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g-HARTREE AB-INITIO CALCULATION OF ATOMIC IONIZATION ENERGY
M. Ohno
To cite this version:
M. Ohno. g-HARTREE AB-INITIO CALCULATION OF ATOMIC IONIZATION ENERGY. Journal
de Physique Colloques, 1987, 48 (C9), pp.C9-509-C9-512. �10.1051/jphyscol:1987983�. �jpa-00227404�
Colloque C9, supplement au n012, Tome 48, 1987
g-HARTREE AB-INITIO CALCULATION OF ATOMIC IONIZATION ENERGY
Physikalisches Institut der Universitdt Bonn, Nussallee 12, 1)-5300 Bonn 1 , F.R.G.
La me'thode g-Hartree e s t appl iqude au c a l c u l d ' e'nergi2s d ' e x c i t a t i o n atomiques. Cette methode e s t comparge en d 6 t a i l avec c e l l e de Hartree-Fock
The g-Hartree method i s a p p l i e d f o r t h e a b - i n i t i o c a l c u l a t i o n o f atomic e x c i t a t i o n energy. A d e t a i l e d -comparison t o t h e Hartree-Fock scheme i s made.
D u r i n g t h e l a s t decade, t h e a b - i n i t i o c a l c u l a t i o n s o f i o n i z a t i o n energies ( p o t e n t i a l s ) by many-body formalism have a t t r a c t e d much i n t e r e s t . I n c o n t r a s t t o C I ( c o n f i g u r a t i o n i n t e r a c t i o n ) and MCSCF(mu1ticonfiguration s e l f - c o n s i s t e n t f i e l d ) methods by which one o b t a i n s t h e i o n i z a t i o n energy as a d i f f e r e n c e i n t o t a l energies. t h e i o n i z a t i o n energy can be obtained d i r e c t l y by many-body formalism. The r e s u l t s a r e o f t e n more accurate than those obtained by c o n v e n t i o n a l methods. T h i s i s mainly because o f a more balanced treatment o f b o t h i n i t i a l and f i n a l s t a t e s .
Recently t h e new systematic scheme f o r t h e a b - i n i t i o c a l c u l a t i o n o f the e x c i t a t i o n energy was proposed w i t h i n t h e framework o f t h e g-Hartree mean f i e l d method [ I ] . The g-Hartreetg-H) method i s a mean f i e l d method which was d e r i v e d e x a c t l y from t h e f u l l r e l a t i v i s t i c QED a c t i o n by u s i n g t h e f u n c t i o n a l method and expanding t h e grand canonical p a r t i t i o n f u n c t i o n around t h e s t a t i o n a r y p o i n t which i s g i v e n by t h e s o l u t i o n o f t h e g-H e q u a t i o n [ 2 ] . The method was extended so t h a t t h e e l e c t r o n energy l e v e l i n t h e mean f i e 1 d . i . e . t h e eigenvalues o f t h e g-H e q u a t i o n a r e equal t o t h e t h e o r e t i c a l l y exact i o n i z a t i o n e n e r g i e s [ l ] .
S t a r t i n g w i t h t h e M$llzr-PlessetIMP) p a r t i t i o n i n g o f t h e g-H Hamiltonian and u s i n g t h e Rayleigh-Schrodinger(RS1 p e r t u r b a t i o n theory, one o b t a i n s t h e g-H expansion o f t h e e x c i t a t i o n energy and optimizes so t h a t the c o r r e l a t i o n and r e l a x a t i o n terms v a n i s h and t h e eigenvalue becomes equal t o t h e i o n i z a t i o n energy. The numerical r e s u l t s by t h e g-H method g i v e s a good agreement w i t h t h e experimental i o n i z a t i o n energies [ 1 , 3 , 4 ] .
