relativistic hf minimal basis set

7/29/2019 Relativistic HF Minimal Basis Set

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Theoret. Chim. A cta (Bed.) 54, 35-5 1 (1979)T H E O R E T I C A C H I M I C A A C T A9 by Springer-Verlag 1979

A p p r o x i m a t e R e l a t i v i s t i c H a r t r e e - Fo c k E q u a t i o n s a n dThe ir So lut ion wi th in a Minimum Bas i s Se t o fS l a t e r - T y p e Fu n c t i o n s *F r a n z R o s i c k yInstitut fiJr theoretische Chemic und Strahlench~mie, Universit~t Wien, W~hringerstr. 17,A-1090 Wien, AustriaF r a n z M a r kIns titu t ftir Strahlenche mie im M ax-Planck -Institut ftir Ko hlenforschun g, Stiftstr. 34-36,D-4330 Mii lheim a.d. Ruh r, Federal R epublic o f Ge rma ny

R e l a t i v is t ic c l o s e d -s h e l l a t o m s a r e t r e a t e d b y t h e u s e o f a sp e c if ic a p p r o x i m a t i o nf o r th e s m a l l c o m p o n e n t o f t h e o n e - e l e c t r o n D i r a c s p i n o rs . I t is a s s u m e d t h a t t h el a r g e a n d t h e s m a l l c o m p o n e n t a r e i n t e r c o n n e c t e d b y a p a r a m e t e r - d e p e n d e n tr e l a t i o n w h i c h i s f o r m a l l y a n a l o g o u s t o t h a t o f t h e o n e - e l e c t r o n s y s t e m . S u b j e c tt o t h i s c o n s t r a in t , t h e t o t a l e n e r g y i s v a r i e d w i t h r e s p e c t t o t h e l a r g e c o m p o n e n t s .T h e r e s u l t i n g e i g e n v a l u e e q u a t i o n s f o r t h e l a r g e c o m p o n e n t s c o n t a i n o n l yr e g u l a r p o t e n t i a l t e r m s a n d r e d u c e t o t h e f a m i l i a r H a r t r e e - F o c k e q u a t i o n s i nt h e l i m i t o f in f in i te v e l o c i t y o f li gh t. A n a l y t i c a l s o l u t i o n o f th e s e a p p r o x i m a t er e l a t i v i s t i c H a r t r e e - F o c k e q u a t i o n s i s a c h i e v e d u s i n g a m i n i m u m b a s i s s e t o fS l a t e r - ty p e f u n c t i o n s f o r th e e x p a n s i o n o f t h e r a d i a l p a r t o f t h e l a r g e c o m p o -n e n t s. T o t a l r e l a t iv i s ti c e n e r g ie s , o r b i t a l e n e r g i e s , o r b i ta l e x p o n e n t s a n d m e a nr a d i i a re c a l c u l a t e d fo r t h e g r o u n d s t a te s o f H e , B e , N e , M g , A r , K r , X e a n dC u + .

K e y w o r d s : R e l a ti v is ti c H a r t r e e - F o c k - C l o s ed - sh e ll a t o m s

1 . Introduct ionN u m e r i c a l s o l u ti o n s o f t h e re l at iv i st ic H a r t r e e - F o c k ( R H F ) e q u a t i o n s ( f o r a r e v ie ws e e [ 1 ] ) h a v e b e e n o b t a i n e d f o r m a n y a t o m i c s y s t e m s [ 2 - 7 ] , a n d c o m p i l a t i o n s o fr e l e v a n t d a t a e x i s t [ 8 -1 0 ]. H o w e v e r , a n a l y t i c a l s o lu t i o n s o f t h e R H F e q u a t i o n sw i t hi n t h e L C A O a p p r o x i m a t i o n [ 11 -1 3 ] w e re o b t a i n e d o n l y f o r a fe w a t o m s [1 1,13 ]. T h e p r e d o m i n a n t u s e o f t h e n u m e r i c a l i n t e g r a ti o n m e t h o d is m a i n l y d ue t o t h ef a c t t h a t in t h e L C A O - R H F m e t h o d l a rg e b as is s et s a r e n e ed e d s in c e t h e n u m b e r o fo n e - e l e c t r o n s t a te s is a l m o s t d o u b l e d , a n d f u r t h e r m o r e t h e l a r g e a s w e l l a s t h e s m a l l

* Dedicated to Prof. O. E. Polansky on occasion of his 60th birthday.

0040-5744/79/0054/0035/$03.40


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36 F. Rosicky and F. M arkc o m p o n e n t s o f t h e D i r a c o r b i t a l s h a v e t o b e re p r e s e n t e d a d e q u a t e l y . T h u s , K i m [1 1]r e m a r k s " t h a t t h e s o - c a l l e d m i n i m u m b a s i s s e t r e l a t i v i s t i c w a v e f u n c t i o n s l e a d t or e s u l t s s o b a d t h a t i t m a k e s n o s e n se t o d o a n y r e la t iv i s ti c c a l c u l a t io n s w i t h t h e m " .I n t h e f o l l o w i n g , a n a t t e m p t i s m a d e t o r e d u c e t h e d i ff ic u l ti e s a r i s in g f r o m t h e i n -c r e a s e d s iz e o f t h e b a s i s s et b y t a k i n g t h e a p p r o x i m a t i o n o f a n a pr ior i r e l a t i o n s h i pb e t w e e n t h e l a r g e a n d t h e s m a l l c o m p o n e n t o f t h e o n e - e l e c t r o n D i r a c o r b i t a ls . T h i sf u n c t i o n a l r e l a t io n i s, in t h e c a s e o f a m a n y - e l e c t r o n a t o m , c h o s e n in a n a l o g y t ot h a t o n e w h i c h f o r t h e o n e - e l e ct r o n p r o b l e m i s o b t a i n e d b y t h e e l im i n a t i o n m e t h o d[ 14 ]. G i v e n t h i s f u n c t i o n a l r e l a t io n a n i m p l i c i t e i g e n v a l u e p r o b l e m f o r t h e l a r g ec o m p o n e n t i s d e r i v e d b y v a r y i n g t h e I o t a l e n e r g y w i t h r e s p e c t t o a l l l a r g e c o m p o -n e n t s o n l y . A c c o r d i n g t o t h i s a p p r o x i m a t i o n o n l y r e g u l a r p o t e n t ia l t e r m s i n t h eR H F e q u a t i o n s a p p e a r w h i c h i n t h e l i m i t o f i n f in i t e v e l o c i ty o f l i g h t r e d u c e d i r e c t l yt o th e n o n re l a ti v i st i c H a r t r e e - F o c k ( N R H F ) e q u a ti o n s .I n o r d e r t o e s t im a t e t h e n u m e r i c a l a c c u r a c y o f t h e p r o p o s e d a p p r o x i m a t i o n ,c a l c u l a t i o n s f o r s o m e r e l a ti v i s ti c a l ly cl o s ed - s h e ll a t o m s a r e p e r f o r m e d u s i n g am i n i m u m b a s is s e t ( M B S ) o f S l a t e r - t y p e f u n c t i o n s (S T F s ) f o r t h e r a d i a l p a r t o f th el a r g e c o m p o n e n t s . T h e n u m e r i c a l r e s u lt s f o r r e l at i v is t ic to t a l e n e r g y c o r r e c t i o n s a n do r b i t a l en e r g ie s a r e i n f a ir l y g o o d a g r e e m e n t w i t h t h o s e o f n u m e r i c a l R H F c al -c u l a t i o n s .

