Experiments and Model Calculations to Determine Interatomic Potentials

I1

ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS, VOL. 16

EXPERIMENTS A N D MODEL CALCULATIONS TO DETERMINE INTERATOMIC POTENTIALS R. DUREN Max- Planck-Institut f u r Srromungsforschung Gottingen, West Germany

I. Introduction . . . . . . . . . . . . . . . . . . . . . . 55 A. General. . . . . . . . . . . . . . . . . . . . . . . 55 B. Statement of the Problem. . . . . . . . . . . . . . . . . 56 C. Scope and Outline . . . . . . . . . . . . . . . . . . . 57 11. Electronic Model Potentials and Interatomic Potentials. . . . . . . . 58 A . T h e o r y . . . . . . . . . . . . . . . . . . . . . . . 58 B. General Behavior and Actual Forms of Model Potentials . . . . . . 62 C. Determination of Interatomic Potentials. . . . . . . . . . . . 67 111. Experimental Sources . . . . . . . . . . . . . . . . . . . 70 A. Relation to Model Potentials . . . . . . . . . . . . . . . 70 B. Specific Measurements. . . . . . . . . . . . . . . . . . 71 C. Comparison of Interatomic Potentials from Different Experiments. . . 87 IV. Interatomic Potentials Determined with Model Potentials . . . . . . . 91 V. Conclusions . . . . . . . . . . . . . . . . . . . . . . 96 References . . . . . . . . . . . . . . . . . . . , . . . 97 Note Added in Proof . . . . . . . . . . . . . . . . . . . 100

I. Introduction A. GENERAL

During recent years the determination of interatomic potentials has received new momentum by the extension of the subject to electronically excited species of interacting particles. Many phenomena can be interpreted conveniently on the basis of these potentials. Naming a few of the practical applications-multiplet transitions and energy transfer in general, state specific reactions, the field of line broadening, and excimer transitions-demonstrates sufficiently one basis for this interest. For many of these practical applications interatomic potentials of suffi55 Copyright 0 1980 by Academic Press. Inc All rights of reproduction in any form reserved. IESN 0-12-003816-1

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56

cient accuracy especially at large internuclear distances are readily available. But if the range of intermediate internuclear distances and high accuracy, by which we mean a few percent of the well depth, is considered, few examples remain. To these this report is devoted. This implies that a methodological study rather than a summary of potentials will be presented. The new momentum has two sources. First, general advances in experimental technology have been enhanced specifically by the development of lasers. With the unprecedented narrow linewidth and power of this light source, experiments with well-defined initial conditions and with a high degree of differentiation in the exit channel can be performed. This is not only an academic interest; the accuracy of the potentials obtained from the evaluation of such well-defined experiments makes the subject attractive again. The second source of intensified activity is the development of theoretical tools correlated to such experiments, specifically the use of model potentials in calculations of the interatomic potential. In contrast to ab initio calculations, this method also allows the calculation of interatomic potentials for heavier systems with an accuracy comparable to that which can be obtained in experiments. To a certain degree these two sources have developed independently of each other, and it is one of the goals of this article to contribute to bridging the remaining gap. Basically it will be an attempt to demonstrate how model potential calculations and experimental data can be combined to the benefit of a highly accurate determination of interatomic potentials. We hope to propose a conceptual view of the determination of interatomic potentials, which offers significant advantages over the standard phenomenological approach. B. STATEMENT OF

THE

PROBLEM

The standard determination of interatomic potentials is either to calculate a given measured quantity on the basis of a parameterized model for the interatomic potential and vary iteratively the parameters of the model until a best fit to the experimental data is obtained, or to establish an inversion procedure to obtain the interatomic potential from the measured data. In both cases the result remains unsatisfactory because of the phenomenological basis, which implies, for instance, that one measurement concerning one state of the interacting species yields one potential function. As long as only one state is involved in the interaction these methods serve their purpose. Considering however the case where many states are involved, two specific problems (at least) can be formulated which will be covered unsatisfactorily.

