Beam-foil spectroscopy and new atomic structure theory with a survey of results since 1970

NUCLEAR

INSTRUMENTS

AND

METHODS

IIO

0973)

I93-2o9;

©

NORTH-HOLLAND

PUBLISHING

CO.

BEAM-FOIL S P E C T R O S C O P Y AND NEW A T O M I C STRUCTURE T H E O R Y W I T H A SURVEY OF RESULTS SINCE 1970" OKTAY SINANO(~LU

Sterling Chemistry Laboratory, Yale University, New Haven, Connecticut 06520, U.S.A. BFS h a s given s t i m u l u s to new developments in atomic structure theory a n d has provided tests o f the new approaches. These and previous theoretical a p p r o a c h e s to the prediction o f allowed and forbidden transition probabilities are critically surveyed. Systematic calculations, based on accurate atomic structure theory including the correlation effects in both g r o u n d a n d excited states, are given for multiplet oscillator strenghts for both low a n d high Z ' s for nearly all 2s22pn--+2s2p n+l transitions. These c o m p a r e overall with BFS, phase-shift, a n d Hanle experiments

within 11%. N e w results a n d extension o f this theory ( N C M E T by Sinano(glu) to M-shells (Mg, AI, Si, S, CI, ..) by Sinano(glu a n d Beck are c o m p a r e d with recent BFS experiments a n d examined. Predicted lifetimes suggesting BFS experiments on highlystripped heavy ions in accelerators like S U P E R - H I L A C and t a n d e m s are given. Recent developments in theory a n d experiment on multiply-excited spectra are outlined. Lifetimes o f metastable states in [C 1], [O 1], [O I1], [N II], etc. are also presented.

I. Introduction

The theoreticalfE~ values checked by BFS constitute a stringent test of the atomic-structure theory. The reason is that let is sensitive to the details of the manyelectron correlation ( e - c o r r ) in both ground and excited states ~'2) and to the details of how manyelectron matrix-elements between two such states are handled3). ThefEl-test also tests, as mentioned below, the accuracy of the theory for other atomic properties like hyperfine structure (hfs), electric quadrupole moments of excited states, etc. The difficulty in the prediction of fEe'S has been, as BFS experts well know, with the low-Z (Z ~ N) end. At the high-Z end of an isoelectronic sequence in the fE1 VS 1/Z curves, most methods, such as HartreeFock (RHF), hydrogenic Z-expansion (ZEXP), and conventional superposition of configurations (SOC), work quite well. Electron correlation becomes less important at high Z. In fig. 1, pre-1969 NBS values and theoretical values are shown. Just as BFS and phase-shift did on the experimental side, N C M E T consistently displayed for the first time the strong dips infE~ vs 1/Z, with Z ~ N yielding the now familiar curves with the unsymmetric maxima. Since 1970, applications and the scope of the theory (NCMET) have been much expanded and systematic results obtained on: a) All ls22s22p "--, ls22s2p "÷~ in-shell transitions ( K L ~ K L ' ) of the Bel, BI, C I , N I , O1, F I isosequences for any Z (non-relativisticfEl's). b) Extension of (NCMET) theory to third-row atoms (Mg, A1, Si, P, S, CI) in which calculations required are much larger and available experimental data much less. Much new BFS work reported in the

Beam-foil spectroscopy (BFS) has not only provided accurate transition probabilities, but has also given a strong stimulus to new theoretical developments in atomic structure theory in several directions. We would like to summarize some of these developments, in addition to comparing recent theoretical results with experimental data. Various theoretical approaches that have been used in the recent past for the prediction of lifetimes will also be summarized and commented upon. Prior to the Second International Conference on Beam-Foil Spectroscopy we had already presented a then new theoretical approach (NCMET/MET) for the prediction, based on accurate calculations of excited state properties, of transition probabilities of many electron atoms1-3). In 1968-69 the method had been applied 1'3) to a number of K L ~ KL' ["in-shell (An = 0) transitions"] in second-row atoms and ions like N I, N i l , N III, O I I , O I I I , F I I , N e l l , C I , C I I , etc. Whereas previously available theoretical values had often been off by factors of 2-3, the new results agreed with new experiments (beam-foil, phase-shift, Hanle) within 5-20%, i.e., within experimental error. Thus while BFS was making reliable data possible on allowed transition probabilities (A), and dipole oscillator strengths (fEt), as compared to the density-dependent intensity methods of arcs and furnaces, similar accuracy was becoming possible with N C M E T as of 1966. The first N C M E T calculations and new experiments were performed without prior knowledge of each other's results, so the agreement represented significant checks of both theory and experiment. * S u p p o r t e d in part by U.S. N a t i o n a l Science F o u n d a t i o n .

193 III.

FHEORY

194

OKTAY SINANOGLU

TABLE 1 Multiplet oscillator strengths (fro) and transition probabilities (A in units of l09 s 1) calculated by the atomic structure theory with all f m correlations in both upper and lower states [Non-Closed-Shell Many-Electron Theory, NCMET 1) of the author]. The table gives the KL~ KL' (2s22pn~2s2p n+l) transitions systematically (B, C, N, O, F isosequences for low and high Z's). The values between the vertical dark lines (mainly between F and Si) are obtained by interpolation on f m vs l/Z curves drawn from this table. The f-values are all with the dipole-length operator (fr) and with experimental 2's in the calculation o f f r and .4. (fA andf4(rv)) agree very closely withf r lsoelectronic sequence

B 2

B C

N

2s22pzP - 2s2p2 eD 2S _ 2p 2s22p2zp _ 2s2p3 3D _ 3p aS 1D - 1D _ 1p 1S _ 1p 2sZ2p34S _ 2s2p4 4p ~D2D _

F

A

2089 0.036 0.03 1573 (0.083) (0.7) 1378 0.626 2.2

)1, 1335 1036 904 1561 1329 945

N

f 0.121 0.112 0.485 0.077 (0.092) 0.272

~,

f

A

2

0.27 2.1 4.0 0.13 (0.35) 6.1

971 764 685 1085 916 645 776 660 746 1134

0.111 0.079 0.389 0.097 0.133 0.211 0.306 0.286 0.255 0.138

0.53 2.7 5.5 0.33 1.1 10.1 3.4 7.3 1.02 0.24

788 609 554 834 703 507 600 526 598 833 719

2p

_

_

present conference (1972) has concentrated on such atoms, where previous discrepancies with earlier theoretical approaches (for the weaker type rE, < 0.1) were of the order of factors of 10-30. The results with new theory are m u c h closer. c) F o r b i d d e n transitions - lifetimes of metastable states. The observations particularly of the [O I] 5577 auroral green line have been surveyed a n d N C M E T values calculated in 1970 for f E2 -- electric q u a d r u p o l e transitions 4's) with the programs a n d methods of Westhaus a n d Sinano~lu. Recent experimental results by C o r n e y 6) a n d on other types ( s p i n - o r b i t induced E l ) of transitions by J o h n s o n v) are in excellent agreement with N C M E T calculations. d) Doubly-excited states (B**), particularly in Lilike a n d Be-like ions, have now been observed by BFSS), o p e n i n g up yet a n o t h e r area in atomic spectroscopy. Successful calculations too have been made in several laboratories on the three-electron cases9). A new group-theoretical a p p r o a c h to the classification and u n d e r s t a n d i n g of multiply-excited states has also been developedl°), in addition to N C M E T calculations which apply to such states of atoms with sizable N.

O

A

~D 9S _ 2p 2s22p48P_2s2p5 3p 1D _ 1p 1S _ 1p 2s22pS2p_2s2p62 S 2p

O

f

C

797 644 581

2. KL ~

f

A

0.104 0.56 0.065 3.5 0.327 7.1 0.097 0.56 0.122 1.6 0.178 13.9 0.291 5.4 0.208 8.4 0.285 1.77 0.198 0.63 0.141 1.8 _

0.031 0.100 0.007

_

0.19 0.18 0.14

K L ' (ls22sn2p m --+ l s 2 2 s " - 1 2 p m+~)

t r a n s i t i o n s

A comprehensive table of the n = 2, in-shell oscillator strengths including new data is shown in table 1. T r a n s i t i o n s have been calculated by N C M E T ( N o n Closed Shell M a n y Electron T h e o r y ) l - 3 ) , atomicstructure theory, rigorously including electron correlation in g r o u n d a n d excited states. N C M E T has been implemented in fully a u t o m a t i c programs written by Sinano~lu, Oksiiz, Westhaus, Beck a n d L u k e n who have also r u n most of the N C M E T wave functions. Some of the later runs reported have been r u n with the collaboration of A y d i n Ttizeren at C o l u m b i a University, a n d Cleanthis Nicolaides in our laboratory at Yale. With the exception of some Ne and N a cases, m a n y Ne, Na, Mg a n d A1 i o n f E l - v a l u e s in table 1 come from interpolations between our N C M E T calculations for F a n d Si (some Ne, N a also). In the F - S i region, therE1 vs I / Z curves are smooth a n d accurate interpolation is easy. The a u t o m a t e d programs 1-3) take in a few cards consisting of the atomic n u m b e r a n d specification of the states of interest. They yield the complete N C M E T

A SURVEY

OF R E S U L T S

SINCE

1970

195

as s h o w n in the W e s t h a u s - S i n a n o ~ l u paper3). T h e results s h o w n here have been c o m p a r e d with recent BFS, phase-shift, Hanle experiments. T h e overall a g r e e m e n t a n d certainty o f the entries are within :[: 11%. F o r uniformity the W e s t h a u s - S i n a n o ~ l u 3) values have been converted to fr with 2exv- These values m a y differ by + 3 % from ref. 3, b u t the two sets (fr here orf4(rv) in ref. 3) are equally accurate. T h e B 1 2p 2p-2p2 2S, and the C 1 2p 2 3P-2pa 3p cases, s h o w n in parentheses in the table, are reported tentatively and m a y be less accurate due to c o m p u t a t i o n a l details, a l t h o u g h the theory itself ( N C M E T ) still applies accorately.

F

Ne

Na

Mg

Si

AI

2

f

A

2

f

A

).

f

A

~.

