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SHIELDING OF ELECTRIC FIELD GRADIENTS FOR VARIOUS ELEMENTS AND ELECTRONIC
CONFIGURATIONS
S. Lauer, V. Marathe, A. Trautwein
To cite this version:
S. Lauer, V. Marathe, A. Trautwein. SHIELDING OF ELECTRIC FIELD GRADIENTS FOR VARI-
OUS ELEMENTS AND ELECTRONIC CONFIGURATIONS. Journal de Physique Colloques, 1980,
41 (C1), pp.C1-105-C1-106. �10.1051/jphyscol:1980117�. �jpa-00219657�
JOURNAL DE PHYSIQUE Colloque C1, supplkment au n O 1 , Tome 41, janvier 1980, page C1-105
SHIELDING OF ELECTRIC FIELD GRADIENTS FOR VARIOUS ELUvlENTS AND ELECTRONIC CONFIGURATIONS
S . Lauer, V.R. Marathe and A . Trautwein
Universi tdt des Saar Zandes, 66 Saarbrtlcken 11, u-GeFm~%$4.
We have c a l c u l a t e d S t e r n h e i m e r
'
s h i e l d i n g f u n c t i o n s $ ( r ) and c o r e c o r r e c t i o n s R f o r v a r i o u s atoms and i o n s w i t h 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 c l u d i n g and e x c l u d i n g v a l e n c e e l e c t r o n s . (With r e s p e c t t o MO o r band s t r u c t u r e c a l c u l a t i o n s which i n c l u d e o n l y v a l e n c e o r b i t a l s w i t h i n t h e i r b a s i s s e t i t i s n e c e s s a r y t o c o r r e c t t h e c a l c u l a t e d e l e c t r i c f i e l d g r a d i e n t (EFG) f o r c o r e c o n t r i b u t i o n s w i t h a s p e c i f i cX ( r ) which i s d e r i v e d by e x c l u d i n g va- l e n c e e l e c t r o n s . )
r(r)
d e s c r i b e s t h e con- t r i b u t i o n s t o t h e EFG from t h e e l e c t r o n c o r e due t o p e r t u r b a t i o n s of t h e c o r e by o u t e r c h a r g e s ,where r a u ; ~ ) ( r ) and r . u i : ; ( r ) a r e t h e r a - d i a l p a r t s o f t h e u n p e r t u r b e d and p e r t u r - bed c o r e o r b i t a l s , r e s p e c t i v e l y , and c l l , g i v e s t h e r e s u l t o f t h e i n t e g r a t i o n o v e r t h e a n g u l a r p a r t s o f t h e i n t e g r a l s . To perform t h e c a l c u l a t i o n we a p p l i e d f o u r d i f f e r e n t methods. Method ( a ) i s t h e con- v e n t i o n a l p r o c e d u r e d e v e l o p e d by S t e r n - h e i m e r '
".
S i n c e method ( a ) d e s c r i b e s o n l y t h e p e r t u r b a t i o n o f t h e c o r e o r b i t a l s b u t n e g l e c t s t h e i n t e r a c t i o n between them, S t e r n h e i m e r i n t r o d u c e d 3 a method which t a - k e s c a r e of Coulomb i n t e r a c t i o n between them (method ( b ) ).
~ e c e n t l ~ ~ , we i n c o r p o - r a t e d a d d i t i o n a l l y exchange i n t e r a c t i o n between t h e p e r t u r b e d c o r e o r b i t a l s i n t h e a p p r o x i m a t i o n of t h e XK - H a r t r e e - F o c k t h e - o r y (method ( c ) ) . S i n c e i t i s d i f f i c u l t t o a v o i d d i v e r g e n c e f o r n e g a t i v e i o n s w i t h i n method ( c ) we a p p l i e d t h e c o u p l e d H a r t r e e - Fock method (CHF method) a s d e s c r i b e d by~ a l ~ a r n o ' . We i n t r o d u c e d a p p r o x i m a t i o n s i n
c a l c u l a t i n g exchange i n t e g r a l s , a p r o c e d u - r e which i s u n c r i t i c a l f o r e l e m e n t s h e a - v i e r t h a n Na, and which i s f a s t enough t o h a n d l e a l s o l a r g e s y s t e m s l i k e I and P r . T h i s a p p r o x i m a t e d CHF method i s r e f e r e d t o a s method ( d )
.
