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General criteria for assessing the accuracy of approximate wave functions and their densities
Author(s) -
Núñez Marco A.
Publication year - 1995
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560530106
Subject(s) - mathematics , iterated function , norm (philosophy) , limit (mathematics) , conjecture , position and momentum space , uniform convergence , convergence (economics) , position (finance) , wave function , space (punctuation) , mathematical analysis , cauchy distribution , momentum (technical analysis) , physics , pure mathematics , quantum mechanics , bandwidth (computing) , finance , computer network , linguistics , philosophy , political science , computer science , law , economics , economic growth
By means of examples, Löwdin showed that L 2 convergence of approximate wave functions ψ n to the exact ψ using the single limit lim n →∞ 〈ψ n , A ψ n 〉 = 〈ψ, A ψ〉 is not sufficient to compute accurate expectation values. It is shown that L 2 convergence is indeed a sufficient condition to compute accurate expectation values using iterated limits lim m →∝ lim n →∝ 〈ψ n , A ψ m 〉 = 〈ψ, A ψ〉 instead of a single limit. Practical conditions that guarantee the stability of single‐limit calculations are given. It is also shown that the L 2 covergence of wave functions implies the convergence in the L 1 ( R 3 )‐ norm of their corresponding densities. This permits us to prove Weinhold's conjecture that the rate of convergence of densities are greater than that of wave functions. The results are extended to the momentum space, and their equivalence with those of position space is shown. Properties of L p spaces are used to introduce the Cauchy criterion that permits us to check the convergence in norm of approximate wave functions and their densities, as well as to estimate exact errors. This is illustrated by a numerical example. © 1995 John Wiley & Sons, Inc.