2 . A p p r o x i m a t e R e la t iv i s t i c H a r t r e e - F o c k E q u a t io n s2 .1 . The Func t iona l Re la t ion be tween the Large and the Sma l l Com ponen t sF o r a n a t o m w i t h N e l e c t ro n s a n d a n u c l e a r c h a r g e Z t h e H a m i l t o n i a n ( in a .u . ) ist a k e n t o b e

H = ~ h~ + (1)9 . j "'

w h e r e h i r e p r e s e n t s t h e D i r a c - H a m i l t o n i a n f o r t h e i t h e l e c t r o nZ (2)

He re , c i s the ve loc i ty o f l igh t ( c = 137.0373 in a .u . ) and e a nd f i ' a re 4 x 4 ma t r i ce s( 0 a ) f l ' = ( ; 012 ) ( 3 )~ = 0 ' - - 2 '

T h e v e c t o r a h a s t h e 2 x 2 P a u l i s p in m a t r i c e s a s i t s c o m p o n e n t s ; t h e e n e r g y - s c a leh a s b e e n s h i f t e d b y - c 2 t o s u b t r a c t t h e e l e c t r o n i c re s t -e n e r g y .I n p r i n c i p le , s o m e r e l at i v is t ic t w o - p a r t i c l e i n t e r a c t i o n s n e g l e c t e d i n ( 1 ) c o u l d b et a k e n i n t o a c c o u n t b y u s i n g t h e B r e i t o p e r a t o r i n 1 st o r d e r p e r t u r b a t i o n t h e o r y [ 14 ].F o r c l o se d - s h el l s y st e m s t h e N - e l e c t r o n t r i a l w a v e f u n c t i o n tF is t a k e n t o b e a s i n g leS l a t e r d e t e r m i n a n t

1' F - ~ I ,b~(1) ,~(2) . . .~bu(N ) 1 (4 )


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Approximate Relativistic Hartre e-Foc k Equations 37

w h e r e t h e f o u r - c o m p o n e n t D i r a c o r b i ta l s ~b f o r m a n o r t h o n o r m a l s e t

T h u s , t h e t o t a l e n e r g y i s g i v e n b yE = ~" (~b,(1), h(1)~b~(1)} + ~b~(1)~bj(2), - - ~b,(1)~bj(2) (6), r12

w h e r e P u d e n o t e s t h e t r a n s p o s i t i o n o p e r a t o r . N o w , i t i s c o n v e n i e n t to p a r t i ti o n ~bi n t o t w o - c o m p o n e n t P a u li s p in o r s 9 a n d X, t h e l ar g e a n d t h e s m a l l c o m p o n e n t ,r e s p e c t i v e l y ,

~ b = ( ~ ) . (7 )S u b s t i t u t i o n o f (7 ) in ( 6) a n d a p p l i c a t i o n o f t h e v a r i a t i o n a l p r i n c i p le l e a d s t o t h eR H F e q u a t i o n s ( s ee e .g . [ 15 ]) :

_ Z 1 1 - P uc a l p l X ~ ( 1 ) ~ - [ % ( ) + ~ { f d Sx 2q ~7 (2 ) r - - r - ~ 2 ~~+ f d S x 2 x + ( 2 ) 1 - P~___...~, (2)~o,(1) : e~,(1)J r12 )

c%P 19~(1) - {]-f-~ + 2c2}x~ (1+ ~ . { f d 3 x 2 c P ? 2) 1- r12P~ 9j (2) +

= ~ x ~ ( 1 ) .

(8a )

f d 3 x 2 x + ( 2 ) ~ X j ( 2 ) } x ~ ( 1 )( 8 b )

I n o r d e r t o r e d u c e t h e si ze o f t h e b a s is s e t t o b e u s e d i n th e L C A O r e p r e s e n t a t i o no f t h e o r b i t a ls o n e w o u l d t r y t o e l i m i n a t e a ll th e s m a l l c o m p o n e n t s x i n e i t h e r o f t h eE q s . ( 8 ) ; it is h o w e v e r n o t p o s s ib l e t o a c c o m p l i s h t h is i n a n e x a c t a n d c l o s e d w a y .B e r s u k e r e t a l . i1 5] u s e d t h e m e t h o d o f su c c e s si v e a p p r o x i m a t i o n s t o e l i m i n a t e X i n( 8 a) a n d t h u s o b t a i n e d a n e q u a t i o n w h i c h is e x a ct u p t o t h e o r d e r ( l / c ) 2 a n d w h i c ho n l y i n v o l v e s t h e l a r g e c o m p o n e n t s .I n t h e f o l l o w i n g a d i f fe r e n t a p p r o a c h is p r o p o s e d . X~ s a s s u m e d t o b e i n t e r c o n n e c t e dw i t h q~i b y a f i x e d r e l a t i o n o f t h e f o r m

X~ = f~ i~ (9a )w h e r e

a n d(9b)f~ i = ~ c B 7 1 0 1 )

1Bi -- 1 + ~ (ei - qb). (9c )


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38 F. Ros icky and F. M arkI n ( 9 ) t h e f u n c t i o n 9 r e p r e s e n t s a lo c a l p o t e n t i a l a n d t h e p a r a m e t e r s e~ a r e a s s u m e dt o b e r e a l n u m b e r s . F o r a s i n g le e l e c t r o n i n t h e f i e ld o f a n u c l e u s E q . ( 9 ) h o l d se x a c t l y i f 9 i s i d e n t i f ie d w i t h t h e n u c l e a r p o t e n t i a l a n d t h e e ~'s a r e i n t e r p r e t e d a s t h eo r b i t a l e ne r g i e s ( s e e l a t e r Eq . ( 21 ) ) [ 14 ] .2.2. VariationalEquation fo r the Large C omponentsW i t h t h e a b o v e r e l a t i o n ( 9 ) t h e t o t a l e n e r g y ( 6 ) i s g i v e n b y

1E = ~ (% (1 ), h~(1)50~(1)} + ~ ~ ~ ( ,p~(1)50:(2), 2u (1 , 2)50~(1)50:(2)5 (1 0)a n d t h e o r t h o n o r m a l i t y c o n d i t io n ( 5 ) b e c o m e s

~9~, (1 + f2+~:)50:} = 3u. (11)I n E q . ( 1 0 ) w e u s e d t h e a b b r e v i a t i o n s

Z ( Z 2c 2) f)~ (12 )~ = - ] ~ + c op a~ + n + c ~ p + ~ + - I x - ~ -a n d

2u( 1 , 2 ) = 1 - P u + f 2+ ( 1 ) 1 - Pu f2~(1) + -Q+(2) 1 - P u ~ : (2 )r12 r12 / '12

