RESULTS OF THE PARAMETRIC POTENTIAL METHOD APPLIED TO RARE GASES

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RESULTS OF THE PARAMETRIC POTENTIAL METHOD APPLIED TO RARE GASES

M. Aymar, M. Crance, M. Kapisch

To cite this version:

M. Aymar, M. Crance, M. Kapisch. RESULTS OF THE PARAMETRIC POTENTIAL METHOD APPLIED TO RARE GASES. Journal de Physique Colloques, 1970, 31 (C4), pp.C4-141-C4-148.

�10.1051/jphyscol:1970423�. �jpa-00213878�

RESULTS OF THE PARAMETRIC POTENTIAL METHOD APPLIED TO RARE GASES

B Y

M . A Y M A R , M. CRANCE

M.

K A P I S C H Laboratoire Aim6 Cotton, C. N. R. S. 11, 91, Orsay, France

Resume.

On rappelle d'abord Ics fondcnicnts de la niethode du potcntiel paran~ttrique.

Celle-ci consistc h representer par une fonction analytiquc, dependant de quelques parametres, le potentiel central introduit par Slatcr ; cllc pcut etre ainsi opti~iiiscc pour satisfaire certains cri- tkres de qualite.

011

a iitilisC iln crititre spccrroscopiq~~c -

minimisation dc I'bcart q~ladratiquc moyen (e. q. ni.) entre calcul et expCricncc

pour les gaz rares neutrcs Nc I h Kn I, et pour la sequence isotlectro- nique Ne I

V XIV. Aprcs introduction de corrcctions relativistcs ct d'interactions de configu- ration, les e. q. m . sont inferic~~rs

500 em

pour dcs sl~cctrcs calcules d'cnviron 100000 cni-1.

En ce qui conccrne la sdq~~cnce isodlectroniqi~c. nous sommes en mcsure d'intervcrtir les noms attribuds aux niveaux 2 pr 3 d ' P I ct 2 p' 3 d ? D l .

Un critcre variationnel

~ninimisation de l'kncrgic du Ihndamcntal ail premier ordre - donne des resultats tres praches de ceus dc I-lurtrcc-l.ock. et pc~lt scrvir dc point de depart pratique pour des calculs de perturbat~ons.

Abstract.

A summary of tlie ~>iiramctric ~,otcntial ~iietliod is prcscntcd : the central potential introduced by Slatc~. is c~l,rcsscd as a n analytical function of some par;imeters, which can be optinialircd accordilig to several q~l~tlity critcrin. A spect~.oscopic criterion where the r. ni. s.

deviation bct\veen rlicork and cwpeririien~ is ~iiinimali/ccl

uas applied to rare giises Nc I through R n 1. and ro tlic isoelectronic sequcncc K c I lo V XIV. Once I-cla[ivislic corrcctions are introduced, as well as configu~.;ition mixing. I.. m . s. AE arc smallel' tlian 500 cm

for coniputed spectra of c. a. 100.000 cm

As for

lie isoclcctronic scqilcncc. \vc

crc able to iiitcrcliange names ascribed to Icvels 2 p' 3 d 11'1 and :! 3 d 'Dl.

A variational criterion

mininialiration of ground statc first order cncrgy - yields results very near those of Hi~rtrcc-S.ock. and is ~ ~ s c f u l as a starting point for perturbation studies.

I. Introduction.

- Karc gases. beyond their intrinsic spectroscopic interest, lic at the limits ol'tlie rangcs of the two prominent mcthods ofstudying atoms, namely Hartrec-Fock and paranictric methods. This stems out from tlie l i ~ c t that, on one hand their spectral structure is far from the Russel-Saiinders coupling sclienie, and on the other hand, configuration mixing, associated with relatively small number of Icvcls per configura- tion, makes tlic number of involvecl radial integrals greater tlian the total number 01' lcvcls under study.

