We describe a formulation of electronic density functional theory using localized Gaussian basis functions for systems periodic in three dimensions (bulk crystals) or two dimensions (crystal slabs terminated by surfaces). Our approach generalizes many features of molecular density functional methods to periodic systems, including the use of an auxiliary Gaussian basis set to represent the charge density, and analytic gradients with respect to nuclear coordinates. Existing quantum chemistry routines for analytic and numerical integration over basis functions can be adapted to our scheme with only slight modifications, as can existing extended Gaussian basis sets. Such basis sets permit accurate calculations with far fewer basis functions (and hence much smaller matrices to diagonalize) than plane-wave based methods, especially in surface calculations, where in our approach the slab does not have to repeat periodically normal to the surface. Realistic treatment of molecule–surface interactions is facilitated since both molecule and surface can be treated at the same level of theory. Our real-space method also offers opportunities to exploit matrix sparsity, since in a large unit cell many pairs of basis functions will be essentially nonoverlapping and noninteracting. Longer-ranged Coulomb interactions are summed by a form of the Ewald technique that guarantees absolute convergence. We also give a straightforward extension to periodic systems (both two- and three-dimensional) of the usual molecular formalism for analytic nuclear first derivatives (forces).

Jaffe, J. E., & Hess, A. C. (1996). Gaussian basis density functional theory for systems periodic in two or three dimensions: Energy and forces. The Journal of Chemical Physics, 105(24), 10983–10998.