An energy-minimizing mesh for the Schroedinger equation

• Published in: Journal of computational physics Vol. 83:2
• Main Authors: Levine, Z.H., Wilkins, J.W.
• Format: Journal Article
• Language: English
• Published: United States 01.08.1989
• Subjects: 657002 - Theoretical & Mathematical Physics- Classical & Quantum Mechanics; ATOMS; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DIFFERENTIAL EQUATIONS; ELEMENTS; EQUATIONS; FINITE ELEMENT METHOD; FUNCTIONS; HYDROGEN; MESH GENERATION; NONMETALS; NUMERICAL SOLUTION; ONE-DIMENSIONAL CALCULATIONS; PARTIAL DIFFERENTIAL EQUATIONS; POLYNOMIALS; SCHROEDINGER EQUATION; THREE-DIMENSIONAL CALCULATIONS; WAVE EQUATIONS; WAVE FUNCTIONS
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/0021-9991(89)90124-1

A formula is derived which gives the optimum placement of mesh points in the sense of minimizing the error in energy for a given number of degrees of freedom. The wave function is assumed to be given in a finite difference or finite element representation with polynomial completeness to an arbitrary but fixed degree. The result depends explicitly on the wave function, the kinetic energy operator, and the degree of polynomial completeness for the representation but does not depend explicitly on the potential, even in the presence of a Coulomb singularity. The optimum mesh predicted here for the hydrogen atom is compared to the widely used Herman--Skillman mesh. A 1-dimensional example is given in which the calculated error in energy displays a sharp minimum at the predicted optimal mesh density. The critical role of reproducing the analytic structure of the solution is illustrated with an additional example in one dimension. The hydrogen 1{ital s} wave function is considered as a 3-dimensional problem, and an optimal mesh density is calculated. A singular mesh density is required to account for the cusp while retaining the convergence properties of the basis set. A few percent of the available degrees of freedom are devoted to the description of the wave function cusp in the optimal mesh. {copyright} 1989 Academic Press, Inc.