+ D + ( 1) D + ( 2) 1 - P u f ): (2 ) f2~(1 ) = 2 [ 2 + 2~'2 ( 13 )r12

w h e r e t h e t e r m s n o t c o n t a i n i n g P u w e r e c o l l e c t e d in ~'la w h i l e t h o s e c o n t a i n i n g t h et r a n s p o s i ti o n o p e r a t o r b e l o n g t o g~ z.I n o r d e r t o f i n d a s t a t i o n a r y e n e r g y v a l u e a l l t h e l a r g e c o m p o n e n t s i n t h e e n e r g ye x p r e s s i o n ( 1 0 ) a r e v a r ie d s u b j e c t t o t h e c o n s t r a i n t o f t h e o r t h o n o r m a l i t y c o n d i t io n(1 1 ). I n o t h e r w o r d s , t h e v a r i a t i o n s 8X w h i c h , a c c o r d i n g t o t h e v a r i a ti o n a l p r i n c ip l es h o u l d b e i n d e p e n d e n t o f th e v a r i a t i o n s 350 a r e c o m p e l l e d t o h a v e t h e f o r m

3x = f2 350. (14 )S i n c e a ll a r g u m e n t s c o n c e r n i n g t h e d i a g o n a l i z a ti o n o f t h e m a t r i x o f t h e L a g r a n g em u l t i p l i e r s s t i l l h o l d , t h e e i g e n v a l u e e q u a t i o n

{ ~ (1 ) - ~Y~+(1)D~(1) + ~ (9 : ( 2) , 2u(1 , 2)50:(2)}}50~(1) - ~ % (1 ) = e~50~(1)( 15 )

a n d t h e t o t a l e n e r g y e x p r e s s i o n1E = ~ e ~ - ~ ~ ~ (50~(1)50:(2) , 2u (1, 2)50,(1)~,(2)} (16 )

a r e f in a l l y o b t a i n e d . T h e o n e - e l e c t r o n p a r t i n t h e 1 .h .s . o f E q . ( 1 5 ) i s g i v e n e x p l i c i tl yb y

Z 1 1l f (-~x~ 1d ~ p ( ~ - c ) + + 9 ~p . ( 17 )


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Approxim ate Relativistic Hartree-Fock Equations 392 .3 C h o i ce o f P a r a m e t e r sT h e p o t e n t i a l 9 a n d t h e p a r a m e t e r s e~ w e r e n o t o b t a i n e d f r o m t h e v a r i a t i o n a l p r i n -c ip l e a n d t h u s h a v e t o b e d e t e r m i n e d b y s o m e j u d i c i o u s l y c h o s e n c r it e r ia . S p e c if icc h o i c e s a r e t h e f o l l o w i n g : i n a f i r st a p p r o x i m a t i o n qb i s se t e q u a l t o t h e p o t e n t i a l o ft h e b a r e n u c l e u s

Z9 - (18)I x lt h u s

( 1 9 ,, = B ? ) ( e 3 - B ~ e , , - .

T h i s f o r m o f q~ h a s t h e a d v a n t a g e t h a t a f t e r e x p a n s i o n o f t h e r a d i a l p a r t s o f t h e ~o~i n t o S l a t e r -t y p e f u n c t i o n s a ll t h e o n e - a n d t w o - e l e c t r o n in t e g r a l s a p p e a r i n g i n E q .(11 ) an d (16 ) can b e so lved ana ly t i c a l ly ( cf . t he app end ix ) .G i v e n X b y ( 18 ), t w o c h o i c es f o r t h e p a r a m e t e r s e~ h a v e b e e n t r i e d :a) e~ = ei. (20)If e~ i s ident i f ie d w i th th e e ig env alue e~ of E q. (15) , th e op er a to r D~ (17) i s redu cedt o a n o p e r a t o r a~ w e l l k n o w n f r o m t h e e l i m i n a t i o n m e t h o d [ 1 4] i n t h e o n e - e l e c t r o np r o b l e m

Z 1 1a~(e,) = - T ~ + 2 ap ~ ap . (21)F u r t h e r m o r e , t h e e i g e n f u n c t i o n s c o r r e s p o n d i n g t o d i f f e r e n t e i g e n v a l u e s e a r eg u a r a n t e e d t o b e a u t o m a t i c a l l y o r t h o g o n a l i n t h e s e n s e o f ( 11 ) [ 1 6 ]. F o r t h i s c h o i c eo f p a r a m e t e r s t h e q u a s i - re l a ti v i s ti c e q u a t i o n s f o r t h e s p i n o r ~o, w h i c h h a v e b e e nd e r i v ed b y B e r s u k e r et a l . [ 1 5 ] m a y b e o b t a i n e d f r o m ( 15 ) b y s u b s t i t u t i n g th e p o w e rse r i e s expans ion

1B~0)(e,)-z ~ 1 - ~ (~, - (I)) 4 - . . .a n d n e g l e c t in g a l l t e r m s o f o r d e r h i g h e r t h a n O (1 [c2). T h e r e s u l t in g e q u a t i o n a g r e e swi th the quas i - re l a t iv i s t i c HF equa t ion (Eq . (11 ) in [15 ] ) .b ) e, = d, - (~p~, c~(e,)~q)/(rp~, ~0~) = (~o~,h~(e,)cp,>. (22)I f e~ i s i d e n t i fi e d w i t h t h e e x p e c t a t i o n v a l u e o f t h e o n e - e l e c t r o n o p e r a t o r c ~( e0 , t h en u m e r i c a l s o l u t i o n o f E q . ( 15 ) i s si m p l if ie d , s i nc e t h e t w o - e l e c t r o n c o n t r i b u t i o n s d on o l o n g e r d e p e n d o n t h e e i g e n v a l u e s e t o b e d e t e r m i n e d . H o w e v e r , f o r t h e c h o i c e(22) Eq . (15 ) sa t is f ie s the va r i a t io na l p r inc ip le o n ly appr ox im a te ly s ince in de r iv in g(15) t h e f ~ h a v e b e e n a s s u m e d t o b e i n d e p e n d e n t o f th e w a v e f u n c ti o n s . 1

This point has bee n noted by the referee.


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40 F. Rosicky and F. Mark2 .4 . F o r m u l a e f o r M a t r i x E l e m e n t sI n a c e n t r a l f ie l d t h e o r b i t a l s ~ c a n b e s e p a r a t e d b y u s i n g p o l a r c o o r d i n a t e s [1 7]

(23)~ ( x ) = ~ b ~ ,( r, e , q~) = r k i Q n ~ ( r ) x _ ~ , ( e , ~ o ) )w i t h t h e p h a s e c h o s e n i n s u c h a w a y t h a t t h e r a d i a l fu n c t i o n s P a n d Q a r e b o t h r e a l .P r o p e r t ie s o f t h e c o m p l e te a n d o r t h o n o r m a l s e t o f s p i no r s X a r e g i v e n in [ 17 ]. T h eq u a n t u m n u m b e r s x a n d t z a r e in t e r r e la t e d w i t h t h e t o t a l a n g u l a r m o m e n t u m j a n dt h e o r b it a l a n g u la r m o m e n t u m I b y

1J = -

1 Jrt = j +a n d / x = - j , - j + 1 . . . . j - I , j . T h e p r i n ci p a l q u a n t u m n u m b e r n l ab e ls o r b i ta l sw h i c h c a n n o t b e d i s ti n g u i s h ed b y x a n d / ~ . U s i n g

r = r l f f ( _ p , + / ( / + ) r___T__p)_ f , (p , + ; P ) ) (24)w h e r e t h e p r i m e d e n o t e s d i f f e r e n t i a t io n w i t h r e s p e c t t o r , o n e o b t a i n s

f ; ( [~ o,, ~ ( e , ) 9 , ) = d r - Z + ~ 1 _ p p , , + l ( l rm-g--+ 1 ) p 21 Z ( r ) ] )2 e z B r 2 p p , + p 2 (25)

w i th P - P~ ,~ , an d B - B } ~ T h e o r t h o n o r m a l i t y c o n d i t i o n ( 1 1 ) r e s u l t s i nf o ~176 { 1 1 [ - P I P ; ' + l j( ls + 1 ) p ~ p jd r P ~ P j + 4 c ~ B , B j r ~

