TRANSITION PROBABILITIES FROM THE NON-CLOSED SHELL MANY-ELECTRON THEORY OF ATOMIC STRUCTURE

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TRANSITION PROBABILITIES FROM THE

NON-CLOSED SHELL MANY-ELECTRON THEORY OF ATOMIC STRUCTURE

C. Nicolaides, O. Sinanoğlu

To cite this version:

C. Nicolaides, O. Sinanoğlu. TRANSITION PROBABILITIES FROM THE NON-CLOSED SHELL

MANY-ELECTRON THEORY OF ATOMIC STRUCTURE. Journal de Physique Colloques, 1970,

31 (C4), pp.C4-117-C4-122. �10.1051/jphyscol:1970420�. �jpa-00213875�

JOURNAL DE PHYSIQUE

TRANSITION PROBABILITIES FROM THE NON-CLO SED SHELL MANY-ELECTRON THEORY OF ATOMIC STRUCTURE

C. N I C O L A I D E S (*) a n d 0. SINANOGLU ( t )

Sterling C h e m i s t r y L a b o r a t o r y Yale University, N e w H a v e n , C o n n 06520 U. S. A.

(Presented by Cleanthis Nicolaides)

RksumC.

-

Les probabilites d'emission spontanee pour les transitions dipolaires electriques

1

s2 2 s2 2 p' -1P-1 s' 2 s 2 p-1 de la sequence isoelectronique de CI et 1 s" s 2 p IP-1 s' 2 p2 I D d e la sequence isoelectronique de Be I sont calci~lees, ainsi que pour les transitions quadrupolaires Clectriques IS-ID, 'P-'D de N 1, N

11,

0 I, 0 11 et 0

111.

On utilise, une theorie de la structure atomique a plusiei~rs electrons ( N . C. E. M. T.) q i ~ i est valable pour les couches ouvertes. Dans cette theorie, initialenient introduite par Sinanoglu et ses collaborateurs, les correlations electro- niques sont etudiees a la fois dans I'ktat fondanientai et dans les etats excites, et il apparait certains effets de correlation qui doivent Ctre inclus dans les fonctions d'onde utilisees pour calculer les probabilites de transition. Les valeurs calculees pour les transitions (permises) dipolaires elec- triques sont en accord avec les recentes valeurs expkrimentales. Les autres methodes theoriques qui ne peuvent conduire a iln a ~ ~ s s i bon accord sont aussi discutees. II est montre que I'amClioration de la fonction d'onde H-F habituellement obtenue par I'introduction de qi~elques configurations considCrees traditionnellen~ent cornme importantes, est insuffisante pour rendre co~iipte des donnees expkrimentales. On en c o n c l ~ ~ t donc que toutes les correlations

((

non dynamiques

))

qui apparaissent dans la theorie N . C. E. M. T. doivent etre introduites dans la fonction d'onde pour obtenir des valeurs precises de forces d'oscillateur. Les calculs a plusieurs electrons des transitions (interdites) quadrupolaires Clectriques sont les premiers de ce type. II n'y a pas de valeurs experi- mentales precises a comparer avec nos resultats theoriques, niais la comparaison est faite avec des calculs theoriques anterieurs. On trouve qile I'introduction des correlations reduit de 13 a 17 % les valeurs obteni~es a partir des fonctions d'onde H - F pour les raies interdites alors qile I'effet est beaucoup plus grand que pour les raies permises.

Abstract.

-

Transition probabilities are calculated for the electric dipole C 1 s2 2 s' 2 pz SP- 1 s' 2 s 2 p-1 3D and Be 1 s l 2 s 2 p IP-l s' 2 p' ID isoelectronic sequence lines and for the electric quadrupole N I, N 11, 0 I, 0 I[, 0 111, IS- ID, JP-zD lines. Non-Closed Shell Many-Electron Theory (N. C. M. E. T.) of atomic structure developed by Sinanoglu and co-workers, which treats electron correlation accurately in both ground an excited states, predicts certain correlation effects which need to be and are included in the wave-functions ~ ~ s e d to calculate tlie transition probabilities.

