A minimum principle for atomic systems allowing the use of discontinuous wave functions

Abstract

AbstractsIn this paper a variational principle proposed by Hall [I] is shown to be a minimum principle for coulombic systems. Into this principle it is possible to admit a larger class of trial wave functions than is possible in the conventional variational treatment, including wave functions with discontinuities. I t is further shown that the upper bounds given by this treatment are always at least as good as that given by the Rayleigh-Ritz method.The theory is then applied to the hydrogen atom and upper bounds to the energy are calculated for various "cut off" wave functions. It is usually possible to define an optimum "cut off" distance which minimizes the upper bound.On dtmontre qu'un principe variationnel propost par Hall [l] est un principe de minimum pour les systtmes coulombiens. Pour ce principe-ci on peut admettre une classe plus grande de fonctions d'onde d'essai que celle employte dans la mtthode variationelle conventionelle, y inclues des fonctions d'onde avec des discontinuitks. On dtmontre aussi que les bornes suptrieures donntes par cette mtthode sont toujours au moins aussi bonnes que celles donntes par la mtthode de Rayleigh-Ritz.La thtorie a t t t appliquke i l'atome d'hydrogtne et des bornes supkrieures de l'tnergie ont Ctt calcultes pour quelques fonctions d'onde avec "cut off". I1 est possible, en gtnkral, de dtfinir une distance de "cut off" optimale, qui minimise la borne supkrieure.Es wird gezeigt, dass ein von Hall [I] vorgeschlagenes Variationsprinzip ein Minimumprinzip fur coulombische Systeme ist. Mit diesem Prinzip ist es moglich, eine grossere Klasse von Vergleichsfunktionen als in dem konventionellen Variationsverfahren zuzulassen. Diese grossere Klasse enthalt Funktionen die nicht uberall stetig sind. Es wurde auch gezeigt, dass die obere Grenzen die mit dieser Methode erhalten sind, immer mindestens so gut sind wie die, die mit der Rayleigh-Ritzschen Methode berechnet sind.Die hlethode wurde auf das Wasserstoffatom benutzt und obere Grenzen fur die Energie wurden fur verschiedene Funktionen mit "cut off" berechnet. Es ist im allgemeinen moglich einen optimalen "cut off" Apstand zu definieren, der die obere Grenze minimisiert.

On energy bounds derived from the conjugate eigenvalue problem

“…'he variation-iteration techniques for Equations like (2) were first developed by Temple [9]. They have been used quite widely in nuclear problems where potential-strength eigenvalues are of direct interest When V is indefinite it may still be possible to rescue the simpler inequality (3) proposed by Hall if suitable restraints are imposed on the trial function Y , although this is somewhat contrary to the spirit of the method which is to enlarge the class of trial functions by admitting discontinuities.…”

“…Neither M nor V are differential operators, and so discontinuous trial functions can be employed to obtainfo. Application to the hydrogen atom is discussed by Hall, Hyslop and Rees [2].…”

Boundary conditions for augmented plane-wave methods

“…This was proposed by a number of authors 10,11,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39 Among practitioners of APW methods, the prevailing opinion is that discontinuous boundary conditions are legitimate. For example Shaughnessy and coll.…”

A numerical integration scheme for iterative calculations on atomic systems

“…Variational solutions of this equation have been considered by Hall and his co-workers [3] and [4] and the variation-iteration method is the obvious generalization of this approach, as was noted by Robinson and Epstein [5]. The method was appraised by Robinson, Warnock, and Anderson [6] who suggested a recursive numerical integration prescription for the solution of the radial Schrodinger equation and presented numerical results for the screened Coulomb field.…”