  Half-metallic ferromagnets like the full Heusler compounds with formula
X$_2$YZ are supposed to show an integer value of the spin magnetic moment.
Calculations reveal in certain cases of X = Co based compounds non-integer
values, in contrast to experiments. In order to explain deviations of the
magnetic moment calculated for such compounds, the dependency of the electronic
structure on the lattice parameter was studied theoretically. In local density
approximation (LDA), the minimum total energy of Co$_2$FeSi is found for the
experimental lattice parameter, but the calculated magnetic moment is about 12%
too low. Half-metallic ferromagnetism and a magnetic moment equal to the
experimental value of $6\mu_B$ are found, however, only after increasing the
lattice parameter by more than 6%.
  To overcome this discrepancy, the LDA$+U$ scheme was used to respect on-site
electron correlation in the calculations. Those calculations revealed for
Co$_2$FeSi that an effective Coulomb-exchange interaction $U_{eff}=U-J$ in the
range of about 2eV to 5eV leads to half-metallic ferromagnetism and the
measured, integer magnetic moment at the measured lattice parameter. Finally,
it is shown in the case of Co$_2$MnSi that correlation may also serve to
destroy the half-metallic behavior if it becomes too strong (for Co$_2$MnSi
above 2eV and for Co$_2$FeSi above 5eV). These findings indicate that on-site
correlation may play an important role in the description of Heusler compounds
with localized moments.
