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CALCULATION OF INNER CORE ELECTRON BINDING ENERGIES IN METALS
A. Zdetsis, C. Nicolaides
To cite this version:
A. Zdetsis, C. Nicolaides. CALCULATION OF INNER CORE ELECTRON BINDING EN- ERGIES IN METALS. Journal de Physique Colloques, 1987, 48 (C9), pp.C9-1071-C9-1074.
�10.1051/jphyscol:19879195�. �jpa-00227312�
CALCULATION OF INNER CORE ELECTRON BINDING ENERGIES IN METALS
A.D. ZDETSIS and C.A. N I C O L A I D E S *
Department of Physics, University of Crete and Research Center of Crete, PO Box 1527, 711 10 Heraklio, Crete, Greece
'mational Hellenic Research Foundation, GR-501/1 Athens, Greece
A b s t r a c t
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Employing s t a t e s p e c i f i c t o t a l energy d i f f e r e n c e s between t h e c o r e h o l e and t h e ground Hartree-Fock c l u s t e r s t a t e s . we o b t a i n t h e ASCF b i n d i n a energy o f i n n e r c o r e e l e c t r o n s i n m e t a l s . T h i s ASCF value, enhanced by a t o m i c c o r r e l a t i o n and r e l a t i v i t y e f f e c t s can p r o v i d e i n n e r c o r e e l e c t r o n b i n d i n g e n e r g i e s i n v e r y good agreement w i t h experiment, R e s u l t s have been o b t a i n e d t h u s f a r f o r L i , Be, Mg and Na metals.A l a r g e amount o f e x p e r i m e n t a l work has been d i r e c t e d t o t h e s t u d y o f c o r e l e v e l b i n d i n g e n e r g i e s d u r i n g t h e l a s t two decades
rl],
because i t was e a r l y r e a l i z e d t h a t these b i n d i n g e n e r g i e s r e f l e c t t h e chemical environment o f t h e atom. A p r o p e r t h e o r e t i c a l u n d e r s t a n d i n g o f these b i n d i n g e n e r g i e s and t h e i r s h i f t s would e x p l o i t a l l t h e w e a l t h o f s p e c t r o s c o p i c i n f o r m a t i o n a v a i l a b l e . O f s p e c i a l i n t e r e s t and importance a r e t h e c o r e e l e c t r o n b i n d i n g e n e r g i e s i n m e t a l s and t h e i r r e l a t i v e s h i f t f r o m t h e a t o m i c s t a t e , T h e i r a c c u r a t e c a l c u l a t i o n , , measurement and subsequent i n t e r - p r e t a t i o n embodies a b r o a d spectrum o f u s e f u l i n f o r m a t i o n about t h e o r i e s o f e l e c t r o - n i c s t r u c t u r e , bonding and dynamics121.
The t h e o r e t i c a l o n e - e l e c t r o n b i n d i n g energy i s d e f i n e d as:Eb = Et ( N - I )
-
Et ( N ) ( 1 )where t t ( N - I ) and E t ( N ) a r e t h e t o t a l e n e r g i e s o f t h e f i n a l and i n i t i a l s t a t e s o f t h e N e l e c t r o n system r e s p e c t i v e l y . F o r metals, i f t h e system i s l o o k e d a t as a whole, t h e r e s u l t i n g c o m p l e x i t y o f t h e many-body problem i s enormous. T h e r e f o r e , s i n c e t h e e a r l y 19701s, a number o f approximate models have been enployed, d i s c a r d - i n g e l e c t r o n c o r r e l a t i o n and a i a i n g a t some reasonable q u a s i - q u a n t i t a t i v e understand- i n g . Several improved models have appeared i n t h e l a t e 1 9 7 0 ' s - e a r l y 1980's which have been b r i e f l y reviewed o r r e f e r e n c e d i n Refs. [I-31. Our p r e s e n t approach, which has been s u c c e s s f u l l y t e s t e d f o r Be b e f o r e [2], i s based on s t a t e s p e c i f i c
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19879195
C9-1072 JOURNAL DE PHYSIQUE
s e l f - c o n s i s t e n t f i e l d (SCF) c a l c u l a t i o n s a t t h e c l u s t e r Hartree-Fock l e v e l [2], This work i s based on two assumptions:
