PHYSICAL REVIE% A VOLUME 23, NUMBER 5 MAY I g81

complex-coordinate calculations for doubly excited states of two-electron atoms

Y. K. Ho*Department ofPhysics, 8'ayne State University, Detroit, Michigan 48202

(Received 30 June 1980j

Doubly excited states of helium sequence (Z = 1-10j are investigated by a complex-coordinate method. Some

lower-lying 'S' and "P' states below the N = 2 hydrogenic thresholds are calculated by the use of Hylleraas-type

wave functions. Results are compared with recent calculations and experiments.

I. INTRODUCTION

Doubly excited states of two-electron systemshave been intensively studied by both experimen-talists and theorists. On the experimental sidemany of these states have been observed in photo-absorption, "electron impact, "and ion impact"experiments; They have also been studied byexamining the ejected electrons spectra, "and bythe use of beam-foil techniques. " Some of thedoubly excited states in heliumlike ions have beenidentified in a solar flare" and in the solar Cor-ona, ." On the theoretical side these two-electronsystems (three-body problems) represent someof the simplest nontrivial mathematical problems.Nearly all theoretical methods to study atomicresonance phenomena have been applied to studythese resonances as testing cases. One of thepopular methods for resonance calculations is theclose- coupling approximation. Resonance param-eters such as positions and widths, as mell asbackground phase shifts, can be obtained by fittingthe total phase shifts in the vicinity of a reson-ance to the Breit-Wigner one-level formula. Incases of overlapped resonances the Breit-Wignermulti-level formula is used. Other widely usedmethods are different variants of Feshbachformalism. In the Feshbach projection theory thetwo-electron Hilbert space is divided into twoparts, an open space P and a closed space Q. Fortwo-electron systems the operators P and Q canbe readily constructed" and the resonance posi-tions are approximated by the discrete eigen-values of QHQ. For accurate calculations a shiftdue to the interactions between the open andclosed channels must be included. The sign and

magnitude of such a shift, in general, differfrom system to system and also depend on thewave functions used. For example, the shiftturns out to be quite small for the lowest ''resonance in e -H scattering in which Hylleraas-type mave functions are employed. '4 One variantof the Feshbach formalism is the truncated diag-onalization method" " (TDM) in. which the basissets are constructed by using hydrogenic wave

functions but no open-channel components areincluded. This approach simplifies computationalefforts somewhat since the Q space will automati-cally be orthogonal to the P space and no explicitprojections are needed. In addition, since thehydrogenic orbitals from higher members ofhydrogenic wave functions are able to representthe tail of the long range potential quite mell, theTDM has been able to reveal many members ofresonances. " However, the correlation effectsbetween the tmo electrons are not mell representedby the separable hydrogenic wave functions. As aresult, the values of lower-lying resonances, inwhich the two electrons occupying the same orbi-tal space and hence the correlation effects arevery important, seem less accurate. Further-more, the shifts due to the interactions betweenthe P and Q spaces are not easily incorporated.The stabilization method" and the multiconfigura-tion energy bound" (MCEB) method are the othervariants of Feshbach formalism. In the MCEBmethod, the stabilized eigenvalues computed byseparable wave functions are upper bounds" tothe unshifted eigenvalues of QHQ. The stabiliza-tion method in general obeys no bound theoremwhen nonseparable wave functions are used. Sincethe wave functions include components of openchannels, some contributions to the shifts due tothe interaction between the open and closed chan-nels are included in the calculations.

The width calculations by the above methods ingeneral require the use of continuum wave func-tions. In other words, some bound-free if notfree-free integrals must be involved. One methodwhich can avoid the use of bound-free integralsand which only bound-bound integrals are neededis the recently developed complex-rotationmethod. " The merit of this method, which willbe further discussed later in the text, is thatbound states and resonant states are treated onthe same footing such that resonance parameters,both position and width, can be calculated by usingonly bound-state wave functions. 22 ~ Quite ac-curate results have been obtained by such an ap-proach for simple atomic systems. For reson-

2137 O 1981 The American Physical Society

2138 Y. K.

Let us consider the linearly independent solu-tions of the radial wave equation for a potentialscattering

if z() 8 ( )(1+1)

d~2 ~2

which have the asymptotic forms

f (+y r) ~ e i(ssr (lr-/2)]-r- (2)

and k is regarded as a complex variable. Thefunctions of f, are called Jost functions. Thephysical solution of (1) which vanishes at the ori-gin can then be written as

ul =e[fl(l, r)+ (-1)I"S (u)f, (-l, r)].Bound-state solutions will correspond to zeros ofthe ~ matrix on the negative imaginary axis. Aresonant state corresponds to the vanishing ofthe 8 matrix at the lower k plane. At the reson-ance energy E„ the resonant state will only have anoutgoing component in the asymptotic region, i.e.,

u ~e'~ (4)

where k =~

k~e '8, with P = —,

' arg(E„) At the reson. -ant energy the wave function behaves like a di-verging state:

ei lnlnr (8 ei lnlr cos8einlr sin8

However, if we also transform r into~

r~e', with

8& Q and real (for simplicity), then E(l. (2) be-comes

ances in complex atoms24 and molecules2' somesuccesses have also been achieved. The mathe-matical aspects of this method such that the dis-crete complex eigenvalues can be related to re-sonances have also been discussed in the litera-ture. 2'

In this work we report a complex-rotation cal-culation for some lower-lying doubly excited statesfor a helium isoelectronic sequence below then = 2 hydrogenic thresholds. Hylleraas-type wavefunctions are used and some 'P and ''P' statesare calculated. By examining the convergencebehaviors for different expansion lengths, it isbelieved that the results reported in this work arequite accurate. Accurate results can be used tojudge the relative merits of different theories andshould also be useful for experimental energy cal-ibration purposes. In addition, accurate reson-ance energies and widths for these autoioniza-tion states are useful for the interpretation of therecently observed postcollision interaction" (theinteraction between the scattered and the auto-ionizing electrons in, for example, e -He scatter-ing) phenomena.

