Analytic stress tensor with the periodic fast multipole method

An efficient algorithm for the direct space analytic evaluation of the Coulomb stress tensor in 1-, 2-, and 3-dimensional periodic systems is presented. These stress tensor components are required for energy optimizations with respect to unit-cell dimensions. The proposed scheme is incorporated into the periodic fast multipole method and has a small computational cost. Convergence problems arising from the nonzero dipole moment of the unit cell are treated with the help of fictitious charges. The accuracy of the proposed method is such that the stress tensor components for benchmark NaCl and CsCl structures agree to machine precision with those obtained by direct differentiation of the Madelung energy.