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MOLECULAR POTENTIALS FOR SYSTEMS WITH ONE OR TWO ACTIVE ELECTRONS
F. Masnou-Seeuws
To cite this version:
F. Masnou-Seeuws. MOLECULAR POTENTIALS FOR SYSTEMS WITH ONE OR TWO ACTIVE ELECTRONS. Journal de Physique Colloques, 1985, 46 (C1), pp.C1-43-C1-59.
�10.1051/jphyscol:1985105�. �jpa-00224474�
JOURNAL DE PHYSIQUE
Colloque C I , supplement au n°l, Tome 46, Janvier 1985 page Cl-43
MOLECULAR POTENTIALS FOR SYSTEMS WITH ONE OR TWO ACTIVE ELECTRONS F . Masnou-Seeuws
Laboratoire des Collisions Atomiques et Moléculaires*, Bât. 351, Université Paris-Sud, 91405 Orsay Cedex, France
Résumé - Grâce aux méthodes de pseudo-potentiel e t de p o t e n t i e l modèle, des r é s u l t a t s t r è s p r é c i s ont été obtenus pour les p r o p r i é t é s moléculaires des systèmes à un é l e c t r o n a c t i f comme les couples a l c a l i n - g a z r a r e . Pour t r a i t e r l e s dimères a l c a l i n s e t l e u r s c a t i o n s , i l est important de d é c r i r e avec p r é c i s i o n les e f f e t s de p o l a r i s a t i o n de c œ u r .
Abstract - Owing to model potential and pseudo-potential methods, very accu- r a t e r e s u l t s have been obtained for the molecular p r o p e r t i e s of systems with one a c t i v e electron such as a l k a l i rare-gas couples. In order to t r e a t the a l k a l i dimers and t h e i r c a t i o n s , i t is important to perform an accurate des- c r i p t i o n of core p o l a r i s a t i o n e f f e c t s .
I - INTRODUCTION
Owing to l a s e r techniques, much experimental information i s now a v a i l a b l e concerning the ground and excited s t a t e s of molecules and the dynamical or optical couplings between those s t a t e s . In many c a s e s , the molecular energies a r e known with an accuracy b e t t e r than 1 cm~1. Such a s i t u a t i o n i s a challenge for t h e o r e t i c i a n s . For many systems, the standard methods of quantum chemistry, using extensive confi- guration i n t e r a c t i o n , have succeeded in predicting accurate potential curves. However, i t i s generally d i f f i c u l t t o reach an accuracy comparable t o the experimental one.
In the p a r t i c u l a r case of systems with one or two a c t i v e e l e c t r o n s , model potential and pseudo-potential c a l c u l a t i o n s , in which the effect of the rapid core electrons on the motion of the slow o u t e r ( s ) e l e c t r o n ( s ) is simulated by an effective poten- t i a l f i t t e d on accurate atomic data, sometimes provide an e x c e l l e n t accuracy.
Concerning a l k a l i - r a r e gas systems, i t i s seven and f i v e years since a 1 cm"' accu- racy has f i r s t been obtained in the l a s e r spectroscopy of the ground s t a t e of NaNe / 1 / and NaAr / 2 / . In the same period, a similar t h e o r e t i c a l accuracy has been reached by model potential c a l c u l a t i o n s / 3 , 4 / using the method developped by Valiron e t a l . / 5 / .
Fig. 1 - Model potential c a l c u l a t i o n s for the ground s t a t e of NaNe / 3 / compared with l a s e r spectroscopy determination / 1 / (full l i n e , theory ; square and dotted l i n e , experiment)
*Laboratoire associe N° 281 au CNRS
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985105
C1-44 JOURNAL DE PHYSIQUE
I n such a method, t h e non l o c a l e f f e c t s i n t h e e l e c t r o n - r a r e gas i n t e r a c t i o n a r e r e p r e s e n t e d by an e x p l i c i t o r t h o g o n a l i t y c o n d i t i o n t o t h e o u t e r o r b i t a l s o f t h e r a r e gas. I t has g i v e n good r e s u l t s f o r Ne and f o r A r /6/ p e r t u r b e r s . I t m i g h t n o t b e s u f f i c i e n t l y a c c u r a t e f o r p wave s c a t t e r i n g on an h e l i u m p e r t u r b e r . Recent c a l c u l a t i o n s o f Pascale /7/ u s i n g an 1-dependent p s e u d o - p o t e n t i a l f o r t h e e-helium i n t e r a c t i o n have improved t h e agreement w i t h e x p e r i m e n t ( s e e F i q . 2 ) .
R l a u l
F i g . 2
-
Normalised e m i s s i o n spectrum o f t h e Na3p s t a t e i n t h e presence o f He a t 4 0 0 K. Dashed 1 in e , e x p e r i m e n t a l d a t a o f York e t a1 /8/, s o l i d 1 i n e c a l c u l a t i o n so f Pascale /7/, d a s h - d o t t e d 1 i n e r e s u l t s
*i I
o f Hanssen e t a1 /4/ u s i n g t h e method
'""
d e s c r i b e d i n / 5 / . ( f r o m Ref. / 7 / )
\
560 580 600 620 6LO 660 680 700 720 ?LO A Inml
F o r t h e e x c i t e d s t a t e s , t h e accuracy depends upon t h e e s t i m a t i o n o f t h e c o r e - c o r e i n t e r a c t i o n a t s m a l l i n t e r n u c l e a r d i s t a n c e s , a s we s h a l l d i s c u s s l a t e r on.
D u r i n g t h e l a s t t h r e e y e a r s , t h e a c c u r a c y i n t h e d e t e r m i n a t i o n o f t h e w e l l i n t h e ground s t a t e o f t h e a l k a l i d i m e r c a t i o n s has reached a f r a c t i o n o f a wave number, due t o e x t e n s i v e s t u d i e s o f t h e Rydberg s t a t e s o f t h e a l k a l i dimers u s i n g two- photon t e c h n i q u e s ( s e e Ref. / 9 / t o / 1 6 / ) . Such r e s u l t s have s t i m u l a t e d an i n t e n s e t h e o r e t i c a l a c t i v i t y i n t h e f i e l d , and t h e i m p o r t a n c e o f c o r e p o l a r i s a t i o n e f f e c t s f o r t h e o b t e n t i o n o f h i g h q u a l i t y r e s u l t s has been s t r e s s e d /17//18/.
D u r i n g t h e same p e r i o d , t h e l a s e r spectroscopy o f t h e ground s t a t e and o f t h e f i r s t e x c i t e d s t a t e s o f t h e a l k a l i d i m e r s has reached an a c c u r a c y o f a f e w wave numbers ( s e e /19/, /20/ and r e f e r e n c e s t h e r e i n ) . The most r e c e n t d e t e r m i n a t i o n f o r t h e depth De o f t h e ground s t a t e o f L i z /21/ and Na2 /22/ i s 8516.9 + 0.4 cm-I and 6022.6 + 1 cm-1 r e s p e c t i v e l y . The t h e o r e t i c a l r e s u l t s a r e on the-point o f .reaching t h e e x p e r i m e n t a l accuracy : Jeung /23/ o b t a i n s a w e l l d e p t h o f 6020 cm-1 f o r t h e Na2 ground s t a t e , w h i l e Konowalow and F i s h /24/ e s t i m a t e De = 8501 cm-1 f o r L i z . I n b o t h case, t h e i n c l u s i o n o f c o r e p o l a r i s a t i o n e f f e c t s i s c r u c i a l , and t h e two t r e a t m e n t s a r e i n t e r m e d i a t e between ab initio and e f f e c t i v e p o t e n t i a l methods.
My a i m i s t o p r e s e n t some o f t h e r e s u l t s o b t a i n e d r e c e n t l y , and t o d i s c u s s how i n t h e n e x t f u t u r e v e r y a c c u r a t e r e s u l t s c o u l d be o b t a i n e d f o r t h e e x c i t e d s t a t e s .