II. THEORY AND CALCULATIONS

J e =8i ls I

Iris�'

( 8) i lsl lr Icos(8 8)--lsl I r l sin(8 8) -(6)e

For 8&p and (8-p)(z(/2 the resonant state be-haves like a bound state such that it decays ex-ponentially. The result of Eq. (6) leads to theconclusion that resonance parameters can beobtained by using only L' wave functions.

For two-electron systems with Coulomb inter-actions, the rotated Hamiltonian is simply

H(r, 8)=Te ' +Ye '8

where

T=-V —V

2Z 2g 22 +

where Z is the charge of the target atoms/ions.In this work the energy unit is expressed in ryd-bergs. The wave functions are in the form of theHylleraas-type.

N

em[-&(r, +rs)]rzn[rzr:" I'oo(»8. ( )

+r,"r,"'I;.(1)I„{2)],(1Q)

with (k+nz+n) ~ &8, a positive integer. For 8states we have L, =0 and A~ m.

It has been shown" that the spectrum of H(8)consists of, in addition to the discrete complexeigenvalues which correspond to resonances when8 & P, rotated branch cuts which start from eachof the thresholds of the problem (in this case,they are -Zs/ns, n =1,2, 3, .. . ) making an angle-2~ with the real axis, and discrete bound statesof the system (in this case, they are the realeigenvalues which are less than -Z') which areinvariant under rotation. The theorem also statesthat a complex eigenvalue will stay stationaryonce it is exposed by rotated cuts. The computa-tional aspects of this method indicate, however,that complex eigenvalues will stay stationary onlyfor a short range of rotational angles when finiteexpansion wave functions are used. This is dueto the fact that the oscillatory nature of the "ex-posed" wave function [see Eq. (6)] is difficult tobe represented by I ' wave functions. Neverthe-less, accurate results can still be obtained by acareful examination of the slowing-down behaviorof rotational paths together with other conditionswhich will be discussed later in the text. Theslowing-down behavior has been discussed by sev-eral authors as a condition to fulfill the complexvirial theorem. '

The computational aspects of this work can bedivided into three parts. First, we use the sta-

COMPLEX-COORDINATE CALCULATIONS FOR DOUBLY. . . 2)39

bilization method to search for a region of &, thenonlinear parameter in Eq. (10), such that thereal eigenvalues for the unrotated Hamiltonian(8 = 0) below the n =2 thresholds exhibit stabilizedbehavior as 0' changes. It turns out that forlower-lying resonances associated with the n = 2thresholds, the eigenvalues exhibiting the most'stabilized behavior are calculated by using thenonlinear parameters in the vicinity of Z/2. Thesecond step is to apply the complex-rotationmethod (8+ 0) to the stabilized wave functions.It is found that within the stabilized pj.ateau therotational paths meet each other, from differentdirections, at the position of a pole. At the sametime the rotational paths also slow down whenthey come across the resonant pole as 6I increa-ses. The third step is to apply the first two stepsby using different expansion lengths. In parti-cular we examine in detail wave functions with%=95 (&v =8), %=125 (&v =9), and &=161 (~=10)terms for 8 states, and M=84 (&v=6), %=120(+= V), and N= 165 (ur =8) terms for I' states. An

example for the computational procedures con-cerning the first two steps has been fully discus-sed and demonstrated in an electron-positroniumresonance calculation. ' The final results arededuced from the discrete complex eigenvalueswhich fulfill the following conditions as nearly aspossible:

(11a)

(11b)

(11c)

where E„is the complex eigenvalue in which thereal part gives the resonance position and theimaginary part gives half of the width. Equation

(lla) is required to fulfill the complex virialtheorem. " Equation (11b) is required such thata complex eigenvalue is insensitive to the changesof nonlinear parameters (within the stabilizedplateau). This can be considered as a generalizedcomplex stabilization method. Condition (1lc) isfor the conventional convergence behavior as theexpansion length increases.

Table I shows '8' results for Z =1 to 10 and

Tables II and II show 'I" and 'P' resonances, res-pectively. In the following section we will analyzethe H, He, and Li' results in detail. Then thegeneral result& for the whole sequence will bediscussed.

III. RESULTS AND DISCUSSIONS

A. H

There are many calculations for lower-lyingresonances in e -H scattering. For the 'S'(1)resonance we choose to compare only with a Fesh-bach calculation" in which Hylleraas-type wavefunctions are used and a shift between open and

closed channels is included. The Feshbach pro-jection results and the present calculation areshown in Table IV. It appears that the agreement,within the stated errors, turns out to be extremelygood. These results can be considered as the non-relativistic "solution" for the lowest 'S' resonancewith the assumption that the nucleus of the hydro-gen atom is a point charge and is infinitely heavy.The first improvement of these results shouldcome from nuclear mass effects. It would be de-sirable to have the existing experimental errorsreduced considerably so that further improvedtheoretic. al results can be tested. Table V showsthe lowest 'P' resonance in e -H scattering. Herewe also show Feshbach calculations in which theclosed channel is represented by either configura-tion-interaction-type (CI) wave functions" or

TABLE I. S autoionization states of the helium isoelectronic sequence below the n = 2 hy-

drogenic thresholds. Energy unit is expressed in rydbergs.