DETERMINATION OF INTERATOMIC POTENTIALS

57

(1) The first problem is related to the fact that even though the interaction of given atoms in different states is to be described with different interaction potentials, the approximation of the full Hamiltonian of the problem by a model Hamiltonian exhibits their common basis. At least the major part of the difference between the interaction in the ground state and in the excited state is then attributed to the known differences in the respective electronic wave functions. A fortiori this should hold for the difference between the various molecular states related to one common state of the separated atoms. ( 2 ) A similar argument applies to the problem of a series of interacting systems. Consider for instance the interaction of the alkali atoms with, say, one particular rare gas target. Again the differences in the net interatomic potential should be largely due to the known differences in the wave functions of the different alkali atoms. Both problems are increasingly being encountered in the evaluation of experimental data, and we want to point out that the model potential approach of calculating interatomic potentials seems to be a viable solution to both problems. A review of the various theoretical and experimental sources concerned with fitting the parameters of the model potential will show that such attempts are sufficiently advanced to come into widespread use. Even though a definitive solution has not yet emerged, enough contributions have been accumulated, to be summarized in a unifying concept.

C . SCOPEA N D OUTLINE Surveying the literature for systems where experimental material is available and where the theoretical treatment has come to some sufficiently established point, we find that intermediate and large internuclear distances, interactions of “good” model atoms, and the lowest excited states play a dominant role at the present time. Large and intermediate distances translate into thermal collision energies for scattering experiments and into the usefulness of spectroscopic measurements with the respective molecule. The alkali atoms obviously emerge as good model atoms, since the quality of a model potential approach strongly depends upon the possibility to separate the core and the valence electrons, for which the alkali atoms are the standard demonstration system. Fortunately these systems are also well behaved in the experiments: The generally large dipole moments yield good efficiency in spectroscopic measurements, and the detection efficiency in scattering experiments is usually good. For these reasons we will concentrate our attention on intermediati: and

58

R. Duren

large internuclear distances and on interactions involving alkali atoms, specifically the alkali-rare gas interactions. As narrow as these limitations may appear at first, it is because of both its practical aspects and as pilot systems for further developments that a thorough discussion may be useful. To discuss the various aspects of the model potentials we will first (Section 11) discuss the theoretical aspects of their relation to interatomic potentials. There we will summarize the theoretical background of the model potentials (Section 11, A), describe the general form and particular realizations of model potentials (Section 11, B), and describe in detail their use in determining interatomic potentials (Section 11, C). The next section (111) is devoted to the experimental sources for this determination. A general survey of the relation between experiments and model potentials will be given (Section III,A), followed by a description of specific measurements (Section 111, B), and a comparison for a reference system (Section 111,C). In section I V , we will compare the results for this reference system with interatomic potentials determined with the aid of model potentials. Throughout this report atomic units are used with the following conversions: length, 1 a.u.= 0.52917715 X lo-* cm; energy, 1 a.u.= 2. 194668 x lo5 cm-' = 4.35968 x l o - ' ' erg = 27.2107 eV.

11. Electronic Model Potentials and Interatomic Potentials Model potentials have found extensive applications in the calculation of atomic and molecular properties. In this section we will summarize their connection to interatomic potentials. A. THEORY

For a system of valence electron(s), core electrons, and the respective nuclei, we write the Schrodinger equation from the exact Hamiltonian for fixed internuclear distance R as

where H A and H , are the Hamiltonians for the cores A and B, H , is that of the valence electron(s), uABis the interaction of core A and core B, + ( r , R ) is the total molecular wave function depending on electronic coordinates r and the interatomic coordinate R , and E is the total energy. The intera-

DETERMINATION OF INTERATOMIC POTENTIALS

59

tomic potentials V ( R ) for the different electronic states that we want to determine are obtained as eigenvalues of this Schriidinger equation. The basic guideline for the development of the model potential approach is the observation that the interatomic potentials of the complete system can be obtained approximately from the sum of a term determined by the behavior of the valence electrons and a term available from unperturbed core properties. In particular at the large and intermediate distances to which we have concentrated our attention, this approximation should hold. The total wave function can be expanded in terms of products of core and valence functions (without explicitly stating antisymmetrization) as

where +A is the wave function for core A, @B for core B, and @, for the valence electrons. Following the basic guideline stated previously we will now assume that the core functions are independent of the valence functions. Hence the interatomic potential splits into two contributions, namely V , ( R ) from the cores and V , from the valence electron

