MOLECULAR POTENTIALS FOR SYSTEMS WITH ONE OR TWO … · JOURNAL DE PHYSIQUE Colloque CI, supplement au n°l, Tome 46, Janvier 1985 page Cl-43 MOLECULAR POTENTIALS FOR SYSTEMS WITH

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MOLECULAR POTENTIALS FOR SYSTEMS WITHONE OR TWO ACTIVE ELECTRONS

F. Masnou-Seeuws

To cite this version:F. Masnou-Seeuws. MOLECULAR POTENTIALS FOR SYSTEMS WITH ONE OR TWOACTIVE ELECTRONS. Journal de Physique Colloques, 1985, 46 (C1), pp.C1-43-C1-59.�10.1051/jphyscol:1985105�. �jpa-00224474�

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JOURNAL DE PHYSIQUE

Colloque CI, 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 et de potent ie l modèle, des résu l ta t s t rès précis ont été obtenus pour les propr ié tés moléculaires des systèmes à un é lect ron a c t i f comme les couples a lca l in -gaz ra re . Pour t r a i t e r les dimères a l c a l i n s e t leurs ca t ions , i l est important de décr i re avec précis ion les e f f e t s de po la r i sa t i on de cœur .

Abstract - Owing to model potential and pseudo-potential methods, very accura te resu l t s have been obtained for the molecular propert ies of systems with one act ive electron such as a lka l i rare-gas couples. In order to t r e a t the a lka l i dimers and the i r ca t ions , i t is important to perform an accurate descr ipt ion of core polarisat ion e f fec t s .

I - INTRODUCTION

Owing to laser techniques, much experimental information is now avai lable concerning the ground and excited s t a tes of molecules and the dynamical or optical couplings between those s t a t e s . In many cases, the molecular energies are known with an accuracy be t te r than 1 cm~1. Such a s i tuat ion is a challenge for theore t ic ians .

For many systems, the standard methods of quantum chemistry, using extensive configuration in terac t ion , have succeeded in predicting accurate potential curves. However, i t i s generally d i f f i cu l t to reach an accuracy comparable to the experimental one. In the par t icular case of systems with one or two act ive e lec t rons , model potential and pseudo-potential ca lcu la t ions , in which the effect of the rapid core electrons on the motion of the slow outer(s) electron(s) is simulated by an effective potent i a l f i t t ed on accurate atomic data, sometimes provide an excellent accuracy.

Concerning a l k a l i - r a r e gas systems, i t i s seven and f ive years since a 1 cm"' accuracy has f i r s t been obtained in the laser spectroscopy of the ground s ta te of NaNe / 1 / and NaAr / 2 / . In the same period, a similar theoret ical accuracy has been reached by model potential calculat ions / 3 , 4 / using the method developped by Valiron et a l . / 5 / .

Fig. 1 - Model potential calculat ions for the ground s ta te of NaNe / 3 / compared with laser 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

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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 represented 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. It 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/ pe r tu rbe rs . I t m i g h t n o t be s u f f i c i e n t l y accu ra te 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 pseudo-potent ia l f o r t h e e-hel ium i n t e r a c t i o n have improved t h e agreement w i t h exper iment (see F i q . 2 ) .

R l a u l

F i g . 2 - Normal ised emiss ion spectrum o f t h e Na3p s t a t e i n t h e presence o f He a t 400 K. Dashed 1 i n e , exper imenta l da ta 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 s o f Pascale /7/, dash-dot ted 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

'"" desc r i bed i n / 5 / . ( f r om 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 ta tes , t h e accuracy depends upon t h e e s t i m a t i o n o f t h e core-core i n t e r a c t i o n a t smal l i n t e r n u c l e a r d is tances, as we s h a l l d i scuss l a t e r on.

Du r i ng t h e l a s t t h r e e years , t h e accuracy i n t h e de te rm ina t i on 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 imer 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 ex tens i ve 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 d imers u s i n g two- photon techn iques (see Ref. / 9/ t o /16 / ) . 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 impor tance 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/.

