Bound state methods for elastic scattering phase shifts

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CHEhWALPHYS1C.S

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f& Astrophysics,

Received.19

1975

:

,,._; FOR ELASTIC

Hanlaid College OZ138, USA

.: Canlbridge;~~assachusctts ,.,,L

1 April

_I

SCATTERJN~~PHASE ..’

SHiFTi .

:,

Ol’PENljEIMtiR, H..DOYLE and A. DALGARNO,

Ccnrcr

‘.

,,

tiO’&D STATE METHODS :.,

.’ M.

..

&‘rERS~

November :

ObservatorJl’and

‘.

‘_

Smithsonian

Astroplgvsicai

Gbservatory,

: ,’

1974

_-

. ..

.~~~~rR~~odE.~:q’~~~~~.~~.;~3.=“. . E-E’ .” “ti-~~~~~~l.~~$T~~irr~e representation of the scatter&g wavefunction. cnmpared uitlt previous calculations

use 3 bound-state ‘_

Results

scaikng for singlet

ot^eiectrons tiom atoms and ions that pwave scattering by H qd by He+ are

; .‘_

‘-

.,:

.: ,:

P.-Introduction

., ‘.

‘. Conventional varktibfial procedures for the calcu‘, lation of elastic scattering phase shifts require numer.ic?J q&dratures involving two ‘continuum orbitals. Severdl m$&s havebeen proposed [l-3] ,&ich a,$oid these integrals.-Wepresent here twq further s&h‘.m&thods.which follow’from recent work on photoion- : izatidn by balgamo et al. [4]; by Ddyle’et al. [S]‘, and by L&&off and Corcoran [6]. As an alternative to ._. avoldmg the’double continuum integrals, we propose also a s*pIe. technique for their. evaluation which leads to a third method for ctilculatini the phase shifts in which.a’ hound state is used to,repIace one dr both of the continuuril,functicms in ,tie integrnnd. Resuits from all three.‘method$ are presented for the scattering 6f pwiv& electrons by He+ and-by‘H. : ,, ‘.

:. , 2. Theory

_ ,,

;’ ,,

“.

Thk pfiase~shifj s for the-elastic,scatteiing trons is giGen by-. ‘. ,.

” of eleci

the double continuum integ& is the assumption that $(E) can be adequately approxinkted in the matrix Blement of (1) by a square integrable function. ‘For a Z-electron system we diagonal&e the total hamiltonian for the system, ff, in a Hylleraas basis set df,fLnctions of the form xi = WJ12)f;9,<;

eqh+P2)

Y,;i2LM CIQ

such that ’ jn= ccixi

:

(3)

i

and the ci art: determined (D .iFi-E,le,,.,*) ,‘I where .: jyy_;p;_

, 0)

=

by

0 j

~,;_~,-.z+L~1 p2

‘341,

.‘, _’

712

Let S and C bk the properly symmetrized irregular Coulomb or free wave’scattering wavefunctions arid define :‘.

(5) regular and. product

.‘. @J(E)= (d@ c&(s + ctan.6) (6} ._. sin.8 =.~~(@)IVIIJ+)!~, ‘,i.., “‘.. .: .. (1) ,’: .’ where E.= $cz in ‘at&n& units.. ’ ,.Y..’ ..’ $+F) ii’ttie exaci.scaccei-ipg &en&ncSion.di’ For a sufficiently complefa set Xi, eacki ,vect& 8,, : wht the han$ltonian H, and &!Z) is ‘the eigenfutiction of’ ;. .- is an accurate representation of $~(E,)~neai’the nu.sorne hamiltonian HOFi/ 7 V. Bqth $(cEj ‘&d.@(E) : ,cIeus within-:1 nonna’@&on factor;‘s& that ‘.. .. have &gen&IukE a$d’aie.no%alized t.;! unit density.. c e .. ‘. ‘. _, ;‘, .. : ’ ., “ ; IL y:ILq . : “ : ,, ,, ;, ,. ‘ . ., (7) ~he~.&.s+Sof the .boutid stge methods that ‘avoid .:’ : .’,’ . . ._ ‘, ..: ,. -, h_, ‘_ ,.:;. ‘L. ,.:.::.-:,,~.,r.‘:., ‘. :y’,:., ,, ‘\ -., ,’ .;:v ‘: ,-: ,...: ,,.f.‘.,_, ‘... _. . _._,,’ ‘. ‘.’ ;

:

Volume 32, number 1

an isolated real resonance. The total phase jhift 6 is given by the asymptotic fom of(9) [lo]. When 6 is small, satisfactory results may be obtained by taking q(E) in (9) to be (&)-I12S(kr). The normalization constan; C,, in (7) can be determined from the ratio of any two matrix elements of the form

