the method of complex coordinate rotation and its applications to

THE METHOD OF COMPLEX COORDINATEROTATION AND ITS APPLICATIONSTO ATOMIC COLLISION PROCESSES

Y.K. HO

Departmentof Physicsand Astronomy,Louisiana State University, Baton Rouge,Louisiana, 70803,U.S.A.

NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM

PHYSICSREPORTS(Review Sectionof PhysicsLetters)99, No. 1(1983)1—68. North-HollandPublishingCompany

THE METHOD OF COMPLEX COORDINA~-ROTAQANThITSAl’!LICATIONS TO ATOMIC COLLISION PROCESSES

Y.K. HODepartment of Physics and Asironomy,LouisianaStateUniversity,Baton Rouge,Louisiana, 70803,U.S.A.

ReceivedMarch1983

Contents:

1. Introduction 3 4.2. Resonancesin systemsof more than threecharged2. The complexrotationmethodfor atomicresonances 4 particles 40

2.1. The spectrum of the rotated Hamiltonian; bound 5. Applicationsto molecularresonances 51states,resonantstates,andscatteringstates 4 6. Starkeffect in atoms 54

2.2. Relationto theFeshbachprojectionformalism 10 7. Photoionizationcrosssectioncalculations 552.3. Relation to theclosecoupling approximation 12 8. Otherapplications 572.4. Other theoriesfor atomicresonances 14 8.1, Partialwidthscalculations 57

3. Computationalaspectsfor atomicresonances 16 8.2. Scatteringamplitudecalculations 593.1. An examplefor computationaltechnique 16 8.3. Structuresin e—W scattering 603.2. Thecomplexvirial theorem 18 9. Summaryanddiscussions 623.3. Complexeigenvalueproblems 19 References 63

4. Applicationsto atomicresonances 21 Note addedin proof 684.1. Threebodyproblemswith Coulombinteractions 22

Abstract:Therecentdevelopmentsof themethodof complexcoordinaterotationarethesubjectof this review.Thetheoreticalandcomputationalaspects

of this method will be discussed,with emphasison atomic resonancecalculations.Furthermore,applicationsto otherareassuch as molecularresonances,Starkeffect in atoms,photoionizationcrosssectioncalculations,etc., will alsobe discussed.

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YK. Ho, Themethodof complexcoordinaterotation 3

1. Introduction

In recentyearsa methodto calculate atomic resonanceshasattractedconsiderableattention.Thismethodwhich is basedon the mathematicaldevelopmentsof Aguilar, BalsveandCombes[1, 2], andofSimon [3] is often referredto as the methodof complexcoordinaterotation(complexrotation in short)or the method of dilatation analytic continuation.Resonancesare of coursecommonphenomenainvarious areas of atomic physics and chemical physics. They frequently appear in electron—atomscatteringcrosssectionsas well as in photoionizationcrosssections.Severalresonancesin positron—atom scattering, though yet to be confirmed by experiments,have been predicted by theoreticalcalculations.In electron—moleculescattering,resonancesalsoplay an importantrole in the low energyscatteringphenomena,i.e. in the dissociativeattachmentandrecombination,associativeionization,andPenningionization,etc. Otherphenomenonthat can be examinedfrom the resonantviewpoint is thefield ionization of the atomsand molecules.

The advantageof the complex coordinaterotation to investigateresonancesis that resonanceparameterscan be obtainedby usingL2-type (bound-state-type)wave functions.This methodin whichasymptotic wave functions are not necessarilyincluded appearsto have a great computationalad-vantage.Beforewe discussthemathematicalandcomputationalaspectsof the complexrotationmethodlet usbriefly reviewthe generalbackgroundon different methodsto calculateatomicresonances.Fromthe computationalview point, thereareroughly four methodsby which aresonanceis calculated.Thesemethods,which will be discussedin moredetail throughoutthis review, are

(1) from the scatteringview point;(2) from the point of view that a resonanceis a quasi-boundstatein the scatteringcontinuum;(3) from the point of view that the resonanceis relatedto the exponentiallydecayingstate;(4) from the point of view thataresonanceis relatedto the complexeigenvalueof the Hamiltonian.In the first approach,resonancesare relatedto the structuresin the crosssections.Therefore,the

detail of the resonancecan be analysedby using the Breit—Wigner profile oncethe scatteringwavefunctions in the vicinity of the resonanceare obtained.In the secondapproach,Feshback[4] haselegantly developeda formalism that establishedresonancephenomenainto a part of the formalscatteringtheory. The use of this formalism to atomic physics has producedfruitful results. Thecomputationalaspectof such a theory is that the resonanceposition is approximatedby the eigenvalueof QHQ, the closedpart of the Hamiltonian.The width is aresultof the interactionbetweenthe openandclosedchannelsof the wave function, and is relatedto the off-diagonal matrix elementsof PHQ.The calculation of the widths in the Feshbachapproachis done by using a Fermi—Goldenrule typeformula. In the first two methods,the calculationsof the widths require the useof continuumwavefunctions.The third method,which, from the time dependentpoint of view, has attractedrelatively lessattention.The advantagesandthe disadvantagesof this methodfrom the computationalpoint of vieware not yet cleared.The fourth approach,which is a direct calculationof the resonanceeigenvalueofthe Hamiltonian, hasnot receivedenthusiasmamongatomic theoristsbefore the methodof complexcoordinaterotation becameknown. Since the resonancewave function (the so-calledSiegert [5] wavefunction) diverges at the complexresonanceeigenvalue,this direct approachseemedto havegreatcomputationaldifficulty. This methodwas unfortunatelyinterpretedas ‘tried andrejected’(seeref. [6]).

It is howeverat thispoint that the methodof thecomplexcoordinaterotationcomesinto thepicture.It will be shown later in this review, that by the useof the transformation,r—* r exp(iO), the divergentresonancewave function would becomeconvergent.In fact the wave function becomesL2. The ideathat resonanceparameterscan be obtainedby using only L2 (bound statetype) wave functionsand

4 YK.Ho, The methodof complexcoordinaterotation

without the explicit scatteringcalculationsor without the useof the continuumwave functions,is veryattractive.Furthermore,in atomic physics in which particlesare interactingwith Coulombforces, tocarry out the transformationr -+ r exp(iO) computationallyis quite straightforward.The kinetic andpotentialoperatorswould simply scaleas exp(—2i0) andexp(—iO) respectively.As-a result, resonanceparameterscan be obtainedby simplemodificationsof the existentboundstatecalculationalprograms.Becauseof thissimplicity, manyworkershavebeenmotivatedto use sucha methodto calculateatomicresonances.A greatdealof activitieshavebeengeneratedeversince the mathematicalaspectof thismethodbecomeknown, and computationallydemonstratedas an encouragingtechniqueby the earlywork of Nuttall [7], Doolen[8] andtheir coworkers[9—11].

This review addressessuch activities as well as the relateddevelopmentsby the method of thecoordinaterotation. The objectivesof the present review are: (1) to discuss the theoretical andcomputationalaspectsof such a method,as well as the relation to other establishedtheoriessuch asclose coupling and Feshbachmethodsto calculate atomic resonances;(2) to provide a survey of theapplicationsof this method,with emphasiseof the highly successfulatomic resonancecalculationsinvolving simple systemsfor electrons/positronscolliding with atoms/ions;(3) to discussthe extensions-

of the complex rotation method to calculate a many electron resonance.Such extensionsare themethodsof complexwave functionsandSiegertfunctions,etc.

In addition to the applicationsto atomicresonancecalculations,themethodof coordinaterotationorthe use of complex transformationhas also been applied to calculate scatteringamplitudes,pho-toionizationcrosssections,Stark effect in atoms, and resonancein electron—moleculescattering,etc.The recentdevelopmentson theseareaswill alsobe discussed.

We should point out here that over the past severalyearssomecomments,progressreports,andreviews,whichreflect thestateof the art in variousstagesof thedevelopmentof this method,appearedin the literature.Theseincludea numberof articlesin ref. [12], commentsby Nuttall [13]andby TaylorandYaris [14], a progressreport by McCurdy [15] on molecularresonances,andreviews preparedbyJunker[16]on the methodof complexwave functions,andby Reinhardt[17]who gaveanoverviewonvariousaspectsof complexcoordinaterotation.

2. The complex rotation method for atomic resonances

Sincethe most successfulaspectof the complexrotation methodis to calculateatomic resonances,we will devotea greatdeal of the discussionsin this review on atomic resonancephenomena.Thissectiondealswith the theoreticalaspectof the methodof complexcoordinaterotation.We will firstdiscussthe spectrumof the Hamiltonian before and after the rotationof the coordinateis made.Wethen relate the parametersobtained by the coordinaterotation to those obtainedby other com-putationalmethodssuch as the Feshbachprojection techniqueand closecoupling approximation.Atthe endof this sectionwe will alsoprovide a surveyof variousmethodsto calculateatomicresonances.

2.1. Thespectrumofthe rotatedHamiltonian; boundstates,resonantstates,andscatteringstates

Beforewe discussthe spectrumof the complexHamiltonian,let us first recall the usualspectrumofthe untransformedHamiltonian for an (N+ 1) particle system,where N representsthe numberofparticlesof the targetsystem.Fig. 1 showsthe usualstandardlocationsfor the polesof the S-matrix inthe complexk plane. When thesepoles are transformedinto an energyE planeby a two to one

YK. Ho, The methodof complexcoordinaterotation 5

Im(k)

________________ Re(k)+ x

+ x+ X

Fig. 1. Polesof theS matrix in thecomplexk plane.Herewe usetheonechannelproblemfor simplicity.

mapping,their locationsareshownin the top part of fig. 2. In thefigure therearethreedistinctpartsofthe spectrumwhich arereleventto the presentdiscussions;

(1) Boundstatepoles of the (N+ 1) particlesystem(if theyexist) arelocatedat the negativesideofthe real axisin the complexE plane.

(2) A seriesof branchcutsextendsfrom the elasticand all inelasticthresholds(for the N,N — 1,...etc. particlesystems)to theright handinfinity (+oo)of theenergyplane.The choiceof thebranchcutsinsucha manneris of coursebasedon thetraditional convention.For example,in the single channelcase,the upper half k-planeis mappedonto the physical sheet(or called the first E-sheet)of the energyRiemannsurface,andthe lower half k-planeis mappedonto the unphysicalsheet(or called the secondE-sheet).

(3) Resonantpoles are locatednearthe cuts andarehiddenin higher sheetsof Riemannsurface.Herewe arenot interestedin the poles that are far away from the cuts. In such a case,the resonant

~ ~,//THREsHo::s

RESONANCESHIDDEN

8OUND STATES

RESONANCESEXPOSED

ROTATEDCUTS

Fig. 2. The spectrumof H andof H(9) in the energy(E) plane.Underthe complextransformation r -~r exp(iO)bound stateenergiesremaininvariant;cutsthatbeginatthescatteringthresholdswill berotateddownwardwith anangle20; theresonancepoleswill beuncoveredby theCutswhen0is largerthaniArg(E,~).

6 YK.Ho, The methodof complexcoordinaterotation

width is so broad that the contribution to the cross sectionsfrom the resonantand non-resonantbackgroundarenot clearlyseparated. -

To locateresonantpoles,analyticcontinuationsfrom the physicalscatteringregionsto the unphysicalRiemannsurfacescan be used.The methodof coordinaterotationcan be viewedas oneof the analyticcontinuations.In such a method,all the inter-particleradial coordinatesrq are transformedinto

r11—4r~e~8 (2.1.1)

where 0 is real and positive. Although in mathematicalsense,the form of eq. (2.1.1) is not the onlytransformationto exposeresonances(for example,later on in this reviewwewill look at anotherformof transformation,the so-calledcomplextranslation, to studythe Stark effect), for practicalpurposes,however, such a transformation has a great computationaladvantagefor systemswith Coulombinteractions.The kinetic part of the Hamiltonianwould scaleasexp(—2i0), andthe potentialpartof theHamiltonian would simply scale as exp(—iO). Under such a transformation,one just calculatesthekinetic andpotentialmatrix elementsseparately,andthen scalesthem accordingto the abovescheme.Resonancescan be examinedoncethe complexeigenvalueproblemis diagonalized.

Under the transformationof eq. (2.1.1), the spectrumof the (now so called rotated)Hamiltonianistransformedto the following (seefig. 2);

(1) The boundstatepolesremainunchangedunderthe transformation.(2) The cutsarenowrotateddownwardmaking an angleof 20 with the real axis.(3) The resonantpoles are “exposed” by the cuts oncethe “rotational angle” 0 is greaterthan

~Arg(Eres),whereEres is the complexresonanceenergy,i.e.,

Eres= Er — iI’/2 = EI et~ (2.1.2)

with f3 the phasefactor of the complexenergy,andEr andF the usualresonanceposition andwidth,respectively.

The mathematicalproofs for the aboveresultswereprovidedby Aguiler, Balslev andCombes[1, 2]and by Simon [3] for dilatation analytic Hamiltonians(with potentialsthat vanish in the asymptoticregion).Potentialswith Coulombinteractionsareincludedin the theorem.In the following we will firstdiscussthepropertiesof the resonantpoles,andthen through a simple quantummechanicalexample,the propertiesof the boundandscatteringstatesof the complexHamiltonianwill alsobe discussed.

