Volume 65A, number3 PHYSICSLETTERS 6 March1978

SOME COMMENTS ON QUASI-BOUND-STATE CALCULATIONS

J. KILLINGBECKPhysicsDepartment,Universityof Hull, UK

Received13 December1977

Quasi-bound-stateenergycalculationsaremadefor a perturbedoscillator anda perturbedhydrogenatom. Thetheory oftheleast-squaresmethodis analysed.A simplenumericalvariantof the well-known stabilizationmethodis tested,andtheresultsarecomparedwith thosefrom the least-squaresand perturbation-seriesmethods.

When an unboundedperturbingoperator(e.g.the Variousargumentshavebeenpresentedto justifyStark-effectoperatorXz) is addedto a hamiltonian theleast-squaresapproach.but it seemsto us that theirwhichhasboundstates,theboundstatesareall de- essencecan be summarizedasfollows. The work ofstroyedin principle. For small X a time-dependentap- Titchmarsh[9] andothershasshownthat addingaproachsuggeststhat aparticle initially in somebound termsuch asXz or —-~XIrto thehydrogen-atomharnil-statewill only tunnel throughslowly to the “outer” tonian givesa hamiltonianH for which the Green’sregionof thepotential. Severaltime-independentap- function G(x, x’, E), the kernel of theintegralopera-proachesto this kind of problemhavebeensuggested. tor representationof theresolventoperator(H —- E~I

Reinhardt [1] made anumericallysuccessfulapplica- haspolesjust belowthe real L’ axis. To look for thesetion to the Starkeffectof a techniquewhichhadonly poleswe can evaluateR(~,F) = ~~(H—EY210>,vary-beenrigorouslyjustified for dilation analytic poten- ing ~ andF, to maximizeR(~,F). The optimumEwilltials [2], andwhichinvolves finding complexeigen- be the real part of thecomplex-poleenergy,sinceRvaluesfor a “rotated” hamiltonian.Methodsnot in- will havea maximumaswe passby thepole alongthevolving complexeigenvaluesinclude theleast-squares real axis. From theSchwartzinequality, or from aapproach[3,41,thestabilizationmethod usingsecular more elaboratevariationalargument.we find thatdeterminants[5,6], andthemethod of ref. [7], which ~(H—E)2fr/)~ is a lower boundon R(ç5, F). Also,usesan extra stabilizingpotential to convert the quasi- for agiven ~, theoptimumF is clearly (~IHIØ~,so weboundstatesinto true boundstates.The aboverneth- areledto minimizeods leadto calculationsquite similar to thosefor tra-ditional boundstates;in thepresentnote we report ~2(~) = (~H2I~) (~H~)2. (1)someformal andnumericalwork relatingto the least- by varyingci, taking (c5~HI~>astheoptimumF value.

squaresandstabilizationmethods. If theharniltonianH canbe givenaci-dependentpar-In ref. [3] resultsarequoted for hydrogen-atom titioning H = H

0 + V. so that ~ is an eigenfunctionofStark-effecteigenvaluesobtainedby the Rayleigh— H0, then ~

2(ci) takesthesimpleform (~V~I~)Ritz method.We note herethat,evenwith the twosimplebasisfunctionse~”andze°”, varying~ For theperturbedoscillatorhamiltonianof Benderleadsto an eigenvaluetending to minus infinity for andWu [10] thepartitioning givesthefollowing re-

any non-zerovalueof X [8]. To producefinite eigen- sults,with ~ = e_~x2(normalized),valuesamustbe constrained(at leastin the e c.!r

function) andthechoicea= 1 is suggestedby pertur- H= (—D2+ 4a2x2)+ (~Xx4+ [~— 4a2]x2), (2)bation theory.

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Volume 65A, number3 PHYSICS LETTERS 6 March1978

Table 1Perturbed-oscillatorresults.

x 0.05 —0.01 —0.03 —0.05

E(leastsquares) 0.5331 0.4923 0.4750 0.4548E (numerical) 0.5327 0.4922 0.4742 0.4507E (perturbation) 0.5332 0.4922 0.4742 0.4507IdE/dXI (minimum) — (<2 X 10—8) 1.0 X 10~ 5.0 X i0~

Table 2Perturbed-hydrogen-atomresults.

x 0.05 —0.02 —0.04 —0.06

E (least squares) —0.4266 —0.5303 —0.5610 —0.5923E (numerical) —0.4282 —0.5307 —0.5631 —0.5983E(perturbation)a) —0.4281 —0.5307 —0.5630 —0.5979IdE/dXI (minimum) — (<5 X 10~) 4.8X 10_6 40x i0~

a) Fifth order sums.