Dur ing t h e same pe r i od , 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 imers has reached an accuracy o f a few wave numbers (see /19/, /20/ and re fe rences t h e r e i n ) . The most r e c e n t de te rm ina t i on 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 exper imenta l accuracy : Jeung /23/ o b t a i n s a w e l l dep th 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/ es t ima te 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 rea tmen ts 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 im i s t o p resen t some o f t h e r e s u l t s ob ta ined r e c e n t l y , and t o d i scuss how i n t h e n e x t f u t u r e very accu ra te r e s u l t s c o u l d be ob ta ined f o r t h e e x c i t e d s ta tes .

2 - EFFECTIVE POTENTIALS FOR THE ALKALI ATOMS : APPLICATION TO ALKALI-RARE GAS SYSTEMS

The b a s i c idea o f model p o t e n t i a l and pseudo-potent ia 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 Schrad inger equa t i on i n o r d e r t o desc r i be t h e mot ion 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 co res A and B a t a d i s t a n c e R

F i g . 3 - Coord inate system

We therefore must solve the equation : X x $4

H r L = [ ~ + v ~ I r ~ ) + ~ ~ ( r ~ ) + " ~ ( ~ ~ ~ ~ r ~ ) ] ~ i = i I (1

In ( 1 ) T i s the k ine t ic energy, VA(rA) and VB(rB) a r e the e f f e c t i v e po ten t ia l s f o r the in te rac t ion of the electron with the cores A and B respect ively, V3 i s a three- body tensorial term which depends upon t h e angle eRand comes from the addi t ion of the two e l e c t r i c dipoles induced in B by A and e- respect ively.

2.1 .- Ref i n i t i o n of VA(rA), e f f e c t i v e potent ial f o r t h e a l k a l i atoms

In most cases , VA(rA) i s a parametric potent ial f i t t e d t o p roper t i es of the i so la ted e-A system. The atomic o r b i t a l s of the 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 in V A so t h a t the calculated energies ~Y'a re a s c lose a s possible 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 contains v i r tua l core o r b i t a l s besides the valence o r b i t a l s . The atomic funct ions therefore possess the cor rec t number of nodes. In a pseudo-potential treatment /26 / , V A ( r ~ ) i s repulsive in the core 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 representat ion of the Na 3s o r b i t a l in model and i n pseudo- potent ial ca lcu la t ions .

The f i r s t question one could ask i s then : which i s t h e t m e potent ial ? We shal l show here below t h a t i t i s a wrong quest ion.

Indeed, t h e def in i t ion of an e f f e c t i v e potent ial comes from an ad iaba t ic separation between the motion of the rapid (core) e lec t rons and of the slow (valence) e lec t rons . Calling r l ... r s l the posi t ions of the core e lec t rons , and rA the posi t ion of the valence e lec t ron , we may wr i te /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 A

o r - - -2-4 [:A) $js) ( 3 )

In ( 3 ) xo is the wavefunction f o r t h e motion of t h e core e lec t rons f o r a f k e d position rn of the valence electron. The ad iaba t ic equation :

C 1-46 JOURNAL 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 desc r i bes t h e mot ion o f t h e s low e lec - t r o n i n t h e rnecin potential of the core.The p h y s i c a l i deas 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 sepa ra t i on i n mo lecu la r phys ics . The non a d i a b a t i c c o r r e c t i o n : ,.

i s smal l , and l i m i t e d t o t h e c o r e reg ion. I t has been computed through 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 the core region t h e wavefunct ions 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 rom t h e "exac t " wave- f u n o t i o n s . The phys i ca l ideas 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 theory . 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 su re 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 wavefunct ions have a c o r r e c t quantum d e f e c t outside t he core. As i n quantum d e f e c t t heo ry , VA(rA) i s independent o f t h e energy o f t h e l e v e l cons 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 va lence wavefunct ion i s t o vary 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 tomic ene rg ies i n ( 2 ) a r e as c l o s e as p o s s i b l e f rom t h e exper imenta l energ ies . B o t t c h e r and Dalgarno /29/, u s i n g a 2nd o r d e r per - t u r b a t i o n theory , 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 shou ld 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 :

A I n (6), ad 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 be ing 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 ivergence i n t h e co re reg ion, r d and rq be ing 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 shou ld 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 reatment , t h e proper - 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 as an e f f e c t i v e ope ra to r .

