An energy stable numerical scheme is proposed and analyzed for the square phase field crystal (SPFC) equation, a gradient flow to model the crystal dynamics at the atomic scale in space but on diffusive scales in time. In particular, a modification of the free energy potential to the standard phase field crystal model leads to a composition of the 4-Laplacian and the regular Laplacian operators. The Fourier pseudo-spectral approximation is taken in space, so that the summation in parts formulas enable one to study the discrete energy stability for such a high order spatial discretization. In the temporal approximation, a second order BDF stencil is applied in the time direction, combined with an appropriate extrapolation for the concave diffusion term. At a theoretical level, the unique solvability, energy stability are established, and an optimal rate convergence analysis is derived. In the numerical implementation, the preconditioned steepest descent (PSD) iteration is applied to solve for the composition of the highly nonlinear 4-Laplacian term and the standard Laplacian term, and a geometric convergence is assured for such an iteration. A few numerical experiments are also presented.

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