New Version of the Rayleigh–Schrdinger Perturbation Theory(3)

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

pointswheretheconditions(10)and(11)arenotobeyed.

IftheperturbationenergyEniscalculatedfromEq.(12),thecorrespondingperturbationfunction n(En,x)canbefoundfromEqs.(7)–(9).Morede-taileddiscussionofthemethodisgivenin[6,9].TheenergyE1computednumericallyfromtheequation

E1 x0

f0 x0 0(13)

Nowwediscussthe rst-orderperturbationcor-rectiontoadegenerateeigenvalueE0.AssumingthattheenergyE0isd0-timesdegenerateandthecorrespondingzero-orderfunction 0inEq.(6)isreplacedbythelinearcombination

j 1

a

d0

j 0 j 0,

(16)

itfollowsfromEq.(6)thatEq.(7)canbegeneral-izedas

dependsslightlyonthechoiceofthepointx0.Cal-culating 1fromEq.(7)forn 1,weobtainthe

function

f0 x0 F x f0 x F x0

1 E1,x .

0(14)

1 E1,x E1where

j 1

a

d0

j 0F j x

j 1

a

d0

j j 00f x ,(17)

j

x F j x H0 E0 1 0(18)

Itshowsthatthefunction 1calculatedinthisway

equalszeroatthepointx0:

and

j j f0 x H0 E0 1H1 0 x .

1 E1,x0 0.(15)

(19)

Therefore,theusualorthogonalitycondition,

0 1 0,isnotful lledinnumericalcalculations.Itcaneasilybeshownthatthisresultcanbeex-tendedtoallfunctions n.Asshownin[6],suchfunctionscanhaveinsomecasesmoresimpleformthantheusualperturbationfunctions.Ifnecessary,thefunctions ncanbemadeorthogonalto 0bytheusualorthogonalizationprocedure.

Therefore,thepointxusedinthecalculationoftheenergy(12)shouldbesuf cientlydistantfromthepointswherethefunction 0(x)equalszero.Ourmethodisaremarkableexampleofcalculat-ingtheperturbationenergiesEnfromthevaluesofthefunctionsF(x)andfn 1(x)paringwiththestandardformulationoftheperturbationtheory,large-ordercalculationsaresimpleinourmethod.Todeter-mineE1,thevaluesofF(x)andf0(x)atonlyonepointxaresuf cient.TodetermineEnforn 2,3,...,onlythevalueoffn 1(x)atthepointxistobecomputed.Therefore,thismethodofcalculatingEnismuchfasterthantheusualshootingmethod.Wenotethatthezero-orderfunction 0hastobefoundonlyforthestateforwhichtheperturbationcorrectionsarecalculated.Incontrasttotheusualperturbationtheory,otherzero-orderenergiesandwavefunctionsarenotneededinthecalculation.

ItisseenfromEq.(17)that 1(E1,x)dependsonE1linearlyasinthenondegeneratecase.Therefore,byanalogywiththenondegeneratecase,wecande-rivetheformulafortheperturbationenergyE1:

E1

j j 0

¥jd 1a0f0 x 0j 1a0F.(20)

j)(j)(j)

Here,f(0(x)andF(x)areknown,andE1anda0

j)

aretobefound.To ndE1anda(0,weexploitthefactthattheenergyE1isaconstantanduseEq.(20)atd0differentpointsx x1,...,xd0insidetheintegrationregion.Thesolutionoftheequations j j 0

¥jd 1a0f0 x1 0j 1a0F1 j j 0

¥jd 1a0f0 x2 0j 1a0F2 ···

(21)

j j 0

¥jd 1a0f0 xd0 0j 1a0Fd0j)

thenyieldsd0setsofthecoef cientsa(0.Thed0valuesoftheenergyE1aregivenbyEq.(20),andthecorrespondingperturbationfunctionsequal

1

j 1

a

d0

j 0 j 0.

(22)

INTERNATIONALJOURNALOFQUANTUMCHEMISTRY327