)"

A

2

)'

A

656 507 466 678 572 420 491 431 491 658 568 430 656 507 466 607 457 514

0.096 0.055 0.282 0,090 0.108 0.150 0.210 0,180 0.263 0,208 0,131 0.213 0.036 0.085 0.097 0.150 0.235 0.065

0.89 4.3 8.7 0.78 2.2 17.7 5.8 10.8 2.4 1.1 2.7 12.8 0.33 6.6 3.0 2.7 12.5 0.55

561 435 402 571 482 359 416 366 418 543 470 358 522 422 388 490 379 428 461

0.087 0.048 0.250 0.085 0.105 0.135 0.175 0.162 0.240 0.208 0.127 0.191 0.036 0.077 0.098 0.146 0.229 0.078 0.070

1.I 5.1 10.3 1.0 3.0 21.0 6.7 13.4 3.0 1.6 I 3.8 16.6 0.53 8.6 4.3 4.1 17.7 0.95 6.6

490 380 353 493 416 313 361 318 362 461 401 308 445 360 333 410 1320 [ 361 379

0.080 0.044 0,222 0,079 0.096 0,118 0.151 0.141 0,218 0.200 0.120 0.173 0,035 0,071 0.093 0,146 0,221 0,083 0,081

1.3 6.1 12.0 1.3 3.7 24.1 7.7 15.5 3.7 2.1 5.0 20.3 0.71 10.9 5.6 5.81 24.0 1.4 11.3] t

435 338 315 433 366 278 319 281 320 402 349 270 388 315 293 353 277 312 322

0,074 0.039 0.200 0,075 0,088 0.109 0.136 0.126 0.200 0.187 0.112 0.158 0.033 0.065 0.087 0.141 0,207 0,084 0.081

1.6 6.8 13.4 1.6 4.4 28.2 8.9 17.7 4.3 2.6 6.1 24.1 0.89 13.1 6.7 7.5 30.0 !.9 15.6

390 304 283 386 325 248 285 251 287 355 309 240 343 279 260 310 244 275 280

0.070 0.035 0.185 0.070 0.082 0.100 0.117 0.113 0.185 0.176 0.106 0.148 0.032 0.062 0.083 0.138 0.193 0.082 0.079

1.8 7.6 15.4 1.9 5.2 32.5 9.6 19.9 5.0 3.1 7.4 28.6 1.1 15.9 8.2 9.6 36.6 2.4 20.2

wave functions ~Pc, containing all the non-closed-shelltype charge distribution or transition charge affecting correlations (hence we refer to these ff~c as "charge wave functions")'-3). This is the program system M O D O K . The next stage, I S H I K , calculates the /EI'S.

In table l, the highest Z ' s are for silicon ionization stages. At this point on thefE, vs 1/Z curves, one has reached the nearly linear portion, falling off to 1/Z = 0; ( f ° , ( 1 / Z = 0)=0). Thus one has all the other f-values for the high-Z ions as well which can be read off (caution: only non-relativistic of course) from the curves (the curves are not included in the present article). The advantage of thefEl-values of N C M E T in table 1, and of thefE, vS I/Z curves based on them, is that allfE ,'s result from the same uniform treatment. In the excellent previous work of Wiese") and coworkers, thefE , vs 1/Z curves result from the composite of all available data from diverse experimental and theoretical sources, which, however, have different levels of accuracy. The N C M E T values of table 1 have been compared with BFS, phase-shift, and Hanle experiments exten-

f 347 275 258 350 297 225 258 227 260 319 277 216 308 250 239 275 218 246 246

0.065 0.032 0.168 0.066 0.077 0.093 0.104 0.106 0.173 0.170 0.101 0.139 0.031 0.060 0.079 0.135 0.185 0.077 0.076

sively on rE1 VS 1/Z curves drawn. The agreement is good overall (within a range of 11%) for singly- and higher-ionized atoms (and some neutrals like B I, C I and N I). Although neutrals are more sensitive to details of the computation, the N I Westhaus-Sinano~lu a) value and the C I value 12) were in good agreement with experiment. We note from table l some experimentally-interesting cases where different experiments disagreed with one another, while the N C M E T values agreed with only one of the experiments. Such cases are: Ne V p 2 2 p __~ sp3 3 D , N II p2 3p _~ s p 3 3 p and Ne llI p4 3p ~ sp5 3p. The Westhaus-Sinano~lu papers 3) had given the KL ~ K L ' fE,'S or A's calculated by the dipole-length operator ( f ' ) , by the dipole-velocity operator (fv), and also the geometric mean f "/('v). As we have

f~ = beS ~,

(1)

./v = b 1 S v,

(2)

IlL

THEORY

2.1 8.5 16.8 2.2 5.3 36.8 10.4 22.9 5.7 3.7 8.8 33.1 1.3 19.2 9.4 11.9 43.3 2.8 25.2

196

OKTAY

SINANOGLU

with b constant, e the transition energy, a n d the S ' s the a b s o l u t e line strengths, the f o r m f,/(,v)

,f,/(,r)

= bS,/(rv)

(3)

is i n d e p e n d e n t o f ~. The N C M E T results with the f r a n d f v agree very well with each other, within a few p e r c e n t o r better. The present tables 1 a n d 2 r e p o r t only the f ' - v a l u e s . The W e s t h a u s - S i n a n o ~ l u results are included in the tables with, however, the e x p e r i m e n t a l '

I

i,r

, , T

BORON SEQUENCE 2sz 2p2P°-2s 2p22D NCMET o NBS

0.30

~

D Z

'

'

I I

'

7

/~ /

A PHASESHIFT • HANLE EFFECT

EXPANSION v ]

[

/~/Ii

used in eq. (1). A c c o r d i n g to the t h e o r y (cf. next section), N C M E T ~ e ' s give the transition probabilities very well, i.e., one o b t a i n s g o o d values for the line strengths, S .... t ~

sNCMET=

(4a)

S e NCMET, initial

I(~'o

final

IRIS%

2

)1 •

(4b)

Thus in the exact eq. (1), the exact (experimental) energy e remains. If the energies e calculated 13) f r o m the 7~c's, i.e., ec, are used, t h e f { ~ ' s o b t a i n e d with these differ on the average by 3% f r o m those with the e x p e r i m e n t a l e's. The final e q u a t i o n to use for f ~ given by N C M E T is:

/ ///

f[1

/'~)

=

b(~expt =

ec-[-/3u) S ~ ,

(5)

with the S ~ = S~NcMET, eq. (4b). (Otherwise, or if either %xpt or % is n o t available for some new cases to be calculated, the f,/(rv) equation, eq. (3), is very convenient a n d equally reliable)4).

0.20

0.10

3. The Be I 2s2p 1p°-2p2 tD and 2s2p 1P°-2p2 1S isoelectronic sequences 0

~ O

=

I 0.05

Si ill

i i

Ne F I i 0.10

0 i

N II 0.15

C i

B 02.0

t/Z~

Fig. 1. The low-Z and high-Z behavior of oscillator strengths as demonstrated on the B I sequence for the transition 2s22p 2p0_ 2s2p 2 ZD. The figure shows the pre-1969 theoretical approaches and the NBS-tabulated values. It also shows the accurate values from the new atomic structure theory for ground and excited states (Sinano~,lu and co-workers, NCMET, e.g., ref. 1 in text). New experimental values, also shown, agree with the NCMET, showing the now-familiar bending of such curves down towards the neutrals. V RHF (the Hartree-Fock method); Weiss, Phys. Rev. 188 (1969) 119. © Nat. Bur. Std. tables; W. L. Wiese, M. W. Smith and B. M. Glennon, NSRDS-NBS 4, vol. I (U.S. Government Printing Office, Washington, D.C., 1966). 0 z-expansion [hydrogenic method; M. Cohen and A. Dalgarno, Proc. Roy. Soc. London A280 (1964) 258]. A NCMET-New atomic structure theory with correlation; O. Sinano~lu, ref. 1 in text and earlier references therein; P. Westhaus and O. Sinano~,lu, Phys. Rev. 183 (1969) 56 (C, N, O) and our other papers including B, F, Si. • Phase shift; G. M. Lawrence and B. D. Savage, Phys. Rev. 141 (1966) 67. • Hanle effect; A. Hese and H. P. Wiese, Z. Physik 215 (1968) 95. • Beam-foil; L. Heroux, Phys. Rev. 153 (1957) 156. • Beam-foil; J. Bromander, R. Buchta and L. Lundin, Phys. Letters 29A (1969) 523. • Beam-foil; W. S. Bickel, Phys. Rev. 162 (1967) 7. x Beam-foil; I. BergstrOm, J. Bromander, R. Buchta, L. Lundin and I. Martinson, Phys. Letters 28A (1969) 721.

As some new e x p e r i m e n t a l w o r k 14) on the a b o v e Be I sequences, including transitions between two u p p e r configurations, has been r e p o r t e d in this conference, a n d as there have been sizable discrepancies a m o n g the various experimental a n d theoretical results existing previously, we include here tables 2 a n d 3, b a s e d on N C M E T calculations~5). The results agree with B F S well, while also the N C M E T f r a n d f v values are very close. Only in the one neutral case o f Be 1 2s2p tP°-2p2 IS, there remains a sizable discrepancy. One m a y note that for the C 1II, N IV a n d O V cases which are i m p o r t a n t for the solar c h r o m o s p h e r e c o r o n a transition region, as well as for W o l f - R a y e t envelopes, the N C M E T values are very accurate.

4. Some basic aspects of the atomic structure theory N C M E T and features of the f~t calculations In the 1 9 7 0 B F S conference Bashkin asked the questiona6), ls there an approximate wave function

which gives a number of different atomic properties accurately ? The answer has been p r o v i d e d by N C M E T : the " c h a r g e wave f u n c t i o n " ~ c , predicted, then calculated by the theory, gives all the charge-distributionlike p r o p e r t i e s o f any a t o m i c state (including highly excited), i.e., such expectation values, as well as t r a n s i t i o n m a t r i x elements~). The exact wave function o f a m a n y - e l e c t r o n non-closed-shell state as shown by

A SURVEY

OF R E S U L T S

SINCE

197

1970

TABLE 2 a Multiplet absorption oscillator strengths for the transition lsZ2s2p 1p°-lsZ2p2 1D o f the Be I sequence.

