Silverstone deg pert theory 1980

PHYSICAL REVIEWA VOLUME 23,NUMBER 4 APRIL1981 Practical recursive solution ofdegenerate Rayleigh-Schrédinger perturbation theory and application tohigh-order calculations oftheZeemaneffectinhydrogen Harris J.Silverstone and Richard K.MoatsDeparmentofChemistry,TheJohnsHopkinUniversity,Belimor,Marland21218(Received 29April 1980) Withtheaimofhigh-ordercalculations, anewrecursivesolutionforthedegenerateRayleighSchrbdingerperturbation-theory wavefunctionandeneryhasbeenderived.Theialformulas, AS =ROPEKey EO10OEEMIOY+OBEY yA Which involve new Hamiltonian-related operators 3¢/°"*” and SE“, strongly resemble thestandard nondegenerate recursive forms, ASanillsration, theperturbed energy coeficients forthe3-3dy stats of hydropen intheZeeman effect have been calculated recursively through 87th order inthe square ofthe magneticfield.OurtreatmenticomparedwiththatofHirschfelderandCertain[J.Chem.Phys.60,1118(1974),andsome relativeadvantagesofecharepointedout 1.INTRODUCTION that the Hirschfelder-Certain solution obtains all information consistent with agiven order ofthe Despite much work,""* degenerate Rayleigh- perturbed Schr8dinger equation, whereas thepres- Schrédinger perturbation theory (RSPT) isstill entsolution extracts information from higher anintimidating subject. Aselementary aproblem _orders oftheSchrdinger equation toobtain the asthe Zeeman effect inhydrogen requires degen- entire wave function toagiven order. The two erate perturbation theory forexcited states.* Itis methods are compared inSec. I our purpose here togive anew recursive solution ‘The standard recursive solution ofnondegenerate ofdegenerate RSPT, whose formal structure is RSPT expresses theperturbed energy and wave- almostassimpleasfornondegenerate RSPT,andfunctioncoefficients, E”?andyx“,intermsof then toapply ittotheZeeman effect (inaslightly lower-order E™ andy“?. InSec. IIwedevelop nonstandard way—ride infra) similar formulas forthedegenerate case that are The general degenerate problem appears tohave _the natural genevalization ofthe nondegenerate been treated first byHirschfelder? who developed formulas. Along theway, new operators ofthe recursion formulas for operators used togenerate _nature ofeffective Hamiltonians are introduced. the wave function yand energy Efrom the unper- _‘InSec. IVweapply the new formulas tothe Zeeman turbed wave function, Earlier general treatments, effect inhydrogen, The application parallels that particularly byBloch’ and Kato,® didnotobtain ofRefs. 6and 12similarly uses SO(4,2) tocom- true RSPT series for)and E.Silverstone’ de- pute matrix elements." The treatment here is, veloped explicit formulas for yand EoftheHuby’- however, interestingly different inthat ithasbeen Brueckner"® type and Silverstone and Holloway* cast asaneigenvalue problem fortheperturbed developed explicit formulas using Lagrange’s principal quantum number. formula." These treatments have resolved the special difficulties ofdegenerate RSPT—deter- mination ofthe“correct” zeroth-order functions IL.DEGENERATE RAYLEIGHSCHRODINGER and determination ofthe components of9inthe originally degenerate subspace. They are nicely ‘A.Premios suited for discussing the low orders ofperturbation theory, butthey are less suited forhigh-order ByRSPT, wemean that Y,E,and the Hamilton calculations, which inturn areofgreat interest ianHareallexpanded inapower series ina per- because ofrecent analytical and experimental work _turbation parameter A: connected with the Stark and Zeeman effects.