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

TL;DR

This paper derives finite difference formulas for quantum momentum and kinetic energy operators using a grid basis, analyzing their convergence and intrinsic properties in electronic structure calculations.

Contribution

It provides a derivation of finite difference matrix elements for quantum operators and analyzes their convergence behavior in electronic structure methods.

Findings

Finite difference formulas are derived as matrix elements in a grid basis.

Convergence toward the continuum limit depends on grid spacing and approximation order.

Eigenvalues from finite difference methods tend to converge from below.

Abstract

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 using, as a basis set, the discrete coordinate eigenkets $\vert x_n\rangle$ defined on the uniform grid $x_n=na$. Simple closed form expressions of the matrix elements are obtained starting from integrals involving the canonical commutation rule. A detailed analysis of the convergence toward the continuum limit with respect to both the grid spacing and the approximation order is presented. It is shown that the convergence from below of the eigenvalues in electronic structure calculations is an intrinsic feature of the finite difference method.