These circumstances make rare gases a good case for the parametric potential metliod [I], [2]. This study was made in view 01' evaluation 01' transition proba- bilities, the results of' which arc reported clsewhere [3], [4], [5]. T h e present \vork is a summary o f tlie para- metric potential nictllod. and a report of some results covering spectra 01' the neutral rare gases : Ne 1 through Rn I , and of tlie isoelectronic sequence : N e I through V XIV.

11. Theory of the parametric potentiel method. --

T o ensure separation of angular and radial coordinates of each electron, a spherically symetric central field is introduced. splitting the Hamiltonian into three parts

T h e Spin-orbit term has been taken a p a r t t o avoid difficulties in the convergence o f the well known perturbation expansions [see ref. 11 :

1. GENI;RALITII:S.

This method finds its background I f U(ri) satisfies some general conditions, then the in Perturbation Theory, applied t o isolated atoms series converge, and, as their limits are the exact by virtue of Slater's central field hypothesis [6]. solutiOns SchrGdinger Equation

Thus, let us write tlie Hnmiltonian li>r an N-electrons

atom, in atomic units, as : H'/'

E'/J

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970423

C4- 142 M. AYMAR, M. CRA? \ICE A N D M. KAPISCH these do not depend upon a particular form of U(ri).

Thus, this form appears to be arbitrary.

However, it is also clear that successive corrections dP' and t,b"'' are defined recursively, thus depending upon e("' and $'"', which obviously rely heavily upon the function U(ri).

Having in mind that for most atoms: it is nearly impossible to reach anything beyond third order, we conclude that, practically, computed energies and wavefunctions may vary according to the function U(r,>.

This, in turn, implies that if we are able to define optimal approximate wavefunctions according to some quality criterion, then, an optimal central field may be defined. The apparently free character of the choice of U(r) has been thus reduced to the choice of a quality criterion for wavefunctions.

2. QUALITY

WAVEFUNCTIONS.

A quality criterion is a way of judging that a given approximation is sufficiently near to the exact solution, which may be unknown. Quantitatively, this may be formulated by the statement that a quality criterion corresponds to a choice of definition of a distance in the space of wavefunctions. This will be made clear by the example of variational principle, which may be written as follows :

Let C2(R3,) be the Hilbert Space of square inte- grable twice differentiable functions of 3 N real variables, in which lies the exact solution Y of Schro- dinger Equation, and G a subspace of it, of trial wavefunctions cp, not containing Y . (If Y was in G, this would mean that Schrodinger Equation is sol- vable.)

Now, define a functional

which is minimum when cp = Y.

Then, obviously, the best approximation is cpo such that :

It is readily verified that I ^{E(q) - E(Y)} I is a distance in C2 [(I), (7)], and that such distances may be defined with any functional T in place of E. The case of

where A is an observable, being especially interesting.

More pr-ecisely, as E(cp), being non linear, may assume the same value for several distinct (i. e. not propor- tional) tp, the distance above mentioned concerns a space of equivalence classes on C2(R3,). Generally speaking, any functional T decomposes f2(R,,) in a specific set ST of equivalence classes. Hence, there is a great interest in studying several non-

equivalent quality criteria. The importance of this remark stems out from the well known fact that two trial functions giving identical numerical total energy (e. g. with seven digits) may yield very different hyperfine structure contributions (e. g. of opposite sign) [8].

3. EXTENSION

AS natllrally as one chooses a trial set of wavefunctions in a varia- tional method, we have to choose a trial set of poten- tial functions, if we wish to optimalize U(ri). For the sake of simplicity of complex spectra computation, and application of Perturbation Theory, we shall choose U(r,) to remain identical for all individual, occupied or excited, wavefunctions within a given spectrum, and, moreover, we shall represent this function by an analytical formula depending upon a set of parameters (a,, ..., a,) ; so let us write :

with any one value for the set of parameters (a), we are able to compute many excited states wave- functions. It is then convenient to apply quality criteria to a small set of well-choosen wavefunctions of interest, which we shall call a

test-class

Dis- tances may be defined in the space of test classes, which are finite sequences of wavefunctions.