°!?"7 a = =, ’Afewyears agoHirschfelder andCertain® de- a=SH, beSave, 9Daye. veloped arecursive solution fordegenerate RSPT - “= xo suitable forhigh-order calculations. The new ® recursive solution given here differs significant final working equations. Themain difference is BO70 Bon @) 168s ©1981 TheAmerican Physical Society 1616 HARRIS J.SILVERSTONE ANDRICHARD K.MOATS 2 ‘Wedenote theunperturbed eigenfunctions and en- isnot. Weseek generalizations forboth Eqs. (7) exgies by|i)andE; and(8),valid forthedegenerate case. H|i)=|’) @) ‘Thestate being perturbed is|0),andweusesyn- C.Method forsolving thefundamental RSPTequation.cnymously y=[0),EO=22, intdopantsee Letthedegeneracy of£beg.Weassume that w[0),|tyesss|e'=4 arethe“correct” zerothcorger THesolution for49inthedagenerat caseiseigenfunctions, whichmeans thattheysatisly a lessdirect. SinceHP, «0, thecompo-nested sequence ofdiagonalization conditions** nentsx0?#2074+«+4x9dropoutfromtheLeft=thatwediscuss briefly inSec.C4. handsideofBq.(6),leaving onlyx":The|#)canbeclassifiedaccordingtotheorder a en inwhichtheyfirstsplitofffrom[0).Thestates HONS =Lorene @)notdegenerate with 0)form class C,={|g), |g+1), eeseaf:Thestatesthatsplitofffrom[p)infirst graSpeen , (ao) order constitute class C,, and soforth, uptothe class C,,after which nodegeneracy remains ‘Thefundamental REPT Eq.(6)determines x!”but (unless because ofsymmetry). (Weassume that, jotyf" 4s 0, apart from symmetry, Eisanondegenerate level Buried inEq.(6),however isthedeterminationof.)Thestate[0)isinaclassbyitself.The ofx",x,...,Thesubjectofthissubsection4lothercorrect” zeroth-order degenerate cigen- jghowtoextractformulasfortheyi¥-*)fromEq,functions comprise theclasses C,,Cy)-. Cy (6).Thebasic procedure involves taking theP, (some ofwhich maybeempty). Foreach class, projection ofEq.(6),which uncovers axi¥°*, and foneeanform projection andresolvent operators: Substituting inductively obtained formulas for seh git) ve)togetyahexpreased iknee a ASO, LPPXRtogetxSPexp PeDlthROD pat. Eatively'in termsof¢withbN—a1.Alongite we*‘ theway,certain Hamiltonian- related operators Here£™istheath-order energycoefficient for areintroduced, someofwhichdetermine the statef.Notethat1=P,+P, +++»+P,+ 0]. “correct” zeroth-order eigenfunctions. Each x“ can bedecomposed into components belonging totheclasses C,: 1.Determination ofxf) KODEEYP eee, ‘Thedetermination ofx{%~!”fromEq.(6)provides Wrep.x Qv>0). 5) six-step paradigm. Step 1istomultiply Eq.(6)Perea? .0) (©)GyDP.+JONO|andtoisolatethex4"term: Weuse“intermediate” normalization: (0x?)=ye 0=(Srgspxo))amggg ostyFundamental RSPTequation for"? i ve ‘TheSchrédinger equation,Hy=0,yieldsthe +(Leefoxoi)$2Heyran|ayfundamental RSPT equationfcwit) * °ental RSOTcaation forx Step2istousea“correct”-zeroth-order-functionRy =RaedWreaeay 6) condition: thatH“)should bediagonal withinthe xZe Rey (© Gegenerate subopace, ‘Theformtenherestat Inthe nondegenerate case, thefundamental RSPT a ostfcuation yieldsthestandard recursion formulas (QoPa*OX0I)R DP for£7andx"? : -(oP,+(oxo)RP,+H,5(12) £4=(04y+52OEM«rian,oOLA0)+XeOlnvmynrrey allothertermsvanish.ThusEq.(11)onlyin-an volves xi", x2", andlower-order x(&<N x=k AHYX4*) (nondegenerate case), ~2: ) o=PH (OP,x0') bywhich£4andxareexpressed interms a oflower-order E®andx®(reN1),lnte x(100 Bmore) As degenerate case,Eq.(7)isstillvalidbutEq.(8) stsSener as) 23 PRACTICAL DEGENERATE PERTURBATION THEORY... 1oi7 Step 91stouse the“N—1" version ofEq. (10) for takes theform x") inBq.