(I) ResonantstatesLet usexaminea potential scatteringproblemwith ausualtextbooktreatment[18],

u,+ (k2_ U(r)— ~ ~)ut = 0 (2.1.3)

whereU(r) is a short rangepotential.Theradial solutionof eq. (2.1.3),which vanishesat theorigin, hasa form of

u, = C(f,(k, r)+ (—1)’~’S,(k) f1(—k, r)), (2.1.4)

whereC is a complexconstant,S denotesthe S matrix, and f the usualJost function which behaves


asymptoticallyas

f,(±k,r) r’~exp{—i(±kr—(lir/2))}. (2.1.5)

The locations of the S matrix poles in the k planeare shown in fig. 1. For example,bound statessolutionscorrespondto the zerosof the S matrix on thenegativeimaginary axis, anda resonantstatecorrespondsto the vanishingof the S matrix at the lower k plane.Therefore,in the asymptoticregion,the resonancewavefunction becomes

u,(r) r~ooeth. (2.1.6)

In otherwords, only the outgoingcomponentexists for the resonantwave functions.Of course,such atreatmentfor resonanceshas long been discussedby Siegert [5], and the condition (eq. (2.1.6)) of aresonanceis often referredto as Siegertboundarycondition. At a resonancewith momentumk andenergyEyes,we have

k = Iki e~ with f3 = ~Arg(Eres). (2.1.7)

At suchanenergy,eq. (2.1.6)becomes

u, exp{ilklr exp(—if3)} = exp{ilklr cos/3} exp{IkIr sin~3}. (2.1.8)

It is now evident thatthe wavefunction divergesat acomplexresonanceenergy.The physicalpictureofthe asymptoticdivergenceof theresonantstatecan bebestdescribed(seeref. [19])in an analogywith adecaying star. The exponential decay of the intensities of such a star will be outgrown by thegeometricalr~2decreasein distance.As a result, the intensity observedat a far distance(which wasemitted at an earlier time) will be greaterthanthe intensityobservedat a shorterdistancewhich wasemittedat a later time.

The asymptoticdivergenceof the resonantwave function has causeddifficulties in the resonancecalculations(seeref. [20]for example). In the methodof complexrotation,the radial coordinatesaretransformedaccordingto eq. (2.1.1), and the asymptotic form of the resonancewave function nowbecomes

u, -~ exp(i~k~et~IrIe’°)= exp(i~kIri e1~°~)

= exp(iiki ri cos(0— /3)) exp(—ik~in sin(0 — 13)). (2.1.9)

It is now apparentthatwhen ir/2> (0—/3)> 0, thewave function becomesasymptoticallyconvergent.The wavefunction nowdecaysexponentiallywith an overalloscillatoryfactor.The resultsof eq. (2.1.9)lead to the conclusion that the resonantparameters(both resonanceposition and width) can beobtainedby using bound-statetype wave functions(or so calledthe L2 wave functions).The oscillatorytermin eq. (2.1.9) would, however,causesomedifficulties sinceit now requiresa relatively largebasisset in order to obtainaccuratecalculations.Sucha behaviourwas indeeda commonfinding in complexrotationcalculations.

8 Y.K.Ho, Themethodof complexcoordinaterotation

(II) BoundstatesIn order to seehow theboundstateenergiesare invariantunderrotation,let us first look at a one

dimensionalsquarewell problemas an example[211.Before we apply the coordinaterotation, theSchroedingerequationis

—~~~P+U(x)!P=E~I’ with ~P(0)=O (2.1.10)

and

U=0, 0<x<a

=V, a<x<cc.

In the region0 <x <a (denotedasregionI) thewavefunctionhasa regularsolution(which vanishesatthe origin)

~P1(x)= A sin(kx), k = V2E. (2.1.11)

In region a <x <m (denotedasregionII), theSchroedingerequationbecomes

V~1’=E~. (2.1.12)

For V> E (boundstates)the regularsolutionhasa form of

!P11(x)=Ce~~ (2.1.13)

where C is a constantand K2 = 2(V — E). The boundstateenergycondition canbe obtainedby using

the continuation of the logarithmic derivative at the boundary x = a for the wave functions in theregionsI and II. A standardtextbooksolutionfor theboundstateenergyequationis

tan(ka)= —k/K. (2.1.14)

Now if we rotate the Hamiltonian by the use of x -+ yexp(iO) (we usethe notation of y heretoemphasizethat it is the transformedcoordinateof x), the Schroedingerequationbecomes

— e~°~ ~1’+U(y e’°)~P= E~P, (2.1.15)

with !t’(O) = 0. Again in the innerregionwhere V= 0 the regularsolutionfor ~1’(y)is

= A’ sin(ky e’°) and k = \/2E. - (2.1.16)

In regionII whereV(y)= a, the regularsolutionfor !1’ is

= C’ e”~’exp(iO) (2.1.17)


whereA’ and C’ areconstantsand K in principle,could be complex.The boundstateenergyconditionagaincan be obtainedby the continuationof the logarithmic derivativesof thewave functionsat theboundary. But since we have transformedx -~ y exp(iO), the boundary for x = a should now betransformedinto y = a exp(—iO). If we matchthe wave functionand its derivativeat this newcomplexboundarywewill eventuallyobtain thesameenergyconditionsfor theboundstates,i.e.,

tan(ka)= —k/K. (2.1.18)

The conditionsare identical to eq. (2.1.14). This means that the boundstateenergies(if they exist)would be invariant under the coordinaterotation.The wave functionsin the y coordinate,however,are different from those in the x coordinatebecauseof the extraoscillatory factor in eq. (2.1.17).Furthermore,such a factor contains the energyof the bound electron.The larger the bound stateenergy, the more are the oscillations in eq. (2.1.17). This oscillation hasa pronouncedeffect on themany electron resonancecalculations. We will return to this point later when we examine thecalculationsof a manyelectronresonanceby the methodof complexrotation.

(iii) ScatteringstatesLet us recall that while Siegert resonancewave functionscontain only the componentof outgoing

waves,the scatteringstates,on theotherhand, haveboth incoming andoutgoingwavesexp(—ikx)andexp(+ikx). With a simple algebraone can show that the combinationof theseincoming andoutgoingwaves in the asymptoticregion would lead to the form of (for simplicity, here we only considerthesingle channelS-wavecase)

~I’(x)—sin(kx + ,~). (2.1.19)

Again we considerthe exampleof the one dimensionalpotential problem that we discussedin theprevioussection,with ~ the usual definition of phaseshift. After the complex transformationthescatteringwavefunction,within the frameworkof the theoremof Balslev andCombes,containsalso thecombination of incoming and outgoing waves exp(—ik’y) and exp(+ik’y), such that the asymptoticrepresentationsof the scatteringstatesare conserved.Under the transformx —~ y exp(iO) eq. (2.1.19)becomes-

~P(y) —sin(kye’°+ ~). (2.1.20)

In order to preservethe asymptoticbehaviourof the form of eq. (2.1.19),k mustbe transformedintok —+ k’ exp(—iO) andeq. (2.1.20)becomes

- (2.1.21)

which hasthesameform as(2.1.19).Recallthat thenon-resonanceenergyis E~0~= ~(k)2. We now have= ~(k’)

2exp(—2i0). In other words, the rotated scattering energy has a factor of exp(—2i0)multiplying theunrotatedenergy.As a result,theenergycuts(in fig. 2)arerotated-downwardmakinganangleof 20.

10 Y.K.Ho, The methodof complexcoordinaterotation

2.2. Relation to theFeshbachprojectionformalism

Other than the method of coordinaterotations,various theoreticalmethodshave been used tocalculateatomic resonanceparameters.Two commonly usedmethodsare the Feshba~hprojectiontechniqueand the close coupling approximation. In order to comparethe appropriateparametersbetweendifferent methods,wewill briefly discussthe proceduresto calculateresonanceparametersbytheseapproaches.In the Feshbach[4, 22, 23] formalism,we are interestedin the solution of theSchroedingerequationfor the Hamiltonianof the projectileplus a targetsystem,

HW=E~P. - (2.2.1)

Onecan partition the Hubertspaceby -introducingprojectionoperatorsP and0 suchthat

P+Q=1 (2.2.2)

whereP and0 arethe usualopenandclosedchanneloperatorsrespectively.The conditionsfor P and

0 operatorsare,in the asymptoticregion,

~I’ and Q~P—0. (2.2.3)

Furthermore,P and Q havetheusualcharactersof projectionoperators,i.e.,

p2 = p Q2 Q and P0=0. (2.2.4)

Using theseproperties,the Schroedingerequationcan be decoupledinto two parts,onefor openspaceP~1’,and theotherone for closedspaceQ’I’, where

- ~ + IGPHQ~,)(‘~P~QHPxi 225— X E—e

1—41(E) (. .)

iQ~k) (~kQHPtP) (2.2.6)E—e~

In eq. (2.2.5).p~is referredto asthenon-resonantcontinuum, ~I’~and e~arethe jth eigenfunctionandeigenvalueof QHQ, respectively,i.e.

(4’1QHQ~P1)= e~ (2.2.7)

and4(E) is an energydependentshift, i.e. S

4~(E)=(~QHPGPHQ~) (2.2.8-)

with G an energydependentGreen’sfunction. In eq. (2.2.5) P!1-’ is the resonancewave functionassociatedwith thejth eigenvalueof QHQ.- Recall theasymptoticbehaviourof thenon-resonancewavefunction

Y.K.Ho, The methodof complexcoordinaterotation 11

P~ r1 tj~o(x)V2ir/k Yi,m (0, q$)sin(kn — lir/2 + ~°) (2.2.9)

where ~° is the non-resonancephaseshift, and4~the targetwave function. Similarly, the asymptotic

behaviourfor the Green’sfunction is

G(r, X, r’, X’) -~ —r’\/2i~/ksin(kr — li-r/2 + ~°)(P~(r’,X’)i . (2.2.10)

Substituteeqs. (2.2.9)and(2.2.10)into eq. (2.2.5).The asymptoticbehaviourof P1l~becomes

P’!P~~j~t/~o(x) y,.m(T){5~kT ~2+ fb)_~’h1~~~cos(kr_lir/2+ ~o)} (2.2.11)

Fromeq. (2.2.11)we can deducethe total phaseshift in the vicinity of a resonanceas

tlr’\ — OIr’\ —11 ~ i( )fl ~,L)— 77 ~E)+ tan A —

I__si i -

whereF1(E)can be relatedto the off diagonalmatrix elementPHQ,i.e.

F1(E)= 2ki(x(E)iPHQi~)I2. - - (2.2.13)

It is apparentnowthat F,(E) and4(E) areenergydependentparameters.To the first approximationthe eigenvaluesof QHQ, if 0 is properly chosen,would provide a

reasonableestimateon the resonanceposition. However, since the operatorP is not unique (theconditionsthatP hasto fulfill areeqs. (2.2.3)and(2.2.4)),different choicesof P would lead to differentQHQ eigenvalues.Theburdento havea ‘unique’ value of resonancepositionthenrelieson the shift, 4,which representsthe interactionsbetweenthe open and closedchannels.The sign and magnitudeofsuch a shift, in general,differ from systemto systemand alsodependon the wavefunctionsused.Theevaluationof 4 in eq. (2.2.8)usually requiresprincipal integralsin energy.An alternativeapproach,inwhich the principal integralsare not necessary,was proposedby Chung and Chan [24] Interestedreadersarereferredto their paperas well as the oneby Ho, BhatiaandTemkin [25].

Next let us recall the resonanceparametersin the Breit—Wignerform

t — 0 ~+ —lflr wc’ —

‘1 BW — 77 BW ‘ Lan 12’ BWI ~L.~BW

wheren~w and n~ arethe total andbackgroundphaseshifts respectively,andBW denotesthat theparametersarein the Breit—Wignerform. In the Breit—Wignerform EBW and TBW are constantsandthey are- the sameas the constantsin the Siegertparameters(calculatedfrom the methodof complexrotation).Next we rewrite eq. (2.2.12) in the following form:

o~=77~(E)+ tan ‘{~FF(E)/(EF(E)— E)} (2.2.15)

whereEF(E)= e +4(E), whereF denotesFeshbachparameters.Although the two equations(2.2.14)and (2.2.15) aresimilar and the left handsides of theseequations,the total phaseshifts~t,areequalsincethey can be relatedto experimentallyobservedcrosssections,the individual parameterson the

12 YK. Ho, The methodofcomplexcoordinaterotation

right hand sides are not necessarilythe same.The relation betweenthe Breit—Wigner andFeshbachparameters,evaluatedat the Breit—Wigner energy EBW was derived by Drachman [26] and in-dependentlyby Yamabeet al. [27]. The relationsare

EF— EBW = (F’)(F8~/4)+ (F”)(F~~/8)+ 0(F4) (2.2.16)

FF/FBW = 1 — 4’ + O(F~~). (2.2.17)

It is understoodthat F’, F”, etc. are the first and secondderivatives of FeshbachF with respecttoenergy,calculatedat the Breit—Wigner energy(or calculatedat the unshiftedFeshbacheigenvalue,s[25]). From eqs. (2.2.16) and (2.2.17) it is seen that the difference between the FeshbachandBreit—Wignerresonancepositionsdependson F andF’. Similarly, the differencebetweentheFeshbachand Breit—Wigner widths dependson 4’. In ref. [26], Drachmanhasexplicitly demonstratedthat for -

somemodelproblems,substantialdifferenceswould exist betweenthesetwo setsof parameters.In realatomicsystemsin which the resonancewidths areusuallynarrow,thedifferencesbetweenBreit—Wignerand Feshbachparametersusually are small (smaller than the errors due to the useof less elaboratewavefunctions,the omissionof the Feshbachshifts,4, etc.).Of course,if oneis interestedin aprecisecalculationsuchas the onecarriedout by Ho--et-al. 1251 for- the-lowestISC resonancein e—Hscattering,varioustermsin eqs. (2.2.16)and(2.2.17)haveto beexamined.In ref. [25]variousparametersevaluatedat theunshiftedQHQ eigenvalueare

d4/dE= 0.0022, dF/dE= 10~,

4 = 0.000179eV and F = 0.047065eV.

When thesevalueswere substitutedinto eqs. (2.2.16) and (2.2.17),a smallbut appreciabledifferencewas foundin the final resultfor the width. Thedifferencein resonanceenergyis neverthelessvery smallto give EBW = EF in that case.

In most atomicresonancestheenergydependenceof the Feshbachwidth andof theFeshbachshiftwould havea small effect on the final result. Such an effect, however,may be quite pronouncedinmolecularresonancessincethe widths in manymolecularresonancesarequitebroad.The importanceof the energydependenceof F hasof courselong beenrecognized(see[28] for example).However,notmuch work has beendone on the qualitative studiesof the energydependenceof F and 4 in theliterature.