2
-
EFFECTIVE POTENTIALS FOR THE ALKALI ATOMS : APPLICATION TO ALKALI-RARE GAS SYSTEMSThe b a s i c i d e a o f model p o t e n t i a l and p s e u d o - p o t e n t i a l methods i s t o t r e a t an a l k a l i - r a r e gas system by s o l v i n g a one e l e c t r o n S c h r a d i n g e r e q u a t i o n i n o r d e r t o d e s c r i b e t h e m o t i o n o f t h e a c t i v e e l e c t r o n i n t h e f i e l d o f t h e two c o r e s A and B a t a d i s t a n c e R
F i g . 3
-
C o o r d i n a t e systemWe t h e r e f o r e must solve t h e equation :
X
x
$4H r L = [ ~ + v ~ I r ~ ) + ~ ~ ( r ~ ) + " ~ ( ~ ~ ~ ~ r ~ ) ] ~ i =
i I (1In ( 1 ) T i s t h e k i n e t i c energy, VA(rA) and V B ( r B ) a r e t h e e f f e c t i v e p o t e n t i a l s f o r t h e i n t e r a c t i o n of t h e e l e c t r o n with t h e cores A and B r e s p e c t i v e l y , V3 i s a t h r e e - body t e n s o r i a l term which depends upon t h e angle eRand comes from t h e a d d i t i o n of t h e two e l e c t r i c d i p o l e s induced in B by A and e- r e s p e c t i v e l y .
2.1 .- Ref i n i t i o n of VA(rA), e f f e c t i v e p o t e n t i a l f o r t h e a l k a l i atoms
In most c a s e s , VA(rA) i s a parametric p o t e n t i a l f i t t e d t o p r o p e r t i e s of t h e i s o l a t e d e-A system. The atomic o r b i t a l s of t h e a l k a l i atom a r e obtained by solving ( 2 )
The usual procedure c o n s i s t s in f i t t i n g t h e parameters i n V A so t h a t t h e c a l c u l a t e d e n e r g i e s ~ Y ' a r e a s c l o s e a s p o s s i b l e from t h e experimental ones. In a model poten- t i a l treatment /25/ VA(rA) i s a t t r a c t i v e and c o n t a i n s v i r t u a l c o r e o r b i t a l s besides t h e valence o r b i t a l s . The atomic f u n c t i o n s t h e r e f o r e possess t h e c o r r e c t number of nodes. In a pseudo-potential treatment / 2 6 / , V A ( r ~ ) i s r e p u l s i v e in t h e c o r e region, leading t o nodeless pseudo-orbitals a t small rA. This i s i l l u s t r a t e d on Fig. 4
MODEL POTENTIAL PSEUDO POTENTIAL
A
*
Fig. 4 - Schematic r e p r e s e n t a t i o n of t h e Na 3s o r b i t a l in model and i n pseudo- p o t e n t i a l c a l c u l a t i o n s .
The f i r s t question one could ask i s then : which i s t h e t m e p o t e n t i a l ? We s h a l l show here below t h a t i t i s a wrong q u e s t i o n .
Indeed, t h e d e f i n i t i o n of an e f f e c t i v e p o t e n t i a l comes from an a d i a b a t i c separation between t h e motion of t h e rapid ( c o r e ) e l e c t r o n s and of t h e slow (valence) e l e c t r o n s . Calling r l
...
r s l t h e p o s i t i o n s of t h e core e l e c t r o n s , and rA t h e p o s i t i o n of t h e valence e l e c t r o n , we may w r i t e /27,28/ t h e t o t a l wavefunction f o r t h e ground s t a t e of t h e atom Ao
-
r - -2-4 [:A)$js)
( 3 )In ( 3 )
xo
i s t h e wavefunction f o r t h e motion of t h e c o r e e l e c t r o n s f o r a f k e d position rn of t h e valence e l e c t r o n . The a d i a b a t i c equation :C 1-46 J O U R N A L D E PHYSIQUE
d e f i n e s t h e model p o t e n t i a l !d(rA)
-
l / r ~ and d e s c r i b e s t h e m o t i o n o f t h e slow e l e c - t r o n i n t h e rnecin p o t e n t i a l of t h e core.The p h y s i c a l i d e a s i n v o l v e d a r e t h e r e f o r e c l o s e t o t h e Born-Oppenheimer s e p a r a t i o n i n m o l e c u l a r p h y s i c s . The non a d i a b a t i cc o r r e c t i o n :
,.
i s s m a l l , and l i m i t e d t o t h e c o r e r e g i o n . I t has been computed t h r o u g h a v a r i a t i o n a l procedure i n t h e case o f t h e l i t h i u m atom /27/.
The b e s t e f f e c t i v e p o t e n t i a l may t h e r e f o r e be d e f i n e d so t h a t outside t h e core region t h e w a v e f u n c t i o n s s o l u t i o n s o f ( 2 ) a r e as c l o s e as p o s s i b l e f r o m t h e " e x a c t " wave- f u n o t i o n s . The p h y s i c a l i d e a s a r e t h e r e f o r e v e r y s i m i l a r t o quantum d e f e c t t h e o r y . The s h o r t range p o t e n t i a l i s an e m p i r i c a l t o o l f o r making s u r e t h a t a l a r g e number o f e x c i t e d s t a t e w a v e f u n c t i o n s have a c o r r e c t quantum d e f e c t outside t h e core. As i n quantum d e f e c t t h e o r y , VA(rA) i s independent o f t h e energy o f t h e l e v e l c o n s i - dered, b u t i s u s u a l l y 1 -dependent.
One way o f o b t a i n i n g a good v a l e n c e w a v e f u n c t i o n i s t o v a r y t h e parameters i n t h e p o t e n t i a l so t h a t t h e c a l c u l a t e d a t o m i c e n e r g i e s i n ( 2 ) a r e as c l o s e a s p o s s i b l e f r o m t h e e x p e r i m e n t a l e n e r g i e s . B o t t c h e r and Dalgarno /29/, u s i n g a 2nd o r d e r p e r - t u r b a t i o n t h e o r y , have shown t h a t c o r e p o l a r i s a t i o n e f f e c t s s h o u l d be i n c l u d e d i n t h e e f f e c t i v e p o t e n t i a l . T h i s can be done by i n t r o d u c i n g i n t h e p o t e n t i a l an e f f e c t i v e c o r e p o l a r i s a t i o n o p e r a t o r :
I n (6), ad A and aA a r e r e s p e c t i v e l y t h e d i p o l e and quadrupole s t a t i c p o l a r i s a b i l i t i e s o f t h e core, 63 b e i n g t h e dynamic d i p o l e p o l a r i s a b i l i t y -f and g a r e c u t - o f f f u n c t i o n s necessary t o a v o i d t h e d i v e r g e n c e i n t h e c o r e r e g i o n , r d and rq b e i n g c u t - o f f parameters t y p i c a l o f t h e c o r e dimensions. Two comments s h o u l d be added:
1 ) The c o r e e l e c t r o n s being not expZicitZy included i n t h e t r e a t m e n t , t h e p r o p e r - t i e s o f t h e c o r e such as i t s p o l a r i s a t i o n appear a s an e f f e c t i v e o p e r a t o r .
2) The b e h a v i o u r o f t h e system d u r i n g t h e s h o r t p e r i o d o f t i m e when t h e e l e c t r o n e n t e r s t h e c o r e r e g i o n i s t r e a t e d as a " b l a c k box". The parameters rd and rq must t h e n b e f i t t e d on e x p e r i m e n t a l data.
I n f a c t , t h e c a l c u l a t e d e n e r g i e s a r e n o t v e r y s e n s i t i v e t o t h e c h o i c e o f such parameters /30/. V a r i o u s a u t h o r s u s i n g a l a r g e v a r i e t y o f a n a l y t i c a l f o r m u l a s f o r t h e p o t e n t i a l , have succeeded i n r e p r o d u c i n g t h e o p t i c a l spectrum o f t h e a l k a l i atoms w i t h a good a c c u r a c y / S t / .
I n c o n t r a s t , t h e energy v a r i a t i o n o f t h e p h o t o i o n i s a t i o n c r o s s - s e c t i o n s o f t h e a l k a l i atoms i s v e r y s e n s i t i v e t o t h e c h o i c e o f t h e c u t - o f f parameters /18/. A p o s s i b l e improvement i n t h e d e t e r m i n a t i o n o f t h e e f f e c t i v e p o t e n t i a l would c o n s i s t i n f i t t i n g d i r e c t l y t h e e x p e r i m e n t a l p h o t o i o n i s a t i o n c r o s s - s e c t i o n s . The i n f l u e n c e of such a choice w i l l be discussed l a t e r on.