(13): : = P+ O|)RE’D) P,<page (35,+400) (EP.+mo)ae», a = +[ool)Rer +P)+P,REP,,(20) SEGempugemunren , (Lerlro)eere+ererRerP, ,@0) . ratherthanEq.(12).(iii)Instep3,intheanalog (4)ofEg. (13), itisnecessary tosubstitute the“N-2” ‘Tomake Eq.(14)look more likeEq.(8),instep _version ofEg.(10)forxs" andtheN~2 version 4wedefine theHamiltonian-related operators ofEq.(18)fory{*"*. seeandRPM: 3.Inductiveequationsforthedeterminationofx= HEMTNOROUM G>0), a‘Afterapplicationofthesixstepso~1times, Rew=eer- Ee, (16) oneobtains formulas fory{",x",y{",..., : X20),ando=1newformulasfor£”,Ineach -p.wongra(SP,+ prol)HReI casetheformulasforx9"and£”involveonly .° (otal)x®withk<a1.Theneattimethrough,17)oneobtainsforx2"thefoliowingintermediate equa-steps5and6aretotaketheP,andj0)components tionsanalogous toEqs.(11)~(17) inthexi"case:ofEq.-(17) togetaformula for‘that resem- instep1, bles Eq.(8)andaformula forE™” thatisagener- : allzation ofBq.(0: o=(Srero)arrnencn pemyornn a2) . iD> +(Er.+bol)Fwgonynen B20fe(9)0)+|e |x").a9) 1)° instep2, 2,Remarkonthedetermination ofx (Serbo)er se, ‘Todetermine ys", onefollows thesame six 7 steps thatledtoy{""?withthree mainadaptations: -(Sp.+ oe po@)Usetheprojection operator 2,"P,+0X0)on =(Learmoleygrereirr, cay Eq. (17) for step 1.(ji)Instep 2thesecond-order “correct” -zeroth-order-eigenfunction condition and ee onpapues (So, peo)(erS gers xem) es) inatep 3, ~Peeensrn'= (35P,+vol)Se(armsxeLarne, pen)aren 5 4) 5————— instep 4(here Ri =U), Insteps 5and6,theP,and[0)components ofEa. stead aetertei (27)yield y*"") andE“ interms involving onlyait =e lower-order x(<N-o-1). These equationssepSEacowgemwoop,as)seteseeraton otae)anCh Rigpasegpen gen, es) reno FEegvengren (2s) and : ext E%=(0642)(0) PR eryihee + enemies ‘extPRP=(FPywxol)SeRemy +(0|Sapiro) 9)(27) = 1648 HARRIS J.SILVERSTONE ANDRICHARD K,MOATS 2 4.Comments onthedetermination ofthe“correct” notHermitian (e.g.,*{).However, itisnottoo erothonder functionsandonthe0") difficulttoshowthat(here)=QL?ofHirsch- Inapractical caiculation oneusually starts not felder,? which isclosely related toX¢" ofSilver withthe“correct” zeroth-order functions {|0), stone,? andQi}and areknown tobeHermi- [1),..-,[g-1}, butwith arbitrary linear com- tian? binalions ofthem. Asdiscussed inmany elemen- tary texts, acondition satisfied by, and used to .Summary offormulasoftherecursivesolution determinethe“correct” zeroth-order functions 8 1mpractical calculation theorderofformulas that thegXgsubmatrixofH"bediagonal: isimportant.£mustbecalculatedbeforex°?, G2 |=6,£ (O<i<g-1, 0¢j<¢-1). because E“?appears intheformula forx. In (g0) Summarizing inthissubsection allformulas needed inpractice, weshift the indices inthe ‘Theslates forwhichBf"¥£{"constitutelassCand £0”formulas(8),(19),28),and(29) [Notethatif£{=£{ 420,thecondition (80)does {0bemoreinlinewithcomputational sequence: doesnotseparate unambiguously |i)from |j),but ‘Gong paennoneoftheperturbation formulasforstate0re- SCSH, (93)quiresuchadistinetion.] Iftherearestatesfor set)cagtorer»pn$2pearteneerWhichB28,thatIo)thedegeneracy withBg?ETMascomyFnewisnot removed completely infirst order, then theoperator H®+H“R®H must bediagonalized (oF1,20) (84) overtheremaining states: Ricease ge) —(¢50,k20) (85) i141 RH|)) GHP) =6,B" leg ,i<g-1, =O,EP (i<g-1,j<g-1, 1€C,,j¢C)). both1,76C,,Cg---Co) (ay 8) ‘Those states forwhich £2’+£2? constitute class . 