- "2c2 B ~ B j r 2 P ~ P j + r P ~ P j = 3~j. (26 )T h e t w o - e l e c t r o n p a r t o f E q . ( 1 6 ) c a n b e e v a l u a t e d a l o n g t h e l i n e s g i v e n b y G r a n t[ 1 ] . T h e C o u l o m b i n t e g r a l s a r e g i v e n b y

(~ (1 )9~ (2 ), g;2cp~(1)c&(2)) = ~ av( ( j , t z)~ , ( j , I z )y )Fv( i , j )w h e r e

v = 0 , 2 , 4 , . . . , M in [ (2 j~ - 1), (2 j j - 1)] (27)a n d t h e e x c h a n g e i n t e g r a l s a r e g i v e n b y

(9,(1)g ,~.(2), g~'29~.(1)~o,(2)) = ~ b~((j , t*) , , (J , t*) i )G ~( , J )-#


7/17

Approximate Relativistic H artre e-F oc k Equations 41w h e r e

K Kje v e n i f ~ T # ]x~[I k -JJl < ~ ~ < J , + J / a n d j , + j , + v = ( 28 )~ tQo d d i f ~ =

T h e r e i n t h e n o t a t i o n sF ~( i , j ) = dr1 dr2U~(1 , 2){ P~ (r l ) + Q~ (r l )} {P~(r2) + Q~(r2)} (29)

j o~( i , j ) = dr1 dr2U~(1, 2){P ~(r l )P j(r l ) + Q~(r~)Qj(r l)}x {P ~(r2)Pj(r2) + Q~ (r2)Qj(r2)} (30)

w i t h2 c B ~1 ( K p )~ P{ + r ~ (31)

a n dV V+Uv(1, 2) = r 1

a r e u s e d . T h e n u m b e r s a v a n d b~ c a n b e t a k e n f r o m t a b l e s [1 ].3 . C a l c u l a t io n s U s i n g a M i n i m u m B a s i s S e t o f S la t e r -T y p e F u n c t i o n sT h e r a d i a l fu n c t i o n s P a r e e x p a n d e d a c c o r d i n g t o

P ,~( r ) = ~ c~r ~ , e x p { - ~ r } . ( 32 )i = l

T h e e x p o n e n t s ~ a r e r e a l n o n l i n e a r v a r i a t i o n a l p a r a m e t e r s . G i v e n ~, t h e c o e f f ic i e n tsc~ a r e c o m p l e t e l y d e t e r m i n e d v i a th e o r t h o n o r m a l i t y c o n d i t i o n ( 26 ). A l l e x p o n e n t sn~ a r e t ake n t o be i n t ege r s n~ = l + i, t ho ug h t he cus p cond i t i on f o r Eq . ( 15 ) r equ i r e sn o n i n t e g e r v a l u e s n ' = [K2 - (Z /c)2 ] 1 /2 .A c t u a l l y , b e c a u s e o f r e l a t i o n ( 3 1) t h e r a d i a l p a r t Q o f t h e s m a l l c o m p o n e n t ise x p a n d e d i n t o a d e f i n i te i n f in i t e s e t o f S T F s ; t h i s b e c o m e s e v i d e n t i f t h e f u n c t i o nB - 1 is e x p a n d e d a s a p o w e r - se r i e s o f r .T h e o r b i t a l e x p o n e n t s ~ w h i c h y i e ld a s t a t i o n a r y v a l u e f o r t h e t o t a l e n e r g y (1 6 ) h a v eb e e n d e t e r m i n e d f o r t h e a t o m s H e , B e , N e , M g , A r , K r , X e a n d f o r t h e io n C u + .O p t i m a l v a l u e s ~ w e r e c a l c u l a t e d it e r a t i v e l y : i n th e k t h c y c l e e a c h e x p o n e n t ~ w a sv a r i e d b y s m a l l a m o u n t s T- 8~ a n d c o r r e c t e d a f t e r w a r d s a c c o r d i n g t o

~ k + l , = ~ k ) + w T { k ) (33)w i t h a d a m p i n g f a c t o r w 4 1 a n d

8, [E({ + 8 ,) - E ({ - 8 , ) ] / [E({ + 8 ,) + E ({ - 8 ,) - 2E ({) ] . (34), = - 7


8/17

4 2 F . R o s i c k y a n d F . M a r kC o n v e r g e n c e w a s a s s u m e d i f a l l q u o t ie n t s [T~[ b e c a m e l es s t h a n a t h r e s h o l d v a l u e0 . 05 f o r X e a n d 0 .0 0 01 f o r a ll o t h e r a t o m s .T h e t o t a l r e l a t i v i s t i c e n e r g i e s E R s z a r e r e c o r d e d i n T a b l e 1. E n e r g i e s c a l c u l a t e da c c o r d i n g t o b o t h c h o i c e s o f p a r a m e t e r s e~ ( i. e . e~ = e~ a n d e~ = d 0 a r e a l m o s t e q u a lf o r lo w a t o m i c n u m b e r s b u t d i f fe r i n c r e a s i n g l y f o r h e a v i e r a t o m s i n s u c h a w a y t h a tt h e v a l u e s f o r e~ = e~ b e c o m e l o w e r t h a n t h o s e f o r t h e s e c o n d c h o i c e o f e~. T h eM B S e n e r g i e s a r e l es s n e g a t i v e t h a n t h e e n e r g i e s E~H ~ c a l c u l a t e d b y t h e n u m e r i c a lR H F m e t h o d [8 ]. V a l u e s fo r t h e p e r c e n t a g e e r r o r 1 0 0 ( E ~ F - ERsz)/E~nv r a n g ef r o m 0 . 1 1 ~ t o 0 . 5 8 ~ a n d s h o w s t r o n g s c a t t e r in g f o r d i f fe r e n t a t o m s . H o w e v e r , t h ed i f f e re n c e s A E R s z b e t w e e n r e l a t i v i s t i c ( u s i n g e~ = e0 a n d n o n r e l a t i v i s t i c [1 8 , 1 9]s i ng l e -~ to t a l e n e r g i e s f o l l o w c l o s e l y t h e c o r r e s p o n d i n g e n e r g y d i f f e re n c e s A Er~H r o fn u m e r i c a l R H F a n d N R H F [2 0] c a l c u l a t i o n s ( se e c o l u m n s 1 a n d 2 o f T a b l e 2 ) ; t h ev a l u e s A E g s z a n d A E ~ n r a g r e e w i t h i n l es s t h a n 7 ~ .I n t h e c a l c u l a t i o n o f A E R s z = E R s z - ENRsz b o t h t o t a l e n e r g i e s h a v e b e e no p t i m i z e d w i t h r e s p e c t to t h e o r b i t a l e x p o n e n t s ~. I f t h e e x p o n e n t s o f N R H Fs i n g le - ~ w a v e f u n c t i o n s a r e u s e d t o c a l c u l a t e E ~ s z , r e l a t i v i s t i c c o r r e c t i o n s A E ( ~ zr e s u lt w h i c h a re s h o w n i n c o l u m n 3 o f T a b l e 2 . F o r l o w a t o m i c n u m b e r s b o t h A Ev a l u e s a l m o s t a g r e e , b u t w i t h i n c r e a s i n g Z , A E (R ~z b e c o m e s p r o g r e s s i v e l y s m a l l e rc o m p a r e d t o A E ~ sz . R e l a t i v i s t ic c o r r e c t io n s a p p r o x i m a t e l y c o m p a r a b l e t o A E ~ zh a v e b e e n o b t a i n e d b y H a r t m a n n a n d C l e r n e n ti [ 21 ] w h o c a l c u l a t e d r e l a t i v i s t i cf i r s t- o r d e r p e r t u r b a t i o n e n e r g i e s A E ~ c a u s i n g N R H F s in g le - ~ w a v e f u n c t i o n s a n d a