Species

;t(A)

fRHF

fNCMET (This work) b

Be I

7209

0.164 2.06

0.020 0.016

B II

3451

0.324 0.938

0.148 0.139

0.192 + 0 . 0 0 9 d 0.216 4- 0.011 e

0.414 e 0.23 t 0.16g

C III

2297

0.266 0.674

0.165 0.155

0.18 + 0 . 0 2 h 0.14 4-0.01 d 0.16 4-0.011 0.19 + 0.005J 0.18 4-0.01 k 0.181, m

0.311 c 0.206 r 0.18 g

N IV

1718

0.224 0.538

0.171 0.158

0.24 4-0.01 n 0.1561

0.247 e 0.182 r 0.18 °

O V

1371

0.192 0.445

0.155 0.139

0.16 + 0.005P 0.15 -4-0.01 q 0.151

0.205 e 0.163 f 0.16g

F VI

1139

0.172 0.369

0.143 0.137

0.14 4- 0.005P 0.14 + 0.005 r

0.174 e 0.146 r

N e VII

973

0.149 0.308

0.131 0.138

0.117 + 0.005 s

0.151 e

Si XI

610

0.099 0.222

0.096 0.092

f r.xr~

f (Other theory)

0.66 e

0.11 t

a T h e restricted H a r t r e e - F o c k values, c o m p u t e d earlier by the a u t h o r with C. Beck a n d C. Nicolaides, are given in c o l u m n 3. T h e results o f the present work, u s i n g the " c h a r g e wave f u n c t i o n s " o f the a u t h o r ' s Non-Closed-Shell M a n y - E l e c t r o n T h e o r y ( N C M E T ) , are given in c o l u m n 4. T h e dipole length (fR) is the u p p e r a n d the velocity result (fv) is the lower quantity. Experimental wavelengths are used. b T h e N C M E T calculations use the c o m p u t e r p r o g r a m s o f Sinanoglu, OskiJz, W e s t h a u s , Beck, a n d Luken. e j. Linderberg, Phys. Letters 29A (1969) 467. d I. Bergstr0m, J. B r o m a n d e r , R. Buchta, L. L u n d i n a n d 1. M a r t i n s o n , Phys. Letters 28A (1969) 721. e T. A n d e r s e n , K. A. Jensen a n d G. Sorensen, Phys. Rev. 188 (1969) 76. f C. L a u g h l i n a n d A. D a l g a r n o , Phys. Letters 35A (1971) 61. g A. W. Weiss, Nucl. Instr. a n d Meth. 90 (1970) 121. h B. C u r n u t t e , W. S. Bickel, R. G i r a r d e a u a n d S. Bashkin, Phys. Letters 27A (1968) 680. i E. H. P i n n i n g t o n a n d C h i n - C h a n Lin, J. Opt. Soc. A m . 59 (1969) 780. J D . J . Pegg, E. L. C h u p p a n d L. W. Dotchin, Nucl. Instr. a n d Meth. 90 (1970) 71. k D. L. Mickey, Nucl. Instr. a n d Meth. 90 (1970) 77. 1 p. Ceyzeriat, A. Denis, J. Desesquelles, M. D r u e t t a a n d M. C. Poulizac, Nucl. Instr. a n d Meth. 90 (1970) 103. m M. C. Poulizac, M. D r u e t t a a n d P. Ceyzeriat, J. Q u a n t . Spectrosc. Radiative T r a n s f e r 11 (1971) 1087. n W. S. Bickel, R. G i r a r d e a u a n d S. Bashkin, Phys. Letters 28A (1968) 154. o H. N u s s b a u m e r , M o n t h l y N o t i c e s Roy. A s t r o n . Soc. 145 (1969) 141. P W. S. Bickel, H. G. Berry, I. M a r t i n s o n a n d R. M. Schectman, Phys. Letters 29A (1969) 4. q I. M a r t i n s o n , H. G. Berry, W. S. Bickel a n d H. O o n a , J. Opt. Soc. A m . 61 (1971) 519. r H. G. Berry, I. Martinson, R. M. Schectman, W. S. Bickel a n d H. P. Palenius, J. Opt. Soc. A m . 60 (1970) 1461. s G. B e a u c h e m i n , J. A. K e r n a h a n , E. K n y s t a u t a s , D. J. G. Irwin a n d R. D r o u i n , Phys. Letters 40A (1972) 194. t M. C o h e n a n d A. Dalgarno, Proc. Roy. Soc. L o n d o n A280 (1964) 258.

II1. T H E O R Y

198

OKTAY SINANO(~LU TABLE 3a Multiplet 'absorption oscillator strengths for the transition ls22s2p 1p0-1s22p21S of the Be I sequence.

;t(A)

fRHF

fNCM:ET (This work)

Be I

_b

B II

1843

C Ill

1247

N IV

955

OV

795

F VI

651

Ne VII

5521

Si XI

359

0.003 0.251 0.106 0.124 0.086 0.092 0.071 0.075 0.069 0.055 0.061 0.047 0.055 0.040 0.031 0.032

(0.199b) ~0.339 [0.117 0.243 0.239 0.173 0.163 0.135 0.144 0.112 0.129 0.101 0.111 0.092 0.096 0.062 0.064

Species

fExPa:

f (Other theory)

0.112:k0.003 e 0.119={=0.000d 0.163 q-O.011e

0.23 ¢

0.13 :kO.O1g 0.17h

0.17 f 0.15k

0.11 4"0.01i

0.11 f

0.068J

a See footnotes (a) and (b) of table 2. b Experimental assignment uncertain. ). 3455 A turns out to be 2s3p 1P0--->2p3p1p [Hontzeas et al., Phys. Scripta (1972)]. Hence for this case only, the geometric mean [ x/(fnfv)] value given, which is independent of )l, ought to be used. [This neutral Be I value is nevertheless not accurate due to a computational problem. The ( f r ~ 0.34) and (f17 ~ 0.117) differ from each other appreciably. The 2p2 1S state needs further refinement within the ~e itself.] e I. BergstrOm, J. Bromander, R. Buchta, L. Lundin and I. Martinson, Phys. Letters 28A (1969) 721. d T. Andersen, K. A. Jessen and G. Sorensen, Phys. Rev. 188 (1969) 76. e L Martinson, W. S. Bickel and A. Olme, J. Opt. Soc. Am. 60 (1970) 1213. f A. W. Weiss, Nucl. Instr. and Meth. 90 (1970) 121. g I. Martinson and W. S. Bickel, Phys. Letters 31A (1970) 25. h M. C. Poulizac, M. Druetta and P. Ceyzeriat, J. Quant. Spectrosc. Radiative Transfer 11 (1971) 1087. i I. Martinson, H. G. Berry, W. S. Bickel and H. Oona, J. Opt. Soc. Am. 61 (1971) 519. J M. Cohen and A. Dalgarno, Proc. Roy. Soc. London A280 (1964) 258. k H. Nussbaumer, Monthly Notices Roy. Astron. Soc. 145 (1969) 141. 1 G. Tondello and T. M. Paget, J. Phys. B3 (1970) 1757. the t h e o r y ' ) consists of two distinct o r t h o g o n a l parts: I//. . . .

t

= We + Zu-

(6)

The charge wave f u n c t i o n Tc contains all of the specifically non-closed-shell-type correlations. The difficulties with f-values, the strong cancellation effects in CI (configuration-interaction), the highly a n o m a l o u s term-splitting ratios in hole-states '7) like 2s, all result from these. ~g¢ contains, in the CI language, only a group-theoretically fixed, finite n u m b e r of configurations (20-60 d e t e r m i n a n t s in KL-states). Tc contains the usual restricted H a r t r e e - F o c k wave f u n c t i o n ~oRHF, plus the " i n t e r n a l " , the " s e m i - i n t e r n a l " , a n d the " p o l a r i z a t i o n " correlations rigorously given by N C M E T ' - 3 ) . T o avoid repetition we refer the reader for the simple m e a n i n g and definitions of these effects

to the original papers'-3). The " s e m i - i n t " correlations crucial for a c c u r a t e f E l ' S were first f o u n d by N C M E T , having been by-and-large missed in previous atomicstructure methods. Once the finite Tc's are calculated (by the a u t o m a t e d programs, which even find which d e t e r m i n a n t s enter) a n d t a k e n out of the exact T, the r e m a i n i n g parts of ~u a n d E behave just like closed shelts'-3"7). T h u s T , called "all-external correlations" is given by the NCMET/MET i n d e p e n d e n t - p a i r correlation-energies 's) i n t r o d u c e d a n d defined by M E T first for closed shells. These affect mainly the energetics a n d are added on to the energy quantities, while they do n o t affect significantly the charge-like operator [r, r 2, r2(3 cos 2 0 - 1), 17, etc.] matrix elements. A n example of a ~u direct o u t p u t of M O D O K is

A S U R V E Y OF R E S U L T S S I N C E

1970

199

TABLE 4 Example o f a direct computer o u t p u t of a " c h a r g e wave function" ~e of the N C M E T of this writer. The wave function s h o w n is for a highly excited state (2s hole) with a three-determinant R H F . The determinants figured out from N C M E T by the c o m p u t e r give all of the specific, fEl-affecting, correlation effects. Although for simplicity the determinants s h o w orbitals with the notation 3s, 3p, 3d, these are not the usual n = 3 orbitals, but the semi-internal orbitalsfs,)co, f o of N C M E T (cf. text). (This is an exact retyped replica of the actual c o m p u t e r output. It has been retyped as the o u t p u t itself was a bit faint to be photographable.) 07/13/72 COEFFICIENT

F L U O R I N E V 2S 2P2 D O U B L E T S DETHF=A(1SA DETHF=A(1SA DETHF=A(1SA

1SB 1SB ISB

2SA 2SA 2SA

2P-A 2P-B 2POA

2P+B 2P+A 2POB

) ) )

.57735027 -.57735027 -.57735027 CONTRIBUTIONS E.V. A.U.

COEFFICIENT

DET DET DET DET DET DET DET DET DET DET DET DET DET DET DET

(ISA 2=A (lSA 3=A (1SA 4=A (1SA 5=A 6 = A (ISA 7 = A (ISA (1SA 8=A (1SA 9=A 10 = A ( I S A 11 = A ( I S A 12 = A ( I S A 13 = A ( l S A 14 = A (1SA 15 = A (1SA 16 = A ( I S A

1SB 1SB 1SB 1SB 1SB ISB lSB 1SB 1SB lSB 1SB 1SB 1SB 1SB 1SB

2POA 2P+A 2SB 2P-A 2P-A 2P-A 2P-A 2P-A 2SA 2SA 2SA 2SA 2SA 2P-A 2P-A

2P+B 2P+B 2P-A 2P-B 2POA 2POB 2P+A 2P+A 2SB 2P-B 2POA 2POB 2P+A 2P+B 2P+B

3D-A 3D--A) 3P+A 3D++A) 3D+B 3D+A 3SB 3DOB 3SA 3P+A 3POB 3POA 3P-B 3SA 3DOA

) ) ) ) ) ) ) ) ) ) ) ) )

.00173747 -.00245716 -.00783401 -.00245716 .00000000 .00173747 .00000000 .00000000 .04960850 .00472355 -.00311046 -.00472355 .00311046 .00294884 -.00100313

-1.12326473E-05 -2.24652946E-05 -1.28524966E-04 -2.24652946E-05 0. -1.12326473E-05 0. 0. -4.37477534E-03 -3.82703549E-05 -2.58292290E-05 -3.82703549E-05 -2.58292290E-05 -5.86753046E-07 -3.74421577E-06

(1SA (1SA (1SA (1SA (ISA (1SA (ISA (ISA (1SA (1SA (ISA

1SB 1SB 1SB 1SB 1SB ISB 1SB 1SB 1SB 1SB 1SB

2SA 2SA 2SB 2POA 2POB 2P-B 2P-B 2P-B 2SB 2POA 2POA

2P+B 2P-A 2P+A 2P+A 2P+A 2POA 2P+A 2P+A 2POA 2POB 2POB

3P-A 3P+B 3P-A 3D-B 3D-A 3D+A 3SA 3DOA 3POA 3SA 3DOA

E H F (A.U.) =

-9.12749555E+01

E T O T A L (A.U.) =

-0.12800222E+01

) ) ) ) ) ) ) ) ) ) )

.00472355 .00311046 -.00783401 -.00000000 -.00173747 -.00173747 -.00294884 .00100313 .00783401 -.00294884 -.00200626

-

-

-

-

-

CONTRIBUTIONS E.V. A.U.

