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Approximate lower bounds of the Weinstein and Temple variety
Author(s) -
Marmorino M. G.,
Bauernfeind Ryan W.
Publication year - 2006
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.21268
Subject(s) - hartree , upper and lower bounds , ground state , helium atom , variational method , bound state , quantum , physics , energy (signal processing) , mathematics , atomic physics , quantum mechanics , electron , chemistry , mathematical analysis
By using the Weinstein interval or coupling the Temple lower bound to a variational upper bound one can in principle construct an error bar about the ground‐state energy of an electronic system. Unfortunately there are theoretical and calculational issues which complicate this endeavor so that at best only an upper bound to the electronic energy has been practical in systems with more than a few electrons. The calculational issue is the complexity of 〈 H 2 〉 which is necessary in the Temple or Weinstein approach. In this work we provide a way to approximate the 〈 H 2 〉 to any desired accuracy using much simpler 〈 H 〉‐like information so that the lower bound calculations are more practical. The helium atom is used as a testing ground in which we obtain approximate error bars for the ground‐state energy of [−2.904230, −2.903721] hartree using the variational energy with the Temple lower bound and [−2.919098, −2.888344] hartree for the Weinstein interval. For comparison, the slightly larger error bars using the exact value of 〈 H 2 〉 are: [−2.904358, −2.903721] hartree and [−2.919765, −2.887677] hartree, respectively. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007