'S'0-) S (2) S (3) Se ~4)

1 0.297 5532 1.555743 3.811 694 7.066 925 11.321766 16.576 407 22.830 928 30.085 329 38.339 67

10 47.593 98

0.003 4620.009 080.011320.012 540.013300.013 820.014200.014 500.014 730,01492

1.243 8553.260 8786.275 668

10.289 22515.302 12521.314 64028.326 90636.339 01545,351 010

0.000 4320,000 4860.000 5540.000 6020.000 6420.000 6680.000 6880.000 7040,000 717

1.179852.831 155.204 18.299 0

12,116016.655 021.916427.899 834.605 4

0.002 70.004 30.005 30.005 950.006 370.006 70.006 90.007 100,007 25

1.096 182.648 274.916527.904 66

11.6139216.044 8121.197 5727.072 3033.669 08

0.000 090.000 120.000 170.000 200.000 2230.000 2380.000 250.000 260.000 266

2140 Y. K. HO

TABLE II. &Po autoionization states of the helium isoelectronic sequence below the n = 2 hy-drogenic thresholds. Energy unit is expressed in rydbergs.

'P' (2) i+0 (3) P (4)

2 1.386 27 0.002 733 3.51512 0.004 384 6.638 96 0.005 465 10.760 42 0.006 196 15.880 56 0.006 707 21.999 93 0.007 078 29.11878 0.007 399 37.237 35 0.007 62

10 46.355 55 0.007 80

1.1941492.861 0355.250 3108.361 845

12.19561616.751 61222.029 83528.030 27834.752 948

0.000 0130.000 0140.000 0150.000 01630.000 01720.000 018 00.000 018 80.000 0192

1.128 02.722 865,036 508.070 7

11.826 316.303 5

21.502 627.423 834.066 9

0.000 60.001 320.002 000.002 360.002 620.002 820.002 950.003 070.003 15

1.093 852.650 3764.927 1827.925 585

11.645 97816.088 48521.253 16027.140 02533.749 100

0.000 0080.000 0240.000 0460.000 0640.000 0780.000 0940.000 1040.000 114

Hylleraas-type wave functions. " Feshbach shiftsare included in these calculations. Table V alsoincludes a Kohn variational calculation" in whichthe resonance parameters are obtained by fittingthe phase shifts in the vicinity of the resonanceto the Breit-signer formula. The resonanceenergies obtained by these methods agree quitewell. It appears that for the lowest 'P' e -Hresonance the accuracy for the first four digitsare settled. The agreement between the theoreti-cal calculations and an experimental measure-ment, ' within the experimental error, is alsosatisfactory. It should be mentioned that absentfrom Table II is a 'P' resonance for H . Since theconfigurations of the lowest H 'P' resonances is2ssp, it i.s not an easy task to get accurate resultswhen only one nonlinear parameter in Eg. (9) isused. Furthermore, the width of this resonanceis extremely narrow. In solving a complex eigen-value problem, the absolute errors for real and

imaginary parts of a complex eigenvalue areroughly the same. As a result, the error percen-tage for the width could be quite large although theerror percentage for the resonance position is

relatively small. This may reflect the generalshortcoming of the currently employed complex-rotation methodology and should be subjected toimprove in future investigations. Anyway, sincethe convergence behaviors for this state are not asgood as those for the other resonances reportedin this work, we prefer not to show the 'P' re-sults here. The same 'P' resonance, however,has been investigated in another complex-rota-tion calculation'4 in which the basis sets are con-structed by Larquarre functions.

B. He

There are also many calculations for He reson-ances. But most of them do not provide reson-ance widths; hence, we choose to compare onlywith a Feshbach projection calculation" and aclose-coupling calculation. " In the Feshbachcalculation the shifts between open and closedchannels as well as the widths are calculated byusing static-exchange nonresonant continua. Theresonance positions agree with the present cal-culation very well while the agreement for thewidths can be considered as satisfactory. The

TABLE III. 3P0 autoionization states of the helium isoelectronic sequence below the n = 2 hy-drogenic thresholds. Energy unit is expressed in rydbergs.

&'(2)I'a

3@0(4)g b

1 0.284 2732 1.520 9953 3.756 3704 6.991275 11.225 986 16.460 577 22,695 118 29.929 619 38.164 08

10 47.398 53

0.000 4260.000 5940.000 6240.000 6360.000 641 50.000 6440.000 6460.000 6470.000 6480.000 648 5

1.169302.812 545.178238.266 22

12.0764816.608 9521.863 6627.840 6234.539 78

0.000 160.000 210.000 250.000 270.000 280.000 290.000 2950.000 300.000 303

1.158 062.797 0285.158 5108.242 240

12.048 18016.576 33021.826 69027.799 26334.494 050

1.pxlp 5

1.p xlp-51.1 xlp-51.1 x10-51.2 xlp &

1.2 xlp-~1.2x10 '

1.097 682.672 424.9731127.997 216

11.744 04616.21335421.405 03327.31902633.955 304

~ The values of these widths only indicate the order of magnitude." The widths of these states are extremely narrow.

COMPLEX-COORDINATE CALCULATIONS FOR DOUBLY. . . 2141

TABLE IV. The S {1)and P {1)resonances in e--H scattering. Units are expressed inrydbergs unless otherwise specified.

's'(1)

A C74aT75'BR76c

HBT77

Present'

Present (eV)Experiment (eV)S872~

0.297 555+0.000 0040.297 553

+0.000 004

0.003 467+0.000 0020.003 462

+0.000 008

0.284 2320.284 2410.28'4 265

0.284 273

9.738

9.738+0.010

0.000 4370.000 4630.000 43

0.000 426

0.005 57

0.005 6+0.000 5

Ajmera and Chung (Ref. 30); Kohn-Feshbach formalism including shifts; configuration-interaction-type wave functions.