3 A2 1 1 3 x Table3= + — (-4 — 4a2)2 + ~— (4 —4a2), Nodepositionsfor oscillator (X = —0.01). (Integrationstrip

~ a~ ° a2 ~ a3 widthh = 0.02.)(3)

E Nodepositions

(~IHIci)= ~ ~ (64a3 + 4a+ 3X). (4)64 a2 0.4924 4.57, 11.21, 12.090.4923 4.75, 11.21, 12.09

The correspondingresultsfor the perturbedhydrogen- 0.4922 — 11.21, 12.09

atomproblemof Titchmarsh[9], with 0 = e~-”(nor- 0.4921 — 11.21 12.09

malized)are:

H = (_~D2— 1 D — ~‘~+ (xr + ~ ~ (5) Tables1 and2 also includenumericalresultsob-r ri \ r f tamedasfollows. A realE wasguessed,andthe Schrö-

2 dingerequationwas integratedoutfrom the origin by= a2(l — a)2 + X(l —a)+ ~h—, (6) the methodof ref. [11], the nodepositionsbeing

~a2 noted.VaryingEgives resultssuchasthoseof table3,3 ~ in which the first nodeis seento movevery rapidly

(OIHIO)= ~a2 — a +~—. (7) from 4.75 to 11.21 asE changesby 0.0001.Thisshowsthat theenergyis changingvery slowlywith X,

The resultsobtainedby minimizing ~2 are shown in if we solvetheSchrodingerequationfor theDirichlettables 1 and 2. A positive A valueis includedasa condition~ = 0 at x = X. In their discussionof theroughcheckon thetrial functions.For A> 0, (OIHIO) stabilizationmethod,HaziandTaylor [6] arguethat,— ~)givestheTemplelowerboundon the bound- by adding functionsto a basisset until the eigenvaluestateenergy.(For the Starkeffecta few preliminary stabilizes,theyare effectivelydoing the Dirichiet prob-calculationssuggestthat the Green’sfunction pole lem for graduallyincreasingX values.Wehaveherelies within a semi-circleof radius~(Ø)about(OIHIO), solved theDirichiet problemdirectly, which is farbut it is not clearwhetherthis exemplifiesa general more easythan doingthematrix calculation! Thetheorem.) quantitydE/dX is well defined,whereasthe quantity

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Volume 65A, number3 PHYSICS LETTERS 6 March 1978

dE/dN(whereN is thenumberof basisfunctions)is mit theF dependenceof phaseshiftsto be evaluated.not; Hazi andTaylor’s discussionsuggeststhat the Why thevariousmethodsagree,andwhat theygiveminimum value of sucha derivateshould be roughly us (we hopeit to be thereal part of theGreen-func-proportionalto theimaginarypartof thepole posi- tion pole position) is, despitemuch discussion[3,6,9],tion. not completelyclear.We hopethat theideasandre-

If the traditionalRS perturbedfunctions (cio, ~ suits outlinedherewill stimulatefurther discussion.

areusedas thebasisset,we knowthat thepower-seriesexpansion(in A) of the eigenvaluecorre- Referencesspondingto cio must agreeto (2n + l)th orderwiththe RS energyseries.Accordinglywe might expect Ill W.P. Reinhardt,Intern.J. QuantumChem.Sym. 10

theenergyseriesto exhibit a stabilizationeffect. This (1976)359.is just what variousasymptoticseriesdo! Tables1 and 121 B. Simon,Ann. Math. 97(1973)247.2 showtheresults(to four figures)of summingthe 131 P. FroehichandE. Brandas,Phys.Rev. A12 (1975)1.141 F.H. Read,Chem.Phys.Lett. 12 (1972)549.RS seriesup to its smallestterm. (Incidentally,the [51I. Ehiezer,H.S. TaylorandJ.K. Williams, J.Chem.Phys.hydrogen-atomStark-effectseriesbeyondA4 doesnot 47 (1967)2165.seemto havebeenevaluated).The agreementbetween 161 A.U. Haziand H-S. Taylor, Phys.Rev. Al (1970)1109.the resultsof thevariousmethodsis remarkable,if we 171 J.F. Liebman,D.Y. YeagerandJ.Simons,Chem.Phys.

Lett. 48 (1977)227.rememberthe simple form of the least-squarestrial 181 J.Killingbeck, Rept. Progr.Phys.40 (1977)963.function.We also found closeagreementbetweenthe [91 E.C. Titchmarsh,Proc. Roy. Soc.A210 (1952)30.numericalandleast-squareresultsfor thepotential [101C. BenderandT.T. Wu,Phys. Rev. 184 (1969) 1231.

2x2e Kx2, wherethe oscillationsat largex also per- Ill] J. Kihlingbeck,J.Phys.AlO (1977)L99.

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