Tab le 1 . To ta l r e l a t iv i s t i c energ ies [ nega t ive va lues in a .u . ]A tom Z e a ERsz b ER~Fc E r r o r ~

H e 2 e 2.847764 2.8618 0.49dBe 4 14.5594 14.5759 0.11dN e 10 8 127.9494 128.6920 0.58d 199.1638M g 12 d 1 9 9 . 1 6 3 7 1 99 .9 35 2 0 .3 9

e 527.5733A r 18 528.6845 0.21d 527.5727e 2779.9115K r 36 2788.8845 0.32d 2779.89297420.0079X e 54 7447.1605 0.37d 7419.87191646.2005C u 29 1653.2211 0.42d 1646.1948Para me te r e chose n acco rd ing to Eq . ( 20 ) (e = e ) o r E q .(22) (e = d) .b Re lat iv is t ic s ingle-~ calcu lat ion .~ C a l c u l a t e d b y m e a n s o f D e s c l a u x ' s p r o g r a m [7 ].d P e r c e n t ag e e r r o r 1 0 0 ( E ~ F - - ERsz)/Er~Hr.


9/17

App rox imate Relat iv is tic Ha r t ree - Foc k Equa t ionsTable 2. T ota l relativistic energ y cor recti ons (negative values in a.u.)

43

A to m Z AE~sz a AE~aF b AuCl) ~,o AE(1) a E eR S Z H--C1 m a gHe 2 0.000108 0.000133 0.000108Be 4 0.0027 0.0029 0.0027N e 10 0.1373 0.1449 0.1365M g 12 0.3060 0.3205 0.3038A r 18 1.8080 1.8670 1.7781K r 36 35.392 36.830 33.175X e 54 200.2 215.022 174.156C u + 29 14.046 14.493 f 13.457

0.000004 0.000060.0015 0.00070.1152 0.01750.2671 0.03391.6822 0.1434

a Difference between relativistic (e = ~; Eq. (20)) and N R H F [18, 19] single-~ total energy.b Difference be tween numer ica l R H F and N R H F [20] to ta l ene rgy .~ Relativ is t ic energy calculated using the e xponents of N R H F single-~ wavefunctions.a F i rs t o rde r pe r tu rba t ion energy us ing the Brei t opera to r in Paul i approx ima t ion and N R H Fsingle-~ wavefunctions [21].M agne t ic in te rac t ion ca lcu la ted by m eans o f Desc laux ' s p rogram [7] .f NR H F energy ca lcu la ted by means o f Desc laux 's p rogram [7] set ting c = 137-106 .

p a r t o f t h e B r e it o p e r a t o r i n th e P a u l i a p p r o x i m a t i o n ( c o l u m n 4 o f T a b l e 2 ). S i n cet h e v a l u e s A E ~ c l i n c l u d e r e l at i v is t ic e f fe c ts i n th e t w o - e l e c t r o n i n t e r a c t i o n , t h em a g n e t i c c o n t r i b u t i o n E m ~g c a lc u l at e d b y D e s c l a u x 's n u m e r i c a l R H F p r o g r a m [7 ]i s l is t ed i n t h e l a st c o l u m n o f T a b l e 2 . T a k i n g i n t o a c c o u n t t h e s e c o n t r i b u t i o n s t h ev a l u e s o f A E ( ~ z a n d A E ~ c l a r e s im i l ar e x c e p t f o r H e a n d B e .T a b l e 3 c o n t a i n s o r b i t a l e n e r g i e s e f o r K r o b t a i n e d f r o m ( 1 5 ) b y s o l v i n g t h e i m p l i c i teq ua t i on (~o~, ~ o ~ ) = e~(9~ cp~) w i th q~ = - Z / l x I . F o r e i t h e r c h o i c e o f t h e p a r a m -e t e r s t h e v a lu e s e a r e i n g e n e r a l a g r e e m e n t w i t h e x p e r i m e n t a l X - r a y l e v el d a t a a sw e l l a s w i th t h e e i g e n v al u e s o f n u m e r i c a l R H F c a l c u l a t io n s ; t h e l a r g es t d e v i a ti o n sa r e f o u n d f o r t h e e n e r g e t i c a l ly h i g h e s t o r b i t a ls . T h e f i n e s t r u c t u r e s p l it t in gA e = en , 1 _1 /2 - e ,, 1 + 1/2 i s i n f a i r l y g o o d a g r e e m e n t w i t h e x p e r i m e n t a l d a t a .E x c e p t f o r t h e c a s e o f th e i o n C u + (3 d 1~ w h e r e t h e l i m i t a t i o n s o f a n M B S b e c o m ea p p a r e n t , t h e r e su l ts f o r K r a r e r e p r e s e n t a ti v e f o r al l o t h e r a t o m s u n d e r c o n s i d e r a -t i o n . F o r t h e C u + i o n ( T a b l e 4 ) t h e l o w e s t o r b i t a l e n e r g i e s a g r e e r e a s o n a b l y w e l lw i t h e x p e r i m e n t a l b i n d i n g e n e r g ie s f o r m e t a l li c C u [ 23 ], b u t t h e h i g h e s t - l y i n gd - o r b i ta l s a r e d e s ta b i l iz e d t o s u c h a n e x te n t t h a t t h e y b e c o m e a n t i b o n d i n g . T h i sd e f ic i e n cy is d u e t o t h e u s e o f a n M B S o n l y . It a l so a p p e a r s i n t h e c o r r e s p o n d i n gn o n r e l a t iv i s t ic c a l c u l a t i o n a n d c a n b e a v o i d e d b y t a k i n g a 3 d - f u n c t i o n o f d o u b le - ~q u a l i t y .T a b l e 5 s h o w s t h e o p t i m i z e d e x p o n e n t s ~ o f t h e r el a ti v is t ic M B S - S T F s a n d r e la t iv -i s ti c s c r e e n i n g c o n s t a n t s e w h i c h a r e d e fi n e d a s t h e d i ff e r e n c e s b e t w e e n t h e R H F a n dt h e N R H F [ l 8 , 1 9 ] o r b i t a l e x p o n e n t s . T h e v a l u e s ~1~ i n c r e a s e r a p i d l y w i t h i n c r e a s i n ga t o m i c n u m b e r . F r o m C u + o n t h e y ev e n e x ce e d Z , in d i c a t i n g a d e sh i e l d in g o f t h en u c l e us . I n c o n t r a s t, t h e s h ie l d in g ( Z - : ~ l s ) o f t h e n o n r e l a t iv i s t i c I s - f u n c t i o nb e c o m e s l a r g e r f o r h i g h Z ] 18 , 1 9]. T h e r e a s o n f o r t h e s t r o n g c o n t r a c t i o n o f t h er e l a t i v i s t i c l s - o r b i t a l s i s d i s c u s s e d l a te r .