The calculated values for the allowed (dipole) transitions are in agreenient w ~ t h the recent experi- mental results. Other theoretical riiethods which d o not yield such agreement are also discussed.

It is found that the i~silal improvement of tlie H - F wave-function by the inclusion of a few confi- gurations traditionally thought to be important, is not suficient to obtain agreement with expe- riment. It is concluded therefore that all the

((

non-dyna~iiical

1)

correlations indicated by the theory N. C. M . E. T. should be included in the wave-function to obtain accurate multiplet oscillator strengths. The many-body calsi~lations o n forbidden ( q ~ ~ a d r u p o l e ) lines are tlie first of this kind.

There are no accurate experimental values for these, but

0111.

results are compared with previous theoretical calculations. Inclusion of electron correlation results into smaller values than the ones obtained by tlie use of just the H-F wave-functions. by 13-17 ",: for the forbidden lines although the effect is much greater for the allowed lines.

1.

⁽⁽

Allowed

))

Electric Dipole Transitions.

-

T h e phase shift [ I ] m e t h o d a n d especially t h e The m e a s u r e m e n t o r calculation o f accurate transition

⁽⁽

Beam-Foil

))

technique [2] h a v e yielded results o f probabilities f o r allowed transitions in first r o w a t o m s unprecedented accuracy f o r m a n y lines o f neutral has been a n active field o f research f o r s o m e time. It is a n d (even highly) ionized a t o m s . T h e s e experiments o n l y recently however t h a t new developments o f b o t h measure t h e lifetimes o f excited states f r o m which t h e t h e o r y a n d experiment h a v e m a d e it possible t o obtain transition probability o f a particular line c a n , in multiplet oscillator strengths usually within 10-1 5 % m o s t cases, easily be deduced. Details o f these m e t h o d s

accuracy. c a n be f o u n d in ref. 1 a n d 2.

(t) Work supported by a grant from the

U. S.

National H o w d o e s t h e o r y c o m p a r e with experiment ? Science Foundation. It is well k n o w n by now, t h a t , f o r m a n y spectral lines

(*)

Yale University Predoctoral Fellow. which involve t e r m s c o n t a i n i n g equivalent electrons

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970420

C4-118

C. NICOLALDES AN D

0.

SINANOGLU (e. g. the 1 's2 2 s" 2 pn'

-,

1 s2 2 s"-' 2 p r n f l like

transitions in the first row atoms), independent par- ticle models like the Hartree-Fock approximation fail to obtain good results [3]. Similarly, other methods based on tlie 2-expansion approach [4], [5] although constitute an improvement over the Hartree-FOC~

method by considering certain additional configura- tions which

⁽⁽

mix

⁾⁾

strongly with the ground configu- ration, also fail to yield results in agreement with tlie a c c ~ ~ r a t e experimental values recently reported. It is apparent that additional configurations and effects supplementing those nearly degenerate witli the Hartree-Fock wave-function must also be included.

Weiss Iias considered more extensive wave-functions using

⁽⁽

superposition of configurations

^)),

whicli is the expansion of a wave-function in infinitely many Slater determinants, for some transitions in CI and Boron isoelectronic sequences [6]. A large number of configurations is considered-up to forty-expressed in terms of Slater determinants. This method takes into account a larger portion of the correlation energy and in tlie case of oscillator strengths better agreement with experiment is achieved than tlie previously mentioned methods. Tlie method of configuration interaction being a series expansion, is of course, in principle, capable of giving the correct results. Howe- ver, there is tlie crucial problem of which configurations to

⁽⁽

mix

⁾⁾

in order to obtain a good wave-function s ~ ~ i t a b l e for calculations of oscillator strengths. All configurations important for determining the eiiersy are riot of equal importance in computing transition probabilities whicli would seem to depend strongly only upon those configurations needed to specify an accurate charge distribution. Especially because ((super- position of config~~rations

⁾⁾

is in general slowly convergent, there is the likelihood of including many configurations whicli make little difference and missing some which may be important. It is not a physical tlieory with a guiding principle as to significant effects, but a comp~~tational procedure.

A theory then whicli treats the importants correla- tion effects systematically is clearly needed to obtain accurate multiplet oscillator strengths for the types of lines mentioned above. Such a tlieory is the Non- Closed Shell Many Electron Theory (N. C. M. E. T.) developed by Sinanoglu and co-workers [77, [S]. The various kinds of correlation effects are now explicity found and taken into account. They are physically and mathematically indicated by the tlieory. Details on the tlieory can be found in the two articles by 0 . Sinanoglu of these proceedings and in references therein. We shall simply state the main' features of N. C. M. E. T. and show how they are related to calculations of many-electron wave-functions and transition probabilities.

The N. C. M. E. T. indicates three distinct types of correlation effects each witli a different type of depen- dence on the nuclear charge, the number of electrons

and symmetry of the state. The theory shows that the exact N-electron wave-function of an arbitrary (excited or ground) atomic state is of the form

witli all parts orthogonal to one another and

Tlie three types of correlation are called

⁽⁽

inter- nal

⁾⁾

(x,,,),

⁽⁽

semi-internal

⁾⁾

(plus polarization) (x;)

and

⁽⁽

external

⁾⁾

(xu). The first two are highly specific to tlie state under consideration depending upon N, Z and the symmetry of the state. These so called

⁽⁽

non- dynamical

⁾⁾

correlations are thus not transferable from one state t o another ; they can be calculated however by a finite configuration interaction (CI) expansion. The 1, consists mainly of shorter range

((

dynamical

⁾⁾

pair correlations which can be calcu- lated by the methods discussed by Silverstone and Sina~ioglu [7]. These correlatio~is are the only ones remaining in closed shell systems. As discussed below, they are not importaiit in calculating transition probabilities.

All three types are important in calculating energies and related quantities like electron affinities, term splitting ratios, etc. [7], [S]. However, in calculating transition probabilities, only the internal, the semi- internal and polarization correlatio~l effects are taken into account. This is because, in analogy with the closed shell systems where tlie all-external correlatior~s, due to their local. short-range character, do not effect the Hartree-Fock charge distribution, tlie R H F wave- function along with the noii-dynamical correlations can most accurately describe the charge distrib~itioii and therefore can be used to evaluate, to high accu- racy, matrix elements of the transition operators.

Therefore in predicting the wave-functions capable of yielding transition probabilities, the hypothesis is made that the all-extenial correlations are not impor- tant ; a hypothesis borne out by the results. We shall call the wave-functions to be used

⁽⁽

non-dynamical

^)),

and write

:

Such a wave-function can be calculated by a finite

and theoretically determined configuration interac-

tion calculation with no ambiguity as to which confi-

gurations t o choose. S L I C ~ calculations were first

carried out for 113 states of the 1 s2 2s" 2 p m type

of B, C , N, 0 , F, Ne, Na, and their ions by I. 0ksiiz

and 0. Sinanoglu [S]. Various other states have since

been examined [12]. These wave-functions contain the

non-dynamical correlation effects among L-shell

electrons. [The correlations of the K-shell electrons

are predominantly dynamical because of the relatively

large energy separation between the 1 s orbital and the

vacant orbitals in the H-F sea.] They are given as a

finite linear combination of Slater determinants A,

TRANSITION PROBABILITIES FROM THE NON-CLOSED SHELL MANY-ELECTRON

C3-119

constructed from N-single particle functions selected from an orthonormal set of Q orbitals y , ... ^{y ,} ... ^yQ.

The first M spin orbitals define the H-F sea (for the first row M = 10). The remaining ( Q - M ) orbitals with at most four new radial functions, are those that give

A A

the semi-internal j;.j ; 1 and polarization functions Cf,) (cf. article by 0. S.). Due to vector coupling restric- tions imposed by the symmetry of the states

1 s2 2 s" 2 p"',

these new functions have only the s, p, d, f symmetries with radial functions closely approximated by 3 s-like, 3 p-like 3 d-like and 4 f-like functions with optimized exponents. For all states yLSM, Ms all the single- particle functions involved therefore have the form

where Y and X are the normalized spherical harmonics and spin functions. Rli(r) can be expanded as a sum of Slater-typeorbitals (STO) but even one S T 0 with an optimized exponent is quite sufficient.

Using the

cc

nou-dynamical

)>

wave-functions, P. Westhaus and 0. Sinanogl~~ [9] first performed calculations of multiplet oscillator strengths on 29 transitions in the ultraviolet region of the type

Similar and also other kinds of spectral lines have since been investigated and transition probabilities calculated [lo], [l 11, [12].

In our calculations both the dipole length and dipole velocity operator forms are used. The corres- ponding (absorption) multiplet oscillator strengths are then given by : (in atomic units : e

=

nz

= 77 =

1,

c =

137)

N N

where R

=

C ri, the dipole operator, V = z ^{Vi, the}

i = 1 i = 1

velocity operator, and LSM, Ms, L' St ML, Mk the

set of quantum numbers for the lower and upper state correspondingly. A mentioned above, the corres- ponding wave-functions are expanded as a finite linear combination of Slater determinants. Thus, in evaluating expressions (5) and (6) we have to calculate N-body integrals :

Since the radial functions corresponding to a given spin-orbital are determined independently for each yLS, for every two terms considered we obtain two sets of spin orbitals, 4 and FJ which have no simple orthogonality relationship between each other. The overlap then between orbitals

cpi

and cpj associated with states yLS and

y'

L' St is

We explicitly take this

^<(

non-orthogonality

⁾⁾

problem into account by rigorously evaluating the N-electron integrals of eq. (7). The technique used was developed by P. Westhaus and 0. Sinanojjlu [9] who applied a method pat forth by H. F. King, R. E. Stanton and

al.

( I )

[13]. Thus the

⁽⁽

frozen core approximation

D

[14],

[I51 is replaced by a n exact evaluation of N-body integrals. To obtain the final results for eq. (5) or (6), only a single lionvanishing matrix element needs to be calculated. A reduced matrix element, independent of the magnetic quantum numbers ,M, Ms M'Le M&;

may then be deduced and multiplied by simple alge- braic functions of L and S [16] to give all the matrix elements occuring in the sums of eq. (5) and (6).

Results of recent calculations [lo], [ l l ] for spectral lines in the ultraviolet and visible spectrum are given in Tables I and 11. We examine ions of the

isoelectronic sequence of carbon and of the

isoelectronic sequence of Be. Results of other calcula- tions obtained by using S. C . F. wave-functions [IS],

(1)

This method, based on the corresponding orbital trans- formation of Amos and Hall (Proc. Roy. Soc. (London) ,-. A263,

,-, 483 (1961)),

shows how any two sets of spin orbitals Fi and Fj

A A

can be transformed into equivalent sets of

FI and Fj such that

their overlap matrix is diagonal

i.

e.

A A ,-.

<

Fi

1 f i >

^{L- Dii 6ij}

C. NICOLAIDES AND 0 . SINANOGLU

TABLE I

Multiplet Oscillator Strengths for the 3P-3D isoelectronic sequence in C I , N 11, 0 111

Hart ree- Fock

-

CT _,288

(8)

,328 ('I)

N

II ,236

.268 ('I) 0 111 ,200

(a)

.225 (b)

Theory Cohen and Bolotin Dalgarno Weiss and al. p)

(d)

-

-

('i

-

.I02

⁽⁸⁾ 1 1

7 (b)

.I7 ,192

Present Cslc. (f)

-

.080

(.)

.082 (b) .loo

^(a)

.lo5 (b) .loo

^(a)

.I04 (b)

Experiment Lawrence

Boldt Savage Heroux Bickel Pegg. et al.)

-

(9)

- ^(h)

('1

- ⁽⁹

^{- (k)}

-

.09

1

.076 .083

(")

Result obtained using the dipole length formula.

( '

1)

Result obtained using the dipole velocity formula.

(C)

BOLOTIN (A. P.), LEVINSON (I. B.) and LEVIN (L. I.), Soviet Phys. J. E. T. P., 1956, 2, 391.

( 1 ' )

COHEN (M.) and DALGARNO (A.), Proc. Roy. SOC. (Lond.), 1964, A 280, 255.

( e )

WEISS (A. W.), PIlys. Rev., 1967, 162, 71.

(I) Cf.

Text.

(6) BOLDT (G.), Z . Natltrfot.sc/~., 1963, 18a, 1107.

('I)

LAWRENCE

( G .

M.) and SAVAGE (B. D.), Phys. Rev., 1966,141,67.

(')

HEROUX (L.), Phys. Rev., 1967, 153, 156.

(j)

BICKEL (W. S.), PIiys. Rev., 1967,

162,

7.

(k)

PEGG (D. T.), DOTCHIN (L. W.) and CHUPP (F. L.), PI~ysics Letters, 1970,31A, 255.

TABLE I1

Transition Probabilities for the Be 1 s2 2 s 2 p 'P-1 s2 2 p2 'D isoelectronic sequence

Theory (*) Experiment (*)

Present Weiss (I) Linderberg (11) Calculation (**)

-- -- -

- ---

Be I .51 .12

B 11 2.2 1.39 .50 .73 (8), .64 f .03

(b),

C 111 3.6 2.36 1.25 1.35 f .I

^(C),

1.08 f .06

(b),

1.23 rfr .06(*), 1.4("), 1.33 f .05('),

N I V 5.1 3.35 2.32 3.22 f .2 (9, 2.11

(h),

0 V 6.7 4.36 3.31 3.33 f .1 ('),

F VI 8.2 5.37 4.50 4.35 f .1 ('),

(*)

Given in 108 s-1 units.

(**)

Cf. text. Results obtained using the dipole length operator and experimental wavelengths.

(1)

WEISS (A. W.), as quoted in WEISE (W. L.) and al. N. B.

S . Tables (1966).

(11)

LINDERBERG (J.), Plrys, Letts., 1969, 29A, 467.

(n)

ANOERSEN (T.), JESSEN (K. A,) and SORENSEN ((3.1, Phys. Rev., 1969, 188, 76.

(IJ)

BERGSTROM (I.) and

al.,

Plrys. Letts., 1969,

28A,

721.

(['I CURNUTTE (B.) and al., Plrys. Letts., 1968, 27A, 680.

('1) PINNINGTON (E. H.) and CHI-CHAN LIN,

J.

Opt. SOC.

At??.,

1969, 59, 780.

(")

POULIZAC (M. C.), DRUTTA (M.) and CEYZERIAT (P.), Phys. Letts., 1969, 30A, 87.

(')

MICKEY (D. L.), Second International Conference on Beam-Foil Spect. Lysekil, Sweden, June 1970.

("

BICKEL (W. S.), C I R A R D E A U (R.) and BASHKIN (S.), Phys. Letts., 1968, 28A, 154.

(I1) DESESQUELLES (T.), Thesis, University of Lyon (France), March 1970.

(')

BICKEL (W.

S.)

and al., Plij~s. Lrtts., 1969, 29A, 4.

a limited C. 1. [I 91, 2-expansion perturbation theory [4], [20], and

⁽⁽

superposition of configurations

⁾⁾

[6], as well as experimental values are given for compari- son. To eniphasize the importance of the

⁽⁽

non- dynamical

⁾⁾

correlations in calculating oscillator strengths of such types of transitions, we also give results obtained by

LIS

using just the R H F wave- I'unction (Table I). Also in Table I we give results

obtained using both the dipole length and dipole velocity operators.

It is seen that, by using the

⁽⁽

non-dynamical

⁾⁾

wave-functions, the agreement between theory (with

either operator) and experiment is extremely good,

usually better than 10 %,. the average experimental

error. The other theoretical approaches fail to obtain

such agreement, with the exceptioli of Weiss's on

TRANSITION PROBABLLITLES FROM THE NON-CLOSED S H E L L MANY-ELECTRON CJ-I21

CI

3 ~ - 3 ~ ,

which however gives son~ewhat too

large a value. Also note that, considesing the Hal-tree- Fock results for N I1 and 0 I11

3 ~ - %

transitions, agreement between the two forms of dipole operators does not necessarily i n ~ p l ) correctness (i. c. agreement with an accurately determined experimental value). As observed by other authors [6], [17], this criterion of correctness of a theoretical calculation should thus be considered a necessary but not a sufficient one.

The agreement between the experimental and the corresponding N. C. M. E. T. values is an encouraging confirmation of the theory, particularly in light of the substantial role correlation effects play in determini:ig these osci!lator strengths. When contradictory sets of experimental data exist, our I-esults make an unambi- guous preference for one of them (e. g. N 1V ' P

-

ID).

When experimental measurements d o not exist

^-

as in the Re I case

-

o ~ ~ r results constitute theoretical predictions which can be used with confidence in various atomic and a ~ t r o p h y s i c ~ ~ l applications (z).

11.

⁽⁽

Forbidden

⁾⁾

Electric Quadrupole Transitions.

-

Another recent application of the Non-Closed Shell Many Electron Theory is the calculation of electric quadrupole transition probabilities for the oxygen atom auroral green line 'S,-'D2 (= 5 577 A)

and the 2P-2D, 'S-ID lines of oxygen ions, nitrogen and nitrogen ions, all of astrophysical importance [22].

Unlike the case of allowed transitions, hardly any experimental method exists for the accurate measure- ment of metastable state lifetimes and forbidden transition probabilities. The existing experimental values are not accurate and they are lirnited only to the important 0 I 'So-'D, line. By studying several types of rapidly changing aurorae in which competi- tive factors, like collisional deactivation, vary, Omholt [23] has calci~lated an average transition probability of A (5 577 A)

⁼

1.43 + ¹⁴ ",, s - ' . Le- Blanc, Oldenberg and Carleton [24] have studied the radiative decay of 'So in the laboratory. They obtained a value of A

=

1.36 + . 2 0 s - ' . McConkey and Kernahan [25] have made an absolute intensity measurement. They give a value of A

=

1.0 s - ' with however several sources of error making the result reliable only to within a factor of two. A new experi- mental method is currently being applied by A. Corney and collaborators at Oxford [26].

Previous theoretical results on forbidden lines were obtained using an independent particle S. C. F. model in which a one-electron integral was calculated [27], [28], [29]. Furthermore, the approximation of using the same radial function for the 2 p orbital in the upper and lower state was made. An attempt to estimate the effect of configuration interaction on

( 2 ) Previous theoretical predictions of Westhaus a n d Sina-

noglu o n other lines. have recently been verified by the experi- mentalists (2 1 ).

electric quadrupole transition probabilities has been made by Garstang [30]. He considered only the upper state p4 'So and one perturbing configuration, the I s" p6 'So. He found the effect to be only about 2 '%;.

The many-body calculations are presented here the first of this kind. The transition probability for electric quadrupole radiation is given by [31]

where A , is in s - I , i in Angstrom units and Sq is the

((

line strength

⁾⁾

defined by :

and given in atomic units. The electric quadrupole operator Q is a tensor of rank 2 given by :

the summation running over the electrons.

As in the case of allowed transitions, tlle wave- functions of the initial and final states are expressed in LS coupling as a linear combination of Slater determinants. The N-body mat1 ix elements can then be reduced ~ltirnately t o sums of products of N one-body matrix elements which are evaluated analytically.

Again, the problem of nonorthogonality of the two sets of spin-orbitals is explicitly taken into account.

Having obtained an answer for the n ~ a t r i x element evalilation in the LSM, M, scheme, use of the Wigner- Eckart theorem yields a

⁽⁽

reduced matrix element

⁾⁾

Thzn, to obtdin the

⁽⁽

line strength

⁾⁾

S, i n the desired SLJM scheme, the transformations given by Short- ley 1311 are used, which result into

:

where /: is 1/16 the coefficient of G2, H Z or T2 in the table of p. 253 of Condon and Shortley's

((

Theory of Atomic Spectra

^D.

The N. C. M. E. T. results are given in Table 111

and are compared with the N. B. S. results which are

based on theoretical c ~ l c u l a t i o n s using Hartree-Fock

wave-functions. Results obtained by us using just the

Restricted Hartree-Fock wave-functions are also

given. It is seen that inclusion of electron correlation

((pKIjF

+ Xi,,, + X,:) results in values smaller than the

ones obtained by the use of just the H-F wave-

functions by thirteen to seventeen percent. This

C. NICOLAIDES AND

0.

SINANOGLU

TABLE I11

Calculated electric quadrupole transitions in ^the 0 I, 0 11, 0 ^111, N ^I, N I1 ions. Tlze N. C . M. E. T. wave-functions used, include all the non-dynamical correlation eflects. The N. B. S. values are also given for comparison

Transitions

RHF -

1.421 0.106 0 0.060 6 0.044 9 0.090 2 1.824 0.059 0 0.033 7 0.025 1 0.050 4 1.240

N. C. M. E. T.

2 4 ,

RHF +

X n o n - I I

RHF

-

-

1.183 4.567

0.091 5 5.303 9 0.052 3 1.515 4 0.038

8

2.273 1 0.077 9 2.273

1

1.654 1.717

0.048 9 17.081

9

0.027 9 4.880 5 0.020 8 7.320 8 0.041 7 7.320 8

1.082 4.661

('L)

This value is an average of the theoretical value by R. H. Garstang (Ref. 30) and the experimental one by Ornholt (Ref. 23).

effect of electron correlation o n the electric quadrupole H-F wave-functions b y a b o u t 15 %. In a n y case, the transitions is less drastic t h a n the effect o n the allowed results obtained with t h e n e w atomic structure theory ones, a s i t h a s been suspected by Garstang. I t is n o w with electron correlation m a y be taken a s quite reliable possible t o establish with certainty t h a t t h e effect of f o r use i n various applications a s in connection with electron correlation reduces t h e value obtained by t h e nebulae, t h e study of a u r o r a e a n d t h e airglow.

References

[I] LAWRENCE (G. M.) and SAVAGE (B. D.), P11ys. Rev., 1966, 141, 67.

[2] BASHKIN (S.) (ed.) ((Beam-Foil Spectroscopy

1)

Gordon and Breach, Nry York (1968).

[3] See for example the Hartree-Fock results in Table I.

[4] COHEN (M.) and DELGARNO (A,), Proc. Roy. SOC.

(London), 1964, A 280, 258.

[5] LAYZER (D.), A1717. Phys., 1959, 8, 271.

[6] WEISS (A. W.), Phys. Rev., 1967, 162, 71 ; Pl7ys. Rev., 1969, 188, 128.

[7] SILVERSTONE (H. T.) and SINANOCLU (O.), J . Clzern.

Pliys., 1966, 44, 1899, 3608.

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