Variational plane-wave calculations in adaptive coordinates

A variational scheme is proposed to solve the partial differential equations (Poisson’s and Schrödinger’s equations) that appear in plane-wave calculations in adaptive coordinates [F. Gygi, Europhys. Lett. 19, 617 (1992); Phys. Rev. B 48, 11 692 (1993)]. The method gives accurate total energies, because, as a variational procedure, the error for the energy is second order. In a conventional calculation, the error comes from two sources: finite plane-wave expansions and finite numerical integration grids. The present scheme eliminates the second source of error by using arbitrarily large “virtual” numerical integration grids, yet the cost of the calculation has a scaling close to $O\left(N\right).$ Test calculations over atoms and small molecules show that the present scheme is more accurate (by one order of magnitude, and in some cases, two orders) than the conventional, nonvariational procedure.