10/17

44

Table 3. Orbital energies e for K r (negat ive values in a.u.)F. Rosicky and F. Mark

Shell j e~sz ~ essz b e~asz ~ fie ~'a eaEr e Ae ~,a E~ p f Ae f,dIs 1/2 528.9432s 1/2 71.2262p 1/2 64.2972p 3/2 62.4773s 1/2 10.7363p 1/2 8.3233p 3/2 8.0044s 1/2 1.0664p 1/2 0.456

528.709 520.068 529.695 526.4771.070 68.717 72.081 70.6064.120 64.875 63.47

62.625 1.820 1.996 1.9262.303 62.879 61.5510.653 10.133 11.225 10.6

8.250 8.620 8.187.877 0.319 0.307 0.32

7.931 8.313 7.861.051 1.007 1.188 1.0120.444 0.5415 0.539

0.4209 0.024 0.0272 0.0254p 3/2 0.432 0.420 0.5143 0.5143d 3/2 2.948 2.869 3.778

2.943 0.042 0.051 3.273d 5/2 2.906 2.828 3.727

Calc ulat ed with e = e (Eq. (20)).b Calc ula ted wi th e = d (Eq. (22)).~ Calculated by the techniq ue of Root haa n-B agu s [28].d ~8 ~ - 8n ,1 -1 l 2 - - 8n , l+ l l 2.e Calculated by mean s of Desclau x's progra m [7].f X-ray levels fro m Ref. [22].

Table 4. Orbital energies 8 for Cu 1~ (negative values in a.u.)Shell j ~Rsza ~N~szb Aeo e~rH a A~ Ee~p ~ls 1/2 332.610 328.9262s 1/2 41.302 40.2622p 1/2 36.163

35.5142p 3/2 35.4603s 1/2 4.769 4.5233p 1/2 3.577 2.971

332.55 330.1441.958 40.4836.559 35.11

0.703 0.72135.838 34.38

5.471 4.573.7430.110 0.096 2.87

3p 3/2 3.067 3.6473d 3/2 -0 .1 77 0.810

-0. 20 8 01017 0.0123d 5/2 -0 .1 94 0.798

a Calc ulat ed wi th e = e (Eq. (20)).b Calculated by the tech nique of Roo tha an- Bag us [28].c A 8 ~ 8 n , 1 . _ l l 2 _ _ ~n,1+1/2.d Ref. [3]; a finite nucleus is assumed .e Values for metallic Cu as quoted in [23].


11/17

Te5Orbae

~o

avcminmumb

sSaey

uoa

avcs

nc

o~b

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l

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22

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3

32

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~

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09

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96

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28

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69

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12/17

46 F . Ros icky and F . M arkA s o n e w o u l d e x p e c t , f o r a g i v e n a t o m t h e r e l a ti v i s ti c s c r e e n i n g c o n s t a n t s cr d e c r e a s ew h e n e v e r t h e q u a n t u m n u m b e r s n a n d j a r e r a i se d . I n c o m p a r i s o n t o N R H Fc a l c u l a t io n s s o m e o f t h e h i g h e s t l y i n g R H F o r b i t a ls a r e d e s h i e ld e d . T h i s is d u e t oa n i n d i r e c t r e l a t i v is t i c e ff e ct [ 2 4 -2 6 ] . A s a c o n s e q u e n c e o f t h e r e l a ti v i s t ic c o n t r a c t i o no f t h e i n n e r s h el ls th e n u c l e a r c h a r g e f o r th e o u t e r m o s t o r b i t a ls i s sc r e e n e d t o al a r g e r a m o u n t t h a n i n t h e c a s e o f n o n r e l a t i v i s t i c a t o m s .T h e r a d i a l e x p o n e n t s n o f t h e M B S f u n c t i o n s ( 3 2 ) w e r e c h o s e n t o b e in t e ge r s. I nt h e f o l l o w i n g , t h e c a p a b i l i t y o f t h e s e f u n c t i o n s t o a c c o u n t f o r t h e re l a t iv i s t ic m o d i f i -c a t i o n s o f t h e s h a p e o f t h e r a d i a l w a v e f u n c t i o n s i s d i s c u ss e d . S i n c e r e l a ti v i s ti c e f fe c tsa r e l a r g es t f o r t h e i n n e r m o s t o r b i t a ls a s j u d g e d f r o m t h e s c r e e n i n g c o n s t a n t s ~ , t h eI s - f u n c t i o n i s t a k e n a s a n e x a m p l e .T a b l e 6 s h o w s t h e r e l a t iv i s t i c m e a n r a d i i a n d t h e r a t i o s o f t h e i r r e l a ti v i s ti c a n d n o n -r e l a ti v i s ti c v a l u e s f o r t h e l s - o r b i t a l s ; t h e s e q u a n t i t i e s a r e g i v e n f o r t h e n u m e r i c a la n d t h e a n a l y t i c a l s i n g le - ~ R H F w a v e f u n c t i o n s a s w e l l. T h e m e a n r a d i i ?R s z d i f fe rf r o m ~ R aF w i t h i n 1 - 4 ~ , t h e l a r g e st d e v i a t i o n o c c u r r i n g f o r H e . H o w e v e r , i f Zi n c r ea s e s th e r a t i o s f R s z d e c r e a s e m o r e r a p i d l y t h a n t h e r a t i o s f ~ a r . T h e r e a s o n f o rt h e e x c e ss iv e c o n t r a c t i o n o f t h e M B S l s - o r b i ta l s c a n b e i n f e r r e d b y f it ti n g t h en u m e r i c a l R H F a n d N H R F f u n c t i o n s b y a s in g le S T F o f t h e f o r m r r e - % w h e r ei s c o n s i d e r e d a s f i t ti n g p a r a m e t e r i n t h e l e a s t - s q u a r e s p r o c e d u r e . F i r s t ~, is f i x ed b y9 ' = 1 f o r b o t h f u n c t i o n s . I n a s e c o n d s t e p ~, f o r t h e r e la t i v is t i c f u n c t i o n i s s e t e q u a lt o t h e v a l u e f o r th e h y d r o g e n i c D i r a c 1 s - o r bi t a l, 7 ' = [1 - ( Z / c ) 2 ] 1 /2 . T h e v a l u e s o f

a n d t h e R M S e r r o r s a r e g iv e n i n T a b l e 7 .I n t h e n o n r e l a t i v i s t i c c a s e t h e e x p e c t e d r e s u l t ~ < Z i s o b t a i n e d . H o w e v e r , i n t h er e la t iv i s ti c c a s e th e e x p o n e n t s ~ e x c e e d Z i f 7 ' = 1. F u r t h e r m o r e t h e R M S e r r o r sg r o w s t r o n g l y f o r a t o m s w i t h h i g h e r a t o m i c n u m b e r s . B u t , i f ~, is c h o s e n t o s a ti s fyt h e c u s p c o n d i t i o n t h e r e l a t i o n ~ < Z i s f u lf il le d a g a i n a n d t h e R M S e r r o r s a r er e d u c e d t o t h o s e f o r t h e n o n r e l a ti v i s ti c f u n c t i o n s .

