Real-space grid representation of momentum and kinetic energy operators for electronic structure calculations

We show that the central finite difference formula for the first and the second derivative of a function can be derived, in the context of quantum mechanics, as matrix elements of the momentum and kinetic energy operators on discrete coordinate eigenkets jxni defined on a uniform grid. Starting from the discretization of integrals involving canonical com- mutations, simple closed-form expressions of the matrix ele- ments are obtained. A detailed analysis of the convergence toward the continuum limit with respect to both the grid spacing and the derivative approximation order is pre- sented. It is shown that the convergence from below of the eigenvalues in electronic structure calculations is an intrin- sic feature of the finite difference method.