2.2
-
D e f i n i t i o n o f V g ( r g ) and o f VgVg(rg) r e p r e s e n t s t h e i n t e r a c t i o n o f a f r e e e l e c t r o n w i t h a n e u t r a l r a r e gas and i s u s u a l l y f i t t e t i on l o w energy e l e c t r o n - r a r e gas s c a t t e r i n g d a t a /32/ which a t p r e s e n t t i m e a r e known v e r y a c c u r a t e l y . i l e may u s e t h e e x n r e s s i o n :
i n wi7icI1 Vsr i s a s h o r t range t e r m ( a t t r a c t i v e on r e p u l s i v e , see
5
2.11, 39 andc i ' B = aB
-
6sB a r e r h e p o l a r i s a j i l i t i e s o f t h e r a r e gas, pd and pq cu-i-off r a d i i t y p i c a l o f t h e dimensions of t h e c o r e B. The computed p h a s e - s h i f t s a r e u s u a l l y s e n s i t i v e t o t h e c h o i c e o f such parameters. Besides, as was s a i d i n t h ei n t r o d u c t i o n , t h e s h o r t range t e r m Vsr(rg) n:ust be non l o c a l (5,7). The t e n s o r i a l t e r s '13, anc hence t h e ~ o l e c u l a r e n e r g i e s , a l s o depend upon t h e c h o i c e o f t h e c u t - o f f p a r a n e t e r s :
?g (see f i g . 1 ) i s t h e a n g l e
(R,
XB). Peach i33/ has shown t h a t t h e c u t - o f f f u n c t i o n s i n (8) should be t h e square r o o t o f those i n v o l v e d i n ( 7 ) .I t i s i m p o r t a n t t o remark t h a t v ~ ( r ~ ) -as w e l l as t h e sum VE(rB) + V3(t,rB)- i s a s h o r t range terrr, covpareti t o t h e l / r A b e h a v i o u r o f VA(rA). f i e rare gas
3
may then be considered as introducing an additionnu2 R dependent quantum d e f e c t on t h e a l k a l i atomic wavefunctions. On F i g . 5 and 6 we have r e p r e s e n t e d b o t h t h e atomic w a v e f u n c t i m + l i o ( r A ) s o l u t i o n o f ( 6 ) and t h e r o l e c u l a r wavefunction $ & = O ( r A , ~ ) s o l u t i o n o f (I) o a i n e d i n model p o t e n t i a l c a l c u l a t i o n s f o r t h e KaHe and RaNe systems.I 1 . .
20 30 LO i 1 -_
6 L l o 1L62 28 3 0
r A (a.u.)
F i g . 5 ( f r o n Ref. 34)
-
S o l i d l i n e : sodium atorcic w a v e f u n c t i o n +m=O(r ) as a f u n c - t i o n o f t h e d i s t a n c e r A on t h e q u a n t i z a t i o n a x i s . Dashed l i n e : 6' n o f e c u l a r wavefunction I,!J~'O(I-~, R = 6.75 rA/rA) i n case of an heliul;: p e r t u r b e r l o c a t e d a t 6.75 a.u. o f t h e 6Psobium center.%No-Ne i'l
6.: 'L.V 28.3
r A ( a . u . )
F i g . 6
-
Same a s F i g . 5 i n case of a Ne p e r t u r b e r .C1-48 JOURNAL DE PHYSIQUE
I t i s c l e a r f r o m F i g . 5 and 6 t h a t :
a ) t h e m o l e c u l a r w a v e f u n c t i o n i s v e r y s i m i l a r t o t h e a t o m i c one, e x c e p t i n t h e r a r e gas r e g i o n where i t i s s t r o n g l y m o d i f i e d due t o 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 w i t h t h e o c c u p i e d o r b i t a l s.
b) t h e b a s i s s e t used i n model p o t e n t i a l c a l c u l a t i o n s must be f l e x i b l e enough t o deal c o r r e c t l y w i t h t h e m o d i f i c a t i o n o f t h e a l k a l i w a v e f u n c i t o n i n t h e r e g i o n o f t h e r a r e gas c o r e
2.3
-
R e s u l t s f o r t h e i o n i s a t i o n e n e r g i e s : m o l e c u l a r quantum d e f e c t sThe eigenenergy
E!'[
computed i n ( I ) i s t h e i o n i s a t i o n energy, i.e. t h e energy o f a n e l e c t r o n i n t h e f i e l d o f t h e two c o r e s A+ and 6 a t d i s t a n c e R f o r a m o l e c u l a r s t a t e o f symmetry X c o r r e l a t e d t o an a t o m i c I n ~ m>
=l i >
s t a t e .From t h e p r e d i c t i o n s o f t h e Fermi model /35/ and o f t h e a s y m p t o t i c methods /36/, i t has l o n g been known t h a t t h e d i f f e r e n c e between t h e i o n i s a t i o n energy
EIAI
(R) o f a m o l e c u l a r Rydberg s t a t e o f an a l k a l i r a r e gas system and t h e energy Eat o f t h e c o r -responding atomic s t a t e can b e s c a l e d by : 1
where ~c$!(,R)
l 2
i s t h e p r o b a b i l i t y d e n s i t y o f t h e a t o m i c w a v e f u n c t i o n a t t h e r a r e gas cent&-,K(B)
t h e l o c a l de B r o g l i e wavelength a s s o c i a t e d t o it. The q u a n t i t i e s a o ( k ) and a l ( k ) a r e r e s p e c t i v e l y p r o p o r t i o n n a l t o t h e s and p wave phase s h i f t s f o r t h e e l a s t i c s c a t t e r i n g o f an e l e c t r o n w i t h a wavenumber k by t h e r a r e gas B. Such a f o r m u l a i s v a l i d i n a r e g i o n where K(R) i s a s l o w l y v a r y i n g f u n c t i o n o f R. We have r e p r e s e n t e d on F i g . 7 t h e i o n i s a t i o n e n e r g i e s o f t h e u4p and a4p s t a t e s o f t h e KHe, KNe and KAr systems.F i g . 7
-
I o n i s a t i o n e n e r g i e s ( s e e t e x t ) o f t h e o4p and ~ 4 p s t a t e s o f KHe ( d o t t e d l i n e ) , KNe (broken l i n e ) and KAr ( f u l l l i n e ) as a f u n c t i o n o f t h e i n t e r n u c l e a r d i s t a n c e R. /6/.The maxima i n t h e C s t a t e s correspond t o maxima i n t h e atomic w a v e f u n c t i o n SO t h a t the rare gas can be considered as "measuring" t h e atomic probability density a t a given point. The fl i o n i s a t i o n e n e r g i e s a r e p r o p o r t i o n n a l t o t h e square o f t h e gradient of t h e a t o m i c wavefunction, which i s a severe check on t h e accuracy of t h i s wavefunction.
The r e p r e s e n t a t i o n o f t h e r a r e gas B as an a d d i t i o n n a l R-dependent quantum d e f e c t i s i l l u s t r a t e d on F i g . 8, where we have r e p r e s e n t e d as a f u n c t i o n o f t h e i n t e r n u c l e a r d i s t a n c e R t h e m o l e c u l a r quantum d e f e c t s 6?, w i t h
I n ( 1 0 )
3
i s t h e Rydberg c o n s t a n t and n i t h e p r i n c i p a l quantum number a t i n f i n i t y . We c o n s i d e r t h r e e a t c n i c quantum d e f e c t s , 6S, 6P, and 6-0 f o r R a 2.F i g . 8
-
M o l e c u l a r quantum d e f e c t s f o r t h e 4pa t o 9sa s t a t e s o f t h e NaNe and NaAr molecules. The quantum d e f e c t s c u r v e s 6; ( s e e t e x t ) a r e c o n t a i n e d i n t h e shadowed a r e a s /6/,/37/. The 64 and 6r c u r v e s a r e c o r r e l a t e d t o a t o m i c s t a t e s w i t h Ra2.I t i s c l e a r f r o m F i g . 8 t h a t t h e m o l e c u l a r quantum d e f e c t s a r e f a i r l y n-independent.
F o r t h e s t a t e s c o r r e l a t e d t o a t o m i c s t a t e s w i t h Ra2, t h e quantum d e f e c t c u r v e s p l i t s i n t o two c u r v e s , one of which e x h i b i t s an a v o i d e d c r o s s i n g w i t h t h e ( 8 - 1 ) curve. I t has t h e n been p o s s i b l e t o p r e d i c t c o l l i s i o n a l l y i n d u c e d t r a n s f e r s between t h e Na ((n+l ) p ) and Na nd s t a t e s . The e x p e r i m e n t a l measurements c o n c e r n i n g such t r a n s f e r s
/38/ a r e i n e x c e l l e n t agreement w i t h t h e o r e t i c a l c a l c u l a t i o n s /34/ i n which t h e com- p u t e d r a d i a l and r o t a t i o n a l c o u p l i n g m a t r i x elements a r e i n t r o d u c e d i n t h e c o l l i s i o n problem.