COEFFICIENT DET17=A DET 18=A DET19=A DET20=A DET21=A DET22=A DET23=A DET24=A D E T 25 = A D E T 26 = A DET 27 = A

3.05637638E- 04 - 6.11275275E - 04 - 3.49713347E- 03 - 6.11275275E- 04 0. 1.05637638E- 04 0. 0. 1.19036567E- 01 - 1.04132717E-03 7.02807123E- 04 - 1.04132717E- 03 - 7.02807123E- 04 1.59654096E-05 1.01879213E- 04 -

-

-

-

-

-

3.82703549E-05 2.58292290E- 05 1.28524966E- 04 0. 1.12326473E- 05 1.12326473E-05 5.86753046E- 07 3.74421577E- 06 1.28524966E- 04 5.86753046E-07 1.49768631E-05

- 1.04132717E- 03 - 7.02807123E- 04 3.49713347E- 03 0. - 3.05637638E- 04 - 3.05637638E- 04 1.59654096E- 05 1.01879213E- 04 - 3.49713347E- 03 1.59654096E- 05 - 4.07516850E- 04 -

-

-

-

D I F F E R E N C E (A.U.) = - 5.06673572E- 03 EXPONENTS

3S = 2.5620000 3P = 1.8910000 3 D = 2.5620000

RENORMALIZATION FACTOR =

CONTRIBUTIONS

-4.37653560E-03 -5.7787364BE-04 - 1.12326473E-04

.99860425

III. THEORY

200

OKTAY SINANOGLU TABLE5

Electric dipole transition probabilities (AE1 s-1) and non-relativistic multiplet oscillator strengths (fErl~ of highly-ionized heavy atoms of the KL-Shell configurations a. Values are those of Sinano~lu first reported here and calculated from the non-closed-shell Isoelectronic sequence

K 2

B C

2sZ2p2P--+2s2p2 2D ----> 2S ____> 2p 2s22p2 3P--->2s2pa aD __> ap ---> aS 2s22p2 1D 2s2pz 1D _+ 1p 1S_.__> 1p 2s22p34S__>2s2p44p 2D--> ZD __> 2p 2p__> 2D ----> 2S ____> 2p 2s22p4ap--->2s2p53p 1D____> 1p 1S___> 1p 2s22p5 2P--->2s2p62S

241 189 178 233 198 154 175 155 177 209 183 144 203 167 157 178 143 161 156

- -

N

O F

f

Sc A

0.048 3.3 0.024 13.3 0.124 26 0.049 3.6 0.057 9.7 0.069 58 0.077 17 0.078 36 0.128 9.1 0.125 9.6 0.074 15 0.102 55 0.023 2.2 0.044 32 0.058 16 0.100 21 0.136 75 0.057 4.8 0.056 46

2 214 168 158 206 175 137 155 137 157 184 161 127 174 147 138 156 125 142 136

f 0.043 0.021 0.112 0.044 0.051 0.062 0.069 0.071 0.115 0.113 0.067 0.093 0.021 0.040 0.053 0.090 0.123 0.051 0.051

Mn A

2

3.8 15.2 30 4.2 11.2 66 19 42 10.4 11.2 17 64 2.6 37 18 25 87 5.7 55

175 137 130 167 142 112 126 112 128 148 130 103 145 119 112 125 101 114 108

f

Fe A

0.036 4.8 0.018 19.1 0.094 37 0.037 5.3 0.043 14 0.052 84 0.058 25 0.059 53 0.097 13 0.095 14 0.057 22 0.078 82 0.017 3.3 0.034 48 0.044 23 0.076 32 0.104 113 0.043 7.3 0.043 73

2 167 131 124 160 136 107 120 107 122 141 124 98 138 113 107 119 96 109 103

f

0.035 5.0 0.017 20 0.091 39 0.036 5.6 0.042 15 0.050 88 0.056 26 0.057 56 0.093 14 0.092 15 0.054 24 0.075 86 0.017 3.5 0.032 50 0.043 25 0.073 34 0 . 1 0 0 120 0.042 7.8 0.041 77

a The NCMET-line strengths are non-relativistic, the e and 2 are the experimental ones [C. Moore, Atomic energy levels (Nat. Bur. Std.; U.S. Government Printing Office, Washington, D.C., 1958); or extrapolated ones from experiment]. The e and 2 used in A shown in table 4. W e have now these accurate wave functions available for several h u n d r e d states of m o s t atoms a n d ions in the K L - t e r m s a n d various Z ' s . They yield also accurate hfs-constants (only the contact term requires, however, the Tu as well, n o t being a "chargelike o p e r a t o r " ) , fEz'S, magnetic susceptibilities, etc. ( a n d in molecules, the m a i n features of chemical forces, molecular potential energy curves)'9). All Tc's used for fEI'S in tables 1-3 are exactly {L 2, S 2, Lz, Sz} eigenfunctions. The T~'s are " e x a c t " , in the same sense as the ~PRnF are " e x a c t " solutions o f the H a r t r e e - F o c k equations which, however, are n u m e r i c a l in nature. The Tc contains in a d d i t i o n to the R H F - o r b i t a l s a new type of orbital i n t r o d u c e d by N C M E T , the "semiinternal o r b i t a l s " , f. The f are finite in n u m b e r , in the KL-shells being only f~, fd, a n d ff, i.e., with 1 = 0, 1, 2, 3 symmetries a n d radial parts o r t h o g o n a l to 1s, 2s, a n d 2p H a r t r e e - F o c k orbitals.

fo,

A l t h o u g h there are equations 2°) that determine these new orbitals f, we o b t a i n them most conveniently variationally in terms of Slater orbitals with optimized exponents as in the analytic H a r t r e e - F o c k method. High accuracy is o b t a i n e d this way quite easily in ions,

A

although results for fE, become somewhat sensitive to how good theseft(r) have been reached in neutral atoms d u r i n g optimization. F u r t h e r theoretical study o f this effect in neutrals, along with a possible, though not likely role of Tu, would be of interest. I n the same context, some application of the upper a n d lower b o u n d s tofEl-values such as developed by W e i n h o l d 21) would be useful if the b o u n d s were n o t m u c h too conservative. 5. The high Z - low N domain - highly stripped heavy ions A new d o m a i n o f atomic-structure theory a n d experiment is opened up by BFS in heavy-ion accelerators. Experiments here have just begun and have been mostly o n one-, two-, a n d three-electron h i g h - Z ions22'23). I f they t u r n out feasible, study of more electron 4 ~< N < 10 a n d Z / N ~ 2-3 with KL-shells would be of great interest in still a n o t h e r new direction for atomic-structure theory: to test for the i m p o r t a n t relativistic effects as compared to the correlation effects already treated by N C M E T . The K L ~ K L ' f E I calculations are difficult n o n relativistically n o t at high Z, b u t at the low Z end of the

A S U R V E Y OF R E S U L T S S I N C E

1970

201

many-electron theory ( N C M E T : ref. 1 in text) by extrapolation from the uniform N C M E T results of lower Z ' s (cf. text). Particularly beyond Z - - 4 5 - 5 0 several types of relativistic effects including j-j coupling become very important. So, as remarked in text, larger Z-values are given only for a very rough idea of the 2 and r magnitudes to indicate the feasibility of S U P E R - H I L A C experiments.

Br

120 94 89 114 97 77 86 77 88 100 88 70 98 80 76 83 68 77 72

Cd

Ba

Eu

Hg

U

f

A

2

f

A

2

f

A

2

f

A

~

f

A

~.

f

A

0.026 0.013 0.067 0.026 0.031 0.037 0.042 0.042 0.069 0.068 0.040 0.056 0.012 0.024 0.032 0.054 0.074 0.031 0.030

7.3 29 56 8.2 22 127 38 81 20 23 35 127 5.2 75 36 52 180 11.6 120

85 67 64 80 68 54 61 54 62 70 62 49 69 56 54 58 48 54 50

0.019 0.0093 0.049 0.019 0.023 0.027 0.030 0.031 0.051 0.050 0.030 0.041 0.0090 0.018 0.023 0.039 0.054 0.023 0.022

10.5 42 81 12 32 184 55 117 29 34 52 190 7.7 110 54 77 260 17 180

72 57 54 68 58 46 52 46 53 59 52 42 58 48 45 49 40 46 42

0.016 0.0080 0.042 0.017 0.019 0.023 0.026 0.027 0.043 0.043 0.025 0.035 0.0077 0.015 0.020 0.034 0.046 0.019 0.019

12.5 50 96 14 38 220 65 140 35 40 62 220 9.2 130 64 93 320 21 220

64 50 48 60 51 41 45 41 47 52 46 37 51 42 40 43 35 40 37

0.014 0.0071 0.037 0.015 0.017 0.021 0.023 0.024 0.038 0.038 0.022 0.031 0.0069 0.013 0.018 0.030 0.041 0.017 0.017

14.2 57 109 16 44 250 75 160 39 46 71 260 10.6 150 73 107 360 24 250

50 39 48 37 40 32 35 32 36 41 36 29 40 33 31 33 27 31 28

0.011 0.0056 0.029 0.012 0.014 0.016 0.018 0.019 0.030 0.030 0.018 0.024 0.0054 0.011 0.014 0.024 0.032 0.014 0.013