The validity of (7) has been-demonstrated for model interaction potentials [7], for the molecular hydrogen internuclear

potential

[S], and for wavefunctions

used in photoabsorption and emission calculations on H- [6,9]: In order ta fully exploit (7) in the calculation of continuum properties, the normalization factor C,, must be determined. We have explored two methods. In the first method \ve use the fact that the nor-

f@il”Pl$(ER)’ and

(@i10p18J

for any state Qi and operator Op. The photoionization calculations provide

malization is related to the density of pseudo-resonant functions on. By analogy with Langhoff and Corcoran [6] who derived differential oscillator, strengths from .q~bUe iM_eg.&A ~epXe5f3GaGfln,\.w!e er,n_re%% tbE in_&over a small energy region in gral of IWWl$(~P terms of I~~F~l~ie,I*. Thus

where $Q is an initial bound state and D is the dipole

operator. Then

E n+ 1 s Et,

1 April 1975

CHEMICAL PHYSlCS LETTERS

cn = QJ,lDI$(EJIiQ,IDIf?,)

~~E)I~lS(E)~dE~sL,~E~I~lQ(E)~

,

so that

= ;Oc4(E?lvlen)12 + IW3Ivle,~+$I

sin 6(fTn) = -n

.

c0(E,l)lvle,,c~i~19(~~)} -

(@JDl6,)

-

(11)

Unlike the first method, (1 I) determines the sign of the phase shift In the third method, we insert G from (9) directly

By the mean value theorem, this is approximately equal to

into (l), obtaining Hence

where E = i(Ei+Etil).

sin 6 = cos d,

+

Substitution of (8) int’o (1) gives a phase shift at the energy midway between any two adjacent pseudoPrekmably

r&onmces.

the more.densely

spaced the’

undetermined. In our second method we normalize On in (7) using matrix eiements which are available.from our recent work on oscillator strengths 151.We have shown there that $ may.b& written

ed in methods I and RI. Here we sugest a simple pro-cedue for evalllating t!x L-Legs& We expand $[E) in the basis set (2) tiith the power ofrlZ set equa1 to zero and without the (I ‘PI,) operator and obtain the expansion ‘coefficients by diagqnalizing ..

E-E,

”

E-E~~:~~~I!~R~-EI~(E))I

; e:p(-+) ..

(.lo)’

.’ and where 0, is an eigenve,ctor .. _. .’

: .

ofH &hiCh represents :; ‘. ._ : :

;

(12)

I .

The second term on the right-hand side of (12) contains singularities at pseudoresonances, E = E,. In calculating pho toionization IXOSSsections using this representation of $(I!?) we have obtained sati;fLictory results by evaluating the matrix eIements at E = E,, and eliminating the mth contribution to the sum. We have obtained consistent results in the present calculations using the same technique. The leading term on the right-hand side of (12) is the doubie continuum matrix element that was avoid-

eigenvzlues E,, the more accurate will be the phase shifts: However, (8) leaties the sign of the phase shift

co;+&

,wwle”~@,,lvIQ(E)) ;E-E, /i

-,.

.. 1

‘. _. :.

:

Volume 32, numbq

1.

..‘:

,.

..,

CHEMICALDIYSICS LET@Rs

‘. Then rhe noin-hl’ti~titinofthi bcund state e$nve&

,.

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1 April 1975.

.-

:

.. Fig. 1 &qmstm&

of m’ethod III

the &keigence

found by ca+lating the pverlap I.- f0r.p.-wavrsinglet s6atteri@om Hb’(ls) with in: ((1 ~P1-$~Iq5(E)>. Both.the.$sgonalization ofHO and creasing t&S size for 10, 20 and 50 functions_ Re‘.’sli!k fir 30 and 40 functiorqtire &di&gui&able ., -.the.overlap element are calculated by ihe &me corn$uter q$grarns thai determined. the fl;l rind.(9,lVl@(~> :, from the values obtained.with 50 functions. Phase with,minor.modifications. shifts from 50 fun’ctidn sets,and alI three methods are .‘: hxpression (.12) does nothepend on.the,restriction shown in.fig._2,along v&h the resilts of Jacobs [ 1 I] (7) that.8, be ati accurate representatioti of +(E,,) ,’ usiqg 3 state closecoupling, and exchange polarized ; iin< may.lead to m& accu;a,te’results than method II. -, orbital a& static exchange re&l:s of Sloan [ 121. Our .’.in casei where S ii large. Furthermore (12) produces : phsx shift; calculated by methods II and III are in i ctintinuous function 6(E) thr&gh the resonance regood agreement with each other. Method I gives surgidn. Ho&ever,if the backir&ind phase shift is iarge prisinglj, gobd,values ai low energy considering the hy this region, diffiqlty may arise il.&e calculation simplicity ,af the calculations. The increasing scatter dtiti.to the sensitivity of the resonance &figuration for larger energies is due to the wide separation’of the to changes in &basis set [$I; ._higher eigenvaluts. .” ‘. _ Jacobs [I I] rdmarks that the cl&e coupling phase ., : Afts hi& not.converged and the computed close 3. A&&cations ,’-1 coupling v&es ire a ldwkr bound to the true values ,.’ [13]. Our results lie above the close cbupling values Results for all t&ee.methods were obtained for and niay be more’accurate. se~kril distinct basis sets of’up to.50 functions xi The resclnance region, displayed,in fig..3, is in harwith non-linear pakameteti & = 2 and fl= Z/2’such mony.with the previous close coupling calculations .. that U&energy region or’interest was spanned by the I 1141. .eigenvalues. The sets’used satisfied the i’tcmas- -’ Pha,se shiitS for nqq-resonance p-wave H(ls) singlet Reiche-Kuhn sum’rule to bktt& than 1%. H, $as scattering based on our, calculated photoionization : diagonali~kl tb dete.&ni& the square integrable cross sedtions [S] for-H_ are shown in fig. 4 with both representation of O(E) with (Y= Z, D‘==Z!2 and u = 1. exchange +larized o&l [ 121 and 6.state_close cou: tor ,Bi; ofHO

at .E is

‘.

