Variational solution of Poisson's equation using plane waves in adaptive coordinates

A procedure for solving Poisson's equation using plane waves in adaptive coordinates (u) is described. The method, based on Gygi's work, writes a trial potential ξ as the product of a preselected Coulomb weight μ times a plane-wave expansion depending on u. Then, the Coulomb potential generated by a given density ρ is obtained by variationally optimizing ξ, so that the error in the Coulomb energy is second-order with respect to the error in ξ. The Coulomb weight μ is chosen to provide to each ξ the typical long-range tail of a Coulomb potential, so that calculations on atoms and molecules are made possible without having to resort to the supercell approximation. As a proof of concept, the method is tested on the helium atom and the H_{2} and H_{3}^{+} molecules, where Hartree-Fock energies with better than milli-Hartree accuracy require only a moderate number of plane waves.