13.5 74 141 21 57 330 97 210 51 61 93 330 14 200 96 140 480 31 330

43 34 32 40 34 27 31 27 31 35 31 25 34 28 27 29 24 27 24

0.010 0.0049 0.026 0.010 0.012 0.014 0.016 0.016 0.026 0.026 0.015 0.021 0.0047 0.009 0.012 0.021 0.028 0.012 0.012

21.5 86 64 25 67 380 113 240 60 71 109 390 16 230 112 160 560 36 390

and f, thus include relativistic energies although still LS-multiplet values are given. I f relativistic effects on line strengths turn out small, the values will be good estimates.

fE1 VS 1/Z curves. The N C M E T has been the first theoretical approach to give these non-relativisticfEl'S accurately and systematically (fig. 1 and tables) over the full Z-range. OurfE1 VS 1/Z curves drawn from the uniform N C M E T fEI'S such as the ones in table 1 show that already at Si one is at the nearly linear portion going towards 1/Z~O. Since for the An = 0 , in-shell transitionsfE1 ( I / Z = 0 ) = 0 ) (ref. 24), one has therefore a non-relativisticfE~ curve for all Z. In table 5, we give a few sample fEI'S and A (s -~) values for atoms like Fe, Mn, Sc . . . . isoelectronic with Be I, C I, N I, O I, .... In calculating the A-values either experimental wavelengths or those obtained from C. Moore tables were used. For the higher Z ' s they are extrapolated. We note that fortunately the lifetimes for these theoretically-interesting cases are in the right range to be measurable by beam-foil (in S U P E R - H I L A C , tandem, etc.)25). One should emphasize again, however, that, already beyond Si in these K L - - + K L ' shell transitions, relativistic effects of several different types become important. Beyond Z ~ 50 (Sn), the j-j coupling effects even change the level scheme leaving no semblance of even a meaningful multiplet average.

Nevertheless, the LS-multiplet averages are given in table 5 for an order of magnitude rough idea (rough indeed beyond Z = 45) of the kinds of wavelength regions and lifetimes that may be expected in this uncharted domainS3). 6. New results and extension of theory to third-row atoms, K L M --, KLM' transitions in Mg, AI, Si, P, S and Ci In the third-row atoms, the " H a r t r e e - F o c k sea" {ls, 2s, 2p, 3s, 3p, 3d} of the N C M E T problem has M = 28 occupied or unoccupied R H F spin orbitals for any K L 3s"3pm3d k configuration and their resulting terms, as compared to M = 10 in the KL-shells. The number of N C M E T semi-internal orbitals is also increased. With number of electrons N ,~ 20 as compared to N ~ 6 in the K L problem, the non-closed-shell fEl-accurate wave function (qJc) now contains of the order of 10 3 determinants vs the KL-case of ~ (10-102). Thus until very recently 26) no accurate and theoretically-complete calculations had been available in the literature, although some important work by Zare, Weiss, and others existed27). In the K L M ~ K L M ' cases, therE1 fall roughly into III. THEORY

202

OKTAY

SINANOt~LU TABLE 6

Theoretical multiplet oscillator strengths o f the " W e a k t y p e " (f~,l ~ 0.1) for Si, S, P a n d CI. 2 (A)

1814

Si II

3p 2p0__+ 3p2 2D

_e 1198

S 1I S Ili

3p a 2D°--+ 3p 4 2D 3p2 3p __+ 3p3 aDO

1308

P 1I

3p2 ap__~ 3p3 3p0

1011

C1 ]II 3p a 4 S ° - + 3p 4 4p

f I~HF~

f~BSb

0.454 0.504 0.394 0.389 0.458 0.292 0.256 0.685 0.777

0.0064 0.62 1.2

fNCMSTc,d

0.0137 0.0269 0.0494 0.0358 0.0450 0.0382 0.0436 0.0855 0.1032

fEXYT

0.022 t 0.029g 0.040 h 0.043 i

a T h e restricted H a r t r e e - F o c k ( R H F ) - v a l u e s c o m p u t e d by O. Sinano~lu a n d D. Beck. U p p e r value isfr (dipole-length), lower value f e (velocity operator). b W. L. Wiese, M. W. Smith a n d B. M. Miles, Atomic transition probabilities, Nat. Bur. Std./Nat. Std. Ref. D a t a Ser. - 22 L (U.S. G o v e r n m e n t Printing Office, W a s h i n g t o n , D.C., 1900). T h e Nat. Bur. Std. values are based on previous theoretical approaches. c Calculations (cf. text) of Beck a n d Sinano~lu using S i n a n o ~ l u ' s Non-Closed-Shell M a n y - E l e c t r o n T h e o r y ( N C M E T ) with all chargedistribution-affecting correlation effects ("charge wave f u n c t i o n " 5%) in b o t h upper and lower states. a F o r each case, the first n u m b e r i s f r - v a l u e , s e c o n d f v, a n d third is the e-independent geometric m e a n f4(rv). In fr a n d re-calculations we use the experimental e's as indicated by theory. e Since the u p p e r term h a s n o t been observed a n d Erj (cf. text) was n o t yet available at the time o f these ]El-Calculations, the precise 2 was u n k n o w n . H e n c e only the e-independent f 4(rv) value is reported. H. G. Berry, R. M. Schectman, I. M a r t i n s o n , W. S. Bickel a n d S. Bashkin, J. Opt. Sac. A m . 60 (1970) 335. g L. J. Curtis, I. M a r t i n s o n a n d R. Buchta, Phys. Scripta 2 (1971) 1. h B. D. Savage a n d G. M. Lawrence, A s t r o p h y s . J. 146 (1966) 940. i S. B a s h k i n a n d I. M a r t i n s o n , J. Opt. Sac. A m . 61 (1971) 1686. TABLE 7 E x a m p l e s o f theoretical multiplet oscillator strengths o f the " S t r o n g " type ( f ~ l > 0.1) in Mg, Si a n d P. )l (A)

Transition

fR/:IFa

2780

M g I 3p 3pO__~ 3p2 ap

1196

Si II 3p 2po_+ 3p2 2p

1299

Si III 3p apo__~ 3pZ ap

826

P IV 3p apo__~ 3d aD

0.804 0.504 1.248 0.614 0.651 0.468 0.891 0.777

flgBSb

0.61 0.91 0.564 0.796

flq'CMETp,d (this work)

0.639 0.604 0.878 0.864 0.543 0.569 0.789 0.805

fEXPT

0.50 e 0.53 f 0.74g 0.47 k >0.6 ~

a T h e restriced H a r t r e e - F o c k ( R H F ) - v a l u e s c o m p u t e d by O. Sinano~lu a n d D. Beck. U p p e r value isfr (dipole-length), lower value f v (velocity operator). b W. L. Wiese, M. W. Smith a n d B. M. Miles, Atomic transition probabilities, U. S. Nat. Bur. St./Nat. Std. Ref. D a t a Set. - 22 L (U.S. G o v e r n m e n t Printing Office, W a s h i n g t o n , D.C., 1900). T h e Nat. Bur. Std.-values are based on previous theoretical approaches. e Calculations (cf. text) o f Beck a n d Sinano~lu using S i n a n o ~ l u ' s N o n - C l o s e d Shell M a n y - E l e c t r o n T h e o r y ( N C M E T ) with all charge-distribution-affecting correlation effects ( " c h a r g e wave f u n c t i o n " ~ e ) in both u p p e r a n d lower states. d F o r each case, the first n u m b e r is fr-value, second fv a n d third is t h e e - i n d e p e n d e n t geometric m e a n f 4 ( r v). Infr a n d fv-calculations we use the experimental e's as indicated by theory. e j. B r o m a n d e r , H. G. Berry a n d R. Buchta, Nucl. Instr. a n d Meth. 90 (1970) 55. f T. A n d e r s e n , J. Desesquelles, K. A. Jesson a n d C. Sorensen, J. Quant. Spectrosc. Radiative Transfer 10 (1970) 1143. g H. G. Berry, J. B r o m a n d e r , L. J. Curtis a n d R. Buchta, Phys. Scripta 3 (1971) 125. h L. J. Curtis, 1. M a r t i n s o n a n d R. Buchta, Phys. Scripta 2 (1971) 1. I. M a r t i n s o n , private c o m m u n i c a t i o n .

A SURVEY

OF R E S U L T S

two types2V): (1) the strong fE1 > 0.1; (2) the weak fEX ( < 0.1). In the weak case, R H F or limited SOC-type calculations turn out to be in error by factors of 10-30. For these weak transitions, the complete N C M E T 7~c calculations 26) are needed for any accuracy. The first such extensive fEl-complete calculations have now been carried out by N C M E T and with new third-row programs in collaboration of this writer with Dr. Beck26). The RHF-results for the weak fE1 drop, due to charge-correlations, by factors like 10. A table of these results is shown (table 6). Fortunately much new BFS work has started to concentrate on such atoms at the same time, as evidenced by the many notable papers presented in this conference. The large drop in fE1 from the order of magnitude, 0.5-1.0 to < 0.1 is due to strong cancellation effects noted earlier by Beck and Sinano~lu26), and discussed further in this conference by Aymar and Feneuille. With fEX>0.1, i.e., the strong transitions suffer from no such effects and come out somewhat reasonable already with Hartree-Fock (RHF). Table 7 shows examples of strongfEx cases with R H F and with complete 7Jc-NCMET calculations as well as some experiments. In the third row ( K L M shells) and larger atoms, the 7~c-NCMET calculation still involves a finite number of theoretically-predicted determinants, although this number now is quite large. The qJc wave function is calculated by our new CDC-6600 automatic program system: B E K O K which again does all the group theory, coupling algebra, etc., itself. Rather than diagonalizing a very large matrix, however, to get the exact 7J¢, we have considered three different approximations in getting the coefficients of the same determinants (cf. a sample ~ in table 4). All three are derivable within the mathematical machinery available in the theory METe): l) The VPA ("Varied-Portions" Approach)l'2°). This gives the qJ¢ in several independently-calculated and well-defined segments. 2) The " R E D I A G " procedure used in the recent results reporteda6). The separately (VPA)-obtained segments are put together with a single coefficient in front of each overall segment. The coefficients are determined from an approximate segments-diagonalization. 3) The exact 7'~ calculation as in all of our KL-shell calculations in which every coefficient in the entire 7'c comes from one big overall diagonalization. Approximations (1) and (2) would be sufficient for many properties like energy as well as for the strong f E l ' S . For the weak fEX at least approximation (2)

SINCE

1970

203

(" R E D I A G ' ) is needed. Further refinements with use of exact "full ~¢" diagonalization are presently being studied. [While this paper was being prepared, the author and D. Beck completed an exact "full M-shell diagonalization" [case (3) above] calculation on CI III. The results are very close to the " R E D I A G " values reported earliera6). Thus our third-rowfEl'S are accurate in this respect.] The basic accuracy of N C M E T should be better than 10% for the strongfE1 and maybe 30% or better for the weakfE t . TABLE 8 T h e C1 III 3s23p a (4S°)--+3s3p4 (4p) transition d e m o n s t r a t i n g the need for the inclusion o f electron correlation in both the u p p e r a n d lower states. T h e n e w atomic structure theory N C M E T 1) n o w m a k e s possible the inclusion o f all o f the non-closed shell correlation effects in both g r o u n d a n d excited states. Calculation a

(1) (2) b,e (3) b (4) b

Lower state (4S°) wave function used

U p p e r state wave function used (4p)

fr

fv

RHF NCMET-We RHF NCMET-We

RHF RHF NCMET-qJe NCMET-We

0 685 0.460 0.116 0.0855

0.777 1.54 0.00186 0.1032

a All calculations ( R H F a n d N C M E T ) are by O. Sinano~lu a n d D. Beck. T h e e-experimental is used in b o t h R H F a n d N C M E T values. b T h e 7re-calculations include all o f the LM-shell non-closed shell correlations (1590 a n d 1676 Slater determinants) (cf. text on the calculations.) e N o t e that a l t h o u g h these correlations (We vs q~RHF) have very little effect o n the energy o f the 4S0 state, they change the f ~ l substantially.