" Bhatia and Temkin (Ref. 31); Feshbach formalism including shifts; Hylleraas-type wavefunctions; polarized orbital nonresonance continuum.' Das and Budge (Ref. 32); Kohn variational method; HyQeraas-type wave functions.

~ Ho, Bhatia, and Temkin (Ref. 14); Feshbach formalism including shifts; Hylleraas-typewave functions; full scattering nonresonance continuum.' Complex rotation method; Hylleraas-type wave functions. For the comparison with theexperiment of Bef. 33, the infinite rydberg (1 By=13.605826 eV) is used for energy conver-sion.

~ Sanche and Burrow (Ref. 33).

agreements for the close-coupling calculations"(1S-2S-2P plus 20 Hylleraas-type correlationterms) are also good except for the 'S'(2) state.In Table V'we show the close-coupling results inrydbergs. These results are converted from the

published values which are expressed in eV.The comparison with experiments are made in

Tables VI and VII. It is seen that the agreementwith the photo-absorption experiments is satis-factory within the experimental errors. In parti-

TABLE V. Doubly excited state8 of He below the n =2 hydrogenic threshold. Energy unitsare in rydbergs.

PresentBhatia and

Temkin (Ref. 31)I

Burke andTaylor (Ref. 35)

-Er I

s (1)

se(2)

'S'(3)

's'(4)

&'(1)

&'(2)'P (3)

'&'(4)

1.55574 0.009 08 1,556 07

1.17985

1.09618

1.386 27

l.1g4 149

1.128 0

1.093 85

0.002 7

0.000 09

0.002 73

0.000 6

1.17984

1.096 16

1.386 32

1.19414

1.127 86

1.243 855 0.000432 1.243 88

0.009 19

0.000 49

0.002 85

0.000 177

0.002 668

8.56 x10-6

0.000 735

1,555 79

1.240 32

1.178 51

1.38622

0.009 11

0.000 537

0.002 67

0.002 85

1.16930

1.158 06

1.097 68

0.000 16 1.16925

1.158 01

1.520 995 0.000 594 1.520 98 0.000 654

0.000 lgl 9

3.5g x10-6

1.520 8 0.000 661

2142 Y. K. HO

TABLE VI. Comparison of ~P0 resonances of heliumatoms with the photoabsorption experiments. Both res-onance positions and widths are expressed. in eV andthe positions are measured from the ground state of Heatoms (E=-5.807 448 75 Ry). The reduced rydberg(1Ry=13.603975 eV) is used for energy conversion.

Madden andCodling (Ref. 1) Present

'P'(1)

1Po (2)

Er

63.13 + 0.0150.038 + 0.004

62.758 + 0.01

60.145 60.0371

62.759 2

'P'(3) 63.653 + 0.0070.008 + 0.004

63.6590.005 44

cular we have extremely good agreements for the'P'(2) and 'P'(3) states. It would be desirable tohave the experimental error of the 'P'(1) statereduced so a better comparison between theoriesand experiments can be made. For the compari-son with the photo-absorption experiment we usethe reduced rydberg (1 Ry= 13.603 976 eV} forenergy conversion, and the resonance positionsare measured from the ground state of heliumatoms (E=-5.807 448 75 Ry)." The agreementfor the widths between the present calculationand the photo-absorption experiment is also good

within the experimental errors. Table VII showsa comparison of helium autoionization states withan electron-impact experiment' and with an ejec-ted-electron experiment. ' For comparisons withthese experiments we use, again as indicated inRef. 31, the infinite rydberg for energy conver-sion (1 Ry=13.605826 eV). It is found that theagreements are also satisfactory within the ex-perimental errors. It is further noted that thepresent results are consistently higher than thoseof Hicks and Comer by about 0.02 to 0.03 eV forall resonances shown in Table VII. In the electron-impact experiment of Hicks and Comer' the au-thors used the photoabsorption 'P'(1} reson-ance energy at 60.13 eV as the reference point fortheir energy scale. However, such an energynormalization procedure may not be accurate.According-to, Ref. 31, the ener gy scale for electron-impact experiments is not necessarily the sameas that for the photoabsorption experiments. Inlight of this observation it is interesting to pointout that we wouM have a better agreement be-tween the present calculation and the experiment,of Hicks and Comer if the experimental 'P'(1)resonance energy is normalized to the presenttheoretical value at 60.154 eV. All the experi-mental positions would then be increased by thedifference of 60.154 and 60.13 eV, an amount of0.024 eV. Similarly, we can apply the same in-crement to the ejected-electron experiment of

TABLE VII. Comparison of doubly excited states of helium with an ejected-elec tron experi-ment and with an electron scattering experiment. Both resonance positions and widths are ex-pressed in eV and the positions are measured from the groUnd state of helium atoms (E„= 5.807744875 Ry&. The infinite rydberg (1Ry=13.605 826 eV) is used for energy conversion.

Hicks and Comer (Ref. 4)(electron impact)

Gelebart et al. (Ref. 7)(ejected electrons)

Present

57.82 + 0.04

58.30 + 0.03

60.13~

62.06 + 0.03

62,94 + 0.03

63.07 + 0.03

63.65 +0.03

57.78 + 0.03

58.29 + 0.03

60.13

62.10+ 0.03

63.06 + 0.03

57.848

58.321

60.154

62.092

62,962

63.106

63.668

igg (1)

P'(1)

P'(1)&ge(3)

0.138+0.015

&0.015

0.042 + 0.018

0.041 + 0.010

0.138+ 0.015

=0.01

0.041 + 0.009

0.123 5

0.008 08

0.037 14

0.036 7

The P'(1) resonance is normalized to the resonance energy of the photoabsorption experi-ment at 60.13 eV.

COMPLEX-COORDINATE CALCULATIONS FOR DOUBLY. . .