2) The behav iour 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 en te rs 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 lack box". The parameters rd and rq must then b e f i t t e d on exper imenta l data.

I n f a c t , t h e c a l c u l a t e d energ ies a r e n o t ve ry s e n s i t i v e t o t h e cho i ce o f such parameters /30/. Va r i ous au tho rs 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 fo rmulas f o r t h e p o t e n t i a l , have succeeded i n rep roduc ing 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 accuracy / S t / .

I n c o n t r a s t , t he 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 ross -sec t i ons 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 de te rm ina t i on 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 exper imenta l p h o t o i o n i s a t i o n c ross -sec t i ons . The inf luence 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 Vg ( rg ) and o f Vg

Vg(rg) rep resen ts 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 data /32/ which a t p resen t t i m e a r e known ve ry a c c u r a t e l y . i l e may use t h e exnress ion :

i n wi7icI1 Vsr i s a s h o r t range te rm ( 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 and

c i ' B = aB - 6sB a r e rhe 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 he core B. The computed phase -sh 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 the cho ice o f such parameters. Besides, as was s a i d i n t he i n t r o d u c t i o n , t h e s h o r t range term 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 the ~ o l e c u l a r energ ies , a l s o depend upon t h e cho ice o f t he c u t - o f f paraneters :

?g (see f i g . 1 ) i s t h e ang le (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 impor tan 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 behav iour o f VA(rA). f i e rare gas 3 may then be considered as introducing an additionnu2 R dependent quantum de fec t on the a l k a l i atomic wavefunctions. On F ig . 5 and 6 we have represented bo th t h e atomic wave func t im + 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 wavefunct ion $&=O(rA,~) s o l u t i o n o f (I) o a ined 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.u.) A F ig . 5 ( f r o n Ref. 34) - S o l i d l i n e : sodium atorcic wavefunct ion +m=O(r ) as a func- t i o n o f t h e d i s tance rA on t h e q u a n t i z a t i o n ax i s . Dashed l i n e : 6' n o f e c u l a r wavefunct ion 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 he 6Psobium center.%

No-Ne

i'l

6.: 'L.V 28.3

rA(a .u . )

F ig . 6 - Same a s F i g . 5 i n case of a Ne p e r t u r b e r .


It i s c l e a r f r om F i g . 5 and 6 t h a t :

a ) t h e mo lecu la r wavefunct ion i s ve ry s i m i l a r t o t h e a tomic one, except 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 occup ied 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 wavefunc i ton 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 - Resu l t s f o r t h e i o n i s a t i o n ene rg ies : mo lecu la r quantum d e f e c t s

The 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 an e l e c t r o n i n t h e f i e l d o f t h e two cores A+ and 6 a t d i s t a n c e R f o r a mo lecu la r s t a t e o f symmetry X c o r r e l a t e d t o an a tomic 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 asymp to t i c methods /36/, i t has l ong 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 mo lecu la 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 co r - responding atomic s t a t e can be sca led 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 tomic wavefunct ion 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 assoc ia ted t o it. The q u a n t i t i e s ao (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 fo rmula 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 represented on F i g . 7 t h e i o n i s a t i o n energ ies 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 ene rg ies (see 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 wavefunct ion SO that the rare gas can be considered as "measuring" the atomic probability density a t a given point. The fl i o n i s a t i o n energ ies a r e p ropo r t i onna l t o t h e square o f t h e gradient of t h e a tomic wavefunct ion, which i s a severe check on t h e accuracy of t h i s wavefunct ion.

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 ig . 8, where we have represented 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 mo lecu la r quantum d e f e c t s 6?, w i t h

I n (10) 3 i s t h e Rydberg cons tan 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 cons ide r t h r e e a t c n i c quantum de fec t s , 6S, 6P, and 6-0 f o r R a 2.