In connection with accurate N C M E T f E 1 predictions, we further note in table 8 that full N C M E T charge correlations have to be included in both the lower and upper states. The C1 III 3s 23p 34S°-3s3p44P case displayed in detail shows that inclusion of correlation, i.e., NCMET, ~uc, in either the lower or the upper state alone can give completely wrong results. All of our f~x calculations include the correlations properly in both of the states. By contrast for example the "diagrammatic many-body perturbation theory" ( " M B P T ' ) calculations of Kelly2S), and Altick and Glassgold 29) on a few atoms like Be and oxygen had included some correlation (not systematically indicated forfEx), but only on the ground state. This is because these " m a n y - b o d y " methods 3°) being analogous to quantum field theory apply only to a "non-degenerate vacuum", i.e., single determinantal (PRHF of lowest III.

THEORY

204

OKTAY SINANOGLU

energy state of that symmetry. They do not give a theory of correlation applicable to excited states. The N C M E T theory yields the correlations rigorously for any excited state as well as ground states. One may also note that for the accuratefE1, both the "internal" (mixings only within the M-shell itself among 3sa3pb3d c configurations) and the new "semiinternal" correlations of N C M E T are needed. The NCMETfEl-Values include also the ~u contributions from the inner L-shell part as well26). Other few calculations in the literature such as 27) Zare's, Weiss', and Feneuille and Aymar's on thirdrow K L M ~ K L M ' have included only some configurations (such as 3p3d mixing) belonging to the "internal" (Xi,t) part of ~ . They are not expected therefore to yield accurate fEe'S.

7. A brief guide to various theoretical methods for the calculation of allowed transition probabilities We list the theoretical approaches that have been used for allowed transitions and LS-coupling oscillator strengths fEl ( y L S ~ 7 'L'S') and mention their key features. 7.1.

COULOMB APPROXIMATION - BATES-DAMGAARD TABLES31), ETC.

Applies to out-of-shell d n ¢ : 0 ; A n > l transitions which are basically one-electron in character. Does not apply to the An = 0, in shell cases, the main ones of recent theoretical and BFS interest, in which all electrons of the shell rearrange correlatively during transition. 7.2.

THE CORE-POLARIZATION METHOD FOR ALKALILIKE SPECTRA

In the one-electron model, an outer electron moves in the average field of the core orbitals. A correction arises from the instantaneous polarization of the core by the slow outer electron. A core-polarization potential Vp can be calculated induced by a fixed point charge outside, which in turn determines the motion of the outer. This method has been extended to correct for exchange and penetration effects18'32). Core-polarization is an inter-shell electron-correlation effect I s). It can be derived and used within the full, rigorous context of the N-electron correlation theoryX8'33). It is thus a specialized approximation applicable only to certain electron pair correlations between An ~ 0 spin orbitals of the state. In alkali spectra it corrects thefE~ appreciably34).

7.3. HYDROGENIC MODEL - Z-DEPENDENT THEORY

Applies at high Z, ( 1 / Z ~ 0 ) , where the (l/rij)correlation effects lose importance. Calculations are basically with hydrogenic orbitals35). The theory converges slowly at lower Z and gives the incorrect behavior for Z / N ~ 1 (fig. 1). A higher-order version of this theory, to first order in N

the wave function with respect to the ~ (1/rls) perturi>j

bation was first derived by Sinano~lu36), who showed that the wave function (Z°) contains separately calculated, one-and-two-electrons-at-a-time functions (~°, fo). Each ~oj satisfied a first-order He-atom equation (of appropriate Z). We have pursued our other approach, the more accurate N C M E T theory, in our later work1-3). The hydrogenic 0 n ( Z ) theory which is convenient for heavy ions has been pursued further ~y Dalgarn037), Layzer24), and co-workers [cf, also Dalgarno in these Proceedings (the a°)]. 7.4. THE ONE-ELECTRON CENTRAL FIELD MODEL Here, Hartree-Fock-like wave functions are written for both the lower and upper states. However, the ls, 2s, 2p orbitals of say lsZ2s22p 2 ap and of ls22s2p 3 3D are assumed to be the same (and hence orthonormal, O.N.). Then the line-strength calculation reduces to a one-electron matrix element ((2slrl2p>). This method was the main one in use [cf. Condon-Shortley38)] until the early sixties. 7.5. THE FULL HARTREE-FOCK METHOD The proper Hartree-Fock (H.F.) method for excited states yields a set of 1s, 2s, 2p orbitals which are different for each term of each KL-configuration. The spin orbitals are still restricted to definite Inl rntms> quantum numbers. [Hence the name "Restricted HartreeFock" (RHF)39)]. All R H F results forfEt we calculate and include in our tables for comparison with the accurate ManyElectron Theory (MET/NCMET), are with an R H F solution for each state separately. Thus the { 1s, 2s, 2p} lower state orbitals are not orthonormal to those of the {Is, 2s, 2p} upper state. The line strength involves therefore complicated many-electron matrix elements which do not reduce simply to a (2slrl2p). This non-orthogonality problem is handled efficiently by a mathematical algorithm and its automated computer program Y A T I K devised by Westhaus and Sinano~lu3). The RHF-wave functions are an essential starting input for the full Many-Electron Theory1). Without MET, the R H F wave functions alone yield

A SURVEY OF RESULTS SINCE 1 9 7 0

however incorrect fEl'S for An=O transitions, and some An ~ 0 (e.g., some K L ~ K L ' M in second row) ones by factors as much as 2-30 (cf. fig. 1). Before BFS and NCMET, much of the NBS tables were based on R H F (as on emission data in the experimental case). 7.6. THE MULTI-CONFIGURATIONAL HARTREE-FOCK (MCSCF) The self-consistent field orbitals from two, three configurations from the same shell mixed together (e.g., for the Be I ls22s z ~S state [c~(lsZ2s 2 IS)+c2(ls22p 2 1S)] is used). Developed mainly by Yutsis and co-workers 4°) of the Lithuanian school, the method can reduce the fEl-Values of the R H F method by as much as 50%. Though laborious, it is a good way to get some of the virtual orbitals in neutral atoms containing filled subshells only (2s 2, 3s 2, 3p 6, etc.). The selection of configurations is by and large ad hoc. The ones that have been used are contained in the "internal correlation" (Xl,t) part of the exact N-electron wave function derived in NCMET. For neutrals, the MCSCF wave function [or more completely the "General Restricted Hartree-Fock wave function" ( G R H F ) 2°) based on the full 9R.F+ +Xi.t of NCMET] can be taken as an alternate starting point for the "charge wave function" 7' c calculations in N C M E T containing all the otherfE~correlation effects. 7.7. CONVENTIONALCONFIGURATION-INTERACTION (CI, SUPERPOSITIONOF CONFIGURATIONS, SOC) An attempt is made to improve on R H F by adding some configurations. The difficulty with this approach is well known: as remarked by Garstang41), Slater42), L6wdin43), and others, one has basically an infinite sum, no theoretical guidelines for selecting significant configurations or determinants, and very slow convergence in the infinite SOC sums making it difficult to know where to stop. Prior to 1969 it was very difficult therefore, and expensive, to carry out such calculations. Weiss 44) had carried out such an extensive SOC (CI) on carbon with 40 configurations. (A more accurate value is given by N C M E T with only about 10 configurations. The other configurations in Weiss are of all-external type which do not contribute much.) Configuration-interaction (SOC) is basically a numerical procedure for implementing any theory, rather than a theoretical approach itself, just as Hartree-Fock equations if known, can be solved either numerically, or "analytically", i.e., by expansion in say a Slater-orbital (STO) basis set. Similarly any N-e-

205

wave function ~ ( x x , x2 . . . . . xs) can be expanded in an STO-CI basis:

= ~

CKAr,

(7)

r_>o

where AK are infinitely many, ordered Slater determinants made of STO's directly or a transformed infinite R H F - S T O basis. The mathematical answer as to which wave function and which electron-correlation effects appropriate to fE1 and related properties to expand and calculate, has been provided by the N C M E T theory, the answer ~uc turning out further, to be only a finite sum instead of eq. (7). There are other ways of implementing N C M E T / MET. However, once the theoretically-called-for determinants (or configurations) - and only those - are indicated by NCMET~), CI(SOC) becomes a convenient procedure for carrying out the calculation of the predicted correlation effects (7~). In discussing a calculation therefore, specification of its theoretical basis, rather than the computational procedure alone is needed. 7.8. DIAGRAMMATIC MANY-BODY PERTURBATION THEORY (MBPT)

This approach includes correlation effects in nondegenerate (single determinant, R H F ) ground states, by adding higher and higher orders of perturbations. It has been applied to atoms mainly by Kelly 2s) and Brueckner. It is discussed and compared to other approaches in the recent book by Sinano~lu and Brueckner3°). Like conventional SOC (CI), the MBPT is also openended, in that, rather than predicting the effects needed a priori, it adds more and more diagrams which are however more systematically written down than in the selection of configurations in SOC. MBPT is very useful in exploring higher order effects on ground state properties. When it comes to transitions, however, it has the serious defect demonstrated in table 8. It frequently applies to only one of thf states. Thus correlations cannot be included in the other state which is then limited to RHF. The result potentially can be therefore as much or more in error as the pure RHF-fE 1 itself. 7.9. NON CLOSED-SHELL MANY-ELECTRON THEORY (NCMET); (ATOMIC-STRUCTURE THEORY WITH CORRELATION IN GROUND AND EXCITED STATES)