CO

lQ

O OLQ

Cb

O

, CG

~~

MtD

0

S

bO0'a

II

C)

Q

~~

o

8

O

Q

Cl

g1Ky

cd

0cdO

o

cd

Ocd

4 pQ

8~wL4

4

8~ WQN

t00 M

O O

00O O00 CD

00Cb tCOCb OCO' O00

COO OCD O

+IO

O O

v-I rV ~ ~ M 00Cb e 00 O Ce ALQ ~ O O Cb O

~ ~ ~ 0 ~ ~O O 00 O'K O

00

CQ

OOOO

00

MO O~ ~

O O+I

tQCb

OO OEQ

CbCg M tCQ Cb LQ

CgCO O

~ ~ ~

O CO00

t CO Ot- Cb C0CgCO O LO

O Nt 00

lQ

CD

COOO

C0

t

0 0

O+I

O

t COt Cb

O LQO O

OO@O

00

CgCO

O YJ Cb

O Q OO R O

00

COLQ 00 CO

LQ Cg CD00 O Cb

CO

00 00

Cb Cb00 Cb

00CO CD

rl00

IO

Cat- +CbO'

O Q Cb00

EO

CD

+ItM

t

COOO

00CQ

CbO Cb

Cb M CbO N 00CbO&O~OS CD

~ ~ ~ ~(OOOO00 t

OCb

CO00

COtOCO

II

o~~

0

hG

o

Q)N

00

OfwI

II

4)

0m

C) g)

o+Q

Q7

Q 00Cb

o ~

II

4 M)

IU

Q)4I 9g R

'.8 ":Q

0No o

8 8

'0 "0

Q S

2144 Y. K. HO

Gelebart et al. ' Except for the '8'(2) state whichwould be further away from the present theoreti-cal value, the agreements between. the presenttheory and the experiment are generally better.

C. Li+

The studies of Li' doubly excited states are lessintensive as compared with those of He in both ex-perimental measurements and theoretical cal-culations. Nevertheless, some experimental re-sults and calculations do exist for comparison. InTable VIII we show the results of a photoabsorp-tion experiment by Carroll and Kennedy. By fo-cusing a laser beam on a target with high atomicnumber such as tungsten and uranium, theseauthors were able to produce a uniform continuumfrom which a photoabsorption experiment wascarried out. The experimental study on Li' reson-ances is in analog to that of Madden and Codling'for He doubly excited states. The difference is ofcourse that the background continuum for Hedoubly excited states lies in the ultraviolet regioncompared to the soft x-ray region for Li' ions.Three lowest members of the 2snP+ series, inwhich the configuration mixings are the strongest,have been reported by Carroll and Kennedy. 'These resonances are labeled by, according tothe present notation, 'P'(1), '&'(3), and '&'(6),respectively. The first two members are corn-pared with the present theory in Table VQI and theagreements are found to be extremely good. Forthe comparison with the photoabsorption experi-ment we use the reduced rydberg of lithium forenergy conversion (1 Ry=13.60476 eV). Theagreement for the width of the 'P'(I) resonance,within the quite large experimental error, is seento be adequate. It mould be desirable to have theexperimental error reduced so a better compari-son between theory and experiment can be made.Table VIII also shows the results of a beam-foilexperiment by Zeim et al.' It is found that thepresent results are lower than the experimentalvalues and lie outside the experimental errors.Also the width for the experimental 'P (1) stateis seen to be too small even if the experimentalerror is included. On 'the theoretical side wecompare in Table VIII with a Feshbach projectioncalculation" in which the shifts are calculated byusing static-exchange continua. The agreement issatisfactory except for the 'S'(2) and 'S'(3) statesin which the widths do differ somewhat.

D. General results for the helium sequence

In addition to H, He, and Li', the present cal-culation also includes results up to Z =10 for sev-eral lower-lying 'P and "P' resonances. It wouldbe interesting to discuss the general behavior for

these resonances according to configurations.However, since we are using Hylleraas-type wavefunctions the classification of the resonances is notnot straightforward. Although one could, in prin-ciple, project out the component of various con-figurations from the stabilized wave functions andexamine the dominated configurations, no suchattempt is made here. We choose, instead, toplot the effective quantum number n* vs I/Z inwhich n* is given by the relation

where E is in rydbergs. In Eq. (10) N is the prin-cipal quantum number below which the resonanceslie. In this work, N is equal to 2. The effectivequantum number n* is also related to the quantumdefect by

quantum defect=n -n*,where n is the principal quantum number for theouter electron having values of n =N, N+1, &++ 2, . . . , etc. By plotting n* vs I/P, we can identifythe present resonances with those of previousstudies, from which the configurations can beassigned. Figure 1 shows the 'P results and the'P' and 'P' results are shown in Fig. 2. In Fig. 1the curves starting out from n*=2 at 1/Z =0 rep-resent resonances of those of the two electronsoccupying the same n =2 shell. This is equivalentto having an energy of [-(1/2 ) —1/2 j By in theZ- ~ limit [see Eq. (10)]. Similarly, the curvesstarting out from n* =3 at 1/8 = 0 means one elec-

:l2

0.3

FIG. 1. 1/Z dependence on effective quantum numbern* for ~S~ states. The curves are drawn from the valuesdeduced from Table I.

COMPLEX-COORDINATE CALCULATIONS FOR DOUBLY. . .