Inserting (3) and (2) into (1) and collecting the terms that refer to the cores and the valence electrons we obtain V , and V , from the two independent Schrodinger equations

where c A , c B , e v are the asymptotic energies of core A, core B, and the valence electron, respectively. To obtain the total wave function [Eq. (2)] from the solutions of these equations the orthogonality between the core and valence functions must be maintained:

+,

where describes the state of the valence electron, and +A and GB the core orbitals to which @, must be orthogonal. To determine V c ( R )we may assume that cores A and B d o not overlap significantly and calculate this quantity by a statistical model (Gombas, 1967; Kim and Gordon, 1974): (+A+B

I uAB I + A + d

= V d R1

(7)

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60

On the other hand, scattering experiments of the cores can be used to determine this quantity. In this discussion we will regard this core contribution as known, even though we will see later that in actual calculations its quantitative determination either from a calculation or from an experiment poses a serious problem. To discuss the determination of V v ( R )we write the valence Hamiltonian H, as

where T is the kinetic energy operator, and veA and ueBare the potentials of the interaction of the valence electron with the cores. To introduce the concept of the model potential we note that the assumed independence of core and valence electrons is obviously not correct. In the first place the wave functions are related by the orthogonslity condition (6). In addition the effects of exchange and correlation or polarization are omitted. The central idea of the model potential approach is to attempt to replace the potential of the valence Hamiltonian by a model potential, capable of mimicking these omissions. We therefore write a model Hamiltonian H , for the valence electron

H,

=

T

+ u,,,(r, R )

(9)

instead of Eq. (8), where vm(r,R ) is the model potential, and try to obtain the valence contribution to the interatomic potential V,( R ) from this model Hamiltonian by

instead of Eq. (5). Going through all these steps with care, the restriction which we have imposed for simplicity can be alleviated and the possibility indicated by Eq. (10) can be shown to hold exactly (Weeks et af., 1969; Bardsley, 1974) but with the restriction that then urn,called the pseudopotential, turns out to be a nonlocal potential defined in terms of the core functions. Strictly speaking then the attempt to build the exact solution from Eq. (10) shows it to be a rewriting of the original Schrodinger equation. Thus the solution is not really simplified. On the other hand, a semiempirical approach can be devised. To d o so one chooses the model potential om in most cases as a local one in a suitable parameterized form and determines the parameters from experiments. To this end, perturbation theory and pseudopotential theory is employed, and at least the qualitative aspects to be incorporated in the construction of the model are established. At this point we will assume that the concept of a local model potential

DETERMINATION OF INTERATOMIC POTENTIALS

61

is valid and consider urn as a parameterized potential indicated by u m ( r ,R ) = urn(r,R ; 8 , )

( 1 1)

where 8, describes the set of parameters used in the formulation of the model. With this concept the calculation of interatomic potentials is straightforward: As mentioned previously we can assume the core contribution V , ( R ) to be known in advance. For the valence contribution we have then to solve the one-electron equation [Eq. (lo)] with the model potential [Eq. ( 1 I)] for a given set of parameters 8,. Taking both results together we obtain the interatomic potential by Eq. (3). The experimental sources to determine the parameters in urncan enter in three ways: ( 1 ) The model potential can be formulated to reproduce the asymptotic behavior correctly. Then macroscopic properties, in particular the polarizabilities which may be determined from experiments, will enter the model potential. ( 2 ) We can compare directly the interatomic potential V ( R ) obtained in the calculation with experiments where this potential is involved. The obvious sources for this type of comparison are various scattering experiments and the spectroscopy of the respective molecule. A global fit of the parameters 8, is then achieved. (3) We can further differentiate by noting that urn can be decomposed as z ; ~r (,

R ) = ceA(r , R ) + ueB(r , R )

(12)

by identifying the model and the valence Hamiltonian [Eqs. (8) and (9)]. A further decomposition into terms that depend on the interaction of the valence electron with core A in the absence of core B and vice versa with core B plus a term ulntwhich depends on the interaction yields