By t h e Hartree-Fock(HF) RS 2nd order p e r t u r b a t i o n t h e o r y , t h e i o n i z a t i o n energy o f a s i n g l e c o r e h o l e X i s given by [5,6]
The second t e r m i n eq. ( 1 ) i s a leading term i n
~ ~ ~ A s c F
c a l c u l a t i o n f o r t h e i o n i z a t i o n energy which i s obtained as t h e d i f f e r e n c e between t h e a p p r o p r i a t e HF t o t a l e n e r g i e s f o r t h e N and N-1 e l e c t r o n states. The l a s t two terms d e s c r i b e t h e ground-state c o r r e l a t i o n and t h e hole-hopping(dominate1y d i p o l e ) r e l a x a t i o n [ 7 1 . The i o n i z a t i o n energy can be w e l l approximated by eq. ( 2 ) by i n c l u d i n g t h e h i g h e r o r d e r c o r r e c t i o n s w i t h i n the SCF model.I n t h e case o f g-H method one i n t r o d u c e s t h e g-H p o t e n t i a l as a c e n t r a l f i e l d p o t e n t i a l and o b t a i n s t h e f o l l o w i n g diagrammatic expansions.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987983
C9-5 10 JOURNAL DE PHYSIQUE
Here t h e c r o s s i n g i s t h e propagator o f t h e h o l e which i s created by t h e i n i t i a l c o r e h o l e e x c i t a t i o n . The s o l i d and d o t t e d l i n e s correspond t o t h e e l e c t r o n propagators f o r t h e N and N-1 e l e c t r o n configurations.The l a s t two diagrams a r e same as l a s t two HF terms i n e q . ( l ) . I n the g-H method one determines g such t h a t c o n t r i b u t i o n s o f t h e c o r r e l a t i o n and r e l a x a t i o n vanish;
The f i r s t s i x terms a r e t h e l e a d i n g terms i n theASCF c a l c u l a t i o n s . Then these terms can be w e l l approximated by theASCF r e s u l t .
E:-'
(ASCF,3.) = E-w,
)-
~ ~ ( f v - 1 ~$&I
( 5 )Here ~ f . 4 ~ . g , ) and E (N-1,g,)
ta
are the t o t a l energies o f t h e n e u t r a l N and i o n i c N-1 e l e c t r o n s t a t e s r e s p e c t i v e l y . This k i n d o f approximation w i l l become necessary when t h e number o f e l e c t r o n s increase and m u l t i p l e e x c i t a t i o n energy has t o be evaluated. However, note t h a t i n c o n t r a s t t o t h e HFdSCF energy, t h e g-HASCF energy obtained f o r t h e g-value which i s determined by eq. ( 4 ) has no p h y s i c a l relevance. I n order t o make a comparison w i t h t h e HF SCF energy, one may has t o consider onlyASCF terms and optimizes so t h a t these terms vanish. We leave a numerical i n v e s t i g a t i o n t o t h e f u t u r e .I n t h e case o f u n r e s t r i c t e d HF t h e
A
SCF energy can be w e l l approximated byE:~(&S~F) g ( E ~ + E*I)
( 6 1Here exand a r e t h e eigenvalues of t h e HF equation f o r t h e N and N-1 e l e c t r o n s Hamiltonian. However, i n the case o f g-H method i n s t e a d o f eq. ( 6 ) t h e f o l l o w i n g r e l a t i o n holds.
E ~.(ASCF, 9.)
2(EL Q + &:(p.))-jh ( 2 9 9
< X X U X ~ ~ ( I I n t h e case o f HF scheme, the4SCF energy may be approximated q u i t e w e l l by s o l v i n g an equation o f HF type w i t h some h a l f - i n t e g e r occupation numbers;so c a l l e d t r a n s i t i o n operator(T0) method[8]. However, i n t h e case o f t h e g-H method because o f no cancelations o f the s e l f - i n t e r a c t i o n terms, TO method does n o t p r o v i d e any good agreement w i t h the g - H d S c ~ energy.
W i t h i n t h e sudden approximation, t h e Koopmans energy i s r e l a t e d t o a weighted average energy f o r s i n g l y and m u l t i p l y e x c i t e d s t a t e s i n t h e photoelectron spectrum[9]. W i t h i n t h e HFSCF model t h e f o l l o w i n g r e l a t i o n i s obtained.
Here E ; i s t h e energy f o r t h e e x c i t e d s t a t e s .
n)ri
i s t h e o r b i t a l s of t h e N-1 e l e c t r o n Hamiltonian and i s t h e frozen N-1 o r b i t a l s o f t h e N e l e c t r o n Hamiltonian. I n t h e case o P g - ~ method one can optimize so t h a t t h e g-HASCF energy i s equal t o t h e main i o n i c s t a t e i o n i z a t i o n energy Eo.and r e l a x a t i o n terms beyond t h e d SCF p i c t u r e vanish. A numerical study f o r t h e I s l e v e l o f Ne(Z=lO) shows t h a t t h e eigenvalue o f t h e g-H equation obtained f o r such a g does n o t give any good estimate o f t h e average i o n i z a t i o n energy. T h i s i s because t h e g-value determined f o r the main i o n i c (relaxed) s t a t e d i f f e r e s from those f o r t h e e x c i t e d s t a t e s l i k e normal and conjugate shake up s t a t e s .
L e t us consider t h e i o n i z a t i o n energy of t h e double h o l e X and Y. By the HFRS 2nd order p e r t u r b a t i o n theory one obtains t h e f o l l o w i n g expression.