2.3. Relation to the closecoupling approximation

Another commonly used method to calculate resonanceparametersis the close coupling ap-proximation (see ref. [29]for example). In this method, the total trial wave function of an (N+ 1)electronsystemis expandedin termsof targetstates

(2.3.1)

where ~/,,are various target states,- F, some unknown functions representingthe information for

Y.K. Ho, The methodof complexcoordinaterotation 13

differentscatteringchannels,and A the anti-symmetrizer.It is also understoodthat each term in thesummationwill couple to a- given total angularmomentum.Cross sections-can be deducedoncetheknowledgeof the asymptoticbehaviourfor the channelwave functionsF, are obtained.In practicalcalculations~,only afinite nuinberof states- areincludedin the expansion.These arethe energeticallyopenstates,denotedas P’P in the Feshbachformalism,anda numberof closedchannelstates,denotedas 0’!’, to speedup the convergence.Within the framework of the Feshbachformalism,eq. (2.3.1)becomes

= P’!’ + QV’ (2.3.2)

= A ~ 4~(r1,. . . ,rN) FJ(rN+1)+ A ~ q5k(rl,. . . , rN) Fk(nN±l) (2.3.3)

wherem is the numberof the energeticallyopen states.The closedchannelstatescan be real atomicstates,and/or pseudostates[30—32]that contain contributionsfrom continuum,and/or a number ofcorrelationtermsx that includeshort rangedipole interactionsor longer rangequadrupoleinteractions.Eq. (2.3.3) becomes

- (2.3.4)

wherep is the numberof pseudo-statesand Gk representthechannelwavefunctionsfor pseudo-states.- - - To solv-eeq. (23.4) a Kohn variationalmethodcan---beused--and-aset -of--coispled--equations-for thechannelwave functionscan beobtained.To solve thesecoupledequations,differentmethodshavebeenprovento be quitesuccessful.Someof the computercodesareavailablefor generaluse.Theseincludethe IMPACT codedevelopedby Seatonandcoworkers[33, 34] who useda linear algebraicequationmethodto solve theintegral-differentialequations;theRMATRX code[35,36] asdevelopedby Burke,Robb andcolleagueswho employedthe R-matrix methodof WignerandEisenbud[37];andthe NIEMpackage[38,39], developedby Henry and coworkerswho employeda non-iterativetechniqueto solvethe integralequations,afterthe integral-differentialequationsareconvertedto integralequationsby theuseof appropriateGreen’sfunctions.For a comprehensivediscussionaboutthesecomputerprogramsas well as otheraspectsof the closecoupling anddistordwaveapproximations,readersare referredto areviewby Henry [40].

In addition to thesepublishedcomputerprogramsotherunpublishedprogramswhich arealso shownto be quite effective to solve the set of coupled equationsfor particularphysical problemshavealsobeen discussedin the literature. These include the algebraic variational method (see [41, 42] forexample)andthe linear algebraicequationmethod to solve the integral-differentialequations[43,44].In all of thesemethods,phaseshifts (eigen-phases)can be deducedoncethe scatteringfunctionsareobtained.The resonanceparameterscan be calculatedby examiningthe behaviourof the phaseshifts(sum of the eigen-phases)over a resonance.For a onelevel resonance,theseparameters— backgroundphaseshifts, resonancepositions,andwidths— can beobtainedby fitting the total phaseshifts to theonelevel Breit—Wignerformula,

77’ = n~+ tan1{~F/(E~— E)}. (2.3.5)

14 YK. Ho, The methodof complexcoordinaterotation

whereE~andF are theresonancepositionandwidth respectively.[n the fitting procedureEr andT’aretakento be constants.Therefore,the resonanceparametersdeducedby eq. (2.3.5)are equivalenttothose obtained by the method of complex rotation. In a casethat two or more resonancesareoverlappedovereachother,amulti-level Breit—Wignerformulacan beused.In general,while the closecoupling approachis quite straightforward,this methodmay be time consumingin the processofresonancehunting.Furthermore,whena resonanceis locatedat a position,wheremany channelsareenergeticallyopen (resonancesassociatedwith N 5 or 6 hydrogenicthresholdsin e—Hscattering,forexample),a calculationby thischannel-by-channelapproachwould be extremelytedious.In fact such acalculation has not been done in a close coupling approach.The advantageof the close couplingapproximationis, of course,that completeinformation aboutthe resonancecan be obtained.

2.4. Other theoriesfor atomicresonances

In addition to the methodof complex rotation,the Feshbachformalism, and the close couplingapproximation, other methodsalso exist to calculate resonanceparametersin the literature. Forcompleteness,we herebriefly summarizethesemethodsas well as someof the recentadvancesin theareaof atomicresonancecalculations.Within the frameworkof Feshbachprojectionformalism,varioustechniquesunderdifferent terminologiesarecommonlyused.Forexample,the-stabilization--method~-asdevelopedby Taylor, Hazi andtheir coworkers[6, 45, 46], and independentlyby Holein et al. [47]inthe so called ‘Multiconfiguration EnergyBound’ theory,hasbeenprovento -be a usefultool to- estimateresonancepositions.In fact, aswill beshownlaterin this-review-,sueh-a~eomputaional-teehnique-cam-be~usedas the first stepin the complexrotation method.In this methodonly L2-type wave functions(thethird summationin eq. (2.3.4)) areused.The Hamiltonian is diagonalizedin this L2 basisset and theeigenvaluesareexaminedas a function of certainparametersin the wave functions.Theseparameterscan be the nonlinearparametersin the wavefunctions,or the expansionlength of the basis set.Sincethe amplitudeof the inner part of a resonancewave function is muchlargerthan its outerregion,theresonancewave function is hencemore localized (within atime r = h/F) than thebackgroundscatteringwave functions. As a result, resonanceeigenvalueswould exhibit stabilized behaviour when theparametersarechanged.On thecontrary,the backgroundscatteringwavefunctionswould changemoredrastically since they arenot localized.Furthermore,it can be shown [48] that when separablewavefunctionsareused in the MulticonfigurationEnergyBoundmethodthe resonanceeigenvaluesdeducedin such a procedurewould be upper boundsto the ua-shiftedeigenvaluesof -QHQ. —Calculations- ofwidthsby thesemethods,however,requiretheuseof continuumwavefunctions.Somenewdevelopmentsin this aspecthavebeenmadeby Hazi[491.By usingthestabilizationmethodandthemethodof.Stieltjes(seeref. [50]for example)imagingtechnique,he[49]wasableto obtainresonancewidthswith only L2wavefunctions.

Also within the framework of the Feshbachformalism, a Kohn—Feshbachvariational principle hasbeendevelopedby Chungand Che.n151, 52] as well as the saddlepoint techniqueby Chung[531.TheTruncatedDiagonalizationMethod (TDM) (seerefs. [54, 55]) is also anothervariantof the Feshbachformalism.In thismethod,thebasiswavefunctionsareconstructedby usingclosedchannelcomponentsandno open channelcomponentsare included.As a result, the computationalprocedureis relatiyelyeasyin that no explicit projection is necessary.The shortcomingsof such an approachare first, theFeshbachshift, a result of interactionsbetweenthe open and closed wave functions,is difficult tocalculate,andsecond,the wavefunctionsare lesselaboratebecauseof their infiuxibilities.


The methodsmentionedabovearemainly for calculationsof resonanceparameters(i.e., resonanceposition, width, and backgroundphaseshifts/eigen-phases).Other methodsthat havebeen used tostudy the dynamics of the resonancephenomenaalso exist in the literature. These include thehypersphericalcoordinate approach [56, 57] and a group theoretical approach [58, 59]. In thehypersphericalcoordinateapproach,a hyperradius

R-(4+4)~2 - - -

anda hyperangle

a = tan’’(r2/r1) (2.4.2)

are introduced to replacethe two electron distancecoordinatesr1 and r2. The usual six variables(r1, 0~,~, r2, 02, 42) for the two electronsarenow replacedby (R, 0), where.0 representscollectivelythe five angularcoordinates(a,0~,4)~,02, 4)2). A Born—Oppenheimer-typeseparationbetweenR and(2is subsequentlymadeandpotentialenergycurvesU(R) can be constructed.Once such energycurvesare obtained,boundstates,Feshbachresonances,and shaperesonancescan be convenientlystudied.This approachhasgiven someinsight in the dynamicsof someatomicsystems.However,the numericalresultsobtainedby this methodin general,due to the fact that the interactionsbetweentheopen andclosed channels are not easily incorporated, are not as accurate as other extensivevariationalcalculations.The grouptheoreticalapproachhasbeenappliedmostly to the H andHe doubly excitedresonances[58]. Two new andapproximately‘good’ quantumnumbersK andT were obtained,fromwhich the doubly excited states(resonances)of two electronsystemscan be assigned.Furthermore,thesequantumnumberscan be relatedto molecularquantumnumbersandthe similarity betweenthespectrumof the doubly excitedtwo electronresonancesand that of a triatomicmoleculewaspointedout by Kellman and Herrick [60].Berry and coworkers[61], in a separatestudy, haveexaminedtheconditional probability distribution functions of the two electron resonancewave functions.Theirresults[611tendto supportthe molecular-likecharactersof the doubly excitedtwo electronresonances.More about thesewill be discussedlater in this review whenthe complexrotationresultsfor the twoelectronresonancesareexamined.

Other approachesto studyatomicresonancesincludea time stability theory [62].This method,whichis from a time dependentview point, has provided reasonablenumerical resultsfor some atomicsystems[63,64, 65]. The disadvantageof this techniqueseemsthat the matrix elementsof theoperator112 haveto be calculated.Nevertheless,the time dependentapproachto resonanceshasled to someinterestingdiscussionson non-exponentialdecaysin autoionizations[66,671.

In addition to the above methods, the quantumdefect theory is quite successfulfor studiesofresonancesin electron—ion scattering,especiallyfor the higher membersof the Rydberg series.Furthermore,the useof quantumdefecttheory to estimatethe averagecontributionsof the resonancesto the collision strengthsin electron—ion scatteringhas been proven to be a useful tool. For thedevelopmentsin this areareadersarereferedto reviewsby Seaton[68Jandby Henry [401.Other areasof atomicresonancesthat h-ave attractedconsiderableattention are a random phaseapproximation(RPA) and its relativistic version (RRPA). These methodshave been applied to calculate pho-toionizationcrosssectionswith interestingresults[69].

16 Y.K.Ho, The methodof complexcoordinaterotation

3. Computationalaspectsfor atomicresonances

In this sectionwewill discussthecomputationalaspectof thecomplexrotationmethod.First we willshow an exampleto calculateresonanceparameters(both position andwidth) for anelastic resonance.We will then demonstratehow this straightforwardmethodcan be used to calculatethe resonanceposition and the total width for a multi-channel resonance.We will also discussthe complexvirialtheoremand show how the computationaltechniqueis relatedto such a theorem.At the end of thissection,the computationalaspectsof the complexeigenvalueproblemwill alsobe discussed.

3.1. An examplefor computationaltechnique

To illustrate the computationalproceduresby the method of coordinaterotation with real wavefunctions,we choseto examinethe S-waveresonancesin electron—positroniumscattering.This threebodysystem,which was observedrecently[70]in thelaboratoryundercontrolledconditions,consistsoftwo electronsandonepositron.In manyaspectsthissystemis very similar to thehydrogennegativeion,

a systemwhich hasbeenintensively studiedexperimentallyand theoretically.Both Ps and Hsystemshaveonly oneboundstate,andthe excitedtargetatomicstatesaredegenerate.The mechanismfor resonancesassociatedwith the N = 2 thresholdin Ps is understoodas a result of the degeneratedipole potentialdue to the 2s—2p degeneracy(or 3s—3p—3ddegeneracyfor N = 3 resonances)of thepositroniumatoms[71].Thecomputationaltechniquethatwearegoingto describeherecanbeappliedtocalculatethe knownH- resonances[72, 73].

The wave function usedto representthe Ps systemis of Hylleraastype

iT,_ E ~‘ ~ I i_ vi kIl mj_ ml— ‘—‘kim exp~—a~rip-i- ~ r12 ~r1~r2~-i--r1~r2~

l~m ~O

k~rO

wherenotations1, 2 andp denotethe electron1, electron2 andthe positronrespectively.The first stepof- the calculation---is to--choosean optimized wavefunction for a-particular-resonance.The-stabilizationmethod[6,45] can be usedfor thispurpose.Fig. 3 demonstratessuch aprocedureto estimateresonanceenergiesfor the resonancebelow the N = 2 Ps threshold.In the figure energies(now they areall realsince0 = 0) areplotted asafunction of the nonlinearparametera of eq. (3.1.1).An eigenvaluearoundE = —0.126 Ry was found exhibiting stationarybehaviour.The stabilized plateauhasa rangefroma = 0.22 to a = 0.295.This representsthe lowest S waveresonancein electron—positroniumscattering.In ref. [71]aone-to-onecorrespondencebetweensomelower-lying resonancesin H- andin Ps” werealso found. The physics behind the stabilizationmethod is that since a resonantstatehas a largeramplitude in the inner region comparedwith that for scatteringstates,the resonantstate is muchlocalized. As a result, resonancewave functionswould be less affectedwhen someparametersof thewave functionsarechanged,while the wave functionsfor the scatteringstateschangemoredrastically.The parametersof the wave functionswhich changeare the expansionlengthsand/or the nonlinearparametersof the wave functions,etc.

