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

A formalism for the analytical total energy gradient for Gaussian basis set Hartree-Fock and DFT for periodic systems, with particular emphasis on the applications to two-dimensional systems is given. As such, the formulas include energy gradients with respect to both atomic positions and lattice parameters and the contributions thereof from both short- and long-range Coulomb interaction. The analytical gradients of the total energy are implemented into a program applicable to general 1D and 2D systems with periodic symmetry.