method*

We calculated the spin-wave dispersion and the stiffness constants of three metallic ferromagnetic Heusler alloys: Cu₂MnAl, Ni₂MnSn, and Pd₂MnSn. We determined the ground state by the quasi-particle self-consistent GW (QSGW) method. In conjunction with the Wannier function, we obtain transverse dynamical spin susceptibility based on linear response method. It is found that the ground states within the QSGW are reasonably calculated to reproduce spin-wave dispersion around Gamma point. In Cu₂MnAl, the magnetic moment in QSGW agrees with the experiment, but stiffness constant is underestimated. In Ni₂MnSn, the QSGW overestimates the moment, as seen in the itinerant ferromagnetic case as FCC Ni; however, the stiffness in QSGW agrees with the experiment. In Pd₂MnSn, the QSGW reproduces the spin-wave throughout the Brillouin zone, and the stiffness is close to the experiments. This agreement is due to the reasonable exchange splitting of Mn 3d in accordance with the large screened Coulomb interaction in QSGW.