The secondstep of the computationalprocedureis to apply the complex rotation methodto thestabilized plateau.The results are demonstratedin fig. 4. It is quite clear that within the stabilizedplateauthe rotationalpaths(changesas afunction of 0) meeteachotherat the positionof a pole. Alsoat the positionof a pole, the changeof (resonance)energywith respectto 0 is alsominimized.We thus

YK. Ho, Themethodofcomplexcoordinaterotation 17

-0.06

______________ IIr~~90.20 0.30 -0152065 -0152060 -0152055

ReE(Ry)

Fig. 3. Autoionization statesof Ps” belowthe N = 2 thresholdof Ps Fig. 4. The lowestresonancet5C of Ps” belowtheN = 2 thresholdofatoms,ascalculated(with N = 161 terms)by thestabilizationmethod. the Ps atom, as calculatedby the complex rotation method. TheHeretheeigenvaluesareplotted againstthenonlinearparametersa. nonlinearparameteris shownherenext to eachrotationalpath.TheThenumbershownnextto eacheigenvalueindicatestheorderof that pathsarenearlystationaryfor 9 = 0.2—0.35radwhenthey comeacrosseigenvalue(seeref. [71]). theresonanceposition.Theserotationalpathsarefrom thestabilized

eigenvalueNo. 9 in fig. 3 (seeref. [71]).

summarizetheseproceduresby the following expressions

(ôIErI/30)a.aopt = 0 (3.1.2a)

=0, - - (3.1.2b)

where °optdenotesthe optimized value of 0, etc. In many calculationsinvolving small systems,veryaccurateresultscan be obtainedby using the abovetwo conditionsas well as the usual conditionforconvergencebehaviourwith respectto expansionlengths.This can besymbolizedby the following

(O~ErI/ôN)ø...0op,,a...aopt0. (3.1.2c)

In eq. (3.1.2c), N symbolizesthe expansionlengths.Thesuccessfulapplicationsby theuseof eqs. (3.1.2)or their equivalentto determineatomicresonanceswill be discussedlaterin this review.

For inelastic resonancesthe calculationsof resonancepositionsand total widths are as straightfor-ward as those for elastic resonances.Again we are using the example shown in ref. [71] as ademonstration.In fig. 5 eigenvaluesbelowthe N = 3 positroniumatom are plotted as a function of a,the nonlinearparameterin the wave function. If we interpret the resultsby using the stabilizationmethod,a resonancewould occur at about E= —0.0705Ry. When the complexrotation method isappliedto the stabilizedplateau,one is able to obtainboth the resonanceposition andthe total width,as shown in fig. 6. In such an approach,of course,the partial widths to which the multi-channelresonancedecaysare not determined.Later in this review,we will examinesomerecentadvancesonthe areaof calculationsof partial widths by the methodof complexrotation.

18 - YK. Ho, The methodof complexcoordinaterotation

-0.06 x I0~~

-0.08 1~- - I - -- -

- - ~O~65----0.15 020 e ~

ot. Fig. 6. Thelowestresonance1S~of Ps betweenthe N = 2 andN 3Fig. 5. Autoionizationstatesof P~below the N= 3 thresholdof Ps Ps thresholds,as calculatedby the complex rotationmethod.Theseatoms,ascalculated(with N = 161 terms)by thestabilizationmethod, rotational paths are from the stabilized eigenvaluearound E =

Heretheeigenvaluesareplottedagainstthenonlinearparametersa. —0.0707Ry in fig. 5. This exampledemonstratesthestraight-forwardThenumbershownnext to eacheigenvalueindicatestheorderof that method to calculatethe resonanceposition and total width for aeigenvalue(seeref. [71]). multi-channelresonance(seeref. [71]).

3.2. Thecomplexvirial theorem

In the usualvirial theoremfor Coulombpotentialsthe boundstatewavefun~il~ssatisfy the virialcondition -

2(T)=—(V), (3.2.1)

where T and V are the usual kinetic and potentialenergyoperatorsrespectivelyand () denotestheaveragevalue over the operatorswith real wave functions. Also in the usual virial theorem thescatteringstateswill not fulfill the virial condition(eq.(3.2.1)). In the cornplex virial -theorem,both theboundstatesand resonancestateswill satisfy the virial condition of eq. (3.2.1). The scatteringstates,however,would not satisfythe complexvirial condition either. It can beshown that the condition of aresonancestate,

19~El/90(eq. (3.1.2a)),is related to the fulfillment of the complexvirial theorem(eq.(3.2.1)). Here, we will follow closely the discussionsgiven by Yaris and Winkler [74]. Alternativederivationswhich alsolead to the sameconclusionscan befoundin BrandasandFroelich [75],Yamabeetal. [27], andin Moiseyevet al. [76].

A complexresonanceeigenvalue,denotedby W,exhibitingstationarybehaviourfor smallchangesof 0,would fulfill the condition

W(0)= W(0+ i~0), - - (3.2.2) -

with A0 a small valuein 0. If we expandW(0+ i~0)around0, eq. (3.2.2)becomes

W(0+ A0)= W(0)+ (8W/t90)i~0= W(0)+ W’ &0 + higherorderterms. (3.2.3)


If the condition (3.2.2) is to be fulfilled, we musthave

(r9W/aO)M= 0 or 3W/o0 = 0. (3.2.4)

Next let us recall the complexHamiltonianwith Coulombicinteractions

H(0) = T e21°+ Vet0. (3.2.5)

Such a Hamiltonianat an angle0 + i~0is

H(0 + z~0)= Te2t~°~°~+Ve~~°~°~

= T e21°(1 — 2i z~6+ - - -) + Ve1°(1 — i i~0+---)

= H(0)— i i~s0(2T+ V)+ higherorderterms. (3.2.6)

The expectationvalueof H(O + M) then becomes

~ VI!P). (3.2.7)

Recallingthe definition (~I’IH(0) I~P) W(0),eq. (3.2.7)becomes -

W(0-i-M)= W(0)—iA0{(V-’I2TI~1’)+(~1’I%’1!P)}. (3.2.8)

Now if condition- (3.2.2) is to be fulfilled, the secondtermin eq. (3.2.8)mustvanish.Wehencehave

2(~PITI~P)=—~PIVIV~). (3.2.9)

In otherwords,the stationarycondition i9W/30= 0 leadsto the complexvirial theorem(eq. (3.2.9)).While the complex virial theoremwould helpus conceptuallyto interpret various elementsin the

complexeigenvaluespectrum,the condition (eq. (3.2.9)) aloneshould not be overly emphasizedfromthe computationalview point. In a series of papersMoiseyer and Weinhold [76, 77, 78] haveinvestigatedvarious conditionsbasedon the complexvirial theorem to deducethe bestvalue for theresonanteigenvalues.Theresultsin their papersindicatedthat the useof a- proceduresimilar to thestabilizationcriteria would be more effectivethan the condition (3.2.9) alone. Suchfindings of coursesupportthe examplewe haveshownin section3.1. It should-alsobeemphasizedthatthe analogousrealvirial theorem aloneis not sufficient for accurateresults as can be seenin the exampleof the boundstateenergycalculationby the HartreeFock method.Very often a single configurationHartreeForkwave function would satisfythe virial theoremvery well, but its energymaydiffer substantiallyfrom amulti-configurationHartreeFock value.

3.3. Complexelgenvalueproblems

For the readerswho areinterestedin the computationalaspectsof the complexrotationmethod,wewill briefly discussthesolutionsof thecomplexeigenvalueproblems.As in all variationalcalculations,atrial wave function ~P

5is employedto representthesystem.We thenexpandthe trial wave functionby a

20 YK.Ho, Themethodofcomplexcoordinaterotation

linear combinationof orthonormalbasis functions(i), suchthat

~ C~,y,. (3.3.1)

Substitutingeq. (3.3.1) into the complexHamiltonian,we have

(!P~lH(0)— WI~PS)= ~ CICJHXJ — ~ C1C1&1W= 0 (3.3.2)

where

Htj=(xtIHIxj) and (~I~q)=ô~. (3.3.3)

Again hereW denotesenergy.A variation on W with respectto C, is carriedout. Usingthe stationary

condition81 WII8CL = 0, (3.3.4)

a usualeigenvalueequationis obtained

~ (Hq — WAI)a~= 0 (3.3.5)

- with the WA and a’ the A-th-eigenvalueand--ei~genvectorrespectively,and-I the--identity--matrix.-in -a-- -matrix notationeq. (3.3.5)becomes

H(0)X(0)= W(0)IX(0) - (33.6)

The complexeigenvalueproMerncanbe:solved-for~xample~by-Teducing~the~rmtrix~Hintcr-7an-uppeHessenbergform [79], and the whole energyspectrumcan be obtainedin a single diagonalization.

- Rèpeited~diagónáliiãffonwith differeñt ~wiWgiv~us ãcliiñceto examiiiethe thñëbe1iIãvidtif~of the whole spectrumof eigenvaluesas- 8--changes-.The-advantageof- this approachis tb-at- severalresonancescan be examinedat oncewithin the sameeigenvaluespectrum.Thedisadvantageis that thesize of the eigenvalueproblemwould usuallybe limited.

In a casewhen the trial wave function is expandedby a linearly independentbut not necessaryorthogonalbasis(i.e. Hylleraastypewavefunctions),somemodificationsto the eigenvalueproblem(eq.(3.3.5)) would be made.In sucha caseeq. (3.3.5)becomesa generalizedcomplexeigenvalueproblem,

~ (I-Ia — WANU) a~= 0 (3.3.7)

whereN11 = (xLIx) arethe usualoverlapmatrix elements.In the matrix notationeq. (3.3.7) becomes

- H(0)X(0)= W(O)NX(O). (3.3.8)

YK. Ho. Themethodofcomplexcoordinaterotation 21

To reducethe generalizedeigenvalueproblemof eq. (3.3.8) to the form of eq. (3.3.6)onecan multiplyeq. (3.3.8) by the inverseof N, N”. Eq. (3.3.8)becomes

N”’ H(0)X(0) H’(O) X(0) = W(0)X(0). (3.3.9)

However, such a proceduremay lead to numerical instability, especially,when the dimensionof thecomplexeigenvalueproblem is sufficiently large.Anotherprocedure,which is morestablenumerically,is by the methodof decomposition.Assuming real functions are used in the calculations,one candecomposeN into a productof an uppertriangularmatrix A, andits transposeAT, i.e.,

N=ATA. (3.3.10)

InsertingI = A’A betweenH(0) and X(0) in the left handside of eq. (3.3.8),wehave

H(0)A’~AX(0)= W(0)ATAX(0). (3.3.11)

Multiplying eq. (3.3.11) by (AT)~from the left, we have

(AT)1H(0)A’A X(6)= (AT)1 W(0)ATAX(6). (3.3.12)

Since W(6) is adiagonalmatrix, eq. (3.3.12)can be transformedto the form of eq. (3.3.6), i.e.,

B(0) Y(0)= W(6)Y(6) (3.3.13)

with Y(0)= AX(0) and B(0)= (AT)’~H(O) A1 = (A”)T H(0)A1, a symmetricbut not Hermitianmatrix.

The complexeigenvaluepackagesare usually quite easyto assess.For example, the subroutinesdescribedin IMSL [80] would be adequatefor the generalpurposes.For a specialcasein whichcomplexwave functionsareusedsomeworkersprefer to usean inner projectiontechniqueto simplifythecomplexmatrix calculationsinvolving the kinetic operators.Sucha computationalprocedurewill bediscussedlater in section4. Anyway, the overlapmatrix N would becomplexandfurthermodificationsare needed.In any casethe IMSL routineswould also be able to handle thesegeneralizedcomplexeigenvalueproblems.Interestedreadersarereferredto the original references.

Anotherspecial caseis when the method of sparseiteration is used.This methodwas used in acalculation of e~—Hresonances[81] and in multi-photon processescalculations[82]. By taking theadvantageof the fact that the matrix elementsaresparse,theseauthorswereable to employ very largedimensionsof matrices.The shortcomingof the sparseiteration techniqueis that only one resonanceeigenvalueis examinedat a time, and very often initial values for the resonancepositionsandwidthshaveto beprovidedfor iterationprocedures.

4. Applicationsto atomic resonances

The areato which the complexrotation methodhasbeencommonly applied is atomic resonancecalculations.We will review theseapplicationsaccording to two categories.One is for the simple

22 YK. Ho, Themethodofcomplexcoordinaterotation

systemssuch ase”—H ande”—H, etc.,andthe secondcategoryis for complexsystems(more thanthreeparticles).It has beenfoundthat thedirect applicationsof thecomplexrotationmethodwith real wavefunctionsto simple systemshavebeenvery successful.The extensionto more complexsystemswithonly real wave functions is successfulonly for specialcases.In general,modifications to the directmethodhaveto be madein order to speedup convergence.The modificationsareto usecomplexwavefunctionsor the so-calledSiegertwave functions.

4.1. Threebodyproblemswith Coulombinteractions

In this section,we will discussin detail the resonancephenomenain e—H and &‘—H scatteringaswell as the doubly excitedresonancesin helium iso-electronicsequence.In mostcalculationsof thesesmall but non-trivial systemsa rotated Hamiltonian and real wave functions are used. In thisstraightforwardmannervery successfulresultshavebeenobtained.Theseresultsarenot only for elasticresonancesbut also for multi-channelresonances.Of course in a multi-channelresonanceonly theresonanceposition and the total width were calculated.Partial widths havenot, evenfor thesesmallsystems,been calculatedby the complex rotation method.In the following we will discussthe e’—Hresonancesin differentenergyregions.In the discussionof &‘—H resonanceswe will also-providea briefhistoricaldevelopmentaboutthe investigationof this system.

