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Real‐space grid representation of momentum and kinetic energy operators for electronic structure calculations
Author(s) -
Ninno Domenico,
Cantele Giovanni,
Trani Fabio
Publication year - 2018
Publication title -
journal of computational chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.907
H-Index - 188
eISSN - 1096-987X
pISSN - 0192-8651
DOI - 10.1002/jcc.25208
Subject(s) - discretization , eigenvalues and eigenvectors , mathematics , convergence (economics) , kinetic energy , mathematical analysis , finite difference , coordinate space , physics , quantum mechanics , geometry , economics , economic growth
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 | x n 〉 defined on a uniform grid. Starting from the discretization of integrals involving canonical commutations, simple closed‐form expressions of the matrix elements 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 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. © 2018 Wiley Periodicals, Inc.