Massively parallel computers have been used to perform first principles spin dynamics (SD) simulations of the magnetic structure of γ-FeMn (iron-manganese). The code uses a novel parallel approach to solve the Kohn-Sham equations obtained from density functional theory (DFT). This approach uses limited local communications avoiding the large global communications found in other DFT based methods. This allows our code to scale efficiently to very large processor counts and large system sizes. Computationally our method can be implemented using mainly BLAS3 routines which typically run at a very high percentage of peak speed on most computers. Our large scale quantum mechanical simulations using this method reveal new details of the orientational configuration of the magnetic moments of γ-FeMn that are unobtainable by any other means. The understanding of the magnetic structure of γ-FeMn is important in exchange bias, which involves the use of an antiferromagnetic (AFM) layer such as FeMn to pin the orientation of the magnetic moment of a proximate ferromagnetic (FM). Exchange bias is of fundamental importance in magnetic multilayer storage and read head devices. Here the equation of motion of first principles SD is used to perform relaxations of model magnetic structures to the true ground (equilibrium) state. Our code has achieved a maximum execution rate of 2.46 teraflops on the IBM SP at the National Energy Research Scientific Computing Center (NERSC) and 4.65 teraflops on the HP-Compaq machine at the Pittsburgh Supercomputing Center (PSC).