We are now able t o define precisely optimal central fields :

- Let p(cp, Y) be a given distance in f2(R,,), and Gj;P'(a) = ( cpY)(a), qn(P)(a) , . . ., cpI;P)(a) )

a sequence of n wavefunctions computed up to order p with a given potential U(a, r).

- Now define a distance in the space of test classes as :

( m is a strictly positive integer).

Then we can say that U(a,, r) is optimal for a given test class and a given quality criterion if

where 8:"' is the corresponding test class of exact solutions. In ref. [I] several such quality criteria are studied, and it is shown that trying to optimalize U(a, r) by arguments that cannot be related to such distances actually leads to bad results (i. e. trying to minimize < ^{H, >).} Some suggestions were also made about computation of lower bounds to energies and so on. However, in the present work, we use only two quality criteria :

a) a variational criterion, called HFGS (Hartree-

Fock Generalized to a Spectrum), where the elemen-

tary distance p(tp, Y) is the one of the variationnal

principle stated above. For a test class, the optimal

potential is the one that minimizes, in the mean,

RESULTS OF THE PARAMETRIC POTENTIAL METHOD APPLIED TO R A R E GASES C4- 1-13

first order energies, whicli we sliall describe by a For the step number 6, we used tlie algorithm of

function S(a) : Nelder and Mead [9], by the non linear simplex

1 " method, which does not require derivatives of the

S ( r ) =

~ ! " ( a ) .

II i

function S(a). This enables us to use the same compu- ter code to optitnalize U(a, r), either according to total b) a spectroscopic criterion, where experimental energies, or according to s. deviation.

energy levels are used. It must be recalled that total _-

atomic energies are usually not known, but only energy differences between levels. Hence, we define a new distance in the following way : Let us shift all the individual distances by some constant quantity, so that

p , ( ~ i f a ) , Y,(a')) ⁼ P((P,(~), Yi@')) + I c(a) - j

where c(a) and c(al) are constant, in one to one corres- pondance with each test class. It is then clear that all the properties of distances are preserved, since p, goes to zero as p does. In the particular case of spectros- copic criterion, we choose c(a) to be the mean value of total energies of a test class. We can, then, define an optimal potential as minimizing the function :

5. ANALYTICAL FORMULA. - We must now des- cribe tlie analytical formula used to represent U(a, r).

In previous trials with simple formulae [l, 21 it appeared that a potential corresponding to charge density not reproducing shell structure would not give good results on medium or heavy atoms. The simplest way of achieving this feature is to represent (of course only for purpose of potential's parame- trization) closed shells charge density by nodeless Slater type functions :

AS a consequence [I], the potential for outer electrons can be described by :

where only differences of energies come in, the mean ^- 1

- - ; [:

g(uj, L j , r) + qkf((11, lk,

+ I

value of computed total energies being shifted to

be equal to the mean value of experimental energy _:

I

levels.

This quality criterion is not less theoretical than the preceeding. It only assumes that experimental energy levels are exact eigenvalues of Schrodinger equation, and in that case, tlie vanishing of S;a) indeed implies convergence of the test class towards corresponding test class of exact eigenfunctions of Schrodinger equation.

4. COMPUTATIONAL ALGORITHM.

We have seen that optimal U(a, r) within sonie trial set can be obtained by searching for tlie minimum of a function S(a) of tlie set of parameters (a). Let us summarize the algorithm as follows : choose a quality criterion, a test class and trial set for potentials (i. e. an analy- tical formula, see 9 5 ) .

Then, estimate an initial set (a0) of the parameters ; there is now an iterative computation, of which the step number k is described below :

+1. Compute numerically ~ ( a " r).

2. Solve tlie radial differential equations numerically for R,,,(r).

3. Compute the necessary integrals Fk, Gk, Rk,

t,,,, etc...