Methods ( b )-
( d ) a r e s e l f-
c o n s i s t e n t p r o c e d u r e s w i t h t h e z e r o t h o r - d e r ~ ( r ) b e i n g i d e n t i c a l t o g ( r ) d e r i v e d from method ( a ) . To p r o v i d e s i m p l i c i t y i n t h e c o m p u t a t i o n we r e s t r i c t e d method(d) t o c l o s e d s h e l l s y s t e m s . S i n c e we were m a i n l y i n t e r e s t e d i n z ( r ) c u r v e s which c a n b e u s e d f o r MO o r band s t r u c t u r e c a l c u l a t i o n s t h i s r e s t r i c t i o n i s n o t s e r i o u s . A g e n e r a l t e n d e n c y w i t h r e s p e c t t o & ( = limr,cpj'
( r ) )
,
a p p l y i n g t h e f o u r d i f f e r e n t methods i s t h e f o l l o w i n g :&
c a l c u l a t e d w i t h me- t h o d ( d ) i s t h e most n e g a t i v e , w h i l e i n - t r o d u c i n g Coulomb i n t e r a c t i o n between t h e c o r e o r b i t a l s (method ( b ) ) l e a d s t o more p o s i t i v e -d, - v a l u e s . Method ( c ) y i e l d sifa
- v a l u e s between t h o s e c a l c u l a t e d w i t h methods ( a ) and ( b ) , b e c a u s e exchange and Coulomb i n t e r a c t i o n c o n t r i b u t e t o&
w i t h o p p o s i t e s i g n . For most o f t h e s y s t e m s un- d e r s t u d y we found l a r g e d e v i a t i o n s from t h e c o n v e n t i o n a l p r o c e d u r e ( a ) when a p p l y - i n g t h e s e l f c o n s i s t e n t methods ( b ) a n d ( c ).
The l a r g e s t c h a n g e s we g o t f o r atoms w i t h
one v a l e n c e s e l e c t r o n ( A ~ ~ , C U ~ , N ~ ~ ) , whe- r e t h i s v a l e n c e s e l e c t r o n t e n d s t o s h i e l d t h e o u t e r c h a r g e from t h e e l e c t r o n c o r e . T h i s c a n be s e e n from F i g . 1 , where ~ ( t ) f o r A ~ O i s shown u s i n g methods ( a ) - ( c )
.
As m e n t i o n e d above we u s e d method ( d ) o n l y f o r c l o s e d s h e l l s y s t e m s . I n o r d e r t o check t h e a c c u r a c y o f t h e a p p r o x i m a t i o n s i n c l u d e d i n o u r CHF method, we compare he- r e some of o u r r e s u l t s w i t h t h o s e r e p o r t e d i n t h e l i t e r a t u r e , which a l s o i n c l u d e con- s i s t e n c y and c o r r e l a t i o n e f f e c t s . T h i s c o m p a r i s o n i s shown i n T a b l e I . The s l i g h t
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980117
CI-'106 JOURNAL DE PHYSIQUE
d e v i a t i o n o f o u r v a l u e s compared t o l i t e - r a t u r e v a l u e s i s due t o t h e u s e o f d i f f e - r e n t w a v e f u n c t i o n s (we h a v e u s e d H a r t r e e - Fock f u n c t i o n s from C l e m e n t i 6 ) and o t h e r c a l c u l a t i o n a l methods ( r e f . 7 u s e s a v a r i - a t i o n a l method, r e f . 8 a many body p e r t u r - b a t i o n method, a n d r e f . 9 a method which h a n d l e s c o n s i s t e n c y a s a p e r t u r b a t i o n ) . For t h e c a l c u l a t i o n o f t h e c o r e c o r r e c t i o n R we f o l l o w e d S t e r n h e i m e r f s c a l c u l a t i o n a l
Some o f t h e r e s u l t s a r e shown i n T a b l e 11. The d i s c r e p a n c y o f o u r R v a l u e s f o r
c1°
and B ~ Ocompared t o R v a - l u e s from r e f . 10 i s due t o t e r m s which h a v e b e e n o m i t t e d i n r e f . 10. Because many MBssbauer s p e c t r o s c o p i s t s s t i l l u s e t h e R v a l u e o f 0 . 3 2 f o r i r o n we want t o emphasi- z e , t h a t t h e c o r r e c t v a l u e i s 0 . 1 2 . F o r MO c a l c u l a t i o n s w i t h l i m i t e d b a s i s s e t (Fe3d, 4 s A O f s ) , however, t h e a p p r o p r i a t e v a l u e i s 0 . 0 8 .For r e a d e r s who a r e i n t e r e s t e d i n c u r v e s f o r Li,Na,K,Rb,F,Cl,Br,I,Fe,Cu,Zn,Ag,Sn, Te and P r a d e t a i l e d s u p p l e m e n t i s a v a i - l a b l e from t h e a u t h o r s .