:

‘_. ;.

“. ..

Pi&-i; izotivc&nce’of

.‘-‘-.. methodi

,fG*cfi,ok; *, 50 f~~c~o~~ .. ‘. ,..’ + .8 ,_, : ;, ;: :..:

.

‘. :..

for.pwave

,. ,” :

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1.‘.

., :; ;

..,

E ( Ry.).: ‘.

singlet scattc$&from

,,

.‘_,_‘- :

.

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‘. ; ,..

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Hc~(Ibl.with~~~rea~e.‘~.basis size. x, ‘,

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i0 f+ipns;

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1 April 1975

CHEMICAL PHYSICS LETTERS

Volume 32, number I I

I

I

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---.

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3.6

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STATIC

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1.0

:

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EXCHANGE

I

I

I.4

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1

2.2

1.8

E(Ry)

Fig. 2. Norrresonance p-wave singlet scattering phase shirts from He?10 A, me.thod 1; X, method IL; o, method III; -_-.-, polarized orbital; Q, quantum defect. exci-iange; - - -, 3:stutc closccoupIing; -,

1

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1

static

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0.3

I

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0.4

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0.5

0.6

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E (RY!

Fig. 4. Non-resonance II;

T

po@rked

_, coupled.&qJaGons. :.

scattering for H(h). X, method calcuhtions: - - -, 6-state close-

p~iv%

orb+ ,’

9

,“, -. -.

.. .Volume.32. number : . .,,’ ,: ‘~ ; :., tiling ‘cal&lations [15]. Again, t)e

CHEMICAL

PHYSICS

close coupin~ re-. .m,lts$ay nbt have converged. Because of.the’pradom,:inance. of,ee iong,range polariz$ion and shoit range .e%ch:_ngeiepulsion in singlet fi+e, the polarized orbi:_ tal r&Sult$ probably superior td the clcse,coupling .except neaq!he first ex$t@on thre,&Jid, 0.75 Ryd, :&here the iestinance is important. Our $hase shifts ‘. 1: ag& lie,above the. bound represented by. the cl&e. ~‘... coupling‘results ai&t& &reen?ent between our cal-culation”qd ,the.pplarized orbital rezslts,is ‘striking. ‘-Exte&bn SKthe method to inelastic scattering is in ,progress.,.:.. .’ .‘. ‘;’ : .-; (.,‘_

-:..

:

‘_.

:

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..,

’

The.authors.wish to thank-Ellen L;iviana for assistwifh the cal@]ations, and to acjcnowlc&e4te partial support of the National Science Foundation: I :

:. &ce

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,.,

1 April

\

1,975

.:_ ,,-

“,I. ” [ 11 Fl,Harrin Phys. Rev. Let& and, A. S&o;

[2] W.P..~Reinbzudt

19 (1967) 173. Phys.

Rev. kl

(1970)

.. 1162;,. 1q.P. Reinhardt,

Phys

.4,,t .. ;

Rev. A2 (1970)

:[4] A: Ddgarno,

1967;

..

[?I &J. HeUrr and %A. j’tiani, ‘. 1201, f2119.

44 (1971)

Whys Rev. A9 (1974)

Hi Ddyle and M. Oppenheimer;

Phys. key.

Letters 29 (1972) 1051. [S]

H. Doyle, hi. Oppe&eimer and A. D&arno, Phys. Rev, to be publitied. ;16].P.W. Langhoff and C.,T. Corcorul, J. Cham. Pll*,c h’i (1974) 146. [71 A.V. Haziand H-5. Taylor, Phys R&, Al (1970) 1109.

[l,l]

~fGknowledg&r& i

LET-l-ER.5

[ 81 C.J. L~II afid G.W.F.Drake. Chei. Phys Let& (1972) 32 J. 184 (197” I’;15 ,’ ‘i9] ,G.W.F. Duke, Astrpphys [ 101 U. Fano, I)hys Rev. 124 (1961) 1866

.’

.-.

;

-V. Jncqbs.

.[12] LH. .%a~, [ 131 P.G. Burk: .88 (1966) [ 14‘) P.G. Burke 86 (1965)

Phys

Rev. A3 (1971)

-’

289:

Proc. Roy. Sot 23lA (1664) and-A.J. Taylor. Proc. Phys. 549. -. and D-D: McVicar,

16.

Proc.

Phys

151. SOC.

(London)

SOC. (Lomdon)

989. -:. 1 IS] N.A. Dou~ty, P.A. Fraser and R.P. McEachran, Notices Roy. AsrionSoc. 132 (1966) 255.

Monthly