4. Compute s(ah) (if necessary diagonalize matrices, etc ...)

5. If this value is a minimum go to step 8. >- 6. If not, then evaluate an increment Aaqowering

Xa).

<7. Do a" " ^{ak +} Auk and go to 1.

8. u(uk, r) is optimal ; stop computation.

-

The first sum represents the screening effect to nuclear charge of complete closed n-shells, and the second sum the effect of incomplete shells. I is the degree of ionization plus one.

-

q, or q j is a number of electrons occupying a closed or non closed shell.

-

I, is the orbital quantum number of the non closed-shells.

- L j is the maximum I-number of the closed n-shells.

With this formula, there is a normalization condition to the potential :

C q j + C q , + ~ = z .

i

So that

C4-

M. AYMAR, M. CRAT \ICE AND M. KAPISCH

6. COMPUTER

The programme used is described elsewhere [lo]. It is now in standard FORTRAN l V , and is able to work on any 64 K words memory computer. One noticeable feature of this programme, is tliat i t contains a s ~ ~ b r o u t i n e which automatically gives names to eigenvalues of diagonalized matrices. It is known tliat this work is usually done by hand, and this prevents iterative optimalization in parametric fitting.

We proceed by comparing the array of squared coefficients of eigenvectors to some value r, which is initially set at 0.5. Then a reorganization of lines or columns puts coefficients greater than

on the diagonal.

is then lowered step by step, each time followed by a reorganization of eigenvectors which have not already been moved. This continues till z reaches some value

fixed by tlie user. At each step, il several coefficients greater than r are on one line or column, then steps are made finer, to try to separate the values, till a limit 6, fixed by the user, is reached.

If the limits rmi, or 6 are reached, tlie unnamed eigenvalues are taken out from optimalization process.

Thus the user keeps the control of the algorithm.

111. Results.

A. SPECTROSCOPIC CRITERION. a.

Pure first order : First of all, computations with Spectroscopic criterion were performed on neutral Rare Gases, choosing as test class some 20 lowest levels, which were the only ones in each configuration with same J-value.

(E. g. p5 s 3 ~ 2 , p5 p1 3D3, p5 d 3F4, etc ...) Thus, their energy is independant of the coupling scheme. The total range of computed energies goes from 170,000 cm-' (Ne) to 86,000 cm- ' (Rn). It is seen on figure 1 that the fit on Neon is much better than on other atoms.

1. - The relatively smooth character of the upper curve in figure 1 prompted us to look for an explana- tion, and if possible, to get better. As, in ref. [I],

cm-14 *

*

20 levels

1 ^o

.//

1500-

i 150 levels x I n t . Coup ⁺ Conf.

1nt.R.C.K.E.

we showed that a considerable gain of fit could bt obtained on heavy Alkalis by refining the formuln for U ( a , r ) , we first thought that our formula was still not enough refined for heavy atoms.

However, a plot of the first two parameters (Fig. 2) a , , and a, referring to effect of sliells I s2 and 2 s2 2 p6 shows the great consistency of the parameters. Actually, they have the signifiance of effective Z in the charge density for closed shells, and it is rewarding that they appear to be linear functions of actual 2.

The same behavior is exhibited by other parameters.

In figure 2, the vertical bars shown indicate the varia- tion of the parameters for different quality criteria, or different test classes (e. g. with or without relativis- tic corrections, etc ...). They do not show the sensiti- vity of r. m. s. KE with respect to changes in the para- meters. These uncertainties are usually less than 1 %,

for a variation o f r ~ of 1 %.

2. - Anyway, t o test the formula U(a, r ) , we computed with tlie same criterion, the isoelectronic sequence of Neon I, through Vanadium XIV. The resulting TE is shown in figure 3 (upper curves).