I n c o n c l u s i o n , one may say t h a t the accuracy of t h e i o n i s a t i o n energies i s due t o t h e q u a l i t y of t h e atomic wavefunctions and t o the precision of t h e electron rare gas eZastic phase s h i f t s obtained w i t h t h e e f f e c t i v e p o t e n t i a l s . Such an accuracy i s p o s s i b l e because t h e m o t i o n o f t h e c o r e e l e c t r o n s i s s u f f i c i e n t l y r a p i d t o a l l o w t h e i n t r o d u c t i o n o f e f f e c t i v e p o t e n t i a l s .
CI-50 JOURNAL DE PHYSIQUE
2.4
-
R e s u l t s f o r t h e m o l e c u l a r energy : problem o f t h e c o r e - c o r e t e r mThe ener
zl$l
o f a m o l e c u l a r I h i > s t a t e i s o b t a i n e d by adding t o t h e i o n i s a t i o n energy EEXI t h e c o r e - c o r e energy VAB+ (R). i . e . t h e energy o f t h e ground s t a t e o f t h e AB+ l l o l e c u l a r i o n f o r an i n t e r n u c l e a r d i s t a n c e R.-
VAB+(R) may be expressed as t h e sum o f a p o l a r i s a t i o n term and o f a s h o r t range r e - p u l s i v e t e r m due t o t h e o v e r l a p o f t h e a s y m p t o t i c p a r t o f t h e two c o r e w a v e f u n c t i o n s (Obviously, t h e method i s no more v a l i d when t h e a t o m i c c l o u d s a r e m o d i f i e d by t h e i n t e r a c t i o n o f t h e two c o r e s )
V,,+(R)
I'Vo (R) - (d:/2 6')-(</2 R 6 )
- B
- ,
I 'I f t h e p o l a r i s a b i l i t i e s a d and a: o f t h e c o r e B a r e we1 1 known, t h e quantity
V o ( ~ )
-which i s generally f i t t e d t o ion-atom s c a t t e r i n g data o r t o mobility measurements- i s not knom very accurately, so t h a t t h e 1cm-1 accuracy avaiZabZe for t h e i o n i s a t i o n energies i s obtained f o r t h e moZecuZar energies only a t large distances where VAB+
(R) i s l i m i t e d t o t h e polarisation terms. We have shown on F i g . 1 t h e e x c e l l e n t agreement which e x i s t s between t h e o r y and experiment c o n c e r n i n g t h e w e l l i n t h e ground s t a t e o f t h e NaNe system a t 10 a.u. I n c o n t r a s t ( s e e ( 9 ) ) t h e w e l l o f t h e f i r s t e x c i t e d IIs t a t e i s m a i n l y determined by VAB+(R) and t h e r e f o r e depends markedly upon t h e e s t i m a t i o n of ' V o ( ~ ) . T h i s i s i l l u s t r a t e d on F i g . 9 where we have r e p r e s e n - t e d t h e II4p c u r v e o f KAr f o r two d i f f e r e n t c h o i c e s o f VAB+(R)
I I I
- 1 0 0 - K A r A*IT -
C I
-500
- -
I J.1 I I
5 6 7 R ( a . u . )
F i g . 9
-
M o l e c u l a r energy c u r v e2
( s e e f o r m u l a ( 1 1 ) i n t e x t ) o f t h e A2II s t a t e o f KAr/6/, o b t a i n e d t h r o u g h model d g t e n t i a l c a l c u l a t i o n s w i t h two c h o i c e s f o r t h e c o r e - c o r e term. S o l i d l i n e : I T - o(R) has been taken frorn Ref. /39/ ; dashed l i n e : V A ~ + ( R ) i s taken f,ron Ref. /40/ ; Cross : e x p e r i m e n t a l d e t e r m i n a t i o n o f Dijren and T i s c h e r /41/ f o rtip(^).
However, t h e c o r e - c o r e t e r m i s common t o a1 1 m o l e c u l a r curves. I t s h o u l d be a uable, e s p e c i a l l y when a c c u r a t e e x p e r i m e n t a l i n f o m a t i on i s a v a i l a b l e f o r s e v e r a l
c u r v e s f r o m c o l l i s i o n o r spectroscopy experiments, t o f i t V A ~ + ( R ) d i r e c t l y on expe- r i m e n t a l r e s u l t s f o r t h e AB system. I n t h e work o f D u r e n ' s group /42/ parameters I n V g ( r g ) , Vg(8,rB) and VAB+(R) have been f i t t e d so t h a t a c c u r a t e e x p e r i m e n t a l s c a t t e - r i n g c r o s s - s e a i o n s can be reproduced f r o m c o l l i s i o n c a l c u l a t i o n s i n v o l v i n g t h e f ' $ ' ( ~ ) curves. I t s h o u l d b e v e r y i n t e r e s t i n g t o use a s i m i l a r procedure w i t h t h e c u r v e s E$'(R) computed f r o m ( 1 ) w h i l e VAB+(R) depends upon a d j u s t a b l e parameters.
3
-
GENERALISATION TO OPEN SHELL CORES : CALCULATIONS FOR HeHe*, ~ e r k * , ~ e ~ e *4 -
STUDY OF THE ALKALI MOLECULAR IONSThe e f f e c t i v e p o t e n t i a l method has been g e n e r a l i s e a t o open s h e l l c o r e s /46,33,48/.
F o r such s y s t e n s , s e v e r a l s t a t e s o f t h e
P,B+
m o l e c u l a r i o n a r e i n v o l v e d i n t h e c a l c u l a t i o n , and humpsin t h e e x c i t e d c u r v e s ( I s 5 0- and lu o f Ke2) depend markedly upon t h e e s t i m a t i o n o f t h e c o r e - c o r e i n t e r a c t i o n /18/. The accurate e s t i m t i o n of t h e ABf curves i s an open problem a t present time. (.E, 135000 Ne1Ne4.1 - M o l e c u l a r quantum d e f e c t s
F i g . 9 b i s . Hump i n t h e 0; I s 5 p o t e n t i a l c u r v e o f Ne*-Ne.
S o l i d l i n e , d o t t e d l i n e : model p o t e n t i a l c a l c u l a t i o n s ;48/ f o r two d i f f e r e n t e s t i m a - t i o n s o f t h e c o r e - c o r e tern1
Dashed l i n e : f i t t e d j 4 7 / t o s c a t t e r i n g experiments
I n c o n t r a s t , when we t u r n t o a l k a l i mo1e.cular i o n s such as Li;, ~a;, K+, t h e problem o f t h e c o r e - c o r e t e r m i s l e s s c r u c i a l , due t o t h e pvedoninance o f t h e ?/R term.
N e v e r t h e l e s s , t h e p h y s i c a l s i t u a t i o n d i f f e r s markedly f r o m t h e p r e c e e d i n g one, as t h e e l e c t r o n now moves i n t h e f i e l d o f tuo charged c o r e s . I n t h e a l k a l i r a r e gas problem, we have shown t h a t an atomic w a v e f u n c t i o n i s m o d i f i e d by a R-dependent quantum d e f e c t . I n t h e a l k a l i m o l e c u l a r i o n problem, we may c o n s i d e r t h a t t h e s h o r t range non coulombic t e r m i n t h e two p o t e n t i a l s VA and Vg i n t r o d u c e s two quantuK d e f e c t s on the H$ wavefunctions. Such a s i t u a t i o n i s i l l u s t r a t e d on F i g . 10 where we have r e p r e s e n t e d t h e R - v a r i a t i o n o f t h e e f f e c t i v e quantum number n i - ~ y ( ~ ) f o r t h e C s t a t e s o f t h e Na; i o n :
-134000
r-
, : ,
,! i!:
-1330M) I
F i g . f unc Na
$
6
5
4
3
2
5 10 15
'0Ra.u.
10
-
V a r i a t i o n o f t h e e f f e c t i v e quantum number ni-fA(~) (see EB
( 1 0 ) ) a s a : t i o n o f t h e i n t e r n u c l e a r d i s t a n c e R f o r t h e ground and e x c i t e d X s t a t e s o fm o l e c u l a r i o n /18/ t h e
C1-52 JOURNAL DE PHYSIQUE
Avoided c r o s s i n g s a r e c l e a r l y v i s i b l e , which would be p e r m i t t e d i n t h e H i problem.