) = t ; e A ( r A ) + u e B ( r B ) + u~nt(rA’‘B’

(13)

where we have indicated the absence of the respective other core by dropping the dependence of R in ueA and u e B . By this decomposition we have achieved three things. First we see that the valence Hamiltonian goes to the correct asymptotic limit if we construct u I n , ( r A , r BR, ) such that lim u l n , ( r Ar, B ,R ) = 0

R+crj

Second we (hopefully) can choose uln,to be a small correction to the terms ueA(rA),ueB(rB)which simplifies the calculation. A third aspect refers to the

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62

experimental sources: In general the total set of parameters 0, of urncan be decomposed, according to Eq. (13), into subsets and B,(lnf) ,g = I

{ el("),@ ( W , 4 (In0 1

1

(15)

1

where the individual sets are used to parameterize ueA, ueB and qnlas indicated by

The individual subsets can be fixed in advance by related and experiments: from the atomic spectra of the valence atom (A e - ) and Bl(B) from the scattering of electrons from target B or from macroscopic properties, to name a few examples. It is only the remaining (fewer) parameters (if any) that are to be fixed in the above-mentioned experiments where the interatomic potential is involved.

+

B. GENERAL BEHAVIORAND ACTUALFORMS OF MODEL POTENTIALS The model potential urn as used in the previous section can be considered in the sense of a purely phenomenological approach. In this section we want to discuss the question of how to construct this model potential in order to include general restraints as well as possible and to reduce the phenomenological aspect. As general requirements, we have mentioned before the possibility of differentiating between the various contributions to urn.Second, the orthogonality condition of the valence wave function with respect to the core functions is to be maintained. Finally, the asymptotic behavior yields additional requirements that should be satisfied. The sources from which knowledge of these requirements is introduced are basically pseudopotential theory and perturbation theory for model potentials. Accordingly the actual forms of model potentials may have terms based upon these two approaches. The pseudopotentials have the advantage that one equation for the valence function can be formulated which is exact and which contains the orthogonality condition. In the model potential approach the orthogonality condition must be taken into account implicitly or explicitly. Pseudopotential theory has been reviewed extensively elsewhere (Weeks el af., 1969; Bardsley, 1974; Dalgarno, 1975). In the framework of pseudopotentials one can obtain urnin principle exactly without model parameters (sometimes called ab initio pseudopotentials).. In the context of our prob-

DETERMINATION OF INTERATOMIC POTENTIALS

63

lem the merit of the theory is mostly to provide qualitative information rather than to be used in practice. The obstacle to the application lies in the fact that the computational effort to obtain sufficiently accurate pseudopotentials is comparably high. In the following we will report about the asymptotic behavior of the model potentials. This discussion will be mainly a summary of the outstanding paper of Bottcher and Dalgarno (1974) who have given the basis for a differentiation of the various contributions to the model potential and for many of the model potentials actually used. [For more recent supporting work, see Laurenzi (1978) and Valiron et al. (1979).] T o take the asymptotic form as a guideline is quantitatively in accord with the range of the interaction that we want to consider, namely the intermediate and large internuclear distances. In addition, with the differentiation of the various contributions, it provides the key to the rational inclusion of other experiments into the construction of the model as discussed previously in terms of the subsets of parameters 0, [Eq. (15)]. In order to obtain the perturbation equations we rewrite the exact Hamiltonian for two cores A and B and the valence electron(s) [Eq. (l)] as

He, = H A + H ,

+ ueA+ v,, + uAB+ T

(17)

where H A and H , are the Hamiltonians of the cores; veAand u,, are the interaction of the valence electron(s) with cores A and 9, respectively; vAB is the interaction of the core electrons; and T the kinetic energy of the valence electron(s). Introducing the model Hamiltonian H , yields

He, = H A + H ,

+ H , + Av

with

H,=T+v, and

AU

= ueA

+ o,, + uAB-

U,

(20)

The zeroth-order solution to the respective Schrodinger equation is the product +‘O’(rArJv)

