  We study the changes in the Fermi surface with electron doping in the
LaFeAsO$_{1-x}$F$_x$ superconductors with density-functional supercell
calculations using the linearized augmented planewave (LAPW) method. The
supercell calculations with explicit F substitution are compared with those
obtained from the virtual crystal approximation (VCA) and from a simple rigid
band shift. We find significant differences between the supercell results and
those obtained from the rigid-band shift with electron doping, although quite
remarkably the supercell results are in good agreement with the virtual crystal
approximation (VCA) where the nuclear charges of the O atoms are slightly
increased to mimic the addition of the extra electrons. With electron doping,
the two cylindrical hole pockets along $\Gamma-Z$ shrink in size, and the third
hole pocket around $Z$ disappears for an electron doping concentration in
excess of about 7-8%, while the two elliptical electron cylinders along $M-A$
expand in size. The spin-orbit coupling does not affect the Fermi surface much
except to somewhat reduce the size of the third hole pocket in the undoped
case. We find that with the addition of the electrons the antiferromagnetic
state becomes energetically less stable as compared to the nonmagnetic state,
indicating that the electron doping may provide an extra degree of stability to
the formation of the superconducting ground state.
