## The analytical energy gradient scheme in the Gaussian based Hartree-Fock and density functional theory for two-dimensional systems using fast multipole method

The analytical total energy gradient scheme for the Hartree-Fock and density functional crystalline orbital theory is formulated for infinitely extended periodic systems of general dimensions and implemented for those of two dimensions. Two major differences between the analytical gradient scheme for extended systems and that for molecular systems are described in detail. The first is the treatment of the long-range Coulomb interactions, which arises due to the infinite nature of the systems. The long-range effect is efficiently included by the multipole expansion technique and its extension, the fast multipole method The use of the fast multipole method enables us tomore »