4.1.1. e—H elasticresonancesEver sincean S-waveresonancein e—H scatteringwas revealedin a ls—2s two state[83] and in a

ls—2s—2pthreestate[84] closecoupling calculation,and the subsequentdiscoveryof the resonancebySchulz [85], the resonancephenomenain e—H scatteringhavebeen continuously studiedby bothexperimentalistsandtheorists.From the theoreticalside, this threebody systemis oneof the simplest-non-trivial problemsin which no exact analytic solutions have been found. Nearly all theoreticalmethodsto study atomic resonanceshavebeen usedto investigatethe e—H scatteringproblemas atesting case. For example, different approximationsof close coupling methodshave been used tocalculateelastic e—H resonances.Thesecalculationsincludea threestateplus correlationcalculation[86], a pseudo-statecalculation[87],and an 18 statealgebraicvariationalclosecouplingapproximation[88]. The Kohn variationalmethodhasalsobeenquite frequentlyapplied to study the e—H problem.Theseincludecalculationson S statesby Shimamura[89],on P statesby Das andRudge[90],on some5, P. andD wave resonancesby Registerand Poe [91J There are also Quite a numberof FeshbachProjectioncalculationsto investigatetheresonancephenomenain the e—H system.Thesecalculationsincludethe work by O’Malley andGeltman[22], an extensiveseriesof papersby BhatiaandTemkin([23, 921 andreferencestherein), andthe Kohn—Feshbachvariationalcalculationsby Chung andChen[24].Other theoreticalmethodsto studye—H resonancesincludea TruncatedDiagonalizationMethod(TDM) [93], andthe stabilizationmethod [94]. The earlystudiesof electron—hydrogenresonancesarethe subjectsof several reviews in the literature (Burke [95], Schulz [96], Williams [97], Risley [98],Golden[99],McDowell [100],andCallaway[41,101]).The complex--rotationmethodhasalso.beenusedcontinuouslyto studye—H resonanceseversincethe theoreticalaspectof this methodbecameknown.

In the earlydevelopmentsof the complexrotation methodthe lowestsinglet S-waveresonancewasstudiedby Bardsley and Junker[102], by Doolen, Nuttall and Stagat[11] and by Doolen [8]. Someinsightsof the computationalaspectsweregainedin theseearly investigations.For example,Doolen [8]found that the rotational pathsslowed down at the position of a pole as the rotational angle wasincreased.He also found that the resultswe-re insensitive to small changesin the basis set.Such a


computationalprocedurehasled to amorerefinedstudy by Ho [71,72] who showedcomputationallythe relationshipbetweenthe complexrotation methodand the stabilization method (seeconditions(3.1.2a) and(3.1.2b)).Such a connectionwas alsodiscussedindependentlyby Taylor andYaris [14]andby Junker[16].

In refs. [11] and [72], the wave functions used to study the lowest S-wave resonancein e—Hscatteringby the complexrotationmethodareof Hylleraastype

= ~ ~ exp[—a(r, + r2)] r~’2(r~rT+ r~rr) (4.1.1)

with k + m + n � w, apositiveinteger.Up to atotalof 161terms(w = 10)wereusedin ref. [72].The finalresults deduced [72] by the use of the conditions 81E1/t9a = 0, 3IEI/30= 0 and 81E1/aw= 0 aresummarizedin table1. The positionandwidth of thelowest S waveresonancecomparefavorablywith apreciseFeshbachcalculation by Ho, Bhatia and Temkin [25]. The Feshbachprojectionresultswereobtainedalso by using Hylleraastype wave functions. In addition, the shift due to the interactionbetween the open and closed channelswere calculated by using a full scatteringnon-resonantcontinuum.The resultswithin the statederrorscan be consideredas the non-relativistic‘solution’ forthe loweste—HS-waveresonance.Theseresultsarebasedon the assumptionsthat the nucleusof thehydrogenatom is a point chargeandis infinitely heavy(comparedwith theelectronmass),andthat thespin—orbit interactionsare alsoignored.Sincea protonis only 1836 timesheavierthanan electron,thefirst correctionfor the aboveresultswouldcomefrom the nuclearmasseffect.

Table 1 alsoshowsa comparisonfor the3P°(1)state.Thecalculationby Ho was basedon Hylleraas

typewavefunction

~P’t= ~ exp[—a(r~+ r2)] r~’2{r~r~”~”Yoo(1)Y,0(2)— r~r7” Yon(2)Y10(1)}. (4.1.2)

kmn

Usingthe proceduredescribedin the previoussection(3.1), andemployingwave functionsof up to 165(w = 8) terms,Ho [72]obtainedareasonanceeigenvalueof E = —0.284273Ry andF= 0.000426Ry forthe - lowest triplet P wave resonance.It seemsthat amongvarious theoretical calculationsin theliterature, the Kohn variationalcalculationby Das and Rudge[90], in which the phaseshiftsarefittedto the Breit—Wignerformula,was theonethat hasresonanceparametersclosestto the complexrotation

Table 1Resonancesin e—Hscatteringbelow theN = 2 threshold

lSe(1) ‘P°(l) ‘P°(l)

-E,(Ry) 1(Ry) -E,(Ry) f(Ry) -E,(Ry) r(Ry)

Complexrotation 0.297555’± 0.000004 0.003467± 0.000002 0.284273’ 0.000426 0.252096l~ 3.8 x 10_6Othertheories 0.297553~± 0.000004 O.003462c± 0.000008 0284265d 0.00043 0.2520992~ 2.73X 10-6

‘Ho [721,Hylleraas-typewavefunctions.bwendoloskiand Reinhardt[104j,orthogonalLeguerrefunctions.Ho, BhatiaandTemkin[261,Feshbachprojectionformulism including shifts; Hylleraas-typewavefunctions.

dDas andRudge[901,Kohn variationalmethod;Hylleraas-typewavefunctions.

‘Caliaway[881,18 statealgebraicvariationalclosecoupling.

24 YK. Ho, Themethodof complexcoordinaterotation

results.The complexrotation results also agreevery well with the electronscatteringexperimentofSancheandBurrow[103].

The calculationsof the lowest1P°resonanceusing Hylleraastypewave functionsandwith only onenonlinearparameterwere less successful[72]. However, somesuccesshasbeen made in. a complexrotationcalculationby Wendoloskiand Reinhardt [104]who usedwave functionsconstructedby thetwo electronproductsof orthogonalLaguerrefunctionsof the form

4.~1(Ar)= +2+ ~)!} (~)l4.1 e_AT~~

2L~t’4’2(Ar). (4.1.3)

hi ref. -[104], wave functionsconstructedfrom lOs, -lop and 6d orbitals--were used,and--resonanceparameterswerededucedby examiningthe so-calleddouble‘kick’ pointswith respectto 0 andA. Thisprocedureis equivalentto fulfilling the conditions 8IEI/tdJO= 0 and alEf/alt = 0 simultaneouslyThecomplexrotationresultsappearto be reasonablyaccuratealthoughby compann~withan extended18statealgebraicvariationalclosecoupling(seetable1) calculation, the widths do differ somewhat.

4.1.2. e”—H resonancesbetweenthe N = 2 and N = 3 hydrogenicthresholds- The region betweentheN =-2 andN = 3 hydrogenicthresholdsis quite-interesting.-In -this-region,

thereis a1P°shaperesonancejust abovethe N = 2 threshold,andthe usualinfinite seriesof Feshbachresonances-belowthe N-= 3--hydrogeniethreshold.-For the 1P°resonance,theincomingelectron--in--thee—H scatteringis trappedby a potentialwell formedby the attractivestaticandpolarizationpotentials,andarepulsiveangularmomentumbarrier~Such apoteii~tiàlwell is able--to -s~pportbôth’F b-typeresonancesbelow the N = 2 thresholdas discussedin the last section,andshaperesonancesabovethethreshold. One 1p0 shaperesonancehas indeedbeen observedin experiments.The widths of shaperesonancesarenormally quitebroadsincethe overlapbetweenthe shaperesonancewavefunction andthe channelwavefunction is quite large. But in the presentcase,the 1P°resonanceis locatedjust above

--the threshuld;andth poentiui-harrierthrou-gh-which--the--eleetronwill eventuallytunnel out is quitcthick. As a result,it will taketheelectronlongertime to cascade,andthewidth is subsequentlysmaller.

Fromthe experimentalsidethe 1P°shaperesonancehasbeenstudiedby the New Mexicoteam [105,106, 107]. Otherresonancestructuresin the regionof N = 2 andN = 3 hydrogenicthresholdshavebeenobservedin the ls—2sandls—2p excitationcrosssectionsby McGowanetal. [1081andby-Williams [109].Of course, in theseexperimentsresonancesotherthan the optically allowed ‘P°resonanceswere alsostudied.On the theoreticalside severalmethodshavebeenapplied to study the ‘P°shaperesonance.Thesemethodsinclude differentvariantsof closecouplingapproximations[110,111, 1121; pseudo-stateclose coupling combinedwith a multi-channelJ-matrix technique[1131;and an 18 state(pseudoandreal atomic) algebraic variationalclose coupling calculation [114]).Other methodsto investigatethisshaperesonanceinclude the group theoreticalapproach[58, 115], hypersphericalcoordinatestudies[116],and Kohn—Feshbachvariation calculations[1171.The use of the complex rotation method tostudy this shaperesonancewas carriedout in [104].

In the complexrotation calculation, the two electronwave functions are productsof orthogonalLaguerrefunctionsof the form of eq. (4.1.3) coupledto a given angularmomentumL. In this caseofcourse L = 1. The results in ref. [104] are obtained by the procedureequivalentto fulfilling theconditions31E1/t96= 0 and 8IEI/3A = 0. The condition 8IEI/3N = 0 was howevernot used.Resultsarecomparedwith othercalculationsin table2. It is seenthat the resultsagreewith eachotherquitewellfor the resonanceposition. The width calculatedby the complex rotation method,however,differs

YK. Ho. The methodof complexcoordinaterotation 25

Table 2The t~OshaperesonanceabovetheN = 2 threshold

Method —E,(Ry) F (Ry) Reference

Complexrotation 0.248702 0.00104 WendoloskiandReinhardt[104]Closecoupling (3 statepluscorrelations) 0.248675 0.OOli 1 MacekandBurke liii]Closecoupling(18 state) 0.24879 0.00147 Callaway [1141

somewhatwith an 18 stateclosecouplingcalculation[114].It seemsthatfor the studiesof the ‘P°shaperesonanceby the complexrotationmethod,moreconfigurations,or othertypesof wave functionsareworthwhile to try.

Other interesting featuresin the region associatedwith the N = 2 hydrogenic thresholdare theelectricfield dependenceon the Feshbachandshaperesonancesas observedin the experimentsby theNew Mexico team. A theoreticalstudy on such field dependenceon the shaperesonancewas carriedout by Wendoloskiand Reinhardt[104]by the useof complexrotationmethod.Very interestingresultswere obtainedby theseauthorsevenwith the seeminglysimple form of wave functions(eq. (4.1.3)).Beforewe discusstheir resultswe will first discussthe generalbackgroundof the field dependenceontheseresonances.

In the photodetachmentof H- ions measuredby the New Mexico team, an H” ion beam wasacceleratedto havean energyof 800MeV by the Los AlamosMeson PhysicsFacility (LAMPF). TheH- ion beam was intersectedwith a nitrogen laser(3370A) beamfor which the photo-absorptionexperimentswere then carriedout. Becauseof a largeDoppler shift the energyof thelaser, in the restframeof the H ion beam,would increaseto aboveabout 12eV, the energyregion aroundwhich theFeshbachand shaperesonancescan be studied. In addition, if one also appliesa moderatemagneticfield in the scatteringregion the H” ion beam,again in its restframe,would experiencea strongelectricfield due to Lorentz transformation.This electric field hasa pronouncedeffect on the Feshbachandshaperesonances.The ‘P°(l)Feshbachresonancebelow the N = 2 thresholdwas splitted into threecomponents,and the shaperesonancedecreasesin width for increasingfield strength.The shaperesonanceeventuallydisappearswhenthe field is sufficiently large.

The fact that the width of the shaperesonanceis broadenedby the increasingfield can be explainedas follows: sincethe shaperesonanceis a resultof arepulsiveangularmomentumbarrieras discussedearlier in this review, the height of the barrier would decreasewhen the external electric field isincreased.At the sametime, the thicknessof the barrier, through which the trappedelectronwouldtunnel out, is also decreasing,causing the lifetime of this quasi-boundstateto be shortened.Theresonancewidth would subsequentlybe broader.Furthermore,the 1P°shaperesonancewouldcoupletothe adjucent15eand 1D~continuain the presenceof theexternalfield, causingadditionalchannelsto beopened.This would alsocontributeto a largerwidth [1181.When the externalfield is sufficiently large,it will completelysuppressthe heightof the angularmomentumbarrier. As a result, the potentialwillno longer be able to trap the electron, and the shape resonancewould hence disappear.Thedependenceof externalelectric field on the shaperesonancehasbeenstudied[1041by the methodofcomplexcoordinaterotation.We will delay the discussionsof the generaltheorem for the complexcoordinatemethodunderthe influenceof electricandmagneticfields. At presentwe only highlight theresultsobtainedby theseauthors.Thewave functionsused in ref. [104]areof Laguerrefunctions.Bythe use of 9s-type, 9p-type and 7d-type wave functions, they obtained good results for the field


F’OOc~ -L I~ F~3O~IO

5a~: -~- -

F. 45

E

F’60 ,I6~.u __~/7

OI~242 -0.1243 -0.1244 ‘0.1245

Re (E in au.)

Fig.7. Electricfield dependenceof the‘P°shaperesonancejustabovetheN = 2 hydrogenicthreshold(seeref. [1041).Thecomplexrotationresultsfor theresonancepositionandwidth agreequalitatively with thephoto-absorptionexperimentof ref. [107].

dependenceon the width of the shaperesonance.Their results,as shown in fig. 7, agreequalitativelywith experimentalobservations.

The Stark effect on the lowest‘P°(l)Feshbachresonancebelow the N = 2 hydrogenicthresholdisalso interesting.Since the 1P°(1)state lies at a position very close to the secondmember of the 1

5e

resonance,evena weakexternalelectricfield will causea strongmixing of thesetwo statesandresult insplitting the

1P°(1)stateinto two components.A third weakcomponent,whichwasalsoobservedin theexperiment,is a resultof the mixing of the 1P°(1)resonancewith a nearbylDe(1) state.A theoreticalstudy on such phenomenonwas carried out by Callaway and Rau [119]. These authorsused astabilizationmethodto examinethe Stark effect on the resonances.A qualitativeagreementon theresonancepositions with experimentswas obtained. The electric field dependenceon the 1P°(1)Feshbachresonancehas not, however, been studied by the complex rotation method. Since thestabilizationcalculation did not provide information on widths, a complexrotation study of the fielddependenceon the Feshbachresonanceswouldbe of interest.