L *

_

Ncl NalL Mg A1 St P 5 C1 Ar K Ca Sc T

RESULTS O F THE PARAMETRIC POTENTIAL METHOD APPLlED TO RARE GASES C4- I45

The discontinuity between A1 1V and Si V corresponds to a change in test class, only resonnance levels being known for Si V and up. It is seen tliat still Neon is a specially good case. However, a detailed analysis shows that 17 levels entered the fit in tliis case, while only 8 t o 6 in the other cases, owing to the lack of data.

But most of levels co~iiing in for Neon are liiglily excited states, where wavefunctions are nearly hydro- genic, and for these, difference between theory and experiment is rather small. So, without changing anything, we computed the r. m. s. AE with only 8 levels, homologous to those known in N a 11, and went up from 62 to 98 cm-'. All this shows in fact the sensitivity of AE and U ( a , ^{r )} to the test class.

Paradoxically, in this case, the extension of the test class reduces TE. This is why in the first part of tliis work we defined test classes by well choosen set of energies.

t ^{t r >} en fonction du d a 'ionisation

I t must be noticed, that although E goes u p with Z in figure 3 the ratio of t o t h e range of the spec- trum decreases (lower curves) a n d eventually tends towards a constant value. We wondered if this fact could also be explained by the asymptotically liydro-

genic character of highly ionized atoms. T o make things clear, we plot, n n figure 4 the value of < ^r >

for some wavefunctions. Let us recall that for hydro- genic wavefunctions, this amounts to :

I t is seen that it is only a t high stages of ionization that < ^r >,,, come in hydrogenic order, a n d even then, although being inversely proportional to 2, their ratio is not purely hydrogenic. As a conclusion, this argu- ment of hydrogenic character must be handled with some care.

b. RELATIVISTIC

- The fact that AE grows rapidly with Z (Fig. 1) may be ascribed to some relativistic effects. T o test this possibility, relativistic corrections to kinetic energies were programmed. This was done, following Condon and Shortley ( 1 I), by first order perturbation theory. The contact term was also computed, since, for s electrons it is of the same order of magnitude as the p4 term. These correcticns had n o influence on TE for the isoelectronic sequence (about 1 %), although the optimal parameters were slightly ditTe- rent.

O n the other hand, it is quite clear from figure 1, that TE is smaller from Kr to R n . This examplifies our claim tliat, unlike the classical parametric method, the parametric potential enables one to distinguish effects having the same angular depcndance (here n.

shift of a configuration as a whole).

C. INTERMEDIATE

Having still q ~ ~ i t e a large AE for heavy atoms wit11 relativistic corrections, the obvious effects to study were intermediate coupling and confi- guration interactions. As a great deal of levels could not be computed in pure coupling, these were, at that stage, brought in the computation. That is why we have here (lower curve, Fig. 1) 150 levels instead of 20. As one sees, A E becomes noticeably smaller.

Notice that the potential was optimalized again with all these levels in the test class, but finally the parame- ters changed by c.a 5 %.

The lowering of E has two main reasons :

1. - For levels c o m p ~ ~ t e d in intermediate coupling, the fit is as good as for pure levels, especially for excited configurations. As these levels are many, there is a n averaging effect on E.

2. - Taking into account configuration mixing

greatly improves the fitting, especially for s-d interac-

tions, which grow larger with increasing Z. In parti-

cular, this explains tke Radon case, where all known

p 5 d configurations lie itwide the first p5 s. Thus, the

quasi degeneracy effects (i. e. mixing with near confi-

gurations) can be easily taken into account.

C4-146 M. AYMAR, M. CRANCE AND M. KAPISCH This is not true for interactions with high-lying

configurations. For instance, for the J = 0 levels of p5 p' and J = 1 of p5 d or of the lowest p5 S, the fit remains poor. But these levels are precisely those where there is a large contribution oT exchange Gh integral. In table I, a comparison is made between computed integrals and parametrized values. Notice that with these parametric values, the r. m. s. AZ ^is

smaller than 100 cm-' [3], [I 51.

