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RELATIVISTIC EFFECTS IN THE CALCULATION
OF OSCILLATOR STRENGTHS : ”AB INITIO”
METHODS
J. Desclaux
To cite this version:
RELATIVISTIC EFFECTS I N THE CALCULATION OF OSCILLATOR STRENGTHS : "AB INITIO" t.lETHODS J.P. Desclaux
CEA
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DRF/CPN-
85 X, 38041 SREN3BLE CEDEX, FranceResume
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3n d e c r i t brievement l e s n@thodes u t i l is6es pour i n c l u r e 1 'i n f l u e n c e des e f f e t s r e l a t i v i s t e s G F T e s c a l c u l s des f o r c e s d ' o s c i l l a t e u r s . Les p r i n c i p a l e s m o d i f i c a t i o n s i n t r o d u i t e s par l a p r i s e en compte de ces e f f e t s s o n t 6tudiCes 3 l ' a i d e de quelques exemples emprunt@s aux c a l c u l s l e s p l u s recents A b s t r a c t-
The methods used t o i n t r o d u c e r e l a t i v i s t i c e f f e c t s i n t h e c a l c u l a t i o n o f o S c i l l a t o r strengths are b r i e f l y reviewed. The main m o d i f i c a t i o n s induced by these e f f e c t s are analyzed by c o n s i d e r i n g some t y p i c a l examples taken from r e c e n t c a l c u l a t i o n s .INTRODUCTION considered have now t o be described i n the interme-
d i a t e c o u p l i n g scheme and n o t i n t h e Russel Sanders Systematic s t u d i e s C1,21 o f t h e r e l a t i v i s -
one, consequently " s p i n forbidden" t r a n s i t i o n s no t i c e f f e c t s on atomic p r o p e r t i e s have p o i n t e d o u t
longer exist and they must be introcuded in the t h a t these e f f e c t s a r e s t r o n g l y dependent on t h e
d e s c r i ~ t i o n o f t h e s ~ e c t r a . value o f t h e o r b i t a l quantum number
t .
Consequen-Many r e c e n t t h e o r e t i c a l works have been t l y i t was expected t h a t t h e c a l c u l a t i o n o f o s c i l -
devoted to the study of relativistic effects on l a t o r s t r e n p t h s which i n v o l v e s r a d i a l m a t r i x e l e -
t h e c a l c u l a t i o n s of o s c i l l a t o r strengths. These m n t s between two one-electron o r b i t a l s o f d i f f e -
r e n t o r b i t a l quantum number,
w i
11 be i n f l u e n c e d , a t l e a s t f o r heavy atoms, by t h e i n t r o d u c t i o n o f t h e r e l a t i v i s t i c e f f e c t s . To be s h o r t l e t us j u s t r e c a l l t h a t t h e r e l a t i v i s t i c e f f e c t s on t h e r a d i a l charge d i s t r i b u t i o n can be d i v i d e d i n t o two c o n t r i - b u t i o n s . The f i r s t one i s due t o t h e mass v a r i a - t i o n and t h e o r t h o g o n a l i t y c o n s t r a i n t s and r e s u l t s i n a c o n t r a c t i o n o f t h e charge d e n s i t y towards the nucleus when compared t o n o n - r e l a t i v i s t i c r e s u l t s . Contrary t o t h i s e f f e c t which d i r e c t l y r e s u l t s from t h e use o f t h e D i r a c equation i n s t e a d o f t h e Schradinger one, the second c o n t r i b u t i o n i s an i n - d i r e c t e f f e c t and i s found o n l y i n s e l f c o n s i s t e n t c a l c u l a t i o n s . Due t o t h e more e f f e c t i v e screening o f t h e n u c l e a r charge by t h e i n n e r e l e c t r o n s , t h e charge d e n s i t y o f t h e o u t e r e l e c t r o n s tends t o ex- pand towards t h e o u t e r r e g i o n o f t h e atom. I t i s obvious t h a t these two e f f e c t s w i l l compete and t h a t the n e t r e s u l t on a given o u t e r she1 1 wi 11 g r e a t l y depend on the e l e c t r o n i c c o n f i g u r a t i o n un- der c o n s i d e r a t i o n . F o r a more d e t a i l e d study o f t h e r e l a t i v e importance o f these two c o n t r i b u t i o n s t h e reader i s r e f e r e d t o t h e r e c e n t paper by Rose e t a1 . [ 3 1 . Besides t h e obvious m o d i f i c a t i o n o f t h e t r a n s i t i o n energy, t h e r e l a t i v i s t i c e f f e c t s w i l l a l s o manifest themselves through t h e s p i n - o r b i t i n t e r a c t i o n i n t h e sense t h a t t h e s t a t e s t o beworks may be c l a s s i f i e d according t o t h e method used t o perform t h e c a l c u l a t i o n b u t a l l o f them show t h a t t h e i n c l u s i o n o f t h e r e l a t i v i s t i c e f f e c t s l e d t o r e s u l t s which, along an i s o e l e c t r o n i c se- quence, l a r g e l y d e v i a t e from t h e p r e d i c t i o n s o f the n o n - r e l a t i v i s t i c r e s u l t s . I n t h e f i r s t p a r t o f t h i s paper we review t h e various methods used t o perform these c a l c u l a t i o n s w i t h a p a r t i c u l a r emphasis o f t h e more s o p h i s t i c a t e d completly ab i n i t i o methods. I n t h e second p a r t we w i l l consider some t y p i c a l examples t o p o i n t o u t t h e m o d i f i c a t i o n s induced by t h e r e l a t i v i s t i c e f f e c t s .