1aG. RO:YEO ETpu, Co) Idegeneracy withstate0remains aftersecond reorder,itisnecessary todiagonalize third-or ygnar) Segeeyer , 9)higher-order operators over the remaining states, s ;‘ThegeneralsituationwasdiscussedbyHirsch- Beacopeco)«(0SeRegenftrs)+(9) felder,? Silverstone,® and Hirschfelder and Cer-tain’ Weemphasize againthesimilarstructure ofEqs. Forthereader notfamiliar withthose papers, (88)and(39)ofthedegenerate caseandEqs.(8) ‘wemake thefollowing remark: Equations (12)and nd(7)ofthenondegenerate case. (22)areconsistency requirements necessary to Oneadditional energy formula wehave found make theequations solvable; their resolution is tobemost useful isduetoHirschfelder? andis thesystem ofmatrix eigenvalue equations that valid bothinthenondegenerate anddegenerate determine the“correct” |i): case (with intermediate normalization); Cifee|p=6,B O<i<g-1, 0<j)<g-1; Been(0fear)0) AVECCay Cy)«(82) +DEQinemenoysy .oy ote that, cae, KE=HMHORO_Y Ifx!hasbeen determined uptosome maximum HEOROEY BORO yOROGw} —OFder,sayN,thenEq.(40)permitscalculation ofEthrough order2N+1. 4H 4HOROY , which appears inelementary texts, butthat the ILCOMPARISGN WITH EIRACHPELDER AND braced terms are identically zero inthesubspace CERTAIN'S RECURSIVE WAVE-FUNCTION SCHEME spanned by|0)andC,,C,,...,C,.) One detail, however, ismissing: that thex”) are Hermitian, __Inthis sectionwecharacterize thedifferences with ItisnottrivialtoproveHermiticity fromthere- Hirschfelder andCertain’srecursivewavefunctioncursive definition, Eqs.(1),(2),(15),(16),(25), solution,’ Theprimary focusintherecursiveand(26). Indeed, theie!" (¢>0) areingeneral scheme developed byHirschfelder andCertain is 2 PRACTICAL DEGENERATE PERTURBATION THEORY... 1619 theNth-order fundamental RSPT equation. From aytheNheandlower-order equationsyonlypartofPRYMeyer, «s) x0canbedetermined (asdiscussed inSec.I1C)— an andsimilarly forx", y,..., x". The uneRen eon critXrothehighes-ordes wavefunction that(e denen Rees “ Completely determined attheMth-order equation Rew nemRORORED (4s)Hirschfelder and Certain’s basic recursive unit isthatpartofx”which isdetermined bythe ‘Their recursive formulas are[seeEqs.(141)— RSPTequations 1throughN:thintheirnota- (143)ofRef.5],partlyinournotation, tion, Explicit inthenotation [Njisthatthe“cor- 02g) 290) eREDAeDrect”zeroth-order functions arenotcompletely xSthhine= OeBVA? + “) determined unless N>y,butwedonotneedto O8R.AxS” [Ea.(43)above], an elaborate onthat aspect here. When M~N>y, 8 theoffistheexactx: AMPSROWsLmevyny (as) X= OneOMnet (41) thegreatsimilarity inthey=1caseisthecon WhenM=N<y, oUfhex - sequence ofthecomplete determination ofx!" Incontrast, themain focus ofourrecursive from theNth-order equation. scheme istheMh-order wavefunction x". ALL Theorganizational difference ismore marked additional information needed from equations N+1, when thedegeneracy islifted insecond order: y Ne+2,..., Ney, tocompletely specify x"?is =2.Ourscheme requires twoadditional equations: built into theoperators R/"*"”. Proliferation of intermediate, incomplete wave-function-like quan- Fatitiesisavoided attheexpense ofproliferation IPR YERoy, 49) ofHamiltonian-like quantities. Thus thephilos- ° ophyoforganization iscompletely different. REM AREY ERPROMOMRORMLLY ‘Asecondary feature oftheHirschfelder-Certain (60) formulation concerns R®: R appears only atthe extreme leftofanexpression, which isespecially Their scheme requires instead, thefollowing equa- convenient iftheexpression istobeevaluated by tions [Eqs. (144)- (148) ofRef.5,again, withsome solving aninhomogeneous differential equation liberty taken withtheir notation]: vather than byusing anexplicit formula [such asEq.