New Version of the Rayleigh–Schrdinger Perturbation Theory

ABSTRACT: It has been shown in our preceding papers that the linear dependence of the perturbation wave functions on the perturbation energies makes possible to calculate the exact perturbation energies from the values of the perturbation wave functions co

NewVersionoftheRayleigh–Schro¨dingerPerturbationTheory:Examples

ˇKALHOUS,1L.SKA´LA,1J.ZAMASTIL,1J.CˇI´ZˇEK2MILOS

1

CharlesUniversity,FacultyofMathematicsandPhysics,KeKarlovu3,

12116Prague2,CzechRepublic2

UniversityofWaterloo,Waterloo,OntarioN2L3G1,Canada

Received22January2003;accepted10October2003

Publishedonline9June2004inWileyInterScience().DOI10.1002/qua.20041

ABSTRACT:Ithasbeenshowninourprecedingpapersthatthelineardependence

oftheperturbationwavefunctionsontheperturbationenergiesmakespossibletocalculatetheexactperturbationenergiesfromthevaluesoftheperturbationwavefunctionscorrespondingtoarbitrarilychosentrialperturbationenergies.Theresultingversionoftheperturbationtheoryisverysimpleandcanbeusedatlargeorders.Inthispaper,thismethodisappliedtoafewproblemsanditsnumericalpropertiesarediscussed.©2004WileyPeriodicals,Inc.IntJQuantumChem99:325–335,2004

1.Introduction

0 1 2 2 ···,E E0 E1 2E2 ···,

(3)(4)

I

nthispaper,weareinterestedintheperturba-tiontheoryfortheboundstatesoftheSchro¨d-ingerequation:

H x E x .

(1)

where ingthese

assumptionsintheSchro¨dingerequation(1),weobtainthewell-knownequationsforEnand n,

H0 0 E0 0

(5)

AsusualintheRayleigh–Schro¨dingerperturbationtheory,weassumetheHamiltonian,wavefunction,andenergytobeintheforms

H H0 H1,

(2)

and

H0 n H1 n 1

i 0

E

n

i

n i

,n 1,2,....

(6)

Correspondenceto:L.Ska´la;e-mail:skala@karlov.mff.cuni.cz

InternationalJournalofQuantumChemistry,Vol99,325–335(2004)©2004WileyPeriodicals,Inc.