I. HETHODS OF CALCULATION
Any r e l a t i v i s t i c c a l c u l a t i o n has t o per- formed u s i n g some kind ofapproximated Hamiltonian since no exact one e x i s t i n a closed form, t h i s p o i n t w e l l discussed by Bethe and S a l p e t e r
141
w i l l n o t be considered here. The methods used up t o now t o i n t r o d u c e t h e r e l a t i v i s t c e f f e c t s i n t h e c a l c u l a t i o n s o f o s c i l l a t o r s t r e n g t h s can be d i v i d e d i n t o two classes, one which i s t h e r e l a t i v i s t i c c o u n t e r p a r t o f t h e Hartree-Fock method, time inde- pendent o r n o t and a second one which makes use o f model p o t e n t i a l s . T h i s l a t t e r one i s discussed i n d e t a i l i n an o t h e r paper a t t h i s conference [5] and i s j u s t summarized here : a given form o f t h e p o t e n t i a l i s assumed, t h i s can be o f the S l a t e rc1-110 JOURNAL DE PHYSIQUE
type [61 o r o f some a n a l y t i c a l form [ 7 , 8 1 . When an a n a l y t i c a l form i s used, a c e r t a i n number o f para- meters have t o be determined according t o v a r i o u s c r i t e r i a , f o r example t o minimize t h e energy [ 7 ]
o r t o recover some known experimental r e s u l t s [81.
With t h i s p o t e n t i a l , which may o r n o t be l o c a l o r
l
dependent, a s e t o f one-electron wavefunctions i s generated. These wavefunctions a r e then used as basis f u n c t i o n s t o perform c a l c u l a t i o n s a t v a r i o u s order o f t h e p e r t u r b a t i o n t h e o r y .I.A. The Dirac-Fock method.
I n t h e r e l a t i v i s t i c theory the o n l y good quan- tum numbers are t h e t o t a l angular momentum 3 and the p a r i t y
n.
States w i t h t h e sane 3 and IT a r e f u r - t h e r c l a s s i f i e d by t h e o r d e r i n g o f t h e i r energies. I n i t s mu1 t i c o n f i g u r a t i o n v e r s i o n [91 t h e D i r a c - Fock method proceeds as f o l l o w s : a s t a t e o f given t o t a l angular momentum J and p a r i t y n i s w r i t t e n as a 1 in e a r combination o f wavefunctions having the sameJ
and T quantum numbers b u t a r i s i n g from d i f f e r e n t e l e c t r o n i c c o n f i g u r a t i o n s ( i n the r e l a - t i v i s t i c case a c o n f i g u r a t i o n i s d e f i n e d i n t h e jj c o u p l i n g scheme) :)J
n> = C cilai J a>i 11)
where ai stands f o r a l l quantum numbers necessary t o f u r t h e r s p e c i f y t h e ith c o n f i g u r a t i o n and ci i s the associated weight. I n e q . ( l ) ( a . J T> are a n t i -
1
symmetric products o f one-electron f o u r t h o r d e r Di r a c s p i nors
.