In
t h e Na; case, t h e p o t e n t i a l d e p a r t s f r o m a p u r e coulombic p o t e n t i a l i n t h e c o r e r e g i o n , and t h e supersymetry o f t h e H+ problem i s broken. I t should be worthwhile t o perform on such systems ezperimenta? i n v e s t i g a t i o n s s i m i l a r t o t h e study of Stark e f f e c t on a l k a l i atoms.4.2
-
Ground s t a t e o f t h e a l k a l i d i m e r ~ a t i o n s : c o r e p o l a r i s a t i o n e f f e c t s Model p o t e n t i a l and pseudo p o t e n t i a l c a l c u l a t i o n s have been used s u c c e s s f u l l y f o r many y e a r s i n o r d e r t o compute t h e p r o p e r t i e s o f t h e ground and e x c i t e d s t a t e s o f such systems /25,26/. I t has l o n g been known t h a t t h e r e s u l t s a r e s e n s i t i v e t o t h e i n t r o d u c t i o n o f c o r e p o l a r i s a t i o n e f f e c t s . Ab i n i t i o c a l c u l a t i o n s , t a k i n g account a p p r o x i m a t e l y o f c o r e p o l a r i s a t i o n , g e n e r a l l y o v e r e s t i m a t e t h e bond 1 e n g t h /43/.Recent v e r y a c c u r a t e e x p e r i m e n t a l d e t e r m i n a t i o n s , o b t a i n e d t h r o u g h two step i o n i s a t i o n t e c h n i q u e s / l o - 1 6 / have s t i m u l a t e d new c a l c u l a t i o m L i m i t i n g t h e p o l a r i s a t i o n terms t o t h e d i p o l e terms, one o b t a i n s t h e m o l e c u l a r e n e r g i e s by s o l v i n g t h e S c h r o d i n g e r e q u a t i o n :
V A ( r ) i s t h e p o t e n t i a l d e f i n e d i n § 2.1 and a! t h e p o l a r i s a b i l i t y o f t h e two i d e n t i - c a l cores. The c r o s s p o l a r i s a t i o n t e r m i s now deduced f r o m ( 6 )
( 5 , , [ pjrB/G\j''7
( I 4 )Because o f t h e i m p o r t a n c e o f t h e V3 term, t h e r e s u l t s depend m a r k e d l y upon t h e c h o i c e o f t h e c u t - o f f r a d i u s r d . T h i s i s i l l u s t r a t e d on F i g . 11:
F i g . 11 - P o t e n t i a l c u r v e f o r t h e ground s t a t e of ~ a ; /18/ i n t h e minimum r e g i o n f o r v a r i o u s c h o i c e s o f t h e c u t - o f f r a d i u s r d . Dashed l i n e s : c a l c u l a t i o n s u s i n g ( a ) r d = 2 ( b ) rd=3 ( c ) rd=3.5 a.u. F u l l l i n e : c a l c u l a t i o n s u s i n g W e i s h e i t ' s v a l u e rd=2.35 (see t e x t ) . C i r c l e s : e x p e r i m e n t a l r e s u l t s /13/. Squares : e x p e r i m e n t a l r e s u l t s /14/.
The d e t e r m i n a t i o n o f t h e c u t - o f f appears then as an i m p o r t a n t problem. The
Veal
t e r m i n VA (see ( 6 ) ) depending upon r d , one p o s s i b i l i t y i s t o f i t t h e c u t - o f f r a d i u s so t h a t t h e a t o m i c e n e r g i e s e i g e n v a l u e s o f ( 2 ) a r e as c l o s e as p o s s i b l e f r o m experiment.Such a procedure has been used i n r e c e n t p s e u d o p o t e n t i a l c a l c u l a t i o n s /17/. However, r e c e n t work u s i n g model p o t e n t i a l c a l c u l a t i o n s /18/ h a s shown t h a t t h e a t o m i c
e n e r g i e s a r e n o t v e r y s e n s i t i v e t o t h e c h o i c e o f r d /44/, i n t r o d u c i n g some a r b i t r a r i n e s s i n t h e method. I n c o n t r a s t , however, t h e a t o m i c d i p o l e moments /28/
and t h e p h o t o i o n i s a t i o n c r o s s s e c t i o n s /45/ a r e s e n s i t i v e t o r d . The v a l u e 2.35 a.u.
f i t t e d by W e i s h e i t /45/ i n o r d e r t o reproduce t h e minimum i n t h e p h o t o i o n i s a t i o n c r o s s - s e c t i o n l e a d s t o a s a t i s f a c t o r y p o t e n t i a l c u r v e /18/ f o r t h e ground s t a t e of Na; (see F i g . 11 ). The r e s u l t s o f /17/ and
/ l a /
a r e compared t o experiment on Table 1.
I n b o t h cases, we have i n d i c a t e d i n i t a l i c s t h e v a l u e s o b t a i n e d when n e g l e c t i n g t h e c r o s s p o l a r i s a t i o n term Vg i n t h e e f f e c t i v e h a m i l t o n i a n .
TABLE 1
Ground s t a t e p r o p e r t i e s o f ~ i ; ~ a ; K;
( 0 . 9 2 )
*
K' 0 . 8 2 3
1
7 2 i n c l u d i n g '3 V3 pseudo p o t . /17/1
IL i i
Na;
( 3 . 5 5 ) 3.60 3.60+0.04 3.60T0.05 -
I n t h e f i r s t two l i n e s o f t a b l e 1 a r e r e p o r t e d ab i n i t i o r e s u l t s f o r t h e ground s t a t e o f ~ i $ . When c o r e p o l a r i s a t i o n e f f e c t s a r e n e g l e c t e d /43/ t h e bond l e n g t h i s overestimated. I n c o n t r a s t , ab i n i t i o c a l c u l a t i o n s c o r r e c t e d f o r c o r e p o l a r i s a t i o n / I 2 7 p r o v i d e a bond l e n g t h i n b e t t e r agreement w i t h experiment.
123 118 120.8+0.8 -
( 0 . 9 0 4 ) 0.801 0.794 ( ? ) 0.789 ( ? )
The importance o f t h e i n c l u s i o n o f t h e c r o s s p o l a r i s a t i o n term V3 i s c l e a r a l s o f r o m t a b l e 1 e s p e c i a l l y f o r KS. When t h e a c t i v e e l e c t r o n and t h e c o r e A ( r e s p
B)
s i m u l t a - n e o u s l y p o l a r i s e t h e c o d B ( r e s p ( A ) ) t h e r e s u l t i n g f o r c e i s t h e sum o f t h e twop o l a r i s a t i o n f o r c e s and depends upon Bg (see F i g . 1 ) . When we n e g l e c t t h e t e r m V3, we assume t h a t t h e two f o r c e s a r e c o l 1 in e a r so we overestimate t h e polarisation forces and hence t h e w e l l depth.
Method ab i n i t i o /43/
ab i n i t i o + c o r e p o l a r i s a t i o / 12 n e g l e c t i n g V3
i n c l u d i n g V3 pseudo p o t . /17/
V3 model p o t . /18/
i n c l u d i n g V Q exp / 9 / exp
/ l o /
exp /11/
'3 pseudo p o t . /17/
i n c l u d i n g V3
::zj:i:::g ::
model p o t . /18/exp /13/
exp /14/
The agreement between t h e o r y and experiment i s s a t i s f a c t o r y when V3 i s i n c l u d e d . I n t h e case o f t h e K$ ion, t h e v a l u e o f t h e experimental w e l l depth depends upon t h e w e l l depth i n t h e ground s t a t e o f t h e K2 molecule, which i s n o t known a c c u r a t e l y a present. Nevertheless, t h e experimental accuracy i s not y e t reached, and further work including a more precise f i t t i n g of t h e cut-off radius Pd, possibZy on atomic photoionisation cross s e c t i o n s , i s needed.
we(cm-1 ) 263.5
268 283 262.221.5
26 0 119 De(ev)
1.280 1 .293 ( 1 . 3 2 ) 1.30 ( 1 . 3 0 )
1.28 1.27 -1- 0.02 1.2980-1-0.0007 1 .283620.003 I ( 2 . 0 2 )
0.993
4.44 4.4 4.60 3.127 3.099 ( 3 . 0 6 )
3.08 ( 3 . 0 9 )
3.09 3.11+0.01 - ( 3 . 2 8 )
3.59
7 5 72.5 73.4~0.5
'3 model p o t .