(87)]forR. Incontrast, R's hereare X=ORare= Ong ROA” 5 on‘embedded inthe¢{°**,‘Thepresentformalism Wy2g)4RUDAMNDSUIROAGD 52)isbestsuitedforexplicitR®asintheZeeman- Sika”PetRIMALSSaiABS bdeffect example ofSec.IV.Theformalism could Xe Xe (53) bbedeveloped intoaform directly suitable fora - Fadifferential-equation approach toR,butitwould OURROHOEROxHeyer (54)notlooksosimpleasEqs.(33)~(38), andwedo nadosoberes ee xg—ROH 5 (65) Athird, more subtle difference concerns the A=[ROU SRO +HOROHP, ;(60) so-called 2N+1 rule (Eq. (40)]. The Hirschfelder- weaCertainversion’involves$4),nd$19.9i€I>trad(HgunsHergurns $epom,won),Se ate aes SeegetesBieUSamen(mmettemeretiias 2me corresponding 4or>"isless than thecomplete 62) xorxthatappears intheoriginal Hirsch- weafeverversiosedhere.ButifbahJ>yand Kagen, (neoigpe SSnome)>, the sparer version reduces tothe original oversion. B‘Atanoperational level,thesimilarity ordis~ +P,dEr(MMoties DLmemyre) ; similarity ofthe two methods depends onthe order° yinwhich degeneracy iscompletely lifted. There (68) islittle difference if'y=1; ourrecursive formulas andasimilar equation forA‘y” are ‘Thequestion naturally arisesastowhichmethod :ismore suitable inwhich situation. Asalready co en 442) mentioned, theHirschfelder-Certain method re-ne (42) esnomodifleation tohandleRe”viahommo- 1650 HARRIS J.SILVERSTONE ANDRICHARD K.MOATS 2B geneous differential equations, while ours does. exponential factors e"”’ intheunperturbed hydro- When, however, R® enters explicitly, wefind genic wave functions. Inparticular, thesumma- thepresent method easier toprogram onacom- tion form (37) forR® does nottruncate inpractice puter, butsuch apreference isahighly subjective ‘and isnotuseful. Away around thee~’’" complica- ‘opinion oftheprogrammers. Computational ef= tion isdeveloped inthenext subsection. ficiency inaspecific case might favor one ortheotherorevenanadaptation wherethebaseunit A.Reformation ofthehydrogenicZonmaneffetstanwould beyi". Inpractice, weactually useamix- genvalue problem forthe pertarbed principal, ture for the 7=1 case worked inSec. IVfor the ‘qantum number . i ponent” andgivealltheradialfunctions thecom- adaptation ismuch closer tothe present formula-’ ‘monfactor e,weuseanenergy-dependent scale tionthantoHirschfelder andCertain’s. Inany rranefaroation withseate fastenevent, bothapproaches leadtothesame end,and Mditisnotclearthatthequestion ofsuitability is m=(CRE)! =(26geeuag+2OM) (62) well posed. 4 . 1sessentially the“perturbed principalquantum _[irschfelder andCertainalsodiscussthevari-|umiper”(n=),andthetransformed Bas.(60) ationalcalculation ofthe@{y},-Similarvariational 44(61)endupasaneigenvalue equation forn.techniques could beused here. Thereader isre- Au(0g)cy“4OnCLEUnValernesi/”) ferredtotheirpaper®forthebasicoutline,as mnandthaydropthesubecyipt “new”toobtain wellasforadiscussion ofother aspects ofde- ” generate perturbation theory, including Hirseh- =~ bots brs bytes +9) (63) der" ator solution * brane aytfelder’s original operator solutioi anfrotedreartetsy!) (64) IV,ZEEMANEFFECTINHYDROGEN =n (65) The Zeeman effect inexcited states ofhydrogen _Note thespecification ofthe perturbation param- isa familiar, real problem that requires the use eter A ofdegenerate RSPT. Itisespecially useful asan edn? (66)illustration because thex?canbeobtained in a . simple closed form. Moreover, conservation of After solving Eqs. (64) and (65) forn(A), itis ““m," conservation ofparity, and the use ofgroup _necessary tounravel 7from the implicit equation, theory greatly simplify thetechnical details. The 7)=(J08) (and inthis respect thetreatment here Zeeman-effect perturbation series isnot converg- _parallels theearlier solution oftheStark ef- entbutasymptotic; analysis ofthedependence of fect"), butsuch unraveling isasmall price the energy coefficients onNhas been reported in (Gee. IVD) forthesimplicity gained inEq. (64): Refs. 6and 12. namely, that ,_ineBeemanelfet Hamiltonian, inatomte units H=-byvtedr, (en . soLest capanecon HY=r(etey?), (68)Ranma=—40-7+bi,+0G49") H%20(e>2) (69) =ceeman © (59) thatthespectrum of#®ispurely discrete, and matrix elements ofH“? are non-Here€denotes theperturbed energy, 6=°B/ ee netthattekeweHermitian. th(2m%ce*), andBdenotes thestrength oftheuniform Zero. Wepoint ou naar tedivaden.an usual Cartesian volume element must bedivided static magnetic fieldthatpoints inthezdirection, ual Carsoaith Totus‘Theterm linear in6isconstant, since /,iscon- : : served, leading onetodefine anm-independent Hamiltonian, fi: B.Unperturbed eigenfunctions degeneracy andsymmetry T=praguebm=-3VAA-T EDM) (60) ‘Theunperturbed eigenfunctions ofH®arejust =€=€ceqnan OM» (61) theenergy-scaled hydrogenic radial functionsnen sae timessphericalharmonies ¥7,andtheunper- Equation (42)shows clearly that}0*isthenatural turbed eigenvalues arejusttheprincipal quantum perturbation parameter rather than bitself. umber n: Direct solution oftheRSPT forji,although feas- neyible, iscomplicated bythendependence ofthe Hei q=Neilnim* (70) 23 PRACTICAL DEGENERATE PERTURBATION THEORY... 1651 Tima my a (naRcogenycn ot)eerie Yaam™BuO, 9)5 (ay PREM —Fepomncnnys (78) Rul) =NyreTLSEP Rr)« (73) ‘Thematrixelements Wedonothavetodealdirectly withtheRy,(ex- resem G2+9?) Pain)“WatemeHEgr) pressed here interms ofgeneralized Laguerre ~polynomials 121°)andnormalization constants faveoemworker!otbyBendosingthegroupN,,)because allnecessary matrixelements can so(4, omc barethatolyaaia3 befound inthepaper byBedndi.?° Yarrrme connect aid um ’ ‘Thedegeneracy ofni),isn?.Butbecause mand WyeremeHHWqiq)=O,unlessm=m",7 parity are conserved, the effective degeneracy is bere .muchlower. Forgivenn,m,andparity, there H-u|=00r2, andjpn’les isdegeneracy {(- jm|+1)/2] foreven-m—even- ‘Thus inusing Eq.(81)forRandforRD, only parity orodd-m-odd-parity, and there isdegen- __afinite number ofterms ever give anonzero eracy [(n-|m |)/2] fortheother twocombinations, contribution. Consequently, allx“canbewritten where [x]means thelargest integer <x. (Note, asfinite linear combinations Ofyyy. Note alson=1mustbe>/m|). thattheenergydenominators inR®arethedif- ‘The removal ofthe degeneracy isbydiagonaliza- ferences ofintegers (principal quantum numbers). tion ofH=(x?+y*). Itappears that alldegener- For any particular state, theeventual limitation acy (1,m,and parity being fixed) isremoved in totheorder ofacalculation turns outtobestorage first order. (We have checked allcases with n<20 capacity for the expansion coefficients ofthe numerically, butwedonothave ageneral proof.) _ontheJy, Note that Eq. (40) can beused toget ‘Theonly nonempty classes areC,andC,,andthe BEfork=Naga+1through2Nq,,+1,whereNu, solution isalotsimpler than ifthere were several _isthehighest order towhich x” iscalculated. In Ce anexample tobereported inmore detail inSec. IVE, wecalculated x“? through 43rd order forthe .Afewcomputational details 3s~3d,states, andthen obtained £ through 87thorder. (Anasymptotic analysis ofthiscal- Inthe design ofacomputer program tosolve theRSPT forn(A),thepresence ofonlyC,andC, culation hasbeenreported inRef.6.) andtheabsence ofanyterms inDiy" with 7orders>2[Eq.