Other interesting featuresbetweenthe N = 2 and N = 3 hydrogenicthresholdsare of coursetheFeshbachresonancesconverging on the N = 3 hydrogenic threshold. Again due to the dipoledegeneracybetweenthe 3s—3p—3dhydrogenicstates,infinite seriesof resonancesexist below the N = 3thresholdif we omit the fine structuresof thesestates.Varioustheoreticalmethodshavebeenappliedto study the Feshbachresonancesin this region.Theseinclude a 14 state[120]and an 18 state[1141closecoupling calculations,different variants-of FeshbachProjectiontechniques[121,122, 58], and thehypersphericalcoordinateapproach[123,124]. The complexrotationmethodhasalsobeenappliedto -this regionby Ho [125,1261 who usedHylleraastypewave functions(eqs. (4.1.1)and (4.1.2))for L = 0

andL = 1 resonances,andby Ho and Callaway[127,128] for statesof L � 2, in which wave functionswereconstructedby the productsof Slaterorbitals,

= A ~ ~ Caiaj~ai(Ti)71b1(T2) Y~~,6(l,2) S(ty1, 02) (4.1.4)

l.Ib U


Table 3Feshbachresonancesin e—H scatteringconvergingon theN = 3 threshold

Closecoupling ClosecouplingComplexrotation 14 state[120] 18 state[114]

Er(Ry) F(Ry) -E,(Ry) F(Ry) Er(Ry) F(Ry)

~SC(1) 0.13800’ 0.00284 0.1379 0.00283‘P°(i) O.i

2S42~~ 0.00234 0.12505 0.00242 0.125432 0.002393P°(i) 0.l3580~’ 0.0032 0.13572 0.00331D~(i) 0.13i9’ 0.0032 0.i3i8 0.00324 0.13191 0.003273Dc(1) 0.11796’ 0.00075 0.i1788 0.000755 0.11793 0.000753F°(i) 0.12302’ 0.00022 0.1228 0.00023 0.12303 0.000217

‘Ref. [125].Ref. [1261.Refs. [127,1281.

with -

~1aafr)= r~~aUe_~~1r. (4.1.5)

In eq. (4.1.4) A is the anti-symmetrizingoperator, S is a two particle spin eigenfunction, Y is atwo-particlesphericalharmonic,andthe i~areindividual Slaterorbitals. Complexrotation resultsaresummarizedin table3 and comparedwith close couplingcalculations.It is seenthat the agreementisvery good in spiteof the fact that theseresultswere obtainedby completelydifferent computationaltechniques.Againwithin thenon-relativisticquantummechanicalapproximation,andassumingthat theproton in the hydrogenatom is infinitively heavy, theseresultsare believed to be quite close to the‘solution’ for theresonancesin thisenergyregion. In addition,thegood agreementbetweenthe 18 stateclosecouplingcalculationand the complexrotation resultsfor statesof L � 2 indicatesthat the useofproductsof Slaterorbitals would also give meaningful results for resonancesassociatedwith higherhydrogenicthresholdsandwith high angularmomenta.An explanationis suggestedas the following:while the r

12 factorsplay an importantrole in regionswhere r12 is small (i.e. lower-lying 5- andP-waveresonances),they are less important for stateswith higher angularmomentaand for highly excited(awayfrom the nucleus)states.

The comparisonof the complexrotation result for the lowest Feshbach1f~0resonanceswith the

photo-absorptionexperimentof theNew Mexico team[105,106, 107] is madein table4. The agreementfor the resonanceposition is very good,but the widths do show somedifferences.

Table4_____ TheN=3’~fl)resonanceinH -~~--- ~-~~----------

Complexrotation [1261 Photodetachmentexperiment[106,1071

Er (eV)* F(eV) E,(eV) I’ (eV)

12.6476 0.0318 12.650± 0.004 0.027± 0.0008

* Measuredfrom thegroundstateof H- ions. The reducedrydbergfor H is

usedfor energyconversion.

28 Y.K. Ho. The method~ complexcoordinaterotation

In conclusion,the complexrotation results are amongthe most accuratedata in the literature. Incomparingthe computationalelforts, the complexrotatkn meThod-forimi1ti~chaniieI~esonan~esis ~asstraightforwardas for elastic resonances,while other techniquesinvolving channel-by-channelcal-culationscouldbe quitecumbersome.However, thecomplexrotationmethodhasyet to providepartialwidths aswell as backgroundeigen-phasesandcrosssections.

4.1.3. e—H resonancesconvergingon theN = 4 and N = 5 hydrogenicthresholdsBelow higher hydrogenicexcitationthresholds,therealsoexist infinite seriesof Feshbachresonances

due to again the dipole degeneracyof the target excited states.In a mannersimilar to the N = 3resonances,the complex rotation method was able to calculatesome of theseresonancesby usingHylleraastypewave functionsfor S andP resonances[1291,andSlaterorbital typewave functionsforstateswith L � 2 [1281.Again the stabilizationmethodwas usedas the first computationalstep,andthefinal resultswerededucedfrom conditions(3.i.2a,b,c).

The complexrotationresultsaresummarizedin table5 andcomparedwith a 17 stateclosecouplingcalculationby Hata, Morgan and McDowell [130]for resonancesconvergingon the N = 4 hydrogenicthresholds.In the closecouplingcalculationall the tenexactatomicstatesrangingfrom the is to N = 4states(4s.—4p—4dand4f), as well as 7 pseudo-states,areincludedin the calculation.Theagreementsforthesetwo setsof resultsaregenerallygood with adiscrepancyof no morethan 0.0009 Ry for a givenresonance.The discrepancy,of course,would be reducedif improvementson both calculationsaremade.

Resonancestructuresassociatedwith higher excitationhydrogenicstates(say N = 4 andN = 5 andabove) are much more complicatedcomparedwith resonancesassociatedwith lower hydrogenicthresholds.In the former casesthe target hydrogenatom in one of its higher excited statesis moreloosely bound and henceis more polarizable. As a result, the degeneratedipole potential, whichbehaveslike r~2asymptotically,will havea largecoefficient comparedwith the counterpartsassociatedwith lower excitedhydrogenicstates.For example,Gailitis and Damburg[131,132] haveshown thatsuch an attractiver2 potential for the N = 2 hydrogenic thresholds,combinedwith the repulsiveangularmomentumbarrier, can only support S-, P- andD-wave resonances.As we go to a higherhydrogenicthreshold,the coefficient of the dipole potential is larger as demonstratedin the work ofHataet al. [130].Resonanceswith higher angularmomentawould thereforealso exist.

Table 5Resonances(in rydbergs)convergingon theN = 4 hydrogenic

fl~e~io~

18 stateclosecoupling,Complex-rotation- l4ata,Morganand

method,Ho [1291 McDowell [1301

~Er F Er 1

lSe(1) 0.0792 0.0019 0.0782 0.0029ISC(2) 0.06943 0.00175 0.070 <0.001‘P°(l) 0.0744 0.0021 0.0735 0.0033

— 0:0786- 0:0022 0:G780~ 0:00323P°(2) 0.0686 0.0013


The spectrumof the doubly excitedH- ions is very complicatedand intriguing. The ‘approximate’symmetryof the two electronHamiltonianhasbeenthesubjectof studiesby severalauthors[58,59, 60,133, 134]. In the absenceof the interactionbetweenthe two electronsthe spectrumof the Hamiltonianis simply aproductof two 0(3) groups.Whenwe includethe interactionsbetweenthe two electrons,aswe must in the real physicalsystem,the symmetryis broken.As a result, the individual orbital angularmomentum, 1, for each electron is no longer a good quantumnumber, although the total angularmomentumand the total spin are still constantsof the motions. Approximately “good’ quantumnumbershavebeenobtainedby Wulfman[115,135] andby Herrick andSinanoglu[58] by diagonalizingIA1 — A21, where A1 and A2 are the Runge—Lenzvectors for electronsone and two respectively.Providedthat the two electronsoccupy the sameshell (the so-calledintra-shell),or if oneshell is notmuchhigher than the other, theseauthorswere able to obtain two new quantumnumbersK and T,from which the doubly excitedresonancescan be classified.Theconditionson K and T are [581

T=0,1,...,L forstatesofi~=(_i)L, (4.1.6a)

T=i,...,L forstatesofir=(—l)~1, (4.1.6b)

and

±K=N—T—1,N—T—3,...,Oori, (4.1.7)

where N is the hydrogenic thresholdbelow which resonanceslie. More on this new classificationschemeas comparedwith otherschemessuchas configurationmixing will be discussedlater in the text.

The grouptheoreticalapproachwasfurtherextendedby KellmanandHerrick [59,60]. Furthermore,theseauthorswere able to relatethe quantumnumbersK and T to the vi-rotation quantumnumbersemployedby molecularspectroscopists.A strikingsimilarity betweenthe spectrumof thedoubly excitedtwo electron atom and that of a linear XYX triatomic moleculewas pointed out by Kellman andHerrick [59,60, 136]. Theseauthors,usingthe truncateddiagonalizationmethod in which productsofhydrogenicwavefunctions(noopenchannelcomponentsareincludedin the wave functions)areused,have constructedquite symmetric ‘super-multiplet’ structuresfor the two electron doubly excitedresonances,despitethe fact that the 0(3) symmetryis brokenby the presenceof the electron—electroninteraction.Of course ‘super-multiplet’ structuresare quite commonin various branchesof physics.Theseincludethe Wignersuper-multipletheory [137],Gell-Mann’seightfold way in elementaryparticlephysics[138],andthe super-multipletstructuresin nuclei [139],to namejust afew. The existenceof the‘super-multiplet’structuresin atomicspectrainvolving electronicmotions, however,is relativelynew, inspiteof the fact that the use of grouptheory to simplify matrix elementcalculationsin atomic physicshasa long history.

The highly excited resonanceshave also been calculated by the method of complex rotation.Hylleraastypewave functionswere usedto calculateS andP waveresonances[128,129], andproductsof Slaterorbital type wave functionsfor resonanceswith L � 2 [127, 128]. When resonancestatesofboth parities(~l)’ and( l)L±1are calculated,the so-called‘I’ super-multipletstructuresareshowninfigs. 8 and9 for the intra-shellresonancesconvergingon the N = 4 andN = 5 resonancesrespectively.It is evidentthat the highly excitedtwo electronresonancespectraarevery similar to thoseof a linearXYX molecule.A simplepicture (fig. 10) to describethesemolecular-like two electron spectrais thefollowing: when the two electronsare far away (highly excited) from the nucleusand with equal

30 Y.K. Ho, The methodof complexcoordinaterotation

Ill I I Ill II

HYDROGEN N~4 THRESHOLD1

30

3pO

~ ‘ 3e 0-007- L L

1~0 3

0e

—— 30F

Se 3P0

-008- —

1~O t~I I~2 1’3.009 liii! III I III

3210123 2IO~2 21012 01

T T T T

Fig. 8. Doubly excited N = 4 resonancesof W plotted accordingto the 1’ super-multipletstructureof ref. [127].

Ill 1-1111 111111 liii

-0040 -

~ 30e

L . ~° ~F° — — 3~0 Ge

-- — —

3F° — —

~ 0.045 Se — IDe

3D0 — De 3PO —

— — — 3Fe 1F°-—

0.050 De

1*0 1~I 1’2

11(1111 III!-0.055 432lOI234 32l0l23 21012

T T T

Fig. 9. Doubly excitedN = 5 resonancesof H- plotted accordingto the 1’ super-multipletstructureof ref. [127].

0 + ØROTAT ION

BENDINGVIBRATION

Fig. 10. A simple pictureof thetnatomicmolecular-likecharacterof thedoubly excited intra-shellresonancesof two electronlow Z systems.


distance(intra-shell states),they are kept to locate on the oppositesides of the nucleusby strongangularcorrelationeffects. The two electronclouds,as usuallyinterpretedfrom quantummechanicalpoint of view, now behave very much like two condensedfluid drops, and undergo vibrational,rotationalandbendingmotionsin a mannersimilar to a linear XYX molecule.Of coursein the doublyexcitedphenomenon,oneof thetwo electronswill eventuallybe autoionized.The relationbetweenthemechanismof autoionizationsand the molecular-like electron movementshas been discussedbyHerrick et al. [136].However, acompletepicturebetweenthe autoionizationwidths, which arerelatedto the lifetime of the quasi-boundstate,andthe ro-vibrationmotions of the two electronsarenot fullyunderstood.Since the complexrotationmethodis able to provide the total widths in a straightforwardmannerwith reasonableaccuracy, the results may help to provide a better understandingof thesymmetriesof thesedoubly excited atomic resonances,and the underlyingmechanismof autoioniza-tions.

4.1.4. Doubly excitedresonancesin helium iso-electronicsequenceThe calculationsof the doublyexcitedresonancesfor helium atomsandits iso-electronicsequenceby

the complex rotation methodwould be as straightforwardas those for H-. This seemsto be theadvantageover other conventionalcomputationaltechniquessince againthe asymptoticbehaviourisnot necessarilyused in the complexrotation method.On the contrary, in other methodslike closecoupling approximationand the Feshbachprojection techniquein which the Fermi-goldenrule typeformula is usedto calculatewidths, continuumwavefunctionsmustbeused.The changefrom free wavetype wave functionsin electron—atomscatteringto the Coulombtype wave functionsin electron—ionscatteringis not a straightforwordmatter.In the following we will discussthe complexrotation resultsfor the doubly excitedresonancesin He andin Lit Resonancesin the helium iso-electronicsequence,as well as the current stateof theoreticalcalculationsand experimentalmeasurementsabout thesesystems,will be discussed.