Table 6 . M ean radii f and ra t io s f o f the re la tiv is tic andnonre la tiv i s tic m ean rad i i o f I s -o rb i ta lsA t o m Z /~RSZ /TRttFb f R S Z c f R H F d

H e 2 0.8888 0.9272 0.9999 0.9999Be 4 0.4069 0.4149 0.9936 0.9998N e 10 0.1552 0.1574 0.9913 0.9983M g 12 0.1287 0.1303 0.9899 0.9976A r 18 0.08487 0.08562 0.9858 0.9944K r i 36 0 .04094 0 .04147 0 .9610 0 . 97 71X e 54 0.02592 0.02665 0.9146 0.9470Calculated w ith e = e (Eq. (20)).b Ca lcu la ted b y means o f Desc laux ' s p rogram [7] .~ No nrelat iv is t ic v alues calculated by the techniqu e ofRo o th a a n - Ba g u s [ 28 ].a No nrela t iv is t ic values from Ref. [20].


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Approximate Relativistic Hartree-Fock EquationsTable 7. Orbital exponents ~ and RMS errors F( x 104) obtained by fitting thenonrelativistic~ and the relativistic b ls-orbitals to r~e- ~r

47

(ls)~, (ls)~7 = 1 y = 1 y = (1 - (Z/c)2 ) 1/2Atom Z [ F ~ F ~ F

He 2 1.6883 157 1.6885 123 1.6883 123Be 4 3.682 106 3.684 107 3.682 106Ne 10 9.619 97 9.643 105 9.617 96Mg 12 11.599 97 11.641 110 11.597 96Ar 18 17.554 93 17.702 133 17 .549 93Kr 36 35.499 77 36.761 342 35.473 75Xe 54 53 .470 68 58.077 876 53.403 65Hg 80 79.488 59 97.345 2 5 9 0 79.258 57Wave functions calculated by means of Desclaux's program [7] setting137.106

b Wave functions calculated by means of Desclaux's program [7].C

4. ConclusionThe presented MBS results are affected by two approximations:a) the fixed relations hip (9) between large and small comp onen t,b) the use of integer principal quantum numbers n in the basis functions.The fairly good agr eemen t for total relativistic energy-corrections and orbitalenergies with the results of numerical RH F calculations indicates that approxim a-tion a) does not int roduce large errors, at least not for atoms u p to a tomic nu mb er54 (Xe). Different choices of the parameters e in (9) yield only slightly differentnumeri cal results. On the other hand, conside ration of orbital exponents an d mea nradii shows that the restriction of n to integer values is acceptable for low atomicnum ber s only. For Z > 36 only an MBS of STFs with nonin teger radial exponen tsn is flexible enoug h to accoun t for the relativistic contra ction an d d efor mati on of theradial wavefunctions.

A c k n o w l e d g e m e n t . The authors wish to express their grat itude to Professors O. E. Polansky andP. Schuster for their continuous interest. We thank Miss E. Wechsel for her skillful help indeveloping and testing the computer programs. We are grateful to Dr. P. Hafner for providingus with a copy of Desclaux's program and to H. Gruen for reading the English manuscript.

AppendixAll the one- and two-electron integrals referred to can be reduced to knownelementary integrals and can be calculated by recurrence relations.


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48 F. Rosicky and F. M arkA . 1 . T h e O n e - E l e c t r o n I n t e g r a l sI f t h e f u n c t i o n s P a r e e x p a n d e d a c c o r d i n g t o ( 3 2), a l l th e i n t e g r a l s a p p e a r i n g i nEqs . (25 ) an d (26) a re o f the fo rm

fo ~ e -n X x m m >>. 0S ( m , n ; 7 , a ) = d X ( a + x ) ~ n >1 0 (A1)since [r e - n ~ r m B - ~ = 1 + ~ S ( m + n , n ; "q, a )w i t h

B = 1 + F c 2 e + a nd a = Z / ( 2 c 2 + e ) .T h e i n t e g r a ls S ( m , n ) ( t h e a r g u m e n t s ~ a n d a a r e o m i t t e d f o r th e s a k e o f s i m p l i c it y )c a n b e c a l c u l a t e d r e c u r s i v e l y a c c o r d i n g t o

S ( m , n + 1) = 1 ~3m0 }n ~ - - ~ - - ~ S ( m , n) + m S ( m - 1, n ) (A2)m !~7~+ ~ a S ( m , 1) (A3)( m + 1 , 1 ) - - -

s t a r ti n g f r o mm !S ( m , O ) ~ m + Z (A4)

S(0 , 1) = e n ~ E x ( ~ a ) . ( A 5 )F o r m = 0 t h e l a s t t e r m i n t h e r e c u r r e n c e r e l a t i o n ( A 2 ) h a s t o b e o m i t t e d . T h eexponen t i a l i n t eg ra l Ex i s de f ined by (A20) ( see Sec t . A .2 . ) .

A . 2 T h e T w o - E l e c tr o n I n t e g ra l sA l l in t e g r a l s a p p e a r i n g i n E q s . (2 9) a n d ( 3 0) h a v e t h e b a s i c f o r m

f r oV ( M , N ; ~ , 6 ; i , r e ; j , n ) = d r1 d r 2 U v ( 1 , 2)x r ~ r ~ e - n r i e - ~ r 2 ( 2 c 2 ) - m - ~ B ~ m ( r l ) B [ ~ ( r 2 ) ( A 6 )

w h e r e1 ( e, + Z / r 3 ;B , ( r ~ ) = 1 + ~ c ~

B j ( r 2 ) i s g i v e n a n a l o g o u s l y . T h e a b o v e i n t e g r a l s I v a r e r e w r i t t e nI v = (2c 2 + e0 -m (2c 2 + e j ) - "

x { J ( N + n + v , M + m - v - 1 ; ~ , ~ ; j , n ; i , m )+ J ( M + m + v , N + n - v - 1 ; ~ 7 , ~ ; i , m ; j , n ) } ( A 7 )


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Approxim ate Relativistic Hartree-Fock Equations 49w i t h

f o e - ,V v , e- ~ " u ~ ( 1 8 )( n , m ; ~ , ~ ; a , p ; b, q ) = dV (a + v)------ d u (--~ + u)----~qw h e r e a = Z / ( 2 c 2 + e ,) ancl b = Z / ( 2 e 2 + e j ). I n o r d e r t o s i m p l i fy t h e n o t a t i o n ,t h e a b b r e v i a t i o n J ( n , m ; ~ , ~; a , p ; b , q ) - J ( n , p ; m , q ) wi l l be used .I n t e g r a t i n g b y p a r t s o n e g e t s th e f o l l o w i n g r e c u r r e n c e r e l a t i o n s f r o m ( A 8 )

1J ( n , p ; m , q + 1) = ~ ( - ~ J ( n , p ; m , q ) + m Y ( n , p ; m - 1, q)+ U ( n + m , p , q ) } q >/ 1, m i> 0 (A 9)

