Comparison of variational real-space representations of the kinetic energy operator

We present a comparison of real-space methods based on regular grids for electronic structure calculations that are designed to have basis set variational properties, using as a reference the conventional method of finite differences (a real-space method that is not variational) and the reciprocal-space plane-wave method which is fully variational. We find that a new definition of the finite differences method [P. Maragakis, J. Soler, and E. Kaxiras, Phys. Rev. B \textbf{64}, 193101 (2001)] satisfies one of the two properties of variational behaviour at the cost of larger errors than the conventional finite differences method. On the other hand, a technique which represents functions in a number of plane-waves which is independent of system size, follows closely the plane-wave method and therefore also the criteria for variational behaviour. Its application is only limited by the requirement of having functions strictly localised in regions of real-space but this is a characteristic of most modern real-space methods as they are designed to have a computational cost that scales linearly with system size.