/ l a /
i n c l u d i n g V3 exp /15/
exp /16/
C1-54 JOURNAL DE PHYSIQUE
5
-
TREATMENT OF THE TWO ELECTRON PROBLEM : APPLICATION TO ALKALI DIMERSF i g . 12
-
C o o r d i n a t e systemI n t h e case o f two e l e c t r o n s moving i n t h e f i e l d o f two a l k a l i c o r e s A and B, i t i s necessary t o t r e a t s i m u l t a n e o u s l y
a ) t h e i n t e r a c t i o n of t h e e l e c t r o n s w i t h t h e two c o r e s b ) t h e b i e l e c t r o n i c i n t e r a c t i o n
It i s beyond t h e scope o f t h e p r e s e n t paper t o r e v i e w a l l t h e t h e o r e t i c a l work on t h a t s u b j e c t ; an e x t e n s i v e b i b l i o g r a p h y i s a v a i l a b l e f o r L i z and Na2 /19,20/. I t seems t h a t f o u r k i n d s o f t r e a t m e n t a r e p o s s i b l e nowadays, a c c o r d i n g t o t h e methods used t o deal w i t h t h e two problems i n a ) and b )
5.1 - E f f e c t i v e o p e r a t o r method w i t h c o n f i g u r a t i o n i n t e r a c t i o n
The method d e s c r i b e d i n 54 f o r t h e m o l e c u l a r i o n s has been extended t o two e l e c t r o n systems. The model p o t e n t i a l t r e a t m e n t has been g e n e r a l i s e d by D a l g a r n o ' s group /25, 49/ w h i l e a p s e u d o - p o t e n t i a l t r e a t m e n t o f a l k a l i dimers has been developped f i r s t by B a r d s l e y and h i s coworkers /26,50/, t h e n by Valance e t a l , who t r e a t e d t h e e x c i t e d s t a t e s /51/. The t o t a l h a m i l t o n i a n i s w r i t t e n :
H = h ( 1 ) + h ( 2 ) + v ( l , 2 ) - VAE< ( R )
*
(15) where h ( j ) i s t h e one e l e c t r o n o p e r a t o r d e f i n e d i n (13)
h ( 5 ) =
T
+ VA ( r ~ j ) + V B ( ~ B ~ ) + V B ( ~ , j;Ai, l B i ) + VAB+(R) ( 1 6 ) and d e s c r i b e s t h e m o t i o n o f t h e a c t i v e e l e c t r o n i i n t h e f i e l d o f t h e two cores.Core p o l a r i s a t i o n e f f e c t s a r e i n c l u d e d and i t i t s c l e a r f r o m t h e p e r t u r b a t i v e t r e a t - ment of B o t t c h e r and Dalgarno /29/ t h a t a l l t h e one e l e c t r o n o p e r a t o r s must be r e p l a - ced by e f f e c t i v e o p e r a t o r s , depending of a cut-off r c d i u s t y p i c a l of t h e core
dimensions. F o r i n s t a n c e , t h e d i p o l e moment i s m o d i f i e d /28/
( 1 7 )
The d i e l e c t r i c c o r r e c t i o n , f i r s t i n t r o d u c e d by Chisholm and Opik /53/, comes f r o m t h e non a d d i t i v i t y o f t h e two p o t e n t i a l s f o r t h e p o l a r i s a t i o n o f one c o r e by t h e two e l e c t r o n s s i m u l t a n e o u s l y , and i s a c r o s s p o l a r i s a t i o n t e r m a s V3. I t has t o be i n c l u d e d i n a c c u r a t e c a l c u l a t i o n s o f t h e a f f i n i t y o f t h e a l k a l i atom /54/. The method i s simple because t h e core electrons are not e x p l i c i t l y included : t h i s can be done provided one m y neglect t h e valence e x c i t a t i o n energies compared t o t h e core e x c i t a t i o n energies / 2 9 / . The r e s u l t s depend upon t h e c h o i c e o f an a r b i t r a r y c u t - o f f r a d i u s r d , which has t o be chosen unanbiguously.
A f t e r s o l v i n g t h e S c h r o d i n g e r e q u a t i o n f o r t h e m o l e c u l a r i o n problem :
( 1 9 )
one performs a s t a n d a r d c o n f i g u r a t i o n i n t e r a c t i o n t r e a t m e n t by expanding t h e t o t a l w a v e f u n c t i o n on a n t i s y m m e t r i z e d p r o d u c t s o f monoel e c t r o n i c o r b i t a l s s o l u t i o n s o f ( 1 3 )
Z [dl,)
:2
LL bf Y,"^(Q
?rtb(2) and computing t h e m a t r i x elements o f0
(1,2).Such a method has given good r e s u l t s , b u t i n many e x i s t i n g treatments e i t h e r t h e cross polarisation term V3 or the d i e l e c t r i c t e r n Vdiel i s neglected.
5.2 - E f f e c t i v e o p e r a t o r method u s i n g c o r r e l a t e d o r b i t a l s
R e c e n t l y m e t h o d have been proposed which t r e a t t h e one e l e c t r o n problem as i n t h e p r e c e e d i n g case b u t improve t h e c a l c u l a t i o n o f t h e two e l e c t r o n c o r r e l a t i o n . Preuss' group has devel opped a s o p h i s t i c a t e d t r e a t m e n t o f c o r e p o l a r i s a t i o n e f f e c t s , i n c l u - d i n g V3 and V d i e l / 5 2 / . A non l o c a l p s e u d o p o t e n t i a l i s f i t t e d t o t h e e x p e r i m e n t a l v a l e n c e energy o f t h e i s o l a t e d atoms, and t h e two e l e c t r o n problem i s s o l v e d i n t h e framework o f t h e d e n s i t y f u n c t i o n a l formalism, t h e charge d e n s i t y o f valence e l e c t r o n s w i t h s p i n t and
-
r e s p e c t i v e l y b e i n g n o t n e c e s s a r i l y i d e n t i c a l . fie i n t e - r e s t of such a method i s t h a t ii has e a s i l y been generalised t o a l k a l i c l u s t e r s . A n o t h e r approach /55/ t r e a t s t h e one e l e c t r o n problem as i n t h e p r e c e e d i n g case, b u t improves t h e c a l c u l a t i o n o f t h e two e l e c t r o n c o r r e l a t i o n owing t o a g e n e r a l i s a - t i o n o f P l u v i n a g e ' s t r e a t m e n t f o r t h e he1 ium atom /56,57/. E q u a t i o n ( 2 0 ) i s m o d i f i e di n which u i k ( r 1 2 ) d e s c r i b e s t h e m o t i o n o f t h e two e l e c t r o n s i n t e r a c t i n g t h r o u g h t h e l / r l 2 p o t e n t i a l i n t h e lack of an m t e r n a l f i e l d . Such a method should be valuable for t h e treatment of t h e e x c i t e d s t a t e s . It has a l r e a d y g i v e n p r o m i s i n g r e s u l t s f o r t h e ground s t a t e o f Na2 and K2, u s i n g t h r e e c o r r e l a t e d c o n f i g u r a t i o n s o n l y , b u t o n l y f u t u r e work w i l l t e l l whether t h e improvement compared t o s t a n d a r d c o n f i g u r a t i o n i n t e r a c t i o n t e c h n i q u e s i s s i g n i f i c a n t .
5.3
-
Ab i n i t i o t r e a t m e n tVery s o p h i s t i c a t e d m u l t i c o n f i g u r a t i o n t r e a t m e n t s have been developped f o r t h e a l k a l i dimers, e s p e c i a l l y by Konowalow and coworkers /58,59/. I t seems a t p r e s e n t t i m e t h a t ab i n i t i o methods have converged t o t h e i r o p t i m a l r e s u l t , which, as f i r s t demonstra- t e d b y Rosmus and Meyer /60/, s y s t e m a t i c a l l y o v e r e s t i m a t e s t h e equilibrium distance for t h e ground s t a t e potential curve due t o a n e g l e c t of core poZarisation e f f e c t s .