I n t h e mu1 t i c o n f i g u r a t i o n method, both t h e r a d i a l f u n c t i o n s o f t h e D i r a c s p i n o r s and the w e i g h t i n g c o e f f i c i e n t s ci a r e determined s e l f - c o n s i s t e n t l y u s i n g t h e v a r i a t i o n a l p r i n c i p l e . The zero o r d e r h a m i l t o n i a n used t o perform s e l f con- s i s t e n t f i e l d c a l c u l a t i o n s i s , i n a l l works r e p o r - t e d up t o now, r e s t r i c t e d t o t h e sum o f one-elec- t r o n D i r a c operators p l u s t h e c l a s s i c a l Coulomb r e p u l s i o n between t h e e l e c t r o n s . A f t e r s e l f - c o n - s i s t e n c y has been achieved, t h e energy can be c o r r e c t e d by f i r s t o r d e r p e r t u r b a t i o n theory f o r h i g h e r r e l a t i v i s t i c e f f e c t s (compared t o those i n - cluded i n the one-electron D i r a c - h a m i l t o n i a n ) , i .e. : magnetic i n t e r a c t i o n and r e t a r d a t i o n i n the Coulomb j n t e r a c t i o n between t h e e l e c t r o n s , vacuum p o l a r i z a t i o n and f l u c t u a t i o n . The i n f l u e n c e o f t h e magnetic i n t e r a c t i o n on t h e p o s i t i o n o f t h e f i n e s t r u c t u r e l e v e l s w i l l be discussed i n t h e n e x t sec- t i o n . I f i n e q . ( l ) t h e e l e c t r o n i c j j c o n f i g u r a t i o n s i n c l u d e d a r e r e s t r i c t e d t o those a r i s i n g from a s i n g l e LS non r e l a t i v i s t i c c o n f i g u r a t i o n , t h e mul- t i c o n f i g u r a t i o n method i s then used o n l y t o i n t r o - duce completly ab i n i t i o t h e breakdown o f t h e LS o r j j c o u p l i n g schemes. Stated i n another way i tmeans t h a t t h e v a r i a t i o n a l p r i n c i p l e a l l o w s t o de- termine wavefunctions i n t h e i n t e r m e d i a t e c o u p l i n g scheme. T h i s p o i n t w i l l be i l l u s t r a t e d i n t h e n e x t s e c t i o n when c o n s i d e r i n g t h e B e r y l l i u m i s o l e c t r o - n i c sequence. I f more than t h e above mentionned c o n f i g u r a t i o n s are included, then some c o r r e l a t i o n e f f e c t s a r e taken i n t o account. As t h e m u l t i c o n f i - g u r a t i o n method u t i l i z e a v a r i a t i o n a l p r i n c i p l e t o determine an optimum s e t o f s i n g l e p a r t i c u l e wave- f u n c t i o n s , few c o n f i g u r a t i o n s are g e n e r a l l y r e q u i - r e d t o g i v e accurate r e s u l t s . For example i n t h e B e r y l l i u m sequence [ l o ] , i t was found t h a t consi- d e r i n g o n l y t h e two c o n f i g u r a t i o n s l s 2 2 s 2 and
2 2 2
I s 2p f o r t h e ground s t a t e and t h e s i n g l e I s 2s2p e x c i t e d c o n f i g u r a t i o n
,
mu1 t i c o n f i g u r a t i o n r e s u l t s o n l y d i f f e r by r e l a t i v e l y small amounts from those obtained by Burke e t a1 . [ I l l i n a c o r r e l a t i o n c a l - c u l a t i o n j n c l u d i n g a t o t a l o f 13 c o n f i g u r a t i o n s .1. B. The r e l a t i v i s t i c random phase approximation. Althouah t h e random phase approximation (XPA) was o r i g i n a l l y developped f o r extended systems, i t s a n p l i c a t i o n t o atomic processes had revealed q u i t e successful e s p e c i a l l y f o r t h e c a l c u l a t i o n o f t r a n s i t i o n probabi 1 i t i e s and p h o t o i o n i z a t i o n cross s e c t i o n s 1121. !!bile i n the m u l t i c o n f i g u r a - ti on gethod described above one t r i e s t o o p t i m i z e s e p a r e t l y t h e wavefunctions o f the i n i t i a l and f i n a l s t a t e s , t h e b a s i c i d e a o f t h e RPA approach i s t h a t one concentrates on t h e o p e r a t o r which connect i n i t i a l and f i n a l s t a t e s r a t h e r than on t h e s t a t e s themselves. fi.pproximations are then i n - troduced e i t h e r on t h e form o f t h e o p e r a t o r o r on t h e i n t e r a c t i o n s r e s t r i c t e d t o belong t o a given c l a s s of approximate wavefunctions. The RPA me- thod can be viewed from t h r e e e q u i v a l e n t ways : t h e Green's f u n c t i o n method [131, t h e l i n e a r i z e d equations o f motion method [ I 4 1 and t h e t i m e dependent Hartree-Fock t h e o r y 1151. The l a s t apnroach has been used t o develop t h e r e l a t i v i s - t i c v e r s i o n o f the random phase approximation 1161