Ratio of computed to parmnetric vaiues of Slater irztcgrais

(Spectroscopic criterion)

It is seen that spin-orbit and Fh integrals are quite near the parametric values. This means that these parameters have an (c effective

part which is rather small. But this is not true for Gk integrals. E. Kcenig [3]

showed that this can be explained, at least qualitati- vely, by the sum of interactions with 2 p5 np, with a probable large contribution from continuum states (for Neon). In other words, it is the lack of self- consistency that makes exchange integrals so large, for, if these integrals were included in radial integro- differential equations (as in the Hartree-Fock method), this w o ~ ~ l d give an inhomogeneous term perturbing largely the wavefunctions. This phenomenon was already noticed by Hartree 11121, and more recently

[I 31, on Alkaline-Earths.

TIILIS, the parametric potential method shows evidence for the great importance of these effects, which we may call : cc radial second order effects n.

d. LEVELS

2 p5 3 d

Nevertheless, the ratio of empirical to computed values of these integrals is not larger than 2. This fact enables us to inter- change the names of 2p5 3d 3 D and 'PI for ionized atoms.

In fact, cc the experimental order

is given by Soderqvist [I41 as 3P,, 'PI, 3D1. It is based on the example of Neon, and on line strengths of transitions t o 2 p5 3 p. But, as the latter is far from L. S. coupling the analysis of Soderqvist is questionnable. Actually, a theoretical investigation of line strength was made, showing that the ratio of intensities observed would be better explained by interchanging 3D1 and 'P,.

The first argument, analogy with Neon Spectrum,

was also analysed. It is readily shown that the diffe- rence of energies is

Here, spin orbit coupling has been neglected. Indeed, it is much smaller than electrostatic interactions, as can be seen from the purity of L. S. coupling in this case (Fig. 5). The order of these levels thus depends solely on the ratio F2/G', the critical value being F2/Gl = 1013. Now, assuming the experimental order of Soderqvist, and values of F ~ , to be correct would lead to a ratio of computed to parametrized G1 greater than 4 for Mg 111, which is much larger than in other neutral atoms. But reversing this order would lead to a ratio of 1.2, which is coherent with table I.

In fact the r. m. s. would be three times larger if we would have kept the order of Soderqvist.

Couplage rnoyen d a n s

.

o 2 p 5 3 d ~ = l

The fact that, in the case of Neon 3D1 lies above ' P I is easily explained by figure 6 , where it is seen that the computed ratio of F2/G' is very much larger than 1013 only for Neon. Moreover, this ratio goes smoothly to the hydrogenic value which is 0.9934.

As a by-product, it may be noticed that on figure 5 appears the same phenomenon as was noticed by R. Cowan [16], namely the fact that LS coupling is a good approximation for several members of the series.

B. AB

- These were done to

compare with previous results, t o be able to appreciate

predictions in case of unknown spectra. At first, some

RESULTS O F THE P.-2lli\\l;lETRIC I'OTENTI:\L h1ETIIOI) XPI'I.LC1) TO H;\RL: Ci'\SLS

computations were made using the same test classes as for spectroscopic criterion. These gave the results that, comparing to experiment, all the excited levels were too low, by a nearly constant quantity of about 15,000 cm- ' to 60,000 cm- '. Tliis was explained by the fact that this test class gave too large a weight to excited levels, thereby yielding a ground state which was not at its minimum.

Then, computations were made, where tlie test class was reduced to the only ground state. This gave better results, where excited states are slightly too high, by about 3,000 cm-I, for neutral atoms.

In table 111, total energies of ground states are

Ratio of conzpliterl to paratnetric calues of' Slater integrals

(Ah initio criterion)

compared

Hsrtree-Fock values, obtained by C. Froese-F~slier. TIle striking fcature is that the difference of the two cntrics is roughly constant. and relatively small, considering the n o n self consistent character ol' the parametric potential. Another jnte- resting feature of tliis kind of potentials is the compa- rison of computed integrals to the one obtained by the parametric method (Table 11).