F ig . 8 - Mo lecu la 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 curves 6; (see t e x t ) a r e con ta ined i n t h e shadowed areas /6/ , /37/ . The 64 and 6r curves a r e c o r r e l a t e d t o a tomic s t a t e s w i t h Ra2.

It i s c l e a r f rom F ig . 8 t h a t t h e mo lecu la r quantum de fec 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 tomic s t a t e s w i t h Ra2, t h e quantum d e f e c t cu rve s p l i t s i n t o two curves, one of which e x h i b i t s an avo ided c r o s s i n g w i t h t h e ( 8 - 1 ) curve. I t has then 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 induced t r a n s f e r s between t h e Na ((n+l ) p ) and Na nd s ta tes . The exper imenta l measurements concern ing 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 computed 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 conc lus ion, one may say that the accuracy of the ionisat ion energies i s due t o the qual i ty of the atomic wavefunctions and t o the precision of the electron rare gas eZastic phase s h i f t s obtained with the e f f e c t i v e potentials . Such an accuracy i s p o s s i b l e because t h e mo t i on 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 - Resu l t s f o r t h e mo lecu la r energy : problem o f t h e core-core te rm

The ener z l$l o f a m o l e c u l a r I h i > s t a t e i s ob ta ined by adding t o t h e i o n i s a t i o n energy EEXI t h e co re -co re 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 ecu la 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 rm due t o t h e ove r l ap o f t h e asymp to t i c p a r t o f t h e two c o r e wavefunct ions (Obviously, t h e method i s no more v a l i d when t h e a tomic c l ouds 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 cores)

V,,+(R) I 'Vo (R) - (d:/2 6')-(</2 R6) -

B - , I '

I f t h e p o l a r i s a b i l i t i e s ad and a: o f t h e c o r e B a r e we1 1 known, the quantity V o ( ~ ) -which i s generally f i t t e d t o ion-atom scattering data or t o mobility measurements- i s not knom very accurately, so t ha t the 1cm-1 accuracy avaiZabZe for the ionisat ion energies i s obtained for the moZecuZar energies only a t large distances where VAB+ (R) i s l imited t o the polarisation terms. We have shown on F ig . 1 t h e e x c e l l e n t agreement which e x i s t s between theo ry and exper iment concern ing 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 (see ( 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 II s 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 ( ~ ) . Th i s i s i l l u s t r a t e d on F ig . 9 where we have represen- t e d t h e II4p cu rve o f KAr f o r two d i f f e r e n t cho i ces o f VAB+(R)

I I I

- 1 0 0 - KAr A*IT -

C I -500 - -

I J.1 I I

5 6 7 R(a .u . )

F i g . 9 - Mo lecu la r energy c u r v e 2 (see fo rmu la (11) i n t e x t ) o f t h e A2II s t a t e o f KAr/6/, ob ta ined through 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 cho i ces f o r t h e core-core 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 : exper imenta l de te rm ina t i on o f Dijren and T i sche r /41/ f o r tip(^). However, t h e core-core te rm i s common t o a1 1 mo lecu la r curves. I t shou ld be a uable, espec ia l l y when accu ra te exper imenta l i n f o m a t i on i s a v a i l a b l e f o r severa l cu rves f rom c o l l i s i o n o r spectroscopy exper iments, 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 Du ren ' s group /42/ parameters I n Vg ( rg ) , Vg(8,rB) and VAB+(R) have been f i t t e d so t h a t accu ra te exper imenta 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 rom 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 shou ld be 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 curves E$'(R) computed f rom ( 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 IONS

The e f f e c t i v e p o t e n t i a l method has been gene ra l i sea t o open s h e l l cores /46,33,48/. Fo r such sys tens, severa l s t a t e s o f t h e P,B+ molecu la 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 curves ( 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 co re -co re i n t e r a c t i o n /18/. The accurate e s t i m t i o n of the ABf curves i s an open problem a t present time.