(I) Doubly excitedresonancesin HeTherehavebeena greatdeal of experimentalactivities to study the doubly excitedresonancesin

helium.Thesestudiesinvolve variousprojectilescollidingwith helium atoms.The projectilescan be lowenergy[140]or intermediateenergy[141,142] electrons,photons[143,144], or protons[145],aswell asatomsand ions. The resonanceprofiles were then deducedfrom the scatteredelectrons[146],ejectedelectrons[140,141, 142, 145], or by studyingthe photonsfrom photo-absorptionphotons[143,144], orthe photonsin beam foil techniques[147].The-generalexperimentalaspectsof the doubly excitedresonancesin helium was reviewed by Williams [97] and summarizedby Hicks Ct al. [148].Earlytheoreticaltreatmentsof doubly excitedhelium resonanceswere reviewedby Burke [95], Bransden[149],andMcDowell [100].Recentcalculationsof thesedoubly excitedhelium resonancesare a seriesof papersusingFeshbachprojectionformalism ([23, 122, 1501 andreferencestherein),Kohn—Feshbachvariationalmethod[52],saddlepoint technique[151,152], andthe TruncatedDiagonalizationMethod[93, 153], as well as the multi-configurationenergyboundmethod(stabilization method)[154]. Closecouplingapproximation([155] andreferencestherein)hasalsobeenusedto calculateresonancesin theregion betweenthe N = 2 andN = 3 He~thresholds.

The calculationsof doubly excitedresonancesin helium atomsby the methodof complexrotationhavebeencarriedout by Ho for 5- andP-waveresonances.The complexrotationresultswhich appearin the literatureare resonancesconvergingon the N = 2 [72], N = 3 [156], N = 4 [157], N = 5 and 6[158] hydrogenic thresholds.A comparisonbetweenthe complex rotation results[72] and Feshbachprojectioncalculationsfor the N = 2 resonancesis madein table6.

32 YK. Ho, The methodof complexcoordinaterotation

Table 6Doubly excitedstatesof He belowtheN = 2 hydrogenicthreshold

Complexrotation[721 Feshbachprojection[24]

Er (Ry) I (Ry) ~Er (Ry) I (Ry)

ISC(1) 1.55574 0.00908 1.55607 0.00919ISC(2) 1.243855 0.000432 1.24388 0.00049lSe(3) 1.17985 0.0027 1.17984 0.00285ISC(4) 1.09618 0.00009 1.09616 0.000177

‘P°(l) 1.38627 0.00273 1.38632 0.002668‘P°(2) 1.194149 1.19414 8.56x 106‘P°(3) 1.1280 0.0006 1.12786 0.0007351P°(4) 1.09385

3P°(1) 1.520995 0.000594 1.52098 0.0006543P°(2) 1.16930 0.00016 1.16925 0.00019193P°(3) 1.15806 1.15801 359x 10-63P°(4) 1.09768

The earlier theoreticalstudiesof the doubly excitedresonancesin helium wereof coursestimulatedby a seriesof photo-absorptionexperimentsby Maddenand Codling [143].The optically allowed 1P°resonanceshaverecently beenstudied experimentallyby Woodruff and Samson[159].A comparisonbetweenthe experimental[143,144, 159] andcomplexrotation resultsareshownin tables7 and 8. Theagreementsaregenerallygood exceptfor the widths of severalhigher-lying resonances.In table9 thecomplexrotationresultsarealsocomparedwith otherexperiments[140,1411 for the N = 2 resonances.The energysplittingbetweenvariousexperimental esonancesagrees~~rotationresults,with the absolutevaluesconsistentlydiffering by aboutthe sameamountfor all (exceptthe lSe(2) in ref. [141]) resonancepositions. Since all the experimental resonanceenergieswerenormalized to the ~P°(1)photo-absorptionresonanceposition [1431,and the energy scale forphoto-absorptionexperimentmaynot bethe sameas that for electronimpactexperiments[23],the useof theoreticalresonancepositionsfor experimentalenergynormalizationis an interestingalternative.

In table8 a new setof quantumnumbers(K, T, n) is usedt5 classifyvariôüs resonances~ëe eqs.(4.1.6) and(4.1.7)).The ‘new’ quantumnumbersK and T were discussedin section4.1, andn hastheusual meaningfor a given Rydbergseries.Of course,the group theoreticalapproachis not the onlyschemethat hastried to classify doubly excited resonances.Other configurationinteraction mixing

Table7The 1p0 resonancesin He belowtheN = 2 threshold

Complexrotation[72] Photoabsorptionexperiment[143]

Er (eV)* F (eV) Er (eV) F (eV)

‘P°(l) 60.1456 0.0371 60.13±0.015 0.038±0.004‘P°(2) 62.7592 62.758±0.01‘P°(3) 63.659 0.00544 63.653±0.007 0.008±0.004

* Measuredfrom thegroundstateof He atoms.The reducedrydbergfor He

is usedfor energyconversion(1 Ry 13.603976eV).


Table 8Comparisonof the complex rotationresults(in eV) with photo-absorptionexperi-ments. The resonancepositionsare measuredwith respectto the ground stateofhelium (E = —5.80744875Ry). The reducedrydberg is usedfor energyconversion(1 Ry = 13.603976eV). For eachresonance,the first entry refers to the resonance

position, and thesecondentry refersto thewidth

Complex Woodruffand NBS‘-~‘L” K T n rotation[158] Samson[159] experimentst

N=31P°(1) 1 1 3 69.873 69.917±0.012 69.919±0.007

69.94±0.040.1905 0.178±0.012 0.132

‘P°(4) 1 1 4 71.625 71.601±0.018 71.66±0.010.0843 0.096±0.015

‘P°(8) 1 1 5 72.178 72.181±0.015 72.20±0.010.0327 0.067 ±0.015

‘P°(3) —1 1 3 71.309 71.30±0.040.039 0.07

‘P°(7) —1 1 4 72.1590.0136

N=4*‘P°(l) 2 1 4 73.712 73.66±0.03 73.76±0.02

0.0979‘P°(4) 2 1 5 74.617 74.57±0.03 74.64±0.02

0.0612‘P°(8) 2 1 6 74.907 74.93±0.03 75.10±002

0.01900 1 4 -74.139 -------74.15±0~04

0.1291P°(6) 0 1 5 74.848

0.0558

N=5*‘P°(l) 3 1 5 75.565 75.54±0.04

0.05993 1 6 76.085 76.10±0.03

0.04081~1(2) 75.762 -

0.087

N=6‘P°(l) 4 1 6 76.594

0.0395

t Resultsfor the N = 3 resonancesare from Dhez and Ederer [144],and theN = 4 resonancesare from MaddenandCodling [143].

* For theN = 4 and N = 5 resonancesWoodruff andSamson[159]reportedthe

minimum for eachresonance,andMaddenandCodling[143]reportedthemaximum.

scheme[160],andthe semi-empirical[161]schemehavebeendiscussedin the literature.A comparisonbetweenvarious classificationschemesis demonstratedin table 10 for someof theN = 2 resonancesand in table 11. for N = 3 resonances.Among all of theseclassificationschemesthe grouptheoreticalapproach[58, 59, 60] seemsto havea bettersuccess,especiallyfor resonancesassociatedwith higherhydrogenicthresholds.An extensionof the ‘+‘ and ‘—‘ schemeusing the hypersphericalapproachtohigher hydrogenicthresholdshasjust begun[124].


Table 9Doubly excited resonancesof He belowtheN = 2 He~threshold

Experiment&’

Complexrotation* Hicks andCorner Gelebartet al.[71] [140] [141]

EriSe(1) 57.848 57.82±0.04 57.78±0.033P°(1) 58.321 58.30±0.03 58.29±0.03‘P°(l) 60.154 60.13 60.13iSe(2) - 62.092 - 62.06±0.03 62.10±0,03ISC(3) 62.962 62.94±0.033P°(2) 63.106 63.07 ±0.03 63.06±0.03‘P°(3) 63.668 63.65 ±0.03

FIS*(1) 0.1235 0.138±0.015 0.138±0.0153P°(1) 0.00808 <0.015 ~0.01‘P°(l) 0.03714 0.042±0.018 0.041±0.009lSe(3) 0.0367 0.041±~1~- -

* Resonancesare measuredfrom the ground state of helium atom (E =

-5:80744875Ry):The iflfi ~ydbei~tRy ~13~ 26 was d-- ~energy:~conversion.

6The ‘P°(l) experimental energyW6oiaI~dtothU1)hOtoabSOYptitflrexperimentat 6C~.13eV(143].

- Table 10Different classification schemes-for doubly excited N 2 statesof two~electron

— — ~~__~_systerns ______________ _________

Upskyand - copper,Fano Herrick and lowes nofConneely[55] and Pratt [1601 Sinanolgu[581--the series(N, n, a)~’L” 25~’L~(KT),,

(2, n,a)’S~ (2sns+2pnp)- - ‘S*(1~o),, - - 2(2,n, b)1S~ (2sns— 2pnp) 1S~(~1, 0)~ 2

(2,n,a)’p° (2snp+ns2p) ‘p°(0,1)~ 2(2,n,b)’p° (2snp—ns2p) ‘p°(1,0)~ 3(2,n,c)’p° 2pnd ‘p°(—1,0)~ 3

(2, n, a)3p° (2snp+ ns2p) 3p°(1,0)~- 2(2, n. b~p° (2snp— ns2p) 3p°(0,1)~ 3(2,n, c)3p° (2snp— ns2p)—2pnd 3p°(—1,0)~ 3

(II) Doubleexcitedresonancesin Li~Thecomplexrotation-methodhas~been-used~o--ealeu-late-some-lower-lying--S--and-P-waveresonances

assocated--with theN = 2 [72kN = 3 [156--IandN= 4 [129~.1571 hydogenicthresh Ids wi~b-Hyllcraastypewave functions.In the experimentalside the doubly excitedstatesof Li~havebeenstudiedby

- beamfoil techniques(Bruch et al. 11611,Z~imet al. 11621,and~Rodbroetal~[t63~);aswellasby~Th~-

photo-absorptionexperiment of Carroll and Kennedy [164]. A comparisonbetween- the -complex


Table 11Different classificationschemesfor doublyexcitedN = 3 statesof two-electronatoms

(N, n, a)’L Approximatemixings ~‘L(K, T),ref. [55] ref. [160] ref. [58] Lowest a

(3,na)’S’ 3sns+3pnp1S’(2,0), 3

(3,nb)’S’ (3sns—3pnp)—3dnd ‘S’(O,O), 3(3,nc)ISe (3sns—3pnp)+3dnd ‘S’(—2,0), 3(3, na)’P° (3snp+ns3p)+(3pnd+np3d) ‘P°(l,1), 3(3,nb)’P° (3snp+ns3p)—(3dnf+np3d) ‘P°(—l,1), 3(3,ncYP° (3snp—ns3p)+(3pnd+np3d) ‘P°(2,0), 4(3, nd)’P° (3pnd—np3d)+(3snp—ns3p)+3dnf ‘P°(O,O) 4(3,ne)1P° (3pnd— 3dnf) ‘P°(—2,0) 4

Table 12 Table 13The N = 2 ‘P°resonancesin Li~ Doubly excited statesof Li~below the N = 2 hydrogenic

thresholdComplex Photoabsorptionrotation[72] experiment[164] Complex Feshbach

rotation[72] projection(165]Er (eV)’ F (eV) E,(eV)’ F (eV)

Er (eV)’ F(eV) Er (eV)’ F (eV)‘P°(l) 150.261 0.0596 150.29±0.05 0.075±0.025‘P°(3) 161.039 0.018 161.06±0.10 ‘S’(l) 70.5912 0.154 70.5837 0.157

‘S’(2) 78.0855 0.00661 78.0842 0.0111* Measured from the ground state of Li~ ions. The reduced ‘S’(3) 83.9323 0.05851 83.9398 0.0776

rydbergfor Li is usedfor energyconversion(1 Ry = 13.60476eV). 1S’(4) 86.4205 0.00163‘P°(l) 74.6263 0.05959 74.6277 0.0593‘P°(2) 83.5257 0.000177 83.5260 0.0001631P°(3) 85.4057 0.01796 85.4134 0.0195‘P°(4) 86.3919 0.0001093P°(1) 71.3439 0.00849 71.3449 0.008923P°(2) 84.1855 0.00286 84.1891 0.003173P°(3) 84.3966 84.3970 9.51x iO~

* Measuredfrom thegroundstateof Li~ions.

rotation resultsand the photo-absorptionexperimentsis shownherein table 12. The agreementsaregenerallygood within the experimentalerrors.

Fromthe theoreticalside othermethodsto study doubly excitedresonancesof Li~arethe Feshbachprojection(including shifts andwidths) calculationsby Bhatia[165],andthe truncateddiagonalizationmethod(no widths exceptN = 2 resonances)by Lipsky andcoworkers[55, 93]. A comparisonbetweenthe complexrotation andFeshbachprojectionresultsis shownhere in table 13. Other comparisonsbetweenvarioustheoreticalandexperimentalresultscan be found in refs. [72] and[156].

(III) Helium iso-electronicsequenceThe studiesof autoionizingstatesfor othertwo electronions are lessintensivecomparedwith those

for the He andLi~systems.The complex rotation method has been applied to calculatedseverallower-lying S and P resonancepositionsand widths associatedwith the N = 2 [72], N = 3 [156] andN = 4 [129,157] hydrogenicthresholdsfor Z = 2 to Z = 10 hydrogenicions. Some resultsareshown

36 Y.K.Ho, Themethodofcomplexcoordinaterotation

4 4 pN’2 ‘P° N’2 ~P°

~

2 ~ ~

(a) L~I I f I I I0.1 0.3 0.5 0.1 0.3 0.5

l/z liz

Fig. 11. Selectedcomplexrotationresultsfor thedoubly excitedresonancesin thehelium iso-electronicsequencefor resonancesassociatedwith theN = 2 hydrogenicthresholds.n’ is theeffectivequantumnumber(seeref. [72]).

herein figs. 11 and12 as representative.In thesefiguresthe effectivequantumnumbersn~areplottedagainstlIZ, wheren * is definedby

Z2 Z_12E~=-(~)-( *) (4.1.8)

whereN denotesthe hydrogenic thresholdbelow which the resonanceslie. The effective quantum

numberis alsorelatedto the quantumdefectby

quantumdefect= N—n’. (4.1.9)

In the limit of Z-~~,a resonanceeigenvaluethat approaches

— f1\ flE— ~22) ~22

meansthe two electronswould occupythe N = 2 shell,whereasthe stateswith resonanceenergyof

— /1\ j~lE— \~22) ~32

• ~ ~P~2( __Fig. 12. Selectedcomplexrotationresultsfor thedoublyexcited resonancesin thehelium iso-electronicsequencefor resonancesassociatedwith theN = 4 hydrogenicthresholds.n* is theeffectivequantumnumber(seeref. [157]).