terminant, one consider t h e y e r t u r b a t i o n s wi indu- ced
i n
t h e Dirac-Fock o r b i t a l s ui by t h e e x t e r n a l f i e l d . The wils have t o s a t i s f y a system oC i n t e - g r o - d i f f e r e n t i a l equations q u i t e s i m i l a r t o t h e Dirac-Fock equations f o r t h e ui's and i n t h i s r e s - p e c t we can say t h a t t h e R!??P leads t o a c o p u t a - t i o n a l schene i n which t h e advantagages oF an i n - dependent-parti c l e theory are r e t a i n e d and f o r which standard and we1 l known numerical methods are a v a i l a b l e . But as t h e c o r r e l a t i o m taken i n t o account i n t h e RPA method a r e t r e a t e d simultaneous- l y f o r t h e i n i t i a l and f i n a l s t a t e s one o b t a i n s os- c i l l a t o r s t r e n g t h s which have i d e n t i c a l values i n t h e l e n g t h and v e l o c i t y forms [ I 7 1 p r o v i d e d t h a t t h e f u l l RPA c a l c u l a t i o n i s c a r r i e d o u t . I n p r a c t i - c a l a p p l i c a t i o n s where t h e RPA system o f equations i s g e n e r a l l y truncated, the discrepancy between t h e values obtained u s i n g t h e two forms o f t h e t r a n s i - t i o n o p e r a t o r i s used as an e v a l u a t i o n o f t h e ap- proximations introduced.From t h e above d e s c r i p t i o n of t h e RPPA method i t i s obvious t h a t i t i s r e s t r i c t e d t o systems t h e grodnd s t a t e o f which can be described by a s i n g l e determinant. However t h i s method has, up t o now, been c o n f i n e d t o t h e s t u d i e s o f e x c i t a t i o n s o f a s i n g l e valence e l e c t r o n from t h e ground s t a t e o f closed s h e l l systems. Furthermore s t a t e s s t r o n g l y i n f l u e n c e d by two e l e c t r o n s e x c i t a t i o n s a r e n o t r e l e v a n t o f the RRPA method. As i n t h e mu1 t i c o n f i s * r a t i o n method, t h e i n t e r m e d i a t e c o u p l i n g schene can be handled completly ab i n i t i o . For systems r e l e v a n t t o t h e two methods t h e r e s u l t s obtained a r e q u i t e s i m i l a r .
11. ILLUST%I~IVE EXAMPLES
The i n f l u e n c e o f r e l a t i v i s t i c e f f e c t s on t h e c a l c u l a t i o n o f o s c i l l a t o r s t r e n g t h s i s i l l u s t r a t e d by c o n s i d e r i n g a few number o f s e l e c t e d examples. The trends along an i s o l e c t r o n i c sequence a r e s t u - d i e d i n t h e case o f t h e b e r y l l i u m sequence. We then consider t h e problem o f l e v e l crossings d"e t o t h e t r a n s i t i o n from
to
jj c o u p l i n g scheme along a sequence. F i n a l l y t h e i n f l u e n c e o f h i g h o r d e r r e l a - t i v i s t i c c o r r e c t i o n s , m a i n l y t h e magnetic i n t e r a c - t i o n , on t h e f i n e s t r u c t u r e l e v e l spacings i s d i s - cussed. authors u s i n g e i t h e r t h e m u l t i c o n f i g u r a t i o n D i r a c - Fock method (HCDF) [18,10] o r t h e r e l a t i v i s t i c random phase approximation (RRPA) 1191. We discuss here t h e !lCDF r e s u l t s which are i n q u i t e good agreement w i t h t h e RRPA ones. As p r e v i o u s l y i n d i - cated, c o n s i d e r i n g f o r t h e ground s t a t e t h e two2 2 2 2
I s 2s and 1s 2p c o n f i g u r a t i o n s and f o r t h e e x c i - 2
t e d s t a t e t h e s i n g l e 1s 2s2p c o n f i g u r a t i o n l e a d t o accurate r e s u l t s . The ground s t a t e wavefunctions i s expressed as :