(.E, 135000 Ne1Ne

4.1 - Mo lecu la r quantum d e f e c t s

F ig . 9b i s . Hump i n t h e 0; I s 5 p o t e n t i a l cu rve 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 es t ima- t i o n s o f t h e core-core 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 exper iments

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 co re -co re te rm 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. Never the less , t h e phys i ca l s i t u a t i o n d i f f e r s markedly f rom t h e preceed ing 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 cores . 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 wavefunct ion 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 mo lecu la r i o n problem, we may cons ide r t h a t t h e s h o r t range non coulombic t e rm i n t h e two p o t e n t i a l s VA and Vg i n t roduces 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 represented 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 E ( 1 0 ) ) as a B : 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 f mo lecu la r i o n /18/

t h e


Avoided c ross ings 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 depa r t s f r om a pu re coulombic p o t e n t i a l i n t h e co re reg ion , 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? invest igations similar t o the 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 yea rs 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 ong 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 approx imate ly o f co re p o l a r i sa t i on , genera l l y ove res t ima te t h e bond 1 ength /43/.

Recent v e r y accu ra te exper imenta l de te rm ina t i ons , ob ta ined through two step i o n i s a t i o n techn iques / l o -16 / 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 mo lecu la r energ ies by s o l v i n g t h e Schrod inger equa t i on :

VA( r ) i s t h e p o t e n t i a l de f i ned 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 rm i s now deduced f rom (6 )

( 5 , , [ pjrB/G\j ' '7 ( I 4 )

Because o f t h e impor tance o f t h e V3 term, t h e r e s u l t s depend marked ly 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 ig . 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 cho ices o f t h e c u t - o f f r a d i u s rd . Dashed l i n e s : c a l c u l a t i o n s u s i n g ( a ) rd=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 va lue rd=2.35 (see t e x t ) . C i r c l e s : exper imenta l r e s u l t s /13/. Squares : exper imenta l r e s u l t s /14/ .

The de te rm ina t i on o f t h e c u t - o f f appears then as an impor tan t problem. The Veal t e rm i n VA (see ( 6 ) ) depending upon rd , one p o s s i b i l i t y i s t o f it t h e c u t - o f f r a d i u s so t h a t t h e a tomic ene rg ies e igenva lues o f ( 2 ) a r e as c l o s e as p o s s i b l e f rom exper iment. Such a procedure has been used i n r e c e n t pseudopotent ia 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/ has shown t h a t t h e a tomic

energ ies a r e n o t very s e n s i t i v e t o t h e cho ice 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 con t ras t , however, t h e a tomic 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 ross sec t i ons /45/ a r e s e n s i t i v e t o r d . The va lue 2.35 a.u. f i t t e d by Weishe 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 ross -sec t i on leads t o a s a t i s f a c t o r y p o t e n t i a l curve /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 bo th cases, we have i n d i c a t e d i n i t a l i c s the va lues obta ined when n e g l e c t i n g the c ross p o l a r i s a t i o n term Vg i n t he e f f e c t i v e hami l ton ian.

TABLE 1 Ground s t a t e p r o p e r t i e s o f ~i; ~ a ; K;

(0 .92) * K' 0 . 8 2 3 1 72 i n c l u d i n g '3 V3 pseudo po t . /17/ 1 I

L 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 repo r ted 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 core p o l a r i s a t i o n e f f e c t s a re neg lec ted /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 27 p rov ide 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 -

(0.904) 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 ross 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 rom 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 the co re A ( resp B) s imu l ta - neous ly p o l a r i s e t h e c o d B ( resp ( 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 two

p o l a r i s a t i o n f o r c e s and depends upon Bg (see F ig . 1 ) . When we n e g l e c t t h e term V3, we assume t h a t t h e two f o r c e s a r e c o l 1 i n e a r so we overestimate the polarisation forces and hence the well depth.