I I I

0.010

‘P°(l)

0.005 • -

lizFig. 13. Selectedcomplex.rotationresultsof widthsfor the doubly excited (N = 2) resonancesin thehelium iso-electronicsequence.The lIZ plotindicatesthewidths arenon-zeroeven in the limit of Z—* ~. This is theresultof theCoulomb singularitybetweenthetwo electrons(ref. [72]).

in the 1/Z-.* 0 limit would meanthe two electronsoccupyingthe N = 2 andN = 3 shellsrespectively.Someselectedresultsfor the liZ dependenceon the widths areshownherein fig. 13. It is seenthat

the widthsareneverthelessnon-zeroeven in the limit of Z—t~ This is aresultof Coulombsingularitythat causesone of the electronsto autoionizeeven in the infinite Z limit. Other l/Z studiesfor thehelium iso-electronicsequencefor resonancesassociatedwith the N = 3 hydrogenic thresholdsandobtainedby the complexrotation results(Ho [156])havebeencarriedout by MoiseyevandWeinhold[166].

The studiesof the doublyexcitedtwo electronresonancesarenot only for academicinterest,but alsohavesomepracticalapplications.Forexample,thesedoubly excitedresonanceshavebeenobservedinsolar flare [167] and in solarcorona [168],as well as in the beamfoil spectroscopy.In addition, themulti-channelresonancesin electron—ionscatteringmayhavea significant contributionto theexcitationcrosssections,which in turn, areusedto determinethe ratecoefficientsfor transitionsbetweenvariousionic states.Such contributionscould be quite large especiallyfor the forbiddentransitions(see[40,168b] and referencestherein).The rate coefficientsare used to determinethe physical conditionsforboth astrophysicalplasmasand laboratoryplasmasused in fusion research.To interpret the physicalconditionsof theplasmassuchasthe temperaturesof theplasmas,thenumberdensitiesfor different ions,as well as the energyloss mechanismin the plasmas,accuraterate coefficientsare needed.In thetwo-electronions, Hayes and Seaton[169] haveexaminedthe resonantcontributionsin e—C5~ande—N~scatteringin a threestate+3! 3!’ close coupling calculation.They found that the thermallyaveragedrate coefficientswould increaseby 10% for ls—2p and 20% for ls—2s excitations,respectively.A recent14 statealgebraicclosecouplingcalculationby Abu-Salbi and Callaway[170]estimatedsuchcontributionsas 3% and6% on e—C5~ande—O7~,respectively,at the lowesttemperatureconsideredin their work. In general, the resonancepositions and widths from the algebraic close couplingcalculationsagreevery well with the complexrotation results[156]for resonancesassociatedwith theN = 3 hydrogenicthresholds.The resonanceparametersfor N = 4 resonancesby the complexrotationmethodcan be usedas referencesfor future resonancehunting in closecouplingcalculations.

4.1.5. e~—HresonancesOne of the most successfulapplicationsof the complex rotation method is the calculation of the

e~—Hresonancesfor which mostof othertheoreticalmethodshavenot beenable to provide accurate

38 YK. Ho. The methodof complexcoordinaterotation

and conclusive results.If we recall the main mechanismfor e—H resonances(say below the N = 2

hydrogenic threshold) being the attractive 2s—2p degeneratedipole potential, it is reasonabletoconclude[171]that suchresonanceswould alsoexist in et~Hscatteringsincethe polarizationeffectsarebothattractivefor e~ande interactingwith aneutralatom.Oneof thedifferencesbetweenthesetwo casesis, of course,that thestaticpotentialfor the positronsis now repulsivewhereasit is attractivefor electroncounterpart.Sucha repulsivestaticpotentialwould reducethe depthof the potentialwell and pushtheresonancesto lie at higherpositions.Thiswasconfirmedin athreestate(ls—2s—2phydrogenicstates)closecouplingcalculationby SeilerandCallaway[172],thatpositronresonancesindeedlie closerto theN = 2hydrogenicthresholdthanthosefor electroncounterparts.Anotherdifferencebetweene—Hand e~—Hinteractionsis that the elastic resonancesbelow the N = 2 hydrogenicthresholdin e—H scatteringnowbecometwo channelresonancesin e~—Hcollisions,sincea lower-lying positroniumformationchannelisopenat 6.8eV. Including the positroniumformationchannel,as onemust in the real physicalcase,mayhavea strong coupling effect such that the threestateresonanceswould be pushedover the N = 2

hydrogenicthresholdanddisappear.To examinedifferent approachesby different theoreticalmethods,let us expandthe e~—Hwave

function by the eigenstateexpansion

~ ~(r)F~(r~,)+~ ~(p) G1(o)+~CkXk(r,rF) (4.1.10)

I k

with

15=f—F~and &=~(F+f~)

wherer andr~,representcoordinatesof theelectronandpositronrespectively,p the internalpositroniumcoordinateando~thecentreof massof thepositroniumatomrelativeto thepositionof thetargethydrogennucleus.It is understoodthat eachtermin eq. (4.1.10)is coupledto agiven total angularmomentum.Thefirst summationin eq. (4.1.10) is summedover the target hydrogenicstates,the secondsummationissummedoverthepositroniumstates,andthethirdsummationcontainsshortrangecorrelationtermswhichtakeinto accounttheinteractionsbetweenthethreechargedparticles.In eq.(4.1.10),F andG arevariouschannelwave functionsfor the positronsand positroniumatomsrespectively.Sinceeachof thesethreesummationsformsacompletesetby itself, toincludeinfinitenumberof termsin all threesummationsis notnecessary,and in fact doingso is redundant.Thetechnicalquestionis of coursehow to choosea finitenumberof terms,amongthesethreesummations,torepresentthesystemin themosteffectivemanner.Theminimumnumberof statesresponsiblefor resonancesbelowtheN = 2 hydrogenicthresholdarethethreelowestatomicstates(ls—2s—2p)in thehydrogenatomandthegroundstateof thepositronium(1~).Suchafourstateclosecouplingcalculationisnottrivial, to saytheleast,sincethecentreof massof thepositroniumatom differs from the centreof massof the target atom.Themixing of coordinateswill makeanalyticalresultsquite difficult to obtain. Therefore,much of the necessaryintegralsrely heavily on numericalquadratures.ChanandFraser[43]choseto employonetermeachfrom thefirst two sumsin eq. (4.1.10).Thesearethehydrogengroundstateis, andthepositroniumgroundstatel~.In addition,anumberof shortrange terms in the third summationto representthe 2s—2p hydrogenicdegeneracy,as well as thepositron—electroncorrelationeffects, were also included in their calculations.In such a two statepluscorrelationcalculation,no resonanceswerehoweverfound. In resonancecalculationsby closecouplingmethods,anegativeresultcouldmeanthattheresonancesmaybetoo narrowto detect,or theasymptotic


r2 potentialwasdifficult to besimulatedby alimited numberof shortrangecorrelationfunctions,despitethe fact the 2s—2phydrogenicconfigurationswere purposelytakeninto accountin the correlationterms.

Drachman[173] hasformulateda 2s—2p two-channeloptical potential variationally such that theN = 2 degeneracywould be explicitly included.When the hydrogenicis is turnedon, the resultsof ref.[172]werequalitativelyreproduced.However,when thepositroniumgroundstate1~was turnedon, theresonancesdisappeared.Since Drachman’scalculationwasless generalthan the full ls—1~—2s—2pfourstateclose coupling calculations,the issue of existenceof e~—Hresonanceswas not yet settled.Theresultsin ref. [173],however,do indicatethat the positroniumformationchannelhas a strongeffectonsuch resonances.The strongpositroniumformation couplingeffect was alsodiscussedin a stabilizationcalculationby Shimamura[174].In general,the stabilizationmethodemploysonly termslike those inthe third summationin eq. (4.1.10).In ref. [174],whenthe configurationsof ls—2s—2phydrogenicwavefunctionswere included in the short rangecorrelationterms, the resultsof ref. [172]were onceagainreproduced.However, whenhe addedtermsinvolving positroniumfactors,the stabilized (resonance)eigenvaluesbecamede-stabilized.Thereforepositive conclusionson the existenceof e~—Hresonancescould not be made.

Before we discussthe complexrotation calculation it should be mentionedthat a completetwochannelFeshbachprojectioncalculationon the e~—Hproblemhasnot beencarriedout. This is becausethe hydrogenicgroundstateis not orthogonalto the positroniumgroundstate.A projectionoperatortoprojectout thesetwo statessimultaneously,andstill retain the usualcharactersof a projectionoperator(i.e. P2 = P, Q2 = Q and P0 = 0), has not been constructed(seeref. [175]).In the close couplingapproximation,a six state (ls—2s—2p hydrogenic and 1~—2~—2ppositronium)algebraic variationalcalculation was carried out by Wakid [176]. Some resonanceswere obtained in his calculation.However, sincethe algebraicvariational techniquemay leadto unphysicalresonancesbecauseof theso-calledKohn singularities,togetherwith the fact that theresonancewidths reportedin ref. [176]wereunusuallybroad,which arenot consistentwith the relativelyhigh positionsat which theseresonanceslie, the controversyas whetheror not the two channele~—Hresonancesexist remainedunsettled.

A complex rotation calculation to study resonancesbelow the N = 2 hydrogenic thresholdwascarriedout by Doolen,Nuttall andWherry [81].Thewave function employedby theseauthorswereofPekeristype,

= ~ Ckmn exp{—a(r1+ r2)} L~°~(u)L~(v)L~(w) (4.1.11)

whereu = a(r2+ r12— r2), v = a(r1+ r12— r2), andw = 2a(r1+ r2— r12).HereL(u) arethe Laguerrepolynomials.The matricescalculatedby thesewave functionsaresparse.By taking the advantageof the sparsenessof the matrix elements,in which no morethan57 elementsarenon-zerofor a given row, theseauthorswereable to usebasissetswith up to 680 terms.By solvingthe eigenvalueproblem with the method of sparseiteration technique,they obtained a resonanceeigenvaluebelow the N = 2 hydrogenic threshold by the use of the conditions 31E1/3O 0 and

3IEIi8N = 0. Fromthe good convergenceof their resonanceparameterswith respectedto N, onecouldassumethat the condition 9IEIic9a = 0 is also satisfied.The resonanceparametersfor the lowest e~—Hresonancebelow the N = 2 hydrogenicthresholdare

Er=0.257374Ry and F0.000134Ry.

The existenceof a e~—Hresonancebelow the N = 2 hydrogenicthresholdis henceestablished.The

40 Y.K. Ho, The methodofcomplexcoordinaterotation

Table 14S-wave e’—H resonancescalculated by the

complexrotationmethod

Er(Ry) F(Ry)

—0.257374’ 0.000134—0.222±

0002b 0.010 ± 0.0004_0150318b 0.000668—0.1316± 00002b 0.000174—0.126D -

‘Doolen, Nuttall and Wherry [81].This

resonanceis below the N = 2 hydrogenicthreshold.

b Doolen [178].These resonanceslocatebetween the N = 2 hydrogenic and N = 2positroniumthresholds.

findings in ref. [81]areconsistentwith thoseof Chooet al. [177],whousedan M matrix theory togetherwith a complexrotatedHamiltonian to calculatea generalizedscatteringlength, andshowedthat aninfinite seriesof S-waveresonancesindeedexists.

Additional resonancesbelow the N = 2 positroniumthresholdwere reportedlaterby Doolen [178]with the complexrotationmethodandwith the sparseiterationtechnique.Table 14 summarizesall theS-wavee~—Hresonancesby the methodof complexcoordinaterotation.Noneof theseresonanceshashoweverbeenobservedin experiments.It shouldalso be mentionedthat thereis no resonancebelowthe positronium formation threshold.This was the conclusionmade by Bhatia et al. [179] in anextensiveFeshbachprojectioncalculation.The complexrotationcalculation[81]did not find resonancesin this regioneither.Furthermore,no resonanceotherthan S-wavehasbeencalculatedby the complexrotationmethod.

In conclusion,complexrotationmethodis highly successfulto locateresonancesin e~—Hscattering.The results reportedin refs. [81] and [178] can be treatedas standardvalues.Of course,since theproton is only 1863 timesheavierthan an electron,the six digit accuracy obtainedin ref. [81] wouldchangesomewhatwhenthenuclearmasseffectsareconsidered.Moreover,theunfinishedwork of thesemulti-channelresonancesin e~—Hscatteringareagainthe calculationsof partial widths, as well as thehighly accurateback ground (both elastic,Ps formation, and excitation if energeticallyallowed) crosssections.

4.2. Resonancesin systemsof morethan threechargedparticles

In principle, the extensionof the complex rotation method to a many electronsystemis quitestraightforward,sinceall the inter-electroniccoordinatesarescaledas r —* r exp(iO).The computationalproceduresare the sameas before.But in practice, the numberof expansiontermsin the trial wavefunction is of the orderof 10”, whereN is the numberof electrons.This maylead to slowconvergencein 0. The difficulty comesfrom the boundorbitals of the system.Let us recall the boundstatesolutionof the Schroedingerequationin the modelproblemthat we discussedin section2. In the asymptoticregion (aftercomplexrotation) the boundstatewave function (eq.(2.1.17)) hasa form

1I~(boundstate) C’ exp(—Ky cos0) exp(—iKy sin 0) (4.2.1)

the method of complex coordinate rotation and its applications to

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