=+A(rA)+drd+drv)

(21)

where we have indicated by r the quantum number of the respective part. and +, to be known from As before we assume HA+A(~A)

=

eLr)+A(rA)>

~ , + B ( r e )= E F ) + B ( ~ B )

The molecular wave function to first order +(

I)

=

+(I)

(22)

can be written as

+(O)(rArsrv) + cA,~+(0)(rArBrv)

(23)

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64

with (24) and (25) with

and similar expressions for core B. From these the model potential is obtained to second order as 0,

= (rArB

I 0 + u G A B + uGiB[ ff,,

u]

I rArB)

(27)

To calculate this matrix element the state of the cores is assumed to be unchanged in the interaction, rAand re referring, for instance, to the ground state of the cores. Expanding u in terms of spherical harmonics, the integrals can be simplified by the introduction of the 2'-pole polarizabilities, a(') for the static contribution and p(') for the dynamic contribution. Then one collects the various contributions according to the sum

+ UeB + CAB + DAB + ucc

u, = ueA

(28)

where ueAand ueB refer to the interaction of the valence electron(s) with core A and B, respectively; GAB to the averaged core-core interaction; uAB to the polarization of the cores; o,, to a core interaction correction. Each of these contributions can be visualized to contain parameters analogously to Eq. (15). Collecting terms to the order of l / r 6 , the asymptotic limit for the special case of one valence electron is obtained as

CAB+O

[see Eq. (7)]

(29c)

DETERMINATION OF INTERATOMIC POTENTIALS

65

As we will see later the model potentials are in practice constructed to contain these terms more or less completely so that the asymptotic behavior is reproduced correctly, in particular the van der Waals interaction with c(6)

=

aL”<$ 1 [

+ ‘2(‘A)Ir2

I +)

(30)

At intermediate distances these terms are then truncated by suitable cutoff functions which contain the f i t parameters. Simultaneously the core-core contribution [Eq. (7)] which vanishes in the asymptotic limit [Eq. (29c)l becomes significant. We will now present some model potentials actually used. It will be seen that most of them are applications of the work mentioned previously (Bottcher and Dalgarno, 1974). As a classification of the various terms taken into account we will use the differentiation as given by the asymptotic expansion [Eq. (29)]. Throughout this compilation w K ( r / q )is used to describe the cutoff function wK for the variable r with a cutoff radius rj (without differentiation with respect to the shape of this function). l a . Baylis ( I 969) Model potential: ueA = t i B D (by use of Bates-Damgaard wave functions)

(pA,pB densities of core A and core

B from “simplified Hartree-Fock”

results)

Application.

Alkali atoms-rare gas atoms.

Parameter sources. By the use of Bates-Damgaard wave functions, atomic spectra (Moore, 1949) of the alkali atom are taken into account to determine ueA, the core-core interacfion is taken from Hartree-Fock results (Gombas, 1967), the dipole polarizability ah1)is taken from semiempirical calculations (Dalgarno and Kingston, 1961), the cutoff radius ro is determined by fitting the well depth of the respective ground state interaction (Buck and Pauly, 1968; Diiren el a / . , 1968).

R. Duren

66

J b. Pascale and Vandeplanque ( I 974) The same model has been used and applied to the same interaction systems using the same experimental sources, with computational improvements.

Ic. Duren (1977) Model potential: The same model has been used as in Baylis (1969) but the polarizability has been used as a second free parameter. Application.

Alkali atoms-Hg.

Parameter sources. In addition to the parameter sources mentioned previously, the cutoff radius ro and the “polarizabitity” a ( ’ ) have been determined by a simultaneous fit to ground state (Buck et at., 1972) and excited state scattering experiments (Duren and Hoppe, 1978). 2a. Bottcher el al. (J975) Model potential:

+ (cAo + c A l r A+ c A z r i )exp

+ (cBo + c B l r B+ cB2ri)exp

Application.

Li-He and Na-He.

(

(

- -

lr:

- -

1

1

DETERMINATION OF INTERATOMIC POTENTIALS

67

Parameter sources. Polarizabilities a(')and a(') are used for the atomic potentials together with the spectra to fit c o , c I , c 2 , and r , for the alkali core (ueJ and for the rare gas atom (ueb). No further fit to experimental data is used to determine the interatomic potential.