The basic theory, already discussed above, was IlL THEORY

206

OKTAY SINANOGLU

derived for excited states (NCMET) in 1964 [for ground states (MET) in 1959-61] 1s) by this writer, and first discussed in relation tofEl-Values by LaPaglia and this writer in 196645). The first accurate and extensive fE~ and other excited state properties calculations were reported in 1967-19691-3). The K L ~ K L ' , K L M K L M ' results have already been discussed above. The theory can also be applied to second- and thirdrow An # 0 type transitions ( K L - , [ ( K L ' ) n l ] , n >_ 3, K L M -~ [(KLM') nl], n _> 4) with relative ease. With the automated program systems MODOK, BUYUK, ISHIK and YATIK, the computer time needed is little. No detailed knowledge or experience with much atomic algebra are needed for their use. 8. Forbidden transitions - lifetimes of metastable states

Transitions, forbidden according to the electricdipole selection rules A L = O, -4- 1 ; A S = 0, of [C I], [N I], IN II], [O I], [O II], [O IlI] occur in aurorae, in the night airglow, in gaseous and planetary nebulae, and in the solar corona. They are mainly electricquadrupole (E2) or magnetic-dipole (M1) transitions. Observations on these lines, and theoretical approaches, including and introducing a new accurate one using NCMET, were surveyed recently by Sinano~lu4). Details of the first extensive many-electron calculations analogous to the.fr~ calculations by N C M E T above, using the "charge wave functions" 7' c of Sinano/glu, Oksfiz, and Westhaus, and their computer programs were also reported 3°) for 1 S o - l D 2 , 2 p - z D and ~S-1D. Recently Corney 6) has measured the 5577 A oxygen auroral green line. His results are in excellent agreement with our theoretical value of AQ= 1.183 s -~. Other theoretical E2-probabilities AQ (s -1) from our TABLE 9 Theoretical electric quadrupole (E2) transition probabilities, AQ(S-1) from the new atomic structure theory including electron correlation in ground and excited states (NCMET) (refs. 1 and 2 in text; cf. ref. 3 for experiment in good agreement with theory on the [O I] case). [C I] [N 1I] [O Ill]

1So-ID 2 1S0-1D2 1So-lD2

0.548 1.082 1.654

[N 1]

2p3/2-2D5/2 2P1/u-2D5/2 2P3/2-2D3/2 2p1/2-2D3/2

0.0489 0.0279 0.0208 0.0417

[0 I]

ISo-ID2

1.183

[0 11]

2p3/2-2Ds/2 2p1/2-2Ds/2 2ps/2-2D3/2 2p1/2-2Ds/2

0.0915 0.0523 0.0388 0.0799

N C M E T calculations are shown in table 9. We have used some of these recently to improve solar carbon and oxygen abundances47). Another type of metastable state lifetime, that of O I 1s 22s22p 33s 5S° was measured recently by JohnsonV), who obtained ( 1 8 5 + 1 0 ) x 10 - 6 S. This value differed considerably from Garstang's theoretical one 48) of 591 × 10 -6 S. Garstang 49) and Mizushima 5°) had shown that this state decays to the 3p ground state via spin-orbit allowed electric dipole (SOAED) transitions. Magnetic-quadrupole (MQ) decay also occurs, but is weaker by 105. The spin-obrit effect is calculated quite well just from Hartree-Fock wave functions (although here too the ~/'c-NCMET wave functions would lead to higher accuracy). This is as in the magnetic-dipole (M1) transitions 4) for example of [O I] ~D2 -~ 3P1.2 22 6364, 6300 A, and 1So ~ 3P 1 2 2972 A. In such predominantly M 1 cases calculation depends again only on getting the spin-orbit coefficients which are responsible for nonzero AM1 (2972) (z = 12.8 s) and AMI (6300), AM1 (6364). For M1 probabilities, R H F wave functions are expected to work well, since other than the spin-orbit coefficient, the main matrix elements are of the M I operator L + 2 S given exactly by group theory. Thus Garstang's ~ ) RHF-AMl-Values are generally quite accurate4). In the (SOAED)-transitions, on the other hand, although the spin-orbit part is again presumably given well by just RHF, the main matrix elements are again of thefE~ type. They are as we saw above, affected by factors like 2-3 by the electron-correlation effects of the charge wave function type. The discrepancy between Johnson's experimental and Garstang's R H F lifetimes above, is thus understandably again a factor of about 3. The charge wave functions of N C M E T 1) are again appropriate therefore for the calculation of SOAED lifetimes. Nicolaides of our laboratory has used our automated programs as developed by Sinano~lu, Oks/iz, Westhaus, Beck, and Luken to run the O I Is22s22p 3 3s5S ° state as well, along with our 7J¢'s available already for the 3p2, 1,o O I ground states. (Also C I ls22s2p 3 5S2 and 2s22p z 3p). The calculated lifetimes are Zo~ = 192 × 10 -6 s and Zc~ = 176 × 10 -3 s, the O I in good agreement with JohnsonV). It is clear that wherever correlation effects in ground and excited states are important, in such forbidden as well as allowed transition probabilities, the new theory, N C M E T ~) is capable of providing accurate values. 9. Doubly-excited spectra

Recent observations 8) of doubly-excited spectra in

A SURVEY OF RESULTS SINCE 1 9 7 0

BFS have given a strong impetus to this new direction in atomic-structure theory. At low Z's, doubly-excited terms contain strong mixings of a large number of configurations. Unlike singly-excited spectra, there is usually no one dominant configuration with which to label the terms unambiguously. As the configurations that mix are quite typical of the ones that occur in the "charge wave functions" of NCMET, N C M E T is ideally suited for the radiativelifetime and term energy calculations of multiplyexcited spectra. The application is straightforward for terms below the ionization limit, but can also be carried out for the other, metastable or autoionizing cases (using QHQ-finite subspace projection method combined with NCMET). Experimental BFS observations, mostly on Li I, Be I isosequences are summarized by MartinsonS). Hopefully data on more electron atoms will also be forthcoming and would provide stronger tests of theoretical calculations. Various calculations have been performed in different laboratories on particularly the 2 and 3e- cases by straightforward large CI (i.e., SOC)52), which can still be handled as the number of electrons is small. Various doubly-excited states in the 3 < N < 11 KL atoms of the types lsZ2p "+2 have been calculated (so far, for the non-autoionizing cases) by N C M E T by Sinano~lu, Skutnik, Oks/.iz, and Beck for some time2). Some Be I transitions were already discussed above and are in table 3. Other cases are presently being calculated by Sinano~lu and Herrick. A new group-theoretic approach has also been developed for doubly-excited statesl°), particularly the He** series. A mathematical group and Lie Algebra are found which classify the doubly-excited states with new quantum numbers, predict the CI mixing coefficients observed in large calculations, and which show which series are "weak", which "strong" from the group theory, in agreement with experiment. 10. Conclusion

In concluding this paper, we note that BFS has been and is playing an important role not only in providing accurate transition probability data, but in stimulating advances and providing stringent tests in new areas of atomic-structure theory. New atomic-structure theory with electron correlation accurately included in ground and excited states (Non-Closed Shell Many-Electron Theory) 1) in comparison with other theoretical calculations has received much confirmation from BFS as well as from phase-shift and Hanle experiments on

207

dipole oscillator strengths. Forbidden transitions, doubly-excited spectra, and energies have also been systematically and accurately obtained by NCMET1-3). Several hundred accurate wave functions and properties of states of the KL and K L M configurations have now been calculated for atoms from Be to the irongroup elements and their ions with the use of our automated computer program systems implementing the theory conveniently and economically. The author thanks P. Westhaus, D. Beck, W. Luken and I. Oksfiz, whose collaboration in this project was particularly valuable. He also wishes to thank C. E. Johnson for calling his experimental results to this writer's attention prior to publication. He is grateful to Dr. Weinhold of Stanford and Dr. D. F. Tuan of Kent State for helpful correspondence on the topic of upper and lower bounds, and to W. Luken for help in preparation of table 5. References 1) O. Sinano~lu, Proc. 1st Intern. Conf. At om i c Physics, New York, June 1968 (Plenum Press, New York, 1969), and earlier references to our work there. ~) O. Sinano~lu and I. Oksiiz, Phys. Rev. Letters 21 (1968) 507. 3) p. Westhaus and O. Sinano~lu, Phys. Rev. 181 (1969) 56. These are the results we reported in the previous conference on BFS at Lysekil, Sweden [C. Nicolaides and O. Sinano~lu, Nucl. Instr. and Meth. 90 (1970) 133l. 4) O. Sinano~lu, Comm. At. Molec. Phys. I I (1970) 73. ~) C. Nicolaides, O. Sinanofglu and P. Westhaus, Phys. Rev. A4 (1971) 1400. 0) A. Corney and O. M. Williams, J. Phys. B5 (1972) 186; A. Corney, this conference. 7) C. E. Johnson, Phys. Rev. A5 (1972) 1688. 8) I. Martinson, this conference, and references therein. 9) H. Doyle, M. Oppenheimer and G. W. F. Drake, Phys. Rev. A5 (1972) 26. 10) O. Sinano~lu and D. Herrick, to published. 11) M . W . Smith and W. L. Wiese, Astrophys. J. Suppl. 23 (1971) 103. 12) O. Sinano~lu, in Topics in modern physics - A tribute to E. U. Condon (eds A. Brittin and H. Odabasi; Colo. Assoc. Univ. Press, Boulder, 1971). lz) The NCMET energies, according to the theory, include "all-external" correlation energies eu in addition to the energy with the charge correlations "only" re. Thus eexaet = ee+eu. The effect Ofeu onfE~ is about 3%. 14) C. E. Head and W. A. Roberts, this conference. 15) Work on Be I sequences is in collaboration with D. Beck and C. Nicolaides using machine programs of Sinano~lu, 0ksiiz, Westhaus, Beck and Luken. 18) S. Bashkin, Nucl. Instr. and Meth. 90 (1970) 3. 17) O. Sinano~lu, Comm. At. Molec. Phys. 1 (1969) 116. 18) O. Sinano~lu, J. Chem. Phys. 33 (1960) 1212; Proc. Roy. Soc. (London) A260 (1961) 379; J. Chem. Phys. 36 (1962) 706. [paper 1 of MET series of V papers (1962-1964) for ground state theory of correlation]. IlL THEORY

