A comparison between the local-energy method and the variation principle

Clark, Robert G.; Stewart, Erica

doi:10.1080/00268977100101881

Cited by 8 publications

(4 citation statements)

References 27 publications

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“…It is well known that wave functions optimized by the variation principle [13,16] give better estimates for expectation values than do wave functions optimized by other means. To determine if this is still significant with wave functions which are as near the true functions as some of those discussed here, wave functions ~4, 4J7, ~b8 and ~b9 were also optimized by the variation principle in order to calculate expectation values for comparison.…”

Section: Expectation Valuesmentioning

confidence: 99%

“…Electron positions for R= 2 B were as described previously [13], and those for R = 1 B and 4 B were set up in a similar manner.…”

Section: Wave Functionsmentioning

confidence: 99%

“…The variation principle is at its best for simple wave functions which require a minimum of numerical integration in the calculation of the mean energy. By contrast, the local-energy method [12,13], as it requires no integration, becomes relatively easier to apply as wave functions increase in quality, and is not greatly (i) exp ( -rA) + exp ( -rB) --1" 05377 [8] (ii) exp ( --crA) + exp (--crB) --1" 08651 [8] (iii) exp (--ClrA) + exp ( --clrB) [11] (vi) exp (--clR~) eosh (c2R~7) -1-10244 [8] (vii) exp (-clR~) cosh (c~Rrl)~es -1" 102623 [4] (viii) exp (-ciR,) cosh (c~R~7)(1 + czR~) -1" 1026227 [5] (ix) exp (-ciR,) cosh (c2R:7)…”

Section: Introductionmentioning

confidence: 99%

“…A full description of the local-energy method, including the choice of sets of electron positions, has already been given [13]. All integrals were either evaluated analytically or by using 26 point Gauss-Laguerre or 96 point Gauss-Legendre quadrature.…”

Section: Introductionmentioning

confidence: 99%

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Local-energy calculations on the ground state of the hydrogen molecule ion

Clark¹,

Stewart²

1971

Molecular Physics

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This paper reports some new wave functions for the ground state of the hydrogen molecule ion which give lower values for the mean energy than do previously published functions of comparable complexity. The wave functions were optimized first by the local-energy method and then by the variation principle. With wave functions of as high quality as those discussed here, the two optima were found to be very similar. The variation-principle optima, however, still gave better expectation values for one-electron operators.

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Section: Expectation Valuesmentioning

confidence: 99%

“…Electron positions for R= 2 B were as described previously [13], and those for R = 1 B and 4 B were set up in a similar manner.…”

Section: Wave Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Local-energy calculations on the ground state of the hydrogen molecule ion

Clark¹,

Stewart²

1971

Molecular Physics

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Standard deviation in reduced errors (SDRE): A criterion to assess the relative overall quality of approximate wavefunctions

Zeiss

Whitehead

1983

J Comput Chem

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On the accuracy of the many‐points local procedures

Majorino

Rubino

1973

Int J of Quantum Chemistry

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AbstractsAs until now proposed in the literature, in the local energy calculations we can distinguish "few-points" procedures, in which the number M of configurational points is strictly related to the number N of trial functions used, and statistical "many-points" procedures, in which the number M of points can be arbitrarily increased. In this gaper we demonstrate that the energy errors resulting from a "many-points" calc'ulation M points/N functions ( M > N ) can be connected in a simple way with the errors ofthe partial calculations N pointslN functions.This suggests a possible approach for the problem of the choice of the configurational points to be introduced in the calculation, and leads to a simple interpretation of the numerical meaning of the error associated with the ordinary Ritz energy. Numerical examples on the hydrogen atom are reported. (13Les calculs des energies locales qu'on peut trouver dans la littkrature peuvent &re partages dans des proc6des "a peu de points," dans lesquels le nombre M de points est exactement determine par le nombre N des fonctions utilisees, et dans des procedks statistiques "a plusieurs points" dans lesquels on peut augmenter M a volontt. Dam ce travail nous demontrons que les errcurs dans lee Cnergies qu'on comecte a un caicul *'a plusieurs points" A4 pointsIN fonetions ( M > N ) peuvent etre reliees simplement avec les erreurs des ( : ) calculs "partiaux" N pointslN fonctions.Cette relation porte un nouve approchment au prablkme du choix des points qu'on doit introduire dans les calculs locaux, et permet ainsi une simple interpretation des erreurs assocites & la mtthode bien connue de Ritz. Nous donnons des exemples numkriques sur l'atome d'hydrogkne.Die bisher in der Literatur vorgeschlagenen Lokalenergieberechnungen konnen in "wenige-Punkten"-und statistische "viel-Punkten"-Verfahren eingeteilt werden. In den ersten ist die Anzahl M von Konfigurationspunkten streng mit der Anzahl von benutzten Versuchsfunktionen in Verbindung gebracht, wahrend in den anderen die Anzahl M von Punkten beliebig vergrossert werden kann. In diesem Artikel zeigen wir dass die 676MAJORINO AND RUBINO Dieses Ergebnis deutet einen moglichen Angriffspunkt des Problems mit der Wahl von Konfigurationspunkten an und fuhrt auch zu einer einfachen Interpretation der numerischen Bedeutung des Fehlers in der gewohnlichen Ritz-Energie. Numerische Beispiele fur das Wasserstoffatom werden gegeben.

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A comparison between the local-energy method and the variation principle

Cited by 8 publications

References 27 publications

Local-energy calculations on the ground state of the hydrogen molecule ion

Local-energy calculations on the ground state of the hydrogen molecule ion

Standard deviation in reduced errors (SDRE): A criterion to assess the relative overall quality of approximate wavefunctions

On the accuracy of the many‐points local procedures

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