( 1 ) The c o n t r i b u t i o n s t o t h e e b in equation ( 1 ) come from two sources: F i r s t , from t h e o r b i t a l approximation a t t h e 4SCF Hartree-Fock l e v e l . This implies t h a t , adopting t h e independent p a r t i c l e model, a l l i n t e r a c t i o n s should be computed s e l f - c o n s i s t e n t l y , f o r i n i t i a l and f i n a l m e t a l l i c s t a t e s e p a r a t e l y . In t h i s way, relax- a t i o n e f f e c t s coming from within t h e atom and those coming from t h e s o l i d medium a r e accounted f o r e x p l i c i t l y . Second, from t h e e f f e c t s of e l e c t r o n c o r r e l a t i o n within t h e atom. These e f f e c t s a r e l o c a l i z e d and correspond t o v i r t u a l e x c i t a t i o n s of high energy a s compared t o t h a t of t h e m e t a l l i c e l e c t r o n s . Therefore, t h e i r influence on t h e binding energy of an inner e l e c t r o n i n t h e s o l i d and i n t h e f r e e atom must be almost t h e same.
( 2 ) The s u i t a b i l i t y of a theory depends on t h e physical q u a n t i t y under examina- t i o n . Thus, i n t h e cage of p r o p e r t i e s involving i n n e r e l e c t r o n s , i t i s proposed t h a t even a metal can be approximated well by a s u i t a b l y l a r g e Hartree-Fock c l u s t e r . The c l u s t e r s i z e u s u a l l y i s determined by t h e n e a r e s t and next near neighbors of a r e f e - rence atom i n t h e c r y s t a l geometry of t h e metal.
I t has been i l l u s t r a t e d [4,5] t h a t t h e small c l u s t e r approximation i s adequate enough f o r d e ~ c r i b i n g ~ s e v e r a l l i g h t metals with r e s p e c t t o e l e c t r o n i c , s t r u c t u r a l and phonon q u a n t i t i e s . The I s binding energy of Li besides Be was a l s o c a l c u l a t e d with excel l e n t agreement with experiment 141
.
Here, we p r e s e n t I s binding energy r e s u l t s f o r Be, L i , Na and Mg metals.The ASCF c l u s t e r binding energy i s obtained from ( 1 ) by replacing Et(N-I) and Et(N) with t h e corresponding t o t a l c l u s t e r e n e r g i e s f o r t h e depopulated and t h e normal (ground) s t a t e s r e s p e c t i v e l y . This i s done by employing a modified version of t h e program UHFABK of A.B. Kunz, which i s capable of s e l e c t i v e l y depopulating occupied o r b i t a l s . The c l u s t e r ASCF value accounts f o r symmetry, exchange and complete r e l a x a t i o n of t h e o r b i t a l s . Contributions of atomiclike e l e c t r o n c o r r e l a t i o n and r e l a t i v i t y e f f e c t s , A E a , ( e s p e c i a l l y f o r heavier atoms) can be incorporated separate1 y by simp1 e superposition t o t h e c l u s t e r ASCF r e s u l t s . The f i n a l c o r r e l a t e d binding energy E:, i s thus given by
The b a s i s f u n c t i o n s e t s used i n t h i s c a l c u l a t i o n , t o g e t h e r with detaiTs about t h e c l u s t e r geometry a r e i n p a r t included in r e f . [2,4,5]. Further d e t a i l s w i l l be published elsewhere,
In Table I , we have summarized t h e main r e s u l t s of t h i s work, concerning t h e I s binding e n e r g i e s t o g e t h e r with c a l c u l a t e d work function and Fermi energy values.
The work f u n c t i o n s , (and i n t u r n Fermi e n e r g i e s ) l i s t e d here a r e obtained by ASCF
Comparison o f t h e c a l c u l a t e d and measured I s b i n d i n g e n e r g i e s , work f u n c t i o n and Fermi e n e r g i e s f o r some s i m p l e metals.