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SINANOGLU

19) O. Sinanol~lu, C o m m . At. Molec. Phys. 3 (1971) 53. 2o) H. J. Silverstone a n d O. Sinanoglu, J. C h e m . Phys. 44 (1966) 1899, 3608; in Modern quantum chemistry, vol. II (ed. O. Sinanol~lu; A c a d e m i c Press, N e w York, 1965). 21) F. W e i n h o l d , J. C h e m . Phys. 54 (1971) 1894. 22) R. M a r r u s a n d R. W. Schmieder, Phys. Rev. A5 (1972) 1160. 23) j. B r o m a n d e r , this conference. 24) D. Layzer a n d R. H. G a r s t a n g , A n n . Rev. A s t r o n . A s t r o p h y s . 6 (1968) 449. 25) J o h n R a s m u s s e n a n d D. A. C h u r c h , private c o m m u n i c a t i o n s . 26) D. Beck a n d O. Sinano~lu, Phys. Rev. Letters 28 (1972) 945. 27) R. N. Zare, J. C h e m . Phys. 47 (1967) 3561; A. W. Weiss, J. C h e m . Phys. 47 (1967) 3573; E. Trefftz, Z. A s t r o p h y s i k 28 (1950) 67. 28) H. Kelly, Phys. Rev. 136 (1964) B896. 29) p. L. Altick a n d A. E. Glassgold, Phys. Rev. 133 (1964) 632. a0) See e.g. t h e c o m p a r a t i v e discussion given in O. Sinano~lu a n d K. A. Brueckner, Three approaches to electron correlation in atoms (Yale Press, N e w H a v e n a n d L o n d o n , 1970). 31) D. R. Bates a n d A. D a m g a a r d , Phil. T r a n s . Roy. Soc. ( L o n d o n ) A242 (1949) 101. a2) E. M. M o r t e n s e n a n d O. Sinano~lu, J. C h e m . Phys. 34 (1961) 1078. 3a) O. Sinano/glu, A d v a n . C h e m . Phys. 6 (1964) 615; D. F. T u a n a n d O. Sinano~,lu, J. C h e m . Phys. 41 (1964) 2677. 34) S. H a m e e d , A. Herzenberg, a n d N. C. James, J. Phys. B1 (1968) 822. 05) M. C o h e n a n d A. D a l g a r n o , Proc. Roy. Soc. ( L o n d o n ) A280 (1964) 255; references to D. Layzer papers in ref. 24. 36) O. Sinano~,lu, Phys. Rev. 122 (1961) 491. 37) C. D. H. C h i s h o l m a n d A. Dalgarno, Proc. Roy. Soc. ( L o n d o n ) A292 (1966) 264; Z. H o r a k , in Modern quantum chemistry, vol. II (ed. O. Sinano~,lu; A c a d e m i c Press, N e w York, 1965), a n d later papers by D. Layzer, A. H o r a k et al. 38) E. U. C o n d o n a n d G. H. Shortley, The theory o f atomic spectra, ( C a m b r i d g e University Press, N e w Y o r k , 1951). 39) C. C. J. R o o t h a a n , Rev. M o d . Phys. 32 (1963) 179. 40) A. Yutsis, in 1969 Vilnius Symp. proc. Theory o f electronic shells o f atoms and molecules (Mintis, U S S R , 1971). 41) R. H. G a r s t a n g , Proc. C a m b r i d g e Phil. Soc. 52 (1956) 107. 42) j. C. Slater, Quantum theory o f atomic structure ( M c G r a w Hill, N e w Y o r k , 1960). 43) p. O. LOwdin, A d v a n . C h e m . Phys. 2 (1959) 207. 44) A. W. Weiss, Phys. Rev. 162 (1967) 71. 45) S. LaPaglia a n d O. Sinano~,lu, J. C h e m . Phys. 44 (1966) 1888. 46) A. Corney, this conference. 47) C . / q i c o l a i d e s a n d O. Sinano~,lu, Solar Phys. (1973) in press. 4s) R. H. G a r s t a n g , Proc. C a m b r i d g e Phil. Soc. 57 (1961) 115. 49) R. H. G a r s t a n g , O b s e r v a t o r y 82 (1962) 50. 5o) M. M i z u s h i m a , J. Phys. Soc. (Japan) 21 (1966) 2335. 51) R. H. G a r s t a n g , M6m. Soc. Roy. Sci. Li6ge, Ser. 5 (1969) 17. 52) L. Lipsky a n d A. R u s s e k , Phys. Rev. 142 (1966) 53. 53) While this p a p e r was in press, we have o b t a i n e d relativistic f~.l vs 1/Z curves with inclusion o f b o t h correlation a n d intermediate coupling. T h e curves are very different t h a n t h o s e linear ones relied u p o n in t h e literature in t h e high Z region. T h e n e w lifetimes however m a k e accelerator-BFS experiments even m o r e feasible (O. S i n a n o g l u a n d W. Luken, Phys. Rev. Letters, submitted).

Discussion HEAD: W e have done s o m e preliminary m e a s u r e m e n t s o n the B I transition (the 2p 2 2D state that y o u m a d e reference to) a n d we

tend to disagree with m o s t o f the other experimental values on the lifetimes there which seem to r u n a r o u n d 23 us. W e only have two decay curves, but from those two decay curves we get 17 ns which, I believe, is going to be in better agreement with y o u r results. SINANO(~LU; AS I recall, the a g r e e m e n t is good. MARTINSON: ]~have one example o f this on your recent paper with Dr Beck. Y o u m e n t i o n e d that on one o f the chlorine lines there is s o m e disagreement. After I saw y o u r paper I checked the decay curve a n d f o u n d a b o u t a factor o f two difference between the experiment a n d V a r s a v s k y ' s theory w i t h o u t configuration interaction. F r o m that I think there m i g h t still be cases where the experimental picture looks to m e fairly simple a n d the cascading above this level is n o t too i m p o r t a n t because m o s t o f t h e m are short-lived. So I just w o n d e r h o w accurate y o u think the theory m i g h t be in that particular case? SINANOOLU: W h e n we c o m p a r e o u r new results based o n o u r new theory with correlation in b o t h g r o u n d a n d excited states with the H a r t r e e - F o c k a n d also earlier theories, the i m p r o v e m e n t is by factors like 10-30 or so. In fact, this is the same type o f i m p r o v e m e n t in f-values that y o u r experiments are n o w indicating. Between o u r new theoretical a n d new experimental results on the other h a n d , in s o m e cases the a g r e e m e n t is still within a factor o f two. T h e results we reported in Phys. Rev. Letters 28 (1972) 945 were the first results by a n y theory which fully included correlation in both g r o u n d a n d excited states with all the n o n closed shell effects. Here the wave functions are still finite b u t a n order o f m a g n i t u d e larger t h a n in the KL-shells, where agreem e n t with precise experiments is within a few percent. With these very large effects of correlation, the f-values, due to correlation effects, become very small. T h e initial results for K L M shells which we just recently reported used the theory n o t in the m o s t rigorous form as h a s been done in the KL-shells, b u t they used a m o r e a p p r o x i m a t e procedure for calculation o f these very large wave functions. W e shall have to e x a m i n e these a p p r o x i m a t i o n s to see w h e t h e r any i m p r o v e m e n t within a factor o f two is possible. (Since the October meeting, Beck a n d I have r e m o v e d these a p p r o x i m a t i o n s a n d carried o u t full M-shell calculations. T h e results are still very similar to o u r earlier ones reported. So this increases the confidence in these values. A t this point, therefore, it is still open w h e t h e r theory a n d experiment w o u l d yield s o m e w h a t different values if other effects m a y have to be b r o u g h t in. In any case, even at this stage, c o m p a r e d to the factors o f 10-30 just a year ago, we can consider that there h a s been a big advance in both theory a n d experiment.) WIESE: In the first systematic trend which y o u showed on the beryllium sequence, I noticed that y o u r theoretical points did n o t give a s m o o t h curve. There was one point, I think on N IV, which seemed to be 10 or 20% off the curve. U s u a l l y theory produces very s m o o t h systematic trends. Is there an explanation ? SINANO~,LU: I have looked at the curves y o u mentioned. It t u r n s o u t that s o m e o f the slides I showed o f the f v s 1/Z curves h a d been d r a w n at the last m i n u t e by C. Nicolaides. After the conference I e x a m i n e d these with care a n d f o u n d that the curves h a d n o t been properly drawn. Ac ually the points do fall on the s m o o t h curves a n d t h e published curves a n d n u m b e r s that I give in the Proceedings f r o m m y calculations with a n u m b e r o f people in o u r laboratory, do fall o n s m o o t h curves as they should since they are all calculated with the same theory rather t h a n c o m i n g from diverse sources. I s h o u l d add, however, that in the neutral m e m bers o f s o m e isosequences, great care is needed in calculations to

A S U R V E Y OF R E S U L T S S I N C E 1 9 7 0 be sure that one is obtaining the correct states where there are a n o m a l o u s mixing effects, which become important. CROSSLEY." Just pursuing D r Wiese's point, I noticed the same thing about this curve. It is the carbon member, and the nitrogen is rather high, in the beryllium sequence. ! wondered, D r Sinano~,lu, if you considered doing your calculations, since they can be done so quickly, for non-integral values of Z. SINANO(~LU: That is a very interesting remark. Recently, about last May, 1 looked at the Z curves that Dr Wiese has drawn in his recent Astrophysical Journal supplement. I put on his curves whatever new data we had. I noticed several cases where everything is fine and then there is some experimental point which differs drastically, even though it is a new experiment. It usually happens at the beginning o f a curve at low Z. It is difficult to draw any curves between the first few points anyway as within

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that range the curve goes through a m a x i m u m somewhere. In that context I thought it would be very useful to repeat the calculations for non-integral Z ' s , and that way, get a better idea of how to draw a curve in the low-Z region, I mean just as an interpolation m e t h o d for low-Z. O f course, I c a n ' t help remarking that some years ago in a completely different context we had made some non-integral Z calculations off-values. These were made to see if there are quarks in the sun. Russian physicists had predicted that if quarks existed, there would be some " q u a r k e d carbons Z = 11 ], 11 ~], 12½. 12~" as well as nitrogen, oxygen, etc. in the sun. We calculated , l.e lines these would have and looked for them in the rocket-t.?.~ ned solar far UV spectra o f Tousey [Sinano~lu, Skutnik and Tousey, Phys. Rev. Letters 17 (1966) 785]. So you see fracUonal charge f vs I / Z calculations can be quite fun.

III. THEORY