5.4 - Ab i n i t i o t r e a t m e n t m o d i f i e d f o r c o r e p o l a r i s a t i o n e f f e c t s
The most i m p r e s s i v e p r o g r e s s has been r e a l i s e d i n t h e l a s t two y e a r s t h r o u g h t h e i n t r o d u c t i o n o f c o r e - v a l ence c o r r e l a t i o n e f f e c t s i n t h e ab i n i t i o t r e a t m e n t . I n Toulouse /61,62,23/ t h e p e r t u r b a t i v e t r e a t m e n t o f /29/ has been r e c o n s i d e r e d , and a method proposed i n which t h e v a l e n c e e x c i t a t i o n energy i s no more n e g l e c t e d w i t h r e s p e c t t o t h e c o r e e x c i t a t i o n energy. The d i v e r g e n c e o f t h e r - 4 o p e r a t o r s i s sup- pressed, and i t i s no l o n g e r necessary t o i n t r o d u c e a c u t - o f f r a d i u s . The f i n a l r e s u l t f o r t h e second o r d e r c o r r e c t i o n i s expressed i n terms o f t h e c o r e p o l a r i s a - b i l i t y , t h e e l e c t r i c f i e l d c r e a t e d by t h e s t a t i c and t r a n s i t i o n d i s t r i b u t i o n s (a by p r o d u c t o f t h e g r a d i e n t a l g o r i t h m i n s t a n d a r d m o l e c u l a r programs), t h e v a l e n c e e x c i - t a t i o n e n e r g i e s and a mean energy t y p i c a l o f c o r e e x c i t a t i o n . Such a method has provided e x c e l l e n t r e s u l t s for Nu2 / 2 3 / using an empirical value for t h e core polari- s a b i l i t y s l i g h t l y (7%) smaZler than t h e experimental, one. The e x p e r i m e n t a l c u r v e s i n t h e ground s t a t e and t h e f i r s t e x c i t e d s t a t e s a r e reproduced w i t h i n a few wavenumbers accuracy.
We s h o u l d m e n t i o n a l s o r e c e n t work o f Konowalow and F i s h on L i 2 /24/ i n which
C1-56 JOURNAL DE PHYSIQUE
B a r d s l e y ' s pseudo p o t e n t i a l /26/ has been i n c l u d e d i n an ab i n i t i o t r e a t m e n t , n e g l e c t i n g t h e V3 t e r m and hence o v e r e s t i m a t i n g t h e c o r e p o l a r i s a t i o n e f f e c t s ; composite c u r v e s a r e p r e s e n t e d which a r e i n t e r m e d i a t e between ab i n i t i o c a l c u l a t i o n s /58/ and such r e s u l t s , t h e two c a l c u l a t i o n s b e i n g c o n s i d e r e d as upper and l o w e r l i m i t f o r c o r e p o l a r i s a t i o n c o r r e c t i o n s .
The r e s u l t s o f t h e f o u r k i n d s o f methods, c o n c e r n i n g t h e ground s t a t e o f t h e a l k a l i dimers, a r e summarised on T a b l e 2 , and i t i s c l e a r t h a t t h e agreement between t h e o r y and experiment i s v e r y good.
TABLE 2
M o l e c u l a r c o n s t a n t s f o r t h e ground s t a t e o f t h e a l k a l i d i m e r s
~ e ( 1 0 3 cm-I) ~ e ( 8 we(cm-l) ~ e t h o d * Reference
7.990 2.70 365.8 A /49/
8.06 2.73 349 B /52/
8.297 2.692 347.1 C /58/
8.501 2.672 350 D /24/
8.5214 2.673 350 E /63/
8.5169 E /21/
5.72 3.04 163 A /50/
5.88 2.96 A /51/
6.45 3.05 165 B /52/
5.64 3.07 159 B /55/
5.72 3.17 C /59/
6.020 3.05 D /23/
6.024 E /20/
6.0226 E /22/
3.06 3.55 A /51/
5.16 3.86 99 B /52/
4.27 3.85 9 7 B /55/
3.95 3.74 8 2 D /62/
4.19 3.905 92.12 E /64/
*
A e f f e c t i v e p o t e n t i a l w i t h c o n f i g u r a t i o n i n t e r a c t i o nB e f f e c t i v e p o t e n t i a l w i t h e x p l i c i t t r e a t m e n t o f v a l e n c e c o r r e l a t i o n C ab i n i t i o c a l c u l a t i o n s
D ab i n i t i o c o r r e c t e d f o r c o r e - v a l e n c e c o r r e l a t i o n e f f e c t s E e x p e r i m e n t
CONCLUSION
The i n t e n s e t h e o r e t i c a l a c t i v i t y s t i m u l a t e d by t h e accuracy o f e x p e r i m e n t a l r e s u l t s o b t a i n e d owing t o l a s e r t e c h n i q u e s has l e d t o some p r o g r e s s i n t h e d e s c r i p t i o n o f systems w i t h one and two a c t i v e e l e c t r o n s .
Concerning a1 k a l i - r a r e gas systems, model p o t e n t i a l and pseudo p o t e n t i a l techniques, i n which t h e c o r e e l e c t r o n s a r e s i m u l a t e d by an e f f e c t i v e p o t e n t i a l , p r o v i d e a s a t i s f a c t o r y d e s c r i p t i o n o f t h e p h y s i c s as f a r a s t h e i o n i s a t i o n e n e r g i e s a r e concer- ned. The r a r e gas then a c t s as an a d d i t i o n n a l quantum d e f e c t . A t s h o r t i n t e r n u c l e a r d i s t a n c e s , t h e a c c u r a t e e v a l u a t i o n o f t h e c o r e - c o r e i n t e r a c t i o n i s s t i l l an open problem. The g e n e r a l i s a t i o n o f t h e method t o e x c i t e d r a r e g a s - r a r e gas i n t e r a c t i o n s demands a p r e c i s e t r e a t m e n t o f t h e c o r e - c o r e term.
E f f e c t i v e p o t e n t i a l methods have l o n g been s u c c e s s f u l i n t r e a t i n g a l k a l i m o l e c u l a r i o n s . Core p o l a r i s a t i o n e f f e c t s a r e i n t r o d u c e d t h r o u g h e f f e c t i v e o p e r a t o r s which c o n t a i n a c u t - o f f f u n c t i o n t o a v o i d d i v e r g e n c e a t small r. The r e s u l t s a r e s e n s i t i v e t o t h e c h o i c e o f t h e c u t - o f f r a d i u s , and an unambiguous way o f d e t e r n i n i n g t h i s parameter, p o s s i b l y by f i t t i n g t o p h o t o i o n i s a t i o n c r o s s - s e c t i o n s , s h o u l d b e d e v e l o p p d . The g e n e r a l i s a t i o n t o two a c t i v e e l e c t r o n systems such as a1 k a l i dimers i s performed e i t h e r by c o n f i g u r a t i o n i n t e r a c t i o n t e c h n i q u e s o r by t h e use o f c o r r e l a t e d f u n c t i o n s . A comparison o f t h e two approaches f o r t h e t r e a t m e n t o f e x c i t e d and Rydberg s t a t e s w i l l b e p o s s i b l e i n t h e n e x t f u t u r e . The d e n s i t y f u n c t i o n n a l f o r m a l i s m has been g e n e r a l i s e d t o t h e t r e a t m e n t o f c l u s t e r s .
Ab initio methods g e n e r a l l y p r o v i d e a n o v e r e s t i m a t i o n o f t h e bond l e n g t h , due t o a n e g l e c t o f c o r e p o l a r i s a t i o n e f f e c t s . A m a j o r improvement has been o b t a i n e d i n t h e l a s t two y e a r s w i t h t h e i n t r o d u c t i o n o f an e x p l i c i t second o r d e r t r e a t m e n t o f c o r e - v a l e n c e c o r r e l a t i o n i n t h e s t a n d a r d codes. Such a method a v o i d s t h e d i f f i c u l t i e s r a i s e d by t h e presence o f a c u t - o f f r a d i u s i n t h e e f f e c t i v e o p e r a t o r method. A s p e c t a c u l a r agreement i s o b t a i n e d between t h e o r y and experiment f o r t h e ground and f i r s t e x c i t e d s t a t e s o f Na2. The comparison o f t h i s method w i t h e f f e c t i v e o p e r a t o r t e c h n i q u e s f o r t h e t r e a t m e n t o f t h e h i g h l y e x c i t e d s t a t e s o f t h e dimers and o f t h e c o l l i s i o n a l i o n i s a t i o n o f t h e a l k a l i s appears as an a p p e a l i n g p r o s p e c t f o r f u t u r e work.
ACKNOWLEDGEMENTS
The a u t h o r wishes t o thank Pr. R. M c C a r r o l l and D r . P. V a l i r o n f o r t h e Kany y e a r s - c o l l a b o r a t i o n on t h e model p o t e n t i a l method, and Pr. A. Dalgarno f o r some en1 i g h t e - n i n g d i s c u s s i o n s . C o n t r i b u t i o n s o f A. Chebanier de Guerra, M. P h i l i p p e , M.E. Dolan, D. Hennecart and A.L. Roche t o v a r i o u s p a r t s o f t h e work p r e s e n t e d here a r e g r a t e f u l l y acknowledged. P a r t i c u l a r thanks a r e due t o A. H e n r i e t , M. Aubert-Frecon and C.
