New Version of the Rayleigh–Schrdinger Perturbation Theory(5)

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

NEWVERSIONOFTHEPERTURBATIONTHEORY

TABLEII______________________________________

PerturbationenergiesEnforthegroundstateoftwocoupledMorseoscillators(29);E0istheexactzero-orderenergy.

n0123456789

En

3/20.2500000 0.0696350.03903 0.03660.0920 0.6116.43 89.21550

3.3.BARBANISHAMILTONIAN

AsanotherexampleofusingourmethodwecalculatedtheperturbationenergiesforthegroundstateoftheBarbanisHamiltonian:

2 2

2222

H xx yy xy2.

(30)

bymeansofthedifferenceequationmethodsug-gestedin[11].Itisalsoseenthatthemethodcanbe

usedatlargeorders.Therefore,theseresultsarequitesatisfactory.

3.2.COUPLEDMORSEOSCILLATORSAsasecondtest,wechoseamoredif cultprob-lemwithonlyoneboundstateofthezero-orderHamiltonianwhenthestandardperturbationthe-oryyieldsthe rst-ordercorrectionE1only.TheperturbationenergieswerecalculatedforthegroundstateoftwocoupledMorseoscillators: 2 2

H 1 e x1 2 1 e x2 2

12

1 e x1 2 1 e x2 2.

(29)

Here,weusedthefrequencies x 1and y 1.ThisHamiltonianhasbeenoftenstudiedasasim-plemodelforsystemswiththeFermiresponsessuchastheCO2stretchbendresonance.PotentialinthisHamiltonianisnotboundedfrombelow.

Theground-stateperturbationenergieswerecal-culatedbothnumericallyandanalytically(seeTa-bleIII).

Inthenumericalcalculation,theintegrationre-gionx [ 11,11]andy [ 11,11]wasused.Similarlytotheprecedingexamples,onlythedigitsthatagreeinthecalculationsforthepointsx [0.2,0.5]andx [0.6,0.8]areshown.Again,theener-giesdependonthechoiceofthepointxonlylittle.ThenumericalresultsagreewithanindependentanalyticcalculationmadeinMaple[12]withanaccuracyto6–9digits.DuetotheantisymmetryoftheperturbationpotentialinEq.(30),theodd-orderperturbationenergiesE1,E3,...equalzero.´NON-HEILESHAMILTONIAN3.4.HE

Asthelasttestofourmethod,theHe´non–Heiles

Hamiltonianwasused.Theperturbationenergieswerecalculatedforthegroundstateandthe rstexcitedstateofthezero-orderHamiltonian

1 21221 2122

x yH0

xy

withtheperturbationpotential

H1 yx2 y3.

(32)(31)

Theintegrationregionx1 [ 12,20]andx2

[ 12,20]wasused.Theground-stateperturbationenergiesareshowninTableII.

Onlythedigitsthatagreeinthecalculationsforthepointsx [4,4]andx [5,6]areshown.ThevalueofE1 0.25wasveri edbyanalyticcalcula-tion.DecreasingaccuracyoftheenergiesEnwithincreasingnisduetothefactthatthefunctions nspreadwithincreasingnrapidlyandgetoutoftheintegrationregion.Theseresultsshowthat,incon-trasttothestandardperturbationtheory,ourmethodcanbeusedforcalculatinglarge-orderper-turbationenergiesevenifthereareonlyafewzero-orderboundstates.

Thefrequencies x 2, y 1.3,andtheparameter 1correspondingtoavery atandshallowpotentialminimumwereused.SimilarlytotheBar-banispotential,theperturbationpotentialisnotboundedfrombelow.

TheanalyticenergiesandwavefunctionsoftheHe´non–Heilespotentialcanbecalculatedasfol-lows.

First,weexpresstheeigenfunctionsofH0astheproductsoftheeigenfunctionsoftheharmonicos-

INTERNATIONALJOURNALOFQUANTUMCHEMISTRY329