te, $^{130}$Te,
$^{136}$Xe, and $^{150}$Nd based on state-of-the-art energy density functional
methods using the Gogny D1S functional. Beyond mean-field effects are included
within the generating coordinate method with particle number and angular
momentum projection for both initial and final ground states. We obtain a
rather constant value for the NME's around 4.7 with the exception of $^{48}$Ca
and $^{150}$Nd, where smaller values are found. We analyze the role of
deformation and pairing in the evaluation of the NME and present detailed
results for the decay of $^{150}$Nd.