(69)],makecertainimportant D.Unravelingn=ntjn%ba)practical simplifications. Equation (10) or(38) for The result ofsolving RSPT for(A) istheseries xf”becomes (with #®=—E" fork>2) = & n0)= Doarar. (7) er =norgnon_ 5pampgoryurn |74) 8 = Tosolve the Zeeman problem, ForE™, Eq.(7)[orEq.(39)with =O] issuf- -ficient,andonlyy{¥"»contributes: €coemanmb+COMEDY, (19) 1) pO)|ae =~ Berson?» (75) yeneedtofinethe¢tromthe0!via Forx{*, wecould useEq.(18)or(38). Butto copy Leach pepesavestoringthematrices ofRE,wetakea €aeeaan=~bo7*(b°0*) (80) simplified form from Eq. (13) (with Nincreased ‘AsintheStark effect,” Lagrange’s formula yield byD: yields theexplicit relationship SSF Oe (AN=3)1neaiee (29,0... Mey. CmannonnnDoteeceneyenyGELOMTIGGLoTTMNMH5 - abt (82) —— Also, asintheStark effect,!*" theexplicit for- = mula (81)becomes inefficient asNincreases be- a=Dyes. (63) ‘cause oftherapid increase inthenumber ofparti- ° tions ofN. Arecursion scheme can first beused ‘Then the series (83) can besquared and inverted toconvert 7toaseries in43%; byelementary recursion. 1652 HARRISJ.SILVERSTONE ANDRICHARD K.MOATS 3 z|o/SEERSELRELLLEE LESSLLELLLE SELLERLSEELESS PamEEE EETEETEEEEESELELELTET ELLE ET B|2)SESSSSEARESESESESSSERRETESTSSSSELESSSeS 3 B|SRSSRSESTSETLSSS TRSTRSSSSASSTEASELL SES S||S2gE SReeeScRSES2R2 2522229822852E822288 PePELELEEECECELEEeeeLetteEeeerei CEEeeegy SoeteCe ee en aaaie en eeee ely3 §|Pa]PERREERRERE LLELELEREELELERLEESESSELELS nFPuRCeeECCCCCCCOCCCSSCCCSCOC COSCCCLCCLeeer «£8]SoSSSESASESAS SSESESSEERE SSTESSSNeSeee S|8)SeSSSSeSSERRESESESSSESESSSEESERSEaESa)|]SEEESSHRSGEGUEEESEEECSEESaGE EESSERECSEE &|§|82222282 sRSRGRsseRRSSISS EASARESIS RSSES oeMEIPEEEEEEEEEEEEEEEEEEEEEEEPErEPEEestereter fPARTI T GSR R STIRS ToT eT agaTSggegsass g a zFSSCISSSARSSBSSRSSSSSESSSSSerReeeeeeesss &|=|BeeshbeSbeSesSeSeSeessesssSseSsesseeses BH) gb]SseSescesecsasasscasessacssesanccsseses Go| 2]SeSS2E RsTas eee SRESETS TSPESEREBEnEES3 BESSSESTRESZSSESSTLERSESLSSSSESRSEREESEae BESRSSELSSSISISSSSSERTSRSSSESTISZEEZISSGE)gd)Beers eeesageese SSeS SSEREESESSSSeeCee PPUPRIRCELTEISRISeeeCeeeeeeeteeeeeeeeeeetas BEL EE] Go959598 peq8paqegegegsgegeqeqgsqeqegegea aa/§ eaeatBaa235544332282228422822SS 598sseeseeesE8) Sa] seeeseeeeseeesseeeselseeeseeeseeeseesss gé“3SSERSESSSSSSSSASHS SENSES TESSESSaRas esas Be$)SSRSSSSSSR ESSSSR zSsARSPAS sasssassgsasa abl 3)222382892285222 222352222 2225882832223E Ze] £]ZESSERRSSRESERSSERSCSSSegSeSSERSSEERSS a2S| SAG as iGgigadeyaddidddgGgdddade dvagdgd gad a3 oiELA ie:tee EEEECECEEEEEEELETeeeeeerrrryy zea} ° 5 23 PRACTICAL DEGENERATE PERTURBATION THEORY... 1653 E.Specific example: 3¢-3dy The first case for which degenerate RSPT isa eeTTT necessity consists ofthe3s~3d,sublevel.Diag- 3/ERSR% GeatieatonofHyeldstwocosrectsorbonderBEERS eigenfunctions, |s)and|d): algiage . wana,zi22582 |s)=0.91562901|3s)-0.40202427/3d,), (84) 2oaee fa)=0.40202427/3s) +0.91562901|3d,). (85) B/e838 Wewere able tocalculate y"to43rd order, od staying ina256000-word core onaDECsystem 10KLcomputer, andweobtained £‘*? through 87th order. TheE“”agreed to13significant, PaPeerss figureswithcalculations byAdams,Avron,Cféek, 2/eees andPaldus? through 53rd order onanIBMcom- ag |SSSE2 pater byasignificantly different procedure. Such#|23832 caleulations