E :
; = EX + ZJ + A?+ A: + A?<+ A> - F?% Y,Y)
Here deSeand
d
R a r e t h e ground s t a t e c o r r e l a t i o n and r e l a x a t i o nterms f o r a s i n g l e h o l e and ~ ' ( x , y ) i s t h e bare hole-hole Coulomb r e p u l s i o n i n t e r a c t i o n . I n t h e case o f g-H method one obtains the f o l l o w i n g expression.
E
; ; = Er + A s + El + A* + ( 1 - 2 ~ ) { ~ * @
4- z { g Z m - 3 r r - 0 ) Q-8 -:,(+-0) ;db-o
For t h e sake o f s i m p l i c i t y we denote t h e f i r s t 13 diagrams i n equation ( 3 ) f o r a s i n g l e c o r e h o l e energy by
A .
Here t h e double l i n e w i t h t h e c r o s s i n g i s a h o l e propagator f o r t h e h o l e Y . The s o l i d and d o t t e d l i n e s correspond t o t h e e l e c t r o n propagators f o r t h e N and N-2 e l e c t r o n configurations.0ne determines t h e g-value so t h a t t h e c o r r e l a t i o n and screening terms vanish and t h e double h o l e energy i s a l i n e a r sum o f the h o l e energy o f a s i n g l e hole. Approximating t h e terms which correspond t o t h e leading terms i n the g-HdSCF c a l c u l a t i o n s by t h e g-HASCF r e s u l t we o b t a i nC9-5 12 JOURNAL DE PHYSIQUE
As i n t h e case o f a s i n g l e h o l e energy t h e A S C F energy obtained does n o t have any p h y s i c a l relevance.
The Auger energy can be obtained by c a l c u l a t i n g t h e i n i t i a l and f i n a l h o l e e n e r g i e s s e p a r a t e l y by eqs. ( 3 ) and ( 1 1 ) . The o t h e r way i s t o o b t a i n
t h e Auger energy u s i n g eqs. ( 3 ) and (11) and o p t i m i z i n g so t h a t t h e Auger energy f o r t h e Auger process i"-, j4 kv+ e 4 ( ~ u g e r ) i s g i v e n by
Here
e
a r e t h e eigenvalues o f t h e g-H equation o f t h e N - e l e c t r o n system.Note t h a t because o f one-step optimazation o f t h e Auger energy, t h e eigenvalue o f t h e i n i t a l s i n g l e h o l e and a l i n e a r sum o f t h e eigenvalues o f t h e f i n a l double h o l e a r e n o t any more equal t o t h e corresponding i o n i z a t i o n energy.
So f a r t h e i o n i z a t i o n processes a r e discussed o n l y from a v i e w p o i n t o f t h e e x c i t a t i o n energy. However,the p h y s i c a l p r o p e r t y which i s more d i f f i c u l t t o c a l c u l a t e a c c u r a t e l y i s a l i f e t i m e o f a h o l e created i n atomic l i k e systems.
The l i f e t i m e o f a h o l e ( h o l e s ) i s governed by b o t h r e a l decay processes l i k e Auger .Caster-Kronig and corresponding f l u c t u a t i o n s [see r e f . 10 f o r a r e v i e w o f t h e s u b j e c t and r e f e r e n c e s ] .
I n b o t h HF and g-H expansion f o r t h e i o n i z a t i o n energy, t h e r e l a x a t i o n term [ l a s t t e r m ) a l s o p r o v i d e s t h e ( n o n - r a d i a t i v e ) decay r a t e . I n t h e case o f HFRS 2nd o r d e r p e r t u r b a t i o n t h e o r y , the l i f e t i m e can n o t be w e l l approximated.
The l i f e t i m e can be o f t e n two o r t h r e e times overestimated o r underestimated.
I n t h e case o f t h e g-H method t h e g-value i s determined so t h a t t h e r e a l p a r t o f t h e f l u c t u a t i o n terms vanish. I t would be i n t e r e s t i n g t o see whether t h e imaginary p a r t o f t h e f l u c t u a t i o n terms ( r e l a x a t i o n terms) c a l c u l a t e d f o r t h e g-value which i s o b t a i n e d by eqs(3) and ( 4 ) w i l l provide a reasonable e s t i m a t e o f t h e l i f e t i m e ( n o n - r a d i a t i v e p a r t o f t h e l i f e t i m e 1 .
F u r t h e r d e t a i l e d account o f t h e s u b j e c t s i n c l u d i n g t h e m u l t i p l e h o l e e x c i t a t i o n s and shake up w i l l be g i v e n i n a forthcoming paper.
Acknowledgement T h i s work i s supported by Deutsche Forschung Gemeinschaft.
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