2b. Philippe et a/. ( I 979) Modelpotential. A similar model as in (2a) has been used except that for CAB various core-core potentials (experimental ones extrapolated and

calculated ones) have been applied leading to ambiguities in the interatomic potential (see Section IV). Application.

Na-Ne.

Parameter sources. The polarizability ak') is used and the parameters of ueB are determined from electron scattering; in addition the data from N a + -Ne scattering experiments are used.

We could proceed to give more examples of model potentials but we have selected these few to reflect typical applications within the scope of this article. Similar examples for the application to the excited state of He (Peach, 1978) and for the extensive use of pseudopotential theory have been omitted (e.g., Bardsley et al., 1976; Habitz and Schwarz, 1975; Melius et a f . , 1974). Surveying this compilation we notice that only in group (1) are the measured interatomic potentials used to determine the parameters of the model, while group (2) relies only upon the partial measurements of atomic properties to obtain the interatomic potential. C . DETERMINATION OF INTERATOMICPOTENTIALS

Basically three types of experiments are used as standard experimental sources of information to determine the interatomic potentials: ( a ) the direct observation of differential cross sections and (6) the extreme wing line broadening. (In addition to these the observation of macroscopic properties and the integral cross sections contribute with less resolution.) Similarly to these experiments, (c) the spectra of the respective (diatomic) molecules yield the potential. For the evaluation of these experimental sources either iterative procedures or inversion procedures are used. The straightforward phenomenological approach of the iterative procedure can be summarized as a procedure in the following way. The

68

R. Diiren

“theoretical” counterpart M Ih to a measured quantity M e x pis calculated with a phenomenological form of a potential function V ( R ,0;) depending on R and parameters 0,. Theoretical and experimental results are compared with each other and the parameters of the interaction potential are iteratively varied until the comparison yields satisfactory agreement. This can be represented in the following sketch calculation

V ( R , f ? , ) ---+

T

Mth compare with

iterative variation of

Oi

Mexp

I

Many results have been obtained this way with models ranging from the hard spheres to the Lennard-Jones potentials or the Morse potential and finally to many parameter models of all sorts (Pauly and Toennies, 1965, 1968; Fluendy and Lawley, 1973; Pauly, 1979). The remaining merit of this approach is to provide with a few numbers a simple means of communication of the results. But there are some major problems associated with it, as, for example, the application of objective criteria for the quality of the fit and of the flexibility of the model. These can be overcome with a sufficient broad basis of experimental data for the determination of many parameters in the model function (e.g., Duren and Schlier, 1967; Buck and Pauly, 1968; Duren et af.. 1968). A related problem is that the range of validity of the model mapped out by the range of significance of the underlying experimental material can in practice hardly be determined. These problems have in common that their solution can be obtained if the eventually large effort required is provided. Great improvement concerning this effort is obtained by advances in inversion procedures, in which the interaction potential is obtained directly from the experimental data. This leads to the simple scheme: inversion

Mexp-

V(R)

Besides the eventually large savings in computational effort to obtain the potential from the experimental data, a particular advantage of this method is that the potentials obtained contain the range of validity of the results as given by the experimental data used in their determination. For all three above-mentioned types of experiments probing the interatomic potential, inversion procedures have been developed and successfully realized: for differential scattering processes (Buck, 1974; Shapiro and Gerber, 1976), for line shape measurements (Behmenburg, 1972, 1978), and for spectroscopic measurements (Rydberg, 1931, 1933; Klein, 1932; Rees, 1947: Dunham, 1932; Thakkar, 1975).