I
Work F u n c t i o n Is b i n d i n g (eV)I
Fermi Energy ( e v )I
enerqy ( e v )c a l c u l a t i o n 4.10 3.30
Lit:fcqcc)experiment 1 3.99 1
2.38 59.50 59.48
c a l c u l a t i o n Li(:R) experiment
( )
c a l c u l a t i o n Lijfjc)experiment
c a l c u l a t i o n Li/h;")experiment
c a l c u l a t i o n Be e x p e r i m e n t
c a l c u l a t i o n Be[:; l n L x p e r i n e n t
*
R e s u l t s f r o m A.D. Z d e t s i s r e f . [4].+ The b i n d i n g energy does n o t i n c l u d e t h e a t o m i c l i k e c o r r e c t i o n AEa.
5.0
-
5.1 3.81 3.6
- 15.0 11.4-14.1
c a l c u l a t i o n
Na( experiment
c a l c u l a t i o n
Mg experiment
$ From A.D. Z d e t s i s , t h i s volume.
14,5
--
s t a t e s p e c i f i c d i f f e r e n c e s . The Koopman's v a l u e s a r e w e l l o v e r e s t i m a t e d due t o poor screening. The " e x p e r i m e n t a l " Fermi e n e r g i e s a r e t a k e n from band s t r u c t u r e c a l c u l a - t i o n s i n t h e l i t e r a t u r e . The r e s u l t s f o r L i , p r e s e n t e d i n ref.[4] a r e g i v e n f o r s e v e r a l competing c r y s t a l s t r u c t u r e s . The n o r n a l f o r m o f L i i s t h e c u b i c bcc s t r u c t u r e which a t q u i t e l o w temperatures transforms i n t o a compiex 9R phase [4]
.
The r e s u l t s f o r Be t h i n f i l m have been p r e s e n t e d i n t h i s volume a t a d i f f e r e n t c o n t e x t . AS we move i n t o h e a v i e r atoms, w i t h d - o r b i t a l s (Na, Mg) t h e l a r g e number o f b a s i s f u n c t i o n s makes necessary l a r g e amount o f c o n t r a c t i o n s i n t h e b a s i s s e t , p r o d u c i n g t h u s a p o o r e r agreement w i t h experiment. To improve t h i s agreement, w i t h a moderate s i z e computer, w i l l t a k e a r a t h e r l a r g e amount o f computer t i m e and i s n o t w a r r a n t e d a t t h i s time.
4.1 - 3.8
- 4.0
- 3.5 3.9
3.9 3.23 18.0 14.1
The means o f i m p r o v i n g t h e s e r e s u l t s is,nontheless, s t r a i g h t f o r w a r d and n o t 60.86
- 61.40
- 60.44
- 115.4 115.2-115.6 2.25
--
114.7
-
3.6 2.3 4.9 3.7
1072.2 1074.0 1304.1 1306.7
C9-1074 JOURNAL
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PHYSIQUEe x p e n s i v e , e s p e c i a l l y f o ~ a l a r g e s i z e computer system.Needless t o say, t h a t even w i t h o u t any f u r t h e r improvement, t h e p r e s e n t r e s u l t s a r e r e a l l y i m p r e s s i v e .
F u t u r e work on more c o m p l i c a t e d systems w i l l t e s t t h e v a l i d i t y and p r a c t i c a l i t y o f t h e p r e s e n t scheme.
References
[ll B. Johansson and N. Flartensson, Phys. Rev. 821 (1980) 4427; and r e f e r e n c e s t h e r e i n
[2] C.A. N i c o l a i d e s , A.D. Z d e t s i s and A.N. A n d r i o t i s , S o l i d S t a t e Commun. 42 (1982) 227
[3] P.H. C i t r i n and G.K. Wertheim, Phys. Rev. B27 (1983) 3176 [4] A.D. Z d e t s i s , Phys. Rev. B34 (1986) 7666
[5] A.D. Z d e t s i s , Phys. Rev. B35 (1987) 5868