It is seen, thus, tliat these computations are quite useful to predict positions of configurations, or to obtain a value for Slater integrals.

IV. Conclusion. - I n tliis work, we described tlieoretically a new method of computing wavefunc- tions and studying spectra, and we reported results obtained on rare gases.

Great sensitivity of results with respect to choosen test class (i. c. set of levels on which potential is optimalized) is shown, both for spectroscopic and ab initio criteria. Tliis prooves tliat the formula des- cribing the potential is suficiently flexible and reliable.

About thc use of' spectroscopic criterion to interpret spectra, we must say tliat the r. m. s. A E is much larger witli this mcthod than witli the usual parametric method. This is probably due to radial second order eff'ects. Tlius, i t is still impossible to predict energies of unknown levels with suficient accuracy ( < 500 c ~ i i C ' ) a n d specially when Gqintegrals are involved.

However, this statement is unfair, since, on one hand tlie study of' isoclcctronic sequence ol' Ne is inipossible with the classical method (due ^to lack of' data) but c.e were able to interchange names of' known Ie~els, arid on the other hand, this method has to tackle with energies of different configurations.

Besides tliat, tliis very frict of poorer fit gives a valua- ble informatiori about the effective part of Slater integrals considered as parameters.

Thus, as it stands tllc metl~od of parametric poten- tial is useful in studying some perturbation effects, permitting separation of direrent efTects having same angular dependance. Nevertheless, the project of including radial second order corrections in the optimization is being studied.

Total energies of ground states (a. u.)

Parametric Po-

tential . . . . . . - 128.421 - 526.776 7 - 2,751.917 - 7,232.042 5 - 21,866.61 Hartree-Fock. . - 128.547 - 526.818 5 - 2,752.057 - 7,232.150 - 21,866.79

AE . . . 0.126 0.031 8 0.140 0.108 0.18

M. AYMAR. M. CRANCE A N D M. KAPISCH

References [I] KLAPISCH (M.), These, Orsay 1969.

[2] K L A P I ~ C H (M.), C. R. Acad. Sci., Paris, 1967, 265, 914.

[3] KCENIG (E.), these de 3c Cycle, Orsay 1970.

[4] FENEUILLE (S.), KLAPISCH (M.), KCENIG (E.), LIBER-

MAN

(S.), Physica, 1970, 48, 571.

[5] AYMAR (M.), FENEUILLE (S.), KLAPISCH (M.), 2nd International Conference on Beam Foil Spec- troscopy, Lysekil (Sweden), 1970.

[6] SLATER (J. C.), Phys. Rev., 1929, 34, 1293.

[7] LIUSTERNIK (L.), SOBOLEV (V.), Elements of frirlctional Analysis Ungar, New York, 1961.

[8] NESBET (R. K.), in Hyperfine Str~tcture of Atoms and Molecules, Colloque C . N . R. S., 1966, 164, 85.

and BESSIS (N.), AMBRY (C.), same ref. p. 94.

[9] NELDER (J. A.), MEAD (R.), Comp~iter Journal, 1965, 7. 308.

[lo]

(M.), to be published in Computer Physics Communications.

[I 11 CONDON (E. U.), SHORTLEY (G.), Theory of Atomic Spectra, Cambridge University Press 1963.

[12] HARTREE (D. R.)

and HARTREE (W.), Proc. Roy.

SOC., 1936, 154, 588.

[I31 KLAPISCH (M.), same as (8), p. 227.

[I41 SODERQVIST (J.), Nova Acta Reg. Soc. Sci. Uppsala, 1934, 9, 1.

[15] LIBERMANN (S.), J. Physique, 1969, 30, 53.

[16] COWAN (R. D.), Atomic Spectroscopy Symposium, Washington D. C. 1967.

[17] FROESE-FISHER (C.), Hartree-Fock Parameters for

atoms He to R n (Unpublished).