  The relative thermodynamic stability of surface reconstructions including
vacancies, adatoms and additional layers on both CaO and MnO$_{2}$ terminations
is calculated to predict the surface phase diagram of
($\sqrt{2}\times\sqrt{2}$)$\it{R}$45$^{\circ}$ CaMnO$_{3}$ (001) using
$\it{ab}$ $\it{initio}$ thermodynamics. Stoichiometric and nonstoichiometric
reconstructions are considered. A set of boundary conditions driven by the
binary and ternary sub-phases from CaMnO$_{3}$ defines its bulk stability
region, enclosing the stable surface reconstructions that can be in equilibrium
with the bulk. Most of the surfaces take the magnetic ordering of the bulk
ground state, G-type antiferromagnetic (AFM); however, some of the
MnO$_{2}$-terminated surfaces are more stable when the surface layer spins
flip. At 573 K, the stoichiometric CaO- and MnO$_{2}$-terminated surfaces are
predicted to be thermodynamically stable, as well as a CaO-terminated surface
reconstruction where half the Ca are replaced by Mn. The MnO$_{2}$-terminated
surface reconstructions dominate the phase diagrams at high temperatures,
including phases with MnO and MnO$_{2}$ adatoms per surface unit cell.