N=2 'P

4 I I I I

N=2 P-I3

I I

O. l 03

FIG. 2. 1/Z dependence on effective quantum number n*for p states. The curves are drawn from the values de-duced form Tables II and III.

tron occupies the n = 2 shell while the other oc-cupies the n =3 shell. The energy then gives avalue of [-(1/22) —1/3~] Ry in the Z —~ limit forthese resonances. For 'P states there are twoseries converging on the n =2 thresholds. Only thetwo lowest members for each series are calcula-ted. For P-wave resonances there are three ser-ies for both singlet and triplet states with oddsymmetry. For 'I" resonances we report the twolowest members for one series and only the low-est member for the other two series. For 'I"states four resonances are reported in this work.While the positions and widths for two of themhave been determined accurately, only the orderof magnitude of the widths for the other two re-sonances can be determined. This is partly due tothe narrow widths of these resonances and partlydue to the fact that the use of one nonlinear param-eter in the Hylleraas-type wave functions is notthe most effective way to describe resonances in

which the two electrons occupy different shells.Although we have not provided detailed studies

of the resonance wave functions for classificationpurposes, we briefly summarize the current de-velopments on the studies of two-electron con-figuration mixings. This phenomenon has beeninvestigated ever since the photoabsorptions ex-periments were made. Cooper, Fano, andPrate" (CFP) first realized that the usual classi-fication scheme for singly excited states couldnot adequately describe the doubly excited pheno-

ena, and subsequently employed a + and —nota-tion, depending on the relative (in or out of phase)radial motions of the two electrons, to approxi-mate the strong mixings of the configurations. Analternative notation has been given by Lipsky andco-workers. " By a detailed examination of theresonance wave functions, and by studies of thevalues of quantum defect and reduced widths,Lipsky and Conneely (LC) classified a resonanceaccording to a symbol (N, n, o'. )'~' 'I-', where Nand n have the same meaning as those discussedearlier in the test, S and L are the total spin andorbital angular momentum, respectively, m rep-resents the parity, and & denotes different reson-ance series, and is labeled by a, 0, and e, de-pending on the order of appearance of the lowestmember of the series. Recently these doubly ex-cited states have also been studied by a group

.theoretical SU(4) & SU(4) approach. " In addition tothe usual good quantum numbers for single-parti-cle classifications, Herrick and Sinanoglu (HS), ~'

for example, obtained two nearly "good" quantumnumbers K and T, by diagonalizing &, -&„where&,'and &, are the Runge-Lenz vectors for elec-tron 1 and 2, respectively. These authors (seeTable IX) subsequently classified a resonance bya symbol of 2~ "L'(K, 7)„, where '~"I' and n have

TABLE IX. Comparison of different classification schemes for doubly excited states of bvo-electron systems.

Lipsky andCbnneely (Bef. 17)

Cooper, .Fano,and Prats (Ref. 38)

Herrick andSinanoglu (Bef. 39)

Lowest n ofthe series

Pl, n, o.)2~'~I.~

(2,n, a)'S'

(2,n, b)'S'

(2,n, a)~po

(2,n, b)'p'

(2,n, c)'P'

(2n a)&(2,n, b)3p'

(2,5, C) Po

(2sns + 2pnp)

(2sns —2pnp)

(2snp +ns2p)

(2snp —ns2p)

2pn8

(2snp +ns2p)

(2snp —ns 2p)

(2snp —ns2p) —2pnd

2s+iLg(K y)

's'(1, o)„

's(-1, o)„

p'(o, 1)„'p (1, o)„

po(-1, 0)„3po (1 p)

3po(O, 1)„

3po( 1 p)

2146 Y. K. HO

0.0 IO—

O.OO5

0.005

0.0I

0.5I Z

O. I 0.5 0.5

FIG. 4. 1/Z dependence on width I' for IP'(1) ando&' (3) states. These two resonances are the lowest two

members in (2,n, a) series according to Ref. 17 and(0, 1)„series according to Ref. 39. Also see Table X.

O

C

O. l 0.5

FIG. 3. 1//Z dependence on width I' for ~S' states.(a) 8' {1)and $'(3) states. These two resonances arethe lowest two members in (2,n, ~) series according toBef. 17 and (1,0)„series according to Bef. 39. (b)~8'(2) and ~S'(4) states. These two resonances are thelowest two members in (2,n, b) series according to Ref.17 and (-1,0)„series according to Bef. 39. Also seeTable X. .

the number of 'P series below the N =2 thresholdis two. Similarly, for the "I"resonarices, thepossible values of 1' are 0 and 1. When & has avalue of 1, E can be +1 or -1. When T =1, E canonly have a value of 0. The total number of ser-ies for "P' states below the N= 2 threshold isthus e(lual to three. (Since K and T do not dependon the 2$+ 1 factor, the above argument holds forboth singlet and triplet eases. ) In Table IX, weshow the corresponding classifications betmeenLC and SH, as mell as the approximate configura-tion mixings labeled by CFP.3 A detailed com-parison between the LC scheme and configurationmixings, including higher orbital angular momen-tum resonances, has been given by Lipsky and co-workers. " In Table X we also compare thenotation for the present resonances, mhich aredenoted according to the order of appearance, tothose of LC and SH. The ordering of the reson-

I' I I I

II I I I

the same meaning as those of LC; T is restrictedby the value of I, such that T=6, 1, .. . , L. IfII = (-)~" then T &0 (this case does not apply tothe resonances reported here). The value of Xis governed by N and T, and is given by +E=N-T -1, N-T —3, .. . , 0 or 1. Therefore fora given set of I, N and m, the possible number ofseries can be obtained by the conditions onE andT. For example, for the '8' resonances belowthe N= 2 threshold, the possible value of T is 0and the possible values of E are +1. As a result,

0.0( I I

0.5 I.O

FIG. 5. 1/Z dependence on width I' for &'(1) and& (2) states. These two resonances are the lowest twomembers in (2,n, a) series and (1,0)„series. Also seeTable X.