Method

ab i n i t i o /43/ ab i n i t i o + core 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 po t . /17/

V3 model pot . /18/ i n c l u d i n g V Q

exp / 9 / exp / l o / exp /11/

'3 pseudo pot . /17/ i n c l u d i n g V3

::zj:i:::g :: model pot . /18/

exp /13/ exp /14/

The agreement between theo ry and experiment i s s a t i s f a c t o r y when V3 i s inc luded. I n t h e case o f t h e K$ ion, t h e va lue o f t h e experimental w e l l depth depends upon t h e w e l l depth i n t he ground s t a t e o f t h e K2 molecule, which i s n o t known accu ra te l y a present. Nevertheless, the experimental accuracy i s not yet reached, and further work including a more precise f i t t i n g of the cut-off radius Pd, possibZy on atomic photoionisation cross sect ions, i s needed.

we(cm-1 )

263.5

268

283

262.221.5 26 0

119

De(ev)

1.280 1 .293

(1 .32) 1.30

(1 .30) 1.28

1.27 -1- 0.02 1.2980-1-0.0007 1 .283620.003 I

(2 .02 ) 0.993

4.44

4.4 4.60

3.127 3.099

(3 .06) 3.08

(3.09) 3.09

3.11+0.01 -

(3 .28) 3.59

7 5

72.5 73.4~0.5

'3 model pot . / l a / i n c l u d i n g V3

exp /15/ exp /16/


5 - TREATMENT OF THE TWO ELECTRON PROBLEM : APPLICATION TO ALKALI DIMERS

F i g . 12 - Coord inate system

I 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 cores A and B, i t i s necessary t o t r e a t s imu l taneous 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 co res 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 resen t paper t o rev iew 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 ex tens i ve 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 ea tmen t a r e p o s s i b l e nowadays, acco rd ing 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 ope ra to 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 descr ibed i n 54 f o r t h e mo lecu la 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 ea tmen t has been genera l i s e d by Da lga rno ' s group /25, 49/ w h i l e a pseudo-potent ia l t r ea tmen t o f a l k a l i d imers has been developped f i r s t by Bards ley and h i s coworkers /26,50/, then 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 ope ra to 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) (16)

and desc r i bes t h e mot ion 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 rom 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 ope ra to rs must be r e p l a - ced by e f f e c t i v e operators , depending of a cut-off rcdius typ ica l of the core dimensions. For ins tance, t h e d i p o l e moment i s m o d i f i e d /28/

(17)

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 roduced 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 co re by t h e two e l e c t r o n s s imul taneous ly , and i s a c ross p o l a r i s a t i o n te rm as V3. I t has t o be i n c l u d e d i n accu ra te 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 the 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 the valence exc i ta t ion energies compared t o the core exc i ta t ion energies / 29 / . The r e s u l t s depend upon t h e cho i ce o f an a r b i t r a r y c u t - o f f r a d i u s rd , which has t o be chosen unanbiguously.

A f t e r s o l v i n g t h e Schrod inger equa t i on f o r t h e mo lecu la r i o n problem :

(19)

one performs a s tandard 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 ea tmen t by expanding t h e t o t a l wavefunct ion on ant isymmetr ized products 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 (13)

Z [dl,) : 2 LL b f Y,"^(Q ?rtb(2) and computing t h e m a t r i x elements o f 0 (1,2).

Such a method has given good re su l t s , bu t i n many ex is t ing treatments e i ther the cross polarisation term V3 or the d i e l ec t r i c t e rn Vdiel i s neglected.

5.2 - E f f e c t i v e ope ra to 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

Recent ly me thod 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 preceed ing 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 rea tmen 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 ing V3 and V d i e l / 5 2 / . A non l o c a l pseudopotent ia l i s f i t t e d t o t h e exper imenta l va lence 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 so lved 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 formal ism, 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 be ing n o t n e c e s s a r i l y i d e n t i c a l . fie in te - r e s t of such a method i s tha t ii has eas i ly been generalised t o a l k a l i c lus ters .

Another 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 preceed ing 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 gene ra l i sa - t i o n o f P luv inage ' s t r ea tmen t f o r t h e he1 ium atom /56,57/. Equat ion (20) i s m o d i f i e d

i n which u i k ( r 1 2 ) desc r i bes t h e mot ion 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 th rough t h e l / r l 2 p o t e n t i a l i n the lack of an m te rna l f ie ld . Such a method should be valuable for the treatment of the exci ted s ta tes . It has a l r e a d y g i v e n p rom is ing 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 on 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 tandard c o n f i g u r a t i o n i n t e r a c t i o n techn iques i s s i g n i f i c a n t .