Le Sech f o r t h e v e r y s t i m u l a t i n g c o l l a b o r a t i o n on t h e a l k a l i dimers problem, and t o Pr. P. P l u v i n a g e f o r h i s i n t e r e s t i n t h e work. The a u t h o r i s t h a n k f u l t o S. Sandmeier, M. Soyez, 0 . Chauveau f o r t h e i r h e l p i n t h e p r e p a r a t i o n o f t h e m a n u s c r i p t .
References
1. Ahmad B i t a r R., L a p a t o v i c h W.P., P r i t c h a r d D.E., Renhorn I. Phys. Rev. L e t t . 39 (1977) 1657
2. T e l l i n g h u i s e n J., Ragone A., Myung S.Kim, Auerbach D.J., Smalley R.E., Wharton L., Levy D.H. J. Chem. Phys. 71 (1979) 1283
3. Masnou-Seeuws F., P h i l i p p e M., Val i r o n P., Phys. Rev. L e t t . 41 (1978) 395 4. Hanssen J., M c C a r r o l l R., V a l i r o n P., J . Phys. B 12 (1979) 899
5. Val i r o n P., Gayet R., R c C a r r o l l R., Masnou-Seeuws F., P h i l i p p e M., J. Phys. B 12 (1979) 53
6. Chebanier de Guerra A., Masnou-Seeuws F. Proc. 1 3 t h ICPEAC B e r l i n (1983) 314 and s u b m i t t e d t o J . Phys. B.
7. Pascale J . Phys. Rev. A 28 (1983) 632
8. York G., Scheps f?., G a l l a g h e r A., J. Chem. Phys. 63 (1975) 1C52
C1-58 J O U R N A L DE PHYSIQUE
9 . Mathur B.P., Rothe E.W., Reck G.P., Ligt-tman A.J., Cher. Phys. L e t t . 56 (1978) 336 10. Bernheim R.A., G o l d L.P., T i p t o n T. 11. Chem. Phys. 78 (1983) 2625
11. E i s e l D., DemtrBder W., Chem. Phys. L e ~ i . 8 8 (1982) 481
12. E i s e l D., Demtroder W., M u l l e r Id., Botschwina P., Chemical P h y s i c s 8 0 (1983) 329 13. C a r l s o n N.W., T a y l o r A.J., Schawlow A.L.; Phys. Rev. L e t t . 45 (1980) 1 8
14. M a r t i n S., Chevaleyre J., Bordas C., V a l i g n a t S., B r o y e r M., Cabaud B., Hoareau A. Chem. Phys. L e t t . 8 7 (1982) 235
15. L e u t w y l e r S., Hermann A., W'dste L., Schurnacher E. Chemical P h y s i c s 48 (1980) 253
16. B r o y e r M., Chevaleyre J., D e l a c r e t a z G., M a r t i n S., Woste L. Chem. Phys. L e t t . 99 (1983) 206
17. Fuentealba P., Preuss H., S t o l l H., von S z e n t p a l y L. Chem. Phys. L e t t . 8 9 (1982) 418
18. H e n r i e t A., Masnou-Seeuws, F. Chem. Phys. L e t t . 101 (1983) 535 19. Hessel M.M., V i d a l C.R. J . Chem. Phys. 70 (1979) 4439
20. Verna K.K., Bahns J.T., R a j a e i - R i z i A.R., S t w a l l e y W.C., Zemke W.T. J . Chem. Phys.
76 (1983) 3599
21. Verges J . , B a c i s R.,Barakat P . , C a r r o t P . , Churassy S., C r o z e t P., Chem.Physics L e t t e r s 98 (1983) 203
22. Barrow R.F., Verges J . , E f f a n t i n C., Hussein K., D ' I n c a n J . i b i d 104 ( 1 9 8 4 ) 179 23. Jeung G. H. J. Phys. B 16 (1983) 4289
24. Konowalow D.D., F i s h J.L. Chemical P h y s i c s 8 4 (1984) 463
25. Dalgarno A. Atomic Physics v o l . 4 (New York Plenum Press) (1975) 325
26. B a r d s l e y J.N., Case s t u d i e s i n atomic p h y s i c s v o l . 4 (Amsterdam N o r t h H o l l a n d ) (1974) 299
27. Veselov M.G., S c h t o f f A.V. Opt. Spectrosc. 22 (1967) 457
28. Caves T.C., Dalgarno A. J. Quant. S p e c t r . Fad. T r a n s f 12 (1972) 1539 29. B o t t c h e r C., Dalgarno A. Proc. R. Soc. A 340 (1974) 187
30. H e n r i e t A . , These de 3e c y c l e (1983) Orsay u n p u b l i s h e d 31. K l a p i s c h M. Comput. Phys. Comm. 2 ( 1 9 7 1 ) 239
32. B o t t c h e r C. J . Phys.0 4 ( 1 9 7 1 ) 1140 33. Peach G. J. Phys.B 11 (1978) 2107
34. Val i r o n P., Roche A.L., Masnou-Seeuws F. and Dolan M.E. J. Phys.B (1984) i n p r e s s 35. Fermi E. Nuovo Cim. 11 (1934) 157
36. Smirnov B.M. Sov. Phys. JETP 24 (1967) 314 Ivanov O.K. Opt. Spectrosc.40 (1976) 554
37. Dolan M.E., Masnou-Seeuws F. J . Phys.B 14 (1981) L 583
38. Masnou-Seeuws, F., Boulmer J., M a u r i n T., Roche A.L., V a l i r o n P.;J. Phys.B 15 (1982) 2989
39. Sondergaard N.A., Mason E.A. J. Chem. Phys. 62 (1975) 1299
40. Lanm D.R., Thackston M.G., E i s e l e F.L., E l l i s H.U., T w i s t J.R., Pope U.P., G a t l a n d I . R . , FlcDaniel E.W. J. Che~i. Phys. 74 (1981) 3042 41. Duren R., T i s c h e r H. Chec. Phys. L e t t . 79 (1981) 481
42. Duren R., t i a s s e l b r i n k E., M o r i t z G. Z. Phys. A 307 ( 1 9 8 2 ) l l 43. Konowalow D.D., Rosenkrantz 14.E. Chem. Phys. L e t t . 61 (1979) 489 44. H e n r i e t A. t o be p u b l i s h e d
45. W e i s h e i t J.A. Phys. Rev. A 5 (1972) 1621 46. B o t t c h e r C. J. Phys. B 6 (1973) 2368
47. Colomb de Daunant I., V a s s i l e v G., Baudon J. and S t e r n 0. J o u r n a l de Physique 4 3 (1982) 591
48 Hennecart D., M a s n o u ~ ~ e e u w s F. (1984) s u b m i t t e d t o J . Phys.B
49. Watson D.K., C e r j a n C.J., Guberman S., Dalgarno A. Chem. Phys. L e t t . 50 (1977) 181
B a r d s l e y J.N., J u n k e r B.R., Norcross D.W. Chen. Phys. L e t t . 37 (1976) 502 Valance A. and Mguyen Tuan Q. J. Phys.6 15 (1982) 17
F l a d J., I g e l G., Dolg K . , S t o l l H., Preuss P. Chetxical P h y s i c s 75 (1983) 331 Chisholm C.D.M., % p i k O., Proc. Roy. Soc. 8 3 (1964) 541
N o r c r o s s D.W. Phys. Rev. L e t t . 32 (1973) 192
55. H e n r i e t A., Aubert-Frecon M., Le Sech C., Masnou-Seeuws F. (1984) J . Phys. B i n p r e s s
56. P l u v i n a g e Ph. Ann. Phys. 5 (1950) 145
P l u v i n a g e Ph. J o u r n a l de Physique 1 2 ( 1 9 5 1 ) 789
Aubert-Frecon M . , Le Sech C. J . Chem. Phys. 74 (1981) 2931 Konowalow D.D., Olson
M.L.
J . Chem. Phys. 71 (1979) 450Konowalow D . D . , Rosenkrantz M.E., Olson M.L. J . Chem. Phys. 72 (1980) 2612 Rosmus P., Meyer W. J . Chem. Phys. 65 (1976) 492
Jeung G . H . , Malrieu J . P . , Daudey J.P. J . Chem. Phys. 77 (1982) 3571 Jeung G . H . , Daudey J . P . , Malrieu J.P. J . Phys. B 16 (1983) 699
Huber K . P . , Herzberg G. Constants of Ciatomic Molecules (1979) Van Nostrand Company p 375
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