COMPLEX-COORDINATE CALCULATIONS FOR DOUBLY. ..

Present

Lipsky and . Herrick andConneely (Ref. 17) Sinanoglu (Ref. 39)

(V,g, 0,) (K, T)„

'S'(1)

'S'{2)

S8(3)

iSe(4)

&'0)P (2)

P {3)ipo {4)

(2, 2,a)

(2, 2, b)

(2, 3,a)

(2, 3,b)

(2, 2,a)

(2, 3,b)

(2, 3,a)

(2, 3,c)

(2, 2,a)

(2, 3,a)

(2, 3,b)

(2, 3,c)

(1, 0)2

{-1,0)2

(I, 0)3

(-1,0)3

(0, 1)2

(1, 0)3

(-1.0)3'

(1, 0)2

(1, 0)3

(-1,0),

~ The ~P'(4) state is labeled as (1, 0)4 by Herrick andSinanoglu. See the text for discussions.

TABLE X. Doubly excited states of two-electronatoms calculated in the present work as labeled by theschemes of Lipsky and Conneely and of Herrick andSinanoglu. The order of these states holds for allZ when Z~2.

ances in Table X is valid for all Z when Z~ 2.When Z& 2 some crossings may occur (althoughnot shown in Fig. 2) and the order of resonancesmay be different in the real physical system(Z =1). For example, the work of HS" and the

'present calculation indicate that the 'P'(0, 1), and'P'(1, 0), resonances cross each other near 1/Z= 0.9. This shows why the configuration of thelowest 'I" resonance in H is 2s3P while the low-est 'P' resonance for the rest of the helium se-quence (Z) 2) is 2s2P. Furthermore, not onlydo these two resonances cross each other, butthe Feshbach-type 2s2P resonance for Z) 2 alsobecomes a shape resonance at Z = 1. At thisstage the exact value of Z (say Z') at which theFeshbach resonance becomes a shape resonance,however, has not been accurately determined. Weknow, of cJurse, that the value of S' is found tobe 0.9 &1/Z'& 1. The mechanism of the shaperesonance in e -H scattering is due to the combin-ation of an attractive polarization potential causedby the 2s -2P degeneracy of the target atoms, anattractive short-range static potential, and arepulsive angular momentum barrier. These po-tentials, of which the range are no longer thanr 2 asymptotically, will be dominated, in theasymptotic region, by the attractive -(Z -1)/rCoulomb potential when Z is greater than 1. As

TABLE XI. Comparison of ~P'{1)and P'(1)resonances with a 1//Z expansion calculation(Ref. 40). Hesonance positions and widths are expressed in rydbergs.

PresentP'(1)

1/Z expansion Present'P'(1)

1/Z expansion

9

10

Err

-Err

1.386270.002 73

3.515120.004 38

6.638 960.005 46

10.760 420.006 19

15.880 560.006 70

21.999930.007 07

29.118780.007 39

37.237 350.007 62

46.355 550.007 80

1.387 080.002 69

3.515260.004 35

6.640 300.005 43

10.761 860.006 15

15.882 060.006 68

22.001 460.007 07

29.120 340.007 38

37.238 860.007 62

46.357 120.007 83

0.284 2730.000 426

1.520 9950.000 594

3.756 3700.000 624

6.991270.000 636

11.225 980.000 641 5

16.460 570.000 644

22.695 110.000 646

29.925 610.000 647

38.164 080.000 648

47.398 530.000 648 5

0.284 420.000 667

1.521 140.000 748

3.756 5020.000 729

6.9914000.000 715

11.226100.000 705

16.460 6940.000 698

22.695 2280.000 692

29.929 7220.000 689

38.1641920.000 686

47.398 6420.000 683

2148 Y. K. HO

a result, the shape resonance in H will be pulledback to below the N=2 threshold and become aFeshbach resonance once Z is greater than S'.

Another interesting result shown in Table Xis the ordering of the 'P'(4) and 'P'(5) states. The'P'(4) resonance is labeled here by (4, 0), al-though Herrick and Sinanoglu" indicated that the'P'(4) state should have (1,0), configurations. Ac-cording to these authors, a crossing between(-1,0), and (1,0)4 curves occurs at a value of1/Z slightly smaller than 0.5. However, such acrossing, based on the present analysis and asimplied by the work of LC, would occur only when1/Z &0.5, thus the configurations of 'P'(4) and'p'(5) for Z =2 are (-1,0), and (1,0)», respectively.

In Figs. 3-5 we show the Z dependence on theresonance width. Here we briefly summarize theresults and discuss the implications that the pre-sent results seem to imply: For the resonancesreported in this work the width increases as Zincreases. This also applies to the 'P'(1) curve.The behavior for the 'P'(1) widths found in thiswork contradicts with that of LC" and of a 1/Zexpansion calculation. " In Refs. 17 and 40 the width,which is found first'increasing at low Z, decreasesat high values of Z. Nevertheless, the agreementfor the resonance positions between the pre-sent calculation and the 1/Z expansion 0 seemsquite good, as can be seen in Table XI. Further-more, a better agreement is found for highervalues of Z. This is, of course, an expected be-havior for 1/Z expansion calculations due to thefact that its error becomes smaller when Z islarger.