69

DETERMINATION OF INTERATOMIC POTENTIALS

Both the iteration and the inversion procedure have the common disadvantage of their phenomenological basis. For one thing, this is reflected in the fact that one measurement for one system in one particular state yields one interatomic potential function; extensions to other states and/or other interacting systems are impossible. Second, these methods are strictly related to the measurement of the global interatomic potential. There is no rational way to include in the evaluation of data the knowledge of supplementary measurements which in our picture based on the model potential refers to a subset of parameters. The procedure which includes the use of model potentials for the electronic interaction appears to be more complicated at a first glance. The starting point is the model potential u,(r, R , 8,) which depends on the electronic and the internuclear coordinates and el, a set of paiameters. By a potential calculation one obtains from this model for a given set of parameters the interatomic potentials V y ( R )for various electronic states y. If the experimental interaction potential is known from an inversion procedure, the comparison of the calculation and the experiments to obtain the best fit parameters for the model potential can be performed on this level of interatomic potentials. Otherwise the calculation of the experimental data has to be performed. Summarizing these two possibilities we have the following sketches: u,,,(r, R , 8, )

T

-

potential calculation

V,(R)

calculation of M ( V , )

0,

iterative variation of

M:h comparison M,'"

I

for the case without inversion and om(r , R , 8, )

potential calculation A

inversion

Vih( R ) comparison VYPt---MYp

iterative variation of

0,

I

for the case with inversion. This approach has clearly some advantages compared with phenomenological ones. Its basis lies in the fact that the modeling is transposed to the electronic interaction. Consequently one can determine the parameters of the model potential with a set of experimental data and obtain simultaneously the potentials for many states. The reason for this larger range is that the knowledge of the electronic wave function, which is comparably well established is introduced into the evaluation of the experiment. Similarly the model may

70

R. Duren

be constructed in such a way that the extension to other systems is achieved. Again it is basically the additional knowledge from other experiments that leads to this extension. As another advantage the actual model potentials have fewer free parameters than accurate interatomic model potentials if both are used for evaluations of comparable accuracy, provided that the actual model potential is carefully designed. Finally we notice that supplementary information as provided by experiments which refer to a particular part of the model potential can be incorporated in advance by specifying the respective subset of parameters [see discussion with Eqs. (15) and (16)]. Obviously there are also some severe problems associated with this procedure. Thus we note as a technical problem that an additional effort to establish and perform the potential calculation is required. More severe than this are, however, the difficulties which result basically from the fact that this approach is relatively new. Specifically this means that there is not yet much experience accumulated with respect to the range of validity of the results. Clearly there is a risk of extrapolating the results into ranges (of states and internuclear distances) where they are not supported by the underlying experimental material. T o prevent this from occurring a second inversion procedure would be desirable, and great care must be exercised as long as this is not yet available. With this caution in mind it seems however that the advantages outweigh the disadvantages, and we will present later (in Section IV) results to support this assessment.

111. Experimental Sources A. RELATIONTO MODELPOTENTIALS

According to the decomposition of the model potential [Eq. (13)] confirmed in the asymptotic expansion [Eq. (29)] we can distinguish experimental data in three categories according to their relation to the model potentials. 1. Data that enter the model as constants. These are basically the polarizability a(’) and p (’). 2. Data that determine components of the model potential [Eqs. (15), (16), and (29)]. For this determination the atomic spectra for ueA, the electron scattering data for ueB, and the core-core scattering data for uAB are used. 3. Data that determine the total interatomic potential either by the

DETERMINATION OF INTERATOMIC POTENTIALS

71

determination of f l i ( I n t ) or the set of parameters 0, of om [Eq. (16) or (1 I)]. Such data are found from molecular spectroscopy and from scattering experiments in the ground state and the excited state. The latter experiments can be further differentiated into line shape experiments and scattering experiments (differential and integral) with atomic beams. We will describe these experiments in the following section, but we found it more convenient to order them according to more technical considerations rather than the categories that the previous discussion suggest, namely: polarizabilities, electron scattering, spectroscopy, and heavy particle scattering, Of course some of the experiments use traditional techniques, and we will confine ourself to summarizing results including the most recent ones. Other experiments are the result of more recent developments which will be discussed in greater detail. Special care has been taken to describe completely the route from the measured quantity to the data used in a potential calculation, including the inversion procedures if available.

B. SPECIFIC MEASUREMENTS 1. Polarkabilities We have seen that polarizabilities play a key role in all the model potentials presented. It was understood that the values used there are the “