R e f e r e n c e s :
1 . S t e r n h e i m e r , R., P h y s . Rev.
84
(1951)2. S t e r n h e i m e r , R.M., P h y s . Rev.
146
(1966) 140
3 . S t e r n h e i m e r , R.M., Phys. Rev.
164
(1967) 10
4 . L a u e r , S . , M a r a t h e , V . R . , T r a u t w e i n , A . , Phys. Rev. A ( i n p r e s s )
5 . D a l g a r n o , A . , Advances i n P h y s i c s
11
(1962) 281
6 . C l e m e n t i , E . , R o e t t i , C . , Atomic and N u c l e a r D a t a T a b l e s , New York and London ( 1 974)
7. L a h i r i , J . , M u k h e r j i , A . , Phys. Rev.
155 (1967) 24
-
8 . V a j e d - S a m i i , M . , Ray, S.N., Das, T . P . , P h y s . Rev.
-
BIZ (1975) 45919 . B e r i , A . C . , Ray, S.N., Das, T . P . , S t e r n h e i m e r , R.M., Phys. Rev.
A12
(1975) 1168
1 0 . S t e r n h e i m e r , R.M., P h y s . Rev.
A6
(1972) 1702
1 1 . S e n , K . D . , Narasimhan, P . T . , P h y s .
Rev.
B16
(1977) 107S u p p o r t e d by D e u t s c h e Forschungsgemein- s c h a f t
.
F i g . I : S t e r n h e i m e r s h i e l d i n g f u n c t i o n F ( T ) f o r A ~ O : method ( a ) - s o l i d l i n e ; me- t h o d ( b ) - d a s h e d l i n e ; method ( c ) - d o t t e d l i n e
TABLE I
(method ( d ) ) compared w i t h l i t e r a t u - r e v a l u e s . V a l u e s i n p a r a n t h e s e s c o r r e s - pond t o z e r o t h o r d e r (method ( a ) ) . e l e m e n t ( a ) l i t e r a t u r e N a +
-
4.10{ -
4.64)-
3.81{-
4.41) ( b )K + -14.08 {-17.74) -18.27 ( c )
F- -16.48 {-22.32) -19.29 [-21.57) ( d )
C 1 - - 4 6 . 8 2 (-55.35) -56.58 1-55.08) ( d ) ( a ) t h i s work: method ( d ) ;
( b ) r e f . 8 ; ( c ) r e f . 7 ; ( d ) r e f . 9 . TABLE I I
Core c o r r e c t i o n R f o r some e l e m e n t s . R v a - l u e s e x c l u d i n g t h e v a l e n c e o r b i t a l s a r e g i v e n i n p a r a n t h e s e s . (These v a l u e s c a n be u s e d f o r MO c a l c u l a t i o n s , w i t h a b a s i s s e t of F ( ~ s , ~ P ) , C 1 ( 3 ~ , 3 p ) s B r ( 4 ~ 5 4 ~ ) , 1 ( 5 ~ , 5 ~ ) a n d Fe ( 3 d , 4 s ) .
e l e m e n t 1 0 2 R ( a ) 1 0 2 R l i t e r a t u r e
BO 5 . 0 3
-
4 . 8 0 ( b )F O 8 . 0 6 {2.311 10.07 ( b )
c1°
- 7 . 7 8 1-12.31) 4 . 3 0 ( b )B ~ O -11.42 (-16.34) 4 . 3 0 ( b ) I O - 1 4 . 2 0 1-18.54)
-
~ e ' + 12.62 i 7 . 5 4 ) 1 2 . 1 0 ( c ) 1 1 . 9 0 ( d ) ( a ) t h i s work; (b) r e f . 1 0 ; ( c ) r e f . 2;
( d ) r e f . 1 1 .