J ( n , p ; m , O ) = ~ { U ( n + m , p , O ) + m J ( n , p ; m - 1, 0)} (A10 )

J ( n , p + 1 ; m , q ) = ~ - ~ ? J ( n , p ; m , q ) + n J ( n - 1 , p ; m , q )

v ( . + m , p, q) + s( m , q ) } p 1 , . 0(A11)

J ( n , O ; m , q ) = l { - u ( n + m , O , q ) + n J (n - 1 , 0 ; m , q ) + 3 ~ o S ( m , q ) } .(A12)

H e r e , i n a d d i t i o n t o S ( m , n ) = S ( m , n ; ( 7 + ~ ), b ) d e f i n e d b y ( A 1 ) , a f u r t h e ra u x i l i a r y i n t e g r a l

U ( n , p , q ) = e -( e+ n ) ~ (a + v ) - P ( b + v ) - q v ~ d v ( 1 1 3 )h a s b e e n i n t r o d u c e d . F o r n >/ 0 a n d p , q >/ 1 , U c a n b e r e c u r r e d b y t h e r e l a t i o n

1g ( n , p , q ) = a - b { g ( n , p - 1 , q ) - U ( n , p , q - 1)}. (A14)I f o n e o r b o t h o f t h e i n d ic e s p o r q a r e z e r o , U r e d u c e s t o S .I n o r d e r t o s t a r t t h e r e c u r r e n c e c a l c u l a t i o n o f J ( n , p ; m , q ) o n e n e e d s

1 1 ( A 1 5 )( o , o ; o , o ) - ~ +1J (0 , 1 ; 0 , 0) = ~ V(0, 1 , 0) (A16)

J (0 , 0 ; 0 , 1) = ~ {er - S(0, 1)} (A17)a n d f u r t h e r m o r e J ( 0 , 1 ; 0 , 1 ). T h i s i n t e g r a l

f 0 0 e - r / v ~ o o e - ~uJ ( 0 , 1 ; 0 , 1 ) = d V ( a + v ) j , | d U ( b + u ) (A18)


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50 F . Ros icky and F . M arkc a n b e r e d u c e d t o a s u m o f s in g le i n te g r a ls o f t h e f o r m

ed t t I n & ( t )b y c h a n g i n g t h e v a ri a b l e s a c c o r d i n g t o

t = ~ ( a + v ) + ~ : ( b + u ) a n d x = ( f u + f b ) / t .E x p a n d i n g t h e f u n c ti o n s I n & ( t ) i n t o a p p r o p r i a t e c o n v e r g e n t s e ri es a n d i n t e g r a t in g ,o n e o b t a i n s t h e f o r m u l a

S ( 0 , 1 ; 0 , 1 ) = e ~ ( E l ( K ) I n (K/a~) + E[2~(,

, 7 E , + l ( ~ ) ( ~ + 7 ) 'N \x [~7'H(i + 1 , i ; - , - - e S ) - - ~ ' H ( i + 1, i; ~, ~78)]}}

( A 1 9 )w h e r e K = ~/a + ~ b a n d 8 = b - a . T h e i t h e x p o n e n t i a l i n t e g r a l E~ i s d e f i n e d b y( s e e e . g . [ 2 7 ] )

Ei (x ) = e - t d t x > 0 ( A 2 0 )x

a n d s a t i s f i e s t h e r e c u r r e n c e r e l a t i o n7 7 - E, ( x ) . ( A 2 1 )

T h e a u x i l i a r y i n t e g r a lf ~ : e - tH (i , j; ~c, f ) = - i T ( t + f f ) J d t ( A 2 2 )

c a n b e r e c u r r e d a c c o r d i n g t oH ( i , j ) = H ( i - 1 , j - 1 ) + t z H ( i , j - 1 ) ( A 2 3 )

( t he a r g u m e n t s K a n d / , a r e o m i t t e d ) t o g e t h e r w i t h (A 2 1 ) . E ~2> i s g iv en by [27]E ~ 2 ' ( x ) = d t ( t ) = 8 9 x + C ) 2 + ~ + ( - 1 ) ' i2 -[! ( A 2 4 )

X i = l

w h e r e C is C a t a l a n ' s c o n s t a n t .

References1. Grant , I . P . : Advan. Phys. 19, 747 (1970)2. Coulthard, M. A.: Proc. Phys. Soc. 91, 44 (1967)3. Smith , F . C. , Johnson, W. R.: Phys. Rev. 160, 136 (1967)4. M ann , J . B. , W aber , J . T . : J . Ch em. Ph ys. 53, 2397 (1970)5 . L ibe rm ann , D. A . , Cromer , D. T . , Waber , J . T . : Com p. Phys . Co mm un. 2 , 107 (1971)6. Desclaux, J . P . , Mayers , D. F . , O'Brien, F . : J . Phys. B. 4 , 631 (1971)


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Approximate Relativistic Hart ree-Fock Equations 517. Desclaux, J. P.: Comp. Phys. Commun. 9, 31 (1975)8. Mal3~, J., Hussonnois, M.: Theoret. Chim. Acta (Berl.) 28, 363 (1973)9. Mal~, J., Hussonnois, M.: Theoret. Chim. Acta (Berl.) 31,137 (1973)

10. Desclaux, J. P.: At. Data Nucl. Data Tables 12, 311 (1973)11. Kim, Y.-K. : Phys. Rev. 154, 17 (1967)12. Leclercq, J. M.: Phys. Rev. A1, 1358 (1970)13. Kagawa, T.: Phys. Rev. A12, 2245 (1975)14. Bethe, H. A., Salpeter, E. E. : Quantum mechanics of one- and two-electron atoms. Berlin:

Springer 195715. Bersuker, I. B., Budnikov, S. S., Leizerov, B. A. : Intern. J. Quantum Chem. 6, 849 (1972)16. Rosicky, F., Weinberger, P., Mark, F .: J. Phys. B. 9, 2971 (1976)17. Rose, M. E.: Relativistic electron theory. New York: Wiley 196118. Clementi, E., Raimondi, D. L.: J. Chem. Phys. 38, 2686 (1963)19. Clementi, E., Raimondi, D. L., Reinhardt, W. P.: J. Chem. Phys. 47, 1300 (1967)20. Froese Fischer, C. : The Har tree -Fock method for atoms. New York: Wiley 197721. Hartmann, H., Clementi, E.: Phys. Rev. 133A, 1295 (1964)22. Bearden, J. A., Burr, A. F.: Rev. Mod. Phys. 39, 125 (1967)23. Rosen, A., Lindgren, I.: Phys. Rev. 176, 114 (1968)24. Desclaux, J. P. :lntern. Quantum Chem. Symp. 6, 25 (1972)25. Desclaux, J. P., Kim, Y.-K.: J. Phys. B. 8, 1177 (1975)26. Rose, S. J., Grant , I. P., Pyper, N. C.: J. Phys. B. 11, 1171 (1978)27. Kourganoff, V.: Basic methods in transfer problems. Oxford: Clarendon 195228. Roothaan, C. C. J., Bagus, P. S.: Methods Comp. Phys. 2, 47 (1963)Rece ived Ap ri l 4 , 1979/Au gust 23, 1979

relativistic hf minimal basis set

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