areofimportance inunderstanding g|BEse8 theanalytic behavior of¢zsequ 86@function ofb, Feices whichisreflectedintheasymtotic behavioroftheBS535 €(eeeespeciallyRefs.6and12).Asshownin igESzs Ref.6,the€”aregivenasymptotically by weyrecy pyres2 (848) QN+9/2)1CCIare«@&)“Cia iSlesees x(0+gyteretors)5 (8s)gs} é wherefor0)=|s), thevaluesofCandA,are, “ respectively, 1.1188690267 and ~43.44276, while 3 ceuee for[)=|d) thevalues are0.0061309782718and By J/RELEE =24.65086. Thefactor (2N+9/2)! clearly indi- e] 2)eeee2 cates thedivergent asymptotic character ofthe SSe55 expansion. a/888£3 InTableIwegive€”forthetwostatesfor03|s228E <weat, aE|SSE23 Iv,SUMMARY 5 me ‘Anewrecursive solution hasbeengivenforthe energy and wave function indegenerate Rayleigh- . Schrédingerperturbationtheory(Eqs.(33)-(40)]. :~|RREEE ‘Thefinalequations [(38)and(39)]areverysimilar Ti|ee8e82 informtotheanalogous nondegenerate y‘?and aisaege E°formulas [(8)and(7)].Toachieve thesimil-83335 arity,newHamiltonian-related operators wereBSE23 defined [Eas. (33)~(35)].33538 Becauseofthestrikingsimilarity ofthenewsesca degenerate formulas withthefamiliar nondegen-PaLEEL erateformulas, tobecontrasted withthegreater Te complication ofprevious solutions,****-" wemight opine that atleast translucence has displaced opacity."* The usefulness oftherecursive for- y mulas for high-order calculations was illustrated glasses byacalculation oftheperturbed energycoefficients for the 3s-3d, states ofhydrogen inthe Zee- man effectto87thorderin16+. : 1654 HARRIS J.SILVERSTONE ANDRICHARD K.MOATS 2 ACKNOWLEDGMENT forsupporting thecomputer calculations. We . would also like tothank Professor J.0.Hirsch- Wewould liketothank Professor J.Gfek and felder andProfessor P.R.Certain forvery helpful Professor E.Harrell for helpful and stimulating discussions onthe differences between their meth- conversations and the Johns Hopkins University odand ours. 45,Killingbeck, Rep. Prog. Phys. 40,9630977). p25. 33,0, Hirschfelder, Int.J.QuantumChem.3,731 883,B.Avron,B.G.Adams,J.Clfek,M.Clay,M.L. (969). Glasser, P.Otto, and E.Vescay, Phys. Rev. Leti. 2H.J.Silverstone, J.Chem.Phys.54,2325(1971).The 43,691(1979),andreferences citedtherein. word “exclude” inthefirst sentence ofSec. Iwas ‘,W,HerbstandB.Simon,Phys.Rev.Lett.41,67 ‘meanttobe“exude.”Unfortmately, themisprint do78). was notcaught inproof. ML. Benassi, V.Greechl, E.Harrell, andB.Simon, ‘4H,J,Silverstone and,'T, Holloway, Phys. Rev. A4, Phys. Rev. Loit. 42,704, 1430(8) (1979), andrefer 219i as71). ‘ences cited therein. cant55.0.Hirschfelder andP.R,Certain,J.Chem,Phys.60,"HJ,Silverstone, B.G.Adams,J.Ci%ek,andP.Otto,iis 4974). Phys. Rev. Lett, 43, 1498 (1979), and references cited °B.G. Adams, J.E.Avron, J.Clfek, P.Otto, J.Paldus, therein. R.K, Moats, andHJ. Stiverstone, Phys, Rov. A21, ‘44.J,Silverstone andP.Koch, J.Phys. B12, L837 1914 1980), (1975), and references cited therein, 1c,Bloch, Nucl. Phys. 6,329(1958). ‘Ny,J,Siiverstone, Phys. Rev. A18,1853 (1978). 8T,Kato, ‘Prog. Theor. Phys. (Kyoto) 4,514(1949). 3,Gltek andE.R.Vrscay, Group Theoretical Methods 'R,Huby, Proc. Phys. Soc.London 78,529(1961). inPhysics, Proceedings oftheFifth Intemational 2K.A,Brueckner, Phys.Rev.100,36(1955). Colloquium (Academic, NewYork,1977),p.115.4571, Lagrange, Mm. deI’Acad. deBerlin XXIV 3,Elfek andJ.Paldus,Int.J.QuantumChem.12,875 (1770); sce Oeuvres deLagrange, eilted byM.J.~A. 77). Sorret (Gauthler-Villars, Paris, 1869), Vol.3,Sec.2, ®M.Bednf, Ann. Phys. (N.Y.) 75,905(1973).