5.3 - Ab i n i t i o t r ea tmen t

Very 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 ea tmen ts have been developped f o r t h e a l k a l i d imers, e s p e c i a l l y by Konowalow and coworkers /58,59/. I t seems a t p resen 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 op t imal r e s u l t , which, as f i r s t demonstra- t e d by Rosmus and Meyer /60/ , s y s t e m a t i c a l l y overes t imates t h e equilibrium distance for the ground s ta t e potential curve due t o a neglect of core poZarisation e f f e c t s .

5.4 - Ab i n i t i o t r ea tmen 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 impress i ve progress 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 th rough t h e i n t r o d u c t i o n o f co re -va 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 rea tment . I n Toulouse /61,62,23/ t h e p e r t u r b a t i v e t r ea tmen t o f /29/ has been recons idered, and a method proposed i n which t h e va lence e x c i t a t i o n energy i s no more neg lec ted w i t h respec t t o t h e c o r e e x c i t a t i o n energy. The d ivergence o f t h e r - 4 ope ra to rs 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 core 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 rea ted 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 roduc 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 tandard mo lecu la r programs), t h e va lence e x c i - t a t i o n energ ies 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 exce l len t r e su l t s for Nu2 / 2 3 / using an empirical value for the core polari- sab i l i t y s l i gh t l y (7%) smaZler than the experimental, one. The exper imenta l curves 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 shou ld ment ion 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


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 ea tmen t , n e g l e c t i n g t h e V3 t e rm and hence ove res t ima t i ng 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 ; composi te curves a r e presented 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 be ing cons idered as upper and lower 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, concern ing 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 Tab le 2 , and i t i s c l e a r t h a t t h e agreement between theo ry and exper iment i s v e r y good.

TABLE 2

Molecu la r cons tan ts f o r t h e ground s t a t e o f t h e a l k a l i d imers

~ 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 n

B 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 ea tmen t o f va lence 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 core-va lence c o r r e l a t i o n e f f e c t s

E exper iment

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 exper imenta l r e s u l t s ob ta ined owing t o l a s e r techn iques has l e d t o some progress 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 lec t rons .

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 imu la ted by an e f f e c t i v e p o t e n t i a l , p rov ide 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 phys i cs as f a r as t h e i o n i s a t i o n energ ies 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 de fec t . A t s h o r t i n t e r n u c l e a r d is tances, t h e accu ra te e v a l u a t i o n o f t h e core-core 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 gas- rare gas i n t e r a c t i o n s demands a p r e c i s e t r ea tmen t o f t h e core-core 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 successfu l i n t r e a t i n g a l k a l i mo lecu la r i ons . 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 roduced through e f f e c t i v e ope ra to rs 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 avo id d ivergence a t smal l 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 cho i ce o f t h e c u t - o f f r ad ius , 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 ross -sec t i ons , shou ld be deve loppd.

The genera 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 techn iques o r by t h e use o f c o r r e l a t e d funct ions . A comparison o f t h e two approaches f o r t h e t r ea tmen t o f e x c i t e d and Rydberg s t a t e s w i l l be 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 rma l i sm has been genera l i s e d t o t h e t rea tmen 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 an ove res t ima t i on o f t h e bond l eng th , 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 ma jo r improvement has been ob ta ined 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 ea tmen t o f c o r e - va lence c o r r e l a t i o n i n t h e s tandard codes. Such a method avo ids 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 spec tacu la r agreement i s ob ta ined between theo ry and exper iment 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 techn iques f o r t h e t r ea tmen 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 appea l i ng p rospec 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. McCarro l l and Dr . P. V a l i r o n f o r t h e Kany yea rs - 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 scuss ions . 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 presented 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. Hen r i e t , M. Aubert-Frecon and C. Le Sech f o r t h e very 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 d imers problem, and t o Pr. P. P luv inage f o r h i s i n t e r e s t i n t h e work. The au tho 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 manusc r i p t .

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