Another interesting observation on the widthsis that, within an infinite series, the width of thelower-lying resonance, in general, is larger thanthe widths above. This phenomena is also under-stood. Since the lifetime of the higher resonanceis longer than the lifetimes below (within the sameseries), it will take more time for a higher re-sonance to decay back to the ground state of theHe' ion than that would be needed for a lower-lying state. The widths are also found to havesimilar 2 dependence for resonances within thesame series. This can be seen in the 'P(1) and'S'(3) states as shown in Fig. 3 and in the 'P'(1)and 'P'(3) states as shown in Fig. 4. In addition,as Z increases, a lower-lying resonance seems toincrease faster than a higher-lying resonance.This can be considered as different third-ordereffects on the widths for different resonances. ~The widths are the results of second- and third-order correlation effects between the two elec-trons of the atoms/ions. The second-order effectis a combination of radial correlations (in-and-out motions of the two electrons as first pointedout by Fano'8) and angular correlations (to placethe two electrons on opposite sides of the nucleus. )The third-order effect is then to compensate thesetwo correlation effects. Apparently, such a com-pensation has a more pronounced effect on somestates than on the others.

ACKNOWLEDGMENTI

This research was supported in part by a %ayneState University Faculty Research Award.

*Present address: Department of Physics and Astron-omy, Louisiana State University, Baton Rouge, Louis-iana 70803.

~R. P. Madden and K. Codling, Astrophys. J. 141, 364(1965).

P. K. Carroll and E. T. Kennedy, Phys. Rev. Lett. 38.,1068 (1977).

3N. Oda, F. Nishimura, and S. Tahira, Phys. Rev. Lett.24, 42 (1970); N. Oda, S. Tahira, F. Nishimura andF. Koike, Phys. Rev. A 15, 574 (1977).

P. J.Hicks and J.Comer, J. Phys. B 8, 1866 (1975).5M. E. Hudd, Phys. Rev. Lett. 13, 503 (1964); 15, 580

(1965).~A. Bordenave-lVIontesquieu, A. Gleizes, M. Rodiere,

and P. Benoit-Cattin, J. Phys. B 6, 1997 (1973).~F. Gelebart, R. J. Tweed, and J. Peresse, J. Phys. B

9, 1739 (1976).M. Rodbro, R. Bruch, and P. Bisgaard, J. Phys. B 12,2413 (1979).

9R. Bruch, G. Paul, J. Andra, and L. Lipsky, Phys.Rev. A 12, 1808 (1975).

~ P. Ziem, R. Bruch, and N. Stolterfoht, J. Phys. B 8,

L480 (1975).~~G. A. Doschek, J. F. Meekins, R. W. Kreplin, T. A.

Chubb, and H. Friedman, Astrophys. J. 164, 165(1971).

12A. B.C. Walker, Jr. and H. R-. Rugge, Astrophys. J.164, 181 (1971).Y. Hahn, T. F. 0 Malley, and L. Spruch, Phys. Rev.128, 932 (1962).

~4Y. K. Ho, A. Bhatia, and A. Temkin, Phys. Rev. A 15,1423 (1977).

~5P. L. Altick and E. N. Moore, Phys. Rev. Lett. 15, 100(1965).

6L. Lipsky and A. Russek, Phys. Rev. 142, 59 (1966).L. Lipsky and M. J. Conneely, Phys. Rev. A 14, 2193(1976); L. Lipsky, R. Anania, and M. J. Conneely, At.Data Nucl. Data Tables 20, 127 (1977); M. J. Conneelyand L. Lipsky, J. Phys. B 11, 4135 (1978).

~ A. U. Hazi and H. S. Taylor, Phys. Rev. A 1, 1109(1970).E. Holoien and J. Midtdal, J. Phys. B 4, 32 (1970).J. F. Perkins, Phys. Rev. 178A, 89 (1969).

@For the early developments on the complex-rotation

COMPLEX-COORDINATE CALCULATIONS FOR .DOUBLY. . . 2149

method see the October issue of Int. J. QuantumChem. 14, No. 4 (1978).

~~Y. K. Ho, J. Phys. B 12, 387 (1979).Y. K. Ho, Phys. Rev. A 19, 2347 (1979).

+T. N. Rescigno, C. %. McCurdy, Jr., and A. E. Orel,Phys. Rev. A 17, 1931 (1978).C. W. McCurdy, Jr. and T. N. Rescigno, Phys. Rev.Lett. 41, 1364 (1978); N. Moiseyev and C. Corcoran,Phys. Rev. A 20, 814 (1979};S. I. Chu, J. Chem.Phys. 72, 4772 (1980).B.Simon, Int. J. Quantum Chem. 14, 529 (1978).T. van Ittersum, H. G. M. Heideman, G. Nienhuis,and J. Prins, J. Phys. B 9, 1713 (1976).E. Balslev and J. M. Combes, Commun. Math. Phys.22, 280 (1971).

~OK. Brandas and P. Froelich, Phys. Rev. A 16, 2207-(1977); R. Yaris and P. Winkler, J. Phys. B 8, 1475(1978).

+M. P. Ajmera and K. T. Chung, Phys. Rev. A 10, 1013(1974).

@A. K. Bhatia and A. Temkin, Phys. Rev. A 11, 2018

(1975).J. N. Das and M. R. H. Rudge, J. Phys. 8 9, L131(1976).

33L. Sanche and P. D. Burrow&, Phys. Rev. Lett. 29, 1639(1972).

3~J. J. Wendoloski and W. P. Reinhardt, Phys. Rev. A

17, 195 (1978).3~P. G. Burke and A. J. Taylor, Proc. Phys. Soc. 88,

549 (1966).3~C. L. Pekeris, Phys. Rev. 115, 1216 (1959}.3iA. K. Bhatia, Phys. Rev. A 15, 1315 (1977).@J.W. Cooper, V. Pano, and F. Prats, Phys. Rev.

Lett. 10, 518 (1963).3~D. R. Herrick and O. Sinanoglu, Phys. Rev. A 11, 97

(1975).+G. W. F. Drake and A. Dalgarno, Proc. R. Soc. London

320A, 549 (1971).+N. Moiseyev and F. Weinhold, . Phys. Rev. A 20, 27

(1979).