and i n t h i s case the FlCDF method i s used t o i n t r o - duce c o r r e l a t i o n e f f e c t s due t o t h e s t r o n g m i x i n g between t h e two c o n f i g u r a t i o n s as expected from the hydrogenic degeneracy o f t h e 2s and 2p energy l e v e l s . I n t h e r e l a t i v i s t i c case o n l y t h e 2s and 2plj2 l e v e l s are degenerated and t h e r e s u l t s given i n Table I show, as expected, t h a t t h e weight B o f t h e ( ~ p ~ , ~ ) ' c o n f i g u r a t i o n remains e s s e n t i a l l y constant throughout t h e sequence w h i l e , due t o t h e increase i n the s p i n - o r b i t i n t e r a c t i o n , t h e con- t r i b u t i o n of t h e ( 2 ~ c o n f i g u r a t i o n decreases ~ ~ ~ ) ~ as t h e atomic number Z increases. The i n f l u e n c e o f t h e s p i n - o r b i t i n t e r a c t i o n i s a l s o r e f l e c t e d i n the v a r i a t i o n o f the r a t i o (y/612, f o r Z = 10 t h i s r a t i o i s almost equal t o t h e n o n - r e l a t i v i s t i c va- l u e o f 2 w h i l e f o r Z = 74 i t has decreased t o 0.04
and one has reached t h e jj c o u p l i n g scheme l i m i t .
Table 1 C o n f i g u r a t i o n m i x i n g c o e f f i c i e n t s f o r t h e Be sequence ( c f
.
Eq .2) For t h e e x c i t e d s t a t e t h e wavefunction i s The n o n - r e l a t i v i s t i c allowed t r a n s i t i o n i s1
associated w i t h t h e PI s t a t e f o r whicha
= 0.577c1-112
JOURNAL DE PHYSIQUETable 11, t h e value o f t h e "a" c o e f f i c i e n t r a p i d l y Table 111
deviates from t h e LS l i m i t and f o r h i g h Z-ions one f values f o r t h e b e r y l l i u m sequence
Second J = 1 s t a t e o f t h e Be sequence "a" c o e f f i c i e n t o f Eq.3
reaches t h e jj c o u p l i n g l i m i t . I t i s c l e a r f r o m t h e
When p l o t t e d as a f u n c t i o n o f Z, t h e r e l a t i - v i s t i c r e s u l t s f o r t h e d i p o l e o s c i l l a t o r strengths ( f values) d i s p l a y a q u i t e d i f f e r e n t behaviour than t h e one obtained i n t h e n o n - r e l a t i v i s t i c case.Fhile t h e l a t t e r one gives a monotonically decreasing f u n c t i o n o f Z, the f values obtained i n t h e r e l a t i - v i s t i c c a l c u l a t i o n s show, f o r t h e b e r y l l i u m sequen- ce, a minimum around Z = 35 and then begin t o i n - crease w i t h Z. Furthermore, s t a r t i n o from Z%20 t h e r e l a t i v i s t i c f values are s i g n i f i c a n t l y l a r g e r than t h e n o n - r e l a t i v i s t i c ones and a r e i n b e t t e r agree- ment w i t h the recommended values o f Smith and ).!iese
1 2 0 1 . To analyze t h e r e l a t i v i s t i c e f f e c t s , i t i s convenient t o consider s e p a r a t e l y t h e two f a c t o r s c o n t r i b u t i n g t o t h e f value, i .e. t h e e x c i t a t i o n energy E and t h e l i n e s t r e n g t h S. I n term o f these, f i s d e f i n e d i n t h e l e n g t h form as :
f = E S ( ~ ~ R ) - ~ (4
where R = 13.6 eV and g i s t h e degeneracy o f t h e i n i t i a l s t a t e . Results f o r E, S and f are g i v e n i n Table 111 f o r s e l e c t e d Z values o f the b e r y l l i u m sequence. -It can be seen t h a t , i n the r e l a t i v i s t i c case, E i n c r e a s e much more r a p i d l y w i t h Z w h i l e S
remains c l o s e t o t h e n o n - r e l a t i v i s t i c value throu- ghout the sequence. This a g a i n r e f l e c t s t h e i n f l u - ence of t h e s p i n - o r b i t i n t e r a c t i o n since, f o r the Z-region, t h e e x c i t e d s t a t e i s almost p u r e l y t h e J = 1 s t a t e o f t h e s i n g l e sp c o n f i g u r a t i o n . .3/2 r e s u l t s we have j u s t discussed t h a t t h e i n t e r ~ n e d i a - t e c o u p l i n g scheme must be i n t r o d u c e d t o o b t a i n accurate r e s u l t s e i t h e r by t h e r e l a t i v i s t i c v a r i a - t i o n a l approach we have described o r by a s u i t a b l e p e r t u r b a t i o n method.
Table I1
R and NR stand f o r r e l a t i v i s t i c and n o n - r e l a t i v i s - t i c r e s u l t s E and S a r e t h e t r a n s i t i o n energy and t h e l i n e s t r e n g t h as d e f i n e d by Eq.3.
For t r a n s i t i o n s t o t h e lowest J = 1 s t a t e 3
(corresponding t o t h e P1 s t a t e f o r t h e low Z-re- g i o n ) i t was found t h a t t h e f values a r e s m a l l e r
9
by a f a c t o r o f 10 ( l o w Z-ions) t o 20 ( h i g h Z-ions) compared t o t h e "allowed" t r a n s i t i o n t o t h e upper s t a t e . Thus t h e " f o r b i d d e n " t r a n s i t i o n remains weak even i n t h e jj c o u p l i n g l i m i t f o r t h e b e r y l l i u m l i k e ions. S i m i l a r r e s u l t s where Found f o r t h e ma- gnesium sequence 17,121 and discussed i n terms o f t h e r e l a t i v i s t i c Z-dependent t h e o r y o f many-elec- t r o n atoms [ 2 2 1 . Non monotonous decrease o f t h e f values w i t h t h e atomic number Z where a l s o r e p o r - t e d f o r t h e l i t h i u m [10,181 and sodium [ 2 3 ] i s o - l e c t r o n i c s e r i e s . Z 10 42 74
1I.B. Level crossings.
I n t h i s s e c t i o n we want t o discuss an o t h e r consequence o f the t r a n s i t i o n from LS t o jj cou- p l i n g scheme, i . e . t h e f a c t t h a t the l e v e l orde- r i n g changes between low Z and h i g h Z-region. As an example l e t us consider t h e case o f t h e magne- s i u n sequence s t u d i e d by Cheng and Johnson [ 2 1 ] . The f i r s t f o u r e i g e n s t a t e s w i t h even p a r t y and having J = 2 as t o t a l angular momentum eigenvalue
2
a r i s e from t h e ( 3 ~ ) and (3s3d) c o n f i g u r a t i o n s . F o r low Z-values when t h e s p i n - o r b i t i n t e r a c t i o n i s negligeable, t h e o r d e r i n g o f these l e v e l s was found
2 1 (3s3d) 3 ~ 2 (3s3d) t o b e : (3p) D2, (39) P2, ' D ~ . !:hen s p i n - o r b i t i n t e r d c t i o n
i s dominant, t h e
o r d e r i n g o f t h e (jj') l e v e l s f o l l o w s t h e f o l l o w i n g sequence : (1/2 1/2), (1/2 3/2), (1/2 5/2), ( 3 / 2 3/2)...
Thus s i n c e i n t h e j j c o u p l i n g l i m i t , t h e (3s3dy2) s t a t e i s lower i n energy than t h e (3p )23/2 s t a t e , these two s t a t e s have t o cross a t some va- l u e o f t h e atomic number
2 .
Even i f t r u e crossings a r e f o r b i d d e n,
t h e p r e v i o u s c o n s i d e r a t i o n imp1 i e s t h a t , f o r some g i v e n values o f Z, t h e two l e v e l st a k e place. Such a s t r o n g i n t e r a c t i o n w i l l r e s u l t s t e r v a l s o f t h e f i n e s t r u c t u r e l e v e l s and t h e o r e t i - e i t h e r i n a decrease o f t h e t r a n s i t i o n p r o b a b i l i t y c a l c a l c u l a t i o n s t 2 7 1 have been c a r r i e d o u t t o due t o d e s t r u c t i v e i n t e r f e r e n c e between t h e t r a n - assess t h e r e l i a b i l i t y o f "ab i n i t i o " methods. I t s i t i o n channels o r t o t h e enhancement of t h e t r a n - was found t h a t t h e B r e i t i n t e r a c t i o n ? l a y s a c r u - s i t i o n caused by coherent mixing. F o r t h e Hg c i a 1 r o l e t o o b t a i n good agreement w i t h experiment sequence i t was found t h a t such a s i t u a t i o n a r i s e s
2 around Z = 45 and t h a t t h e m i x i n g o f t h e (3p) and (3s3d) c o n f i g u r a t i o n s i n t h e 3 ~ 2 s t a t e s t r o n g l y supresses, i n t h e h i g h Z-region, t h e t r a n s i t i o n s
3 1
t o t h e P2 and P1 s t a t e s of the ( 3 ~ 3 2 ) c o n f i g u r a - 3
t i o n w h i l e t h e t r a n s i t i o n t o the P1 s t a t e was en- hanced.
Except f o r very simple c o n f i g u r a t i o n s , l e v e l crossings a r e l i k e l y t o happen and w i l l r e s u l t i n i r r e g u l a r i t i e s i n t h e l i n e s t r e n g t h s . I t i s there- f o r e i m p o r t a n t t o consider w i t h g r e a t care the r e - gions o f p o s s i b l e l e v e l crossings when l o o k i n g a t systematic trends i n t h e t r a n s i t i o n p r o b a b i l i t i e s .
and t h a t t h e c o n f i g u r a t i o n averaging was a poor approximation i n t h e cases under consideration,
F o r the f i n e s t r u c t u r e s p l i t t i n g s i n the
2 4 4
l s 2 p ( P) and ls2s2p( P) s t a t e s o f l i t h i u m i o n s t h r e e successive approximations' were used. F i r s t the B r e i t i n t e r a c t i o n was comnletely neglected, i n t h i s case t h e f i n e s t r u c t u r e i n t e r v a l s were found t o f o l l o w c l o s e l y t h e Land6 r u l e as expected since o n l y t h e s p i n - o r b i t i n t e r a c t i o n was included. Se- condly t h e B r e i t i n t e r a c t i o n was c a l c u l a t e d i n t h e c o n f i g u r a t i o n average c o n t r i b u t i o n . Compared t o t h e f i r s t case, a general s h i f t o f the energy l e - v e l s was observed b u t w i t h almost no change i n t h e spacing between them. F i n a l l y t h e magnetic p a r t o f 1I.C. Fine s t r u c t u r e s p l i t t i n g s . the B r e i t term was c a l c u l a t e d c o r r e c t l y and drama-
As mentionned i n
I ,
t h e zero order h a m i l t o - t i c changes i n t h e f i n e s t r u c t u r e occurred. A t t h i s nian, used, i n t h e !.1CDF method t o determine t h e stage, agreement w i t h experiment i s o f t h e o r d e r o f wavefunctions, i n c l u d e s o n l y the s p i n - o r b i t i n t e - 10% and t h e remaining discrepancy should be reduced r a c t i o n which i s e x p l i c i t l y taken i n t o account i n i f t h e r e t a r d a t i o n p a r t o f t h e B r e i t operator and t h e Di r a c o n e - p a r t i c l e hami 1 to n i an. The s p i n - s p i n o t h e r quantum electrodynamic e f f e c t (vacuum p o l a - and s p i n - o t h e r - o r b i t i n t e r a c t i o n s , which come from r i z a t i o n and f l u c t u a t i o n ) a r e a l s o c o r r e c t l y taken the B r e i t operator, a r e t r e a t e d as a f i r s t order i n t o account.p e r t u r b a t i o n and o f t e n c a l c u l a t e d i n t h e average
c o n f i g u r a t i o n approximation 124,251. I f t h e e l e c - ACKNOG!LEDGIENTS
t r o n s belong t o closed s h e l l s o r a t most o n l y one I t i s a pleasure t o thank Dr. Y.K. Kim f o r i s i n an open s h e l l , t h e c o n f i g u r a t i o n averaging many s t i n u l a t i n g discussions and continuous c o l l a - p r o v i d e s t h e e x a c t c o n t r i b u t i o n b u t f o r general b o r a t i o n over more than f i v e years i n the study o f open s h e l l s systems i t i s o n l y an approximation. r e l a t i v i s t i c effects i n atomic s t r u c t u r e c a l c u l a -
t i o n s . References
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