New Version of the Rayleigh–Schrdinger Perturbation Theory(7)

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

0 x,y 00 x,y .(37)

1

Afterasimplebuttediouscalculation,H1 0canbeexpressedasalinearcombinationoftheeigenfunc-tionsoftheunperturbedHamiltonianH0:

H1 0

21

131/453/4y 1200 689y2

26871 507x2 e x 13/20 y.

2

2

(41)

475 01 03 .21

(38)

Higherorderperturbationenergiesandwave

functionscanbeobtainedinasimilarway.Forexample,thesecond-andthird-orderperturbationwavefunctionsequal

The rst-orderperturbationenergyequalszero:

E1 0 H1 0 0.

(39)

2

51

131/453/4 59250448263

5426185752x2 25233511680x2y2 12788016384x2y4 588128112x4

4705024896x4y2 e x 13/20 y

2

2

94848624000y2 45306655680y4 8689293184y6

Thecorrespondingeigenfunctioncanbecalculatedfromtheequation

1 H0 E0 1 E1 H1 0,

withtheresult

(40)

and

(42)

3

251

131/453/4y 267842001350803600 1402374103429392171y2

652300246187820708192 753269605832315520y4 152737767077163840y6 14661974436247424y8 63207691842935367x2 441043571586179016x2y2 183753938493178560x2y4 32367000170583936x2y6 25884928192802328x4

55488811253153616x4y2 23817226540618368x4y4 2190735460104048x6

5841961226944128x6y2 e x 13/20 y.

2

2

(43)

Thesefunctionsagreeverywellwiththenumeri-callycalculatedonesandareshowninFigures1–4.Itisseenthattheground-stateperturbationfunc-tionsgotozeroattheboundariesoftheintegrationregionx [ 11,11]andy [ 11,11].Withincreasingn,thefunctions nspreadoutfromthecentreoftheintegrationregionandtheirabsolutevaluegoesup.

Innumericalcalculations,theintegrationregionx [ 11,11],y [ 11,11]wasused.Thenumer-icalandanalyticalground-stateand rstexcitedstateperturbationenergiesareshowninTablesIV–VII.

Onlythedigitswhichagreeinthecalculationsforthepointsx [0.2,0.5]andx [0.6,0.8]areshown.Dependenceoftheresultsonthechoiceofthepointxissmall,asintheprecedingexamples.Thenumericalresultsagreewiththeanalyticoneswithanaccuracyof6–9digits.

Becauseoftheantisymmetryoftheperturbationpotential(32),theodd-orderperturbation

energies

FIGURE1.Ground-statezero-orderwavefunction 0

fortheHe´non–HeilesHamiltonian(31,32).

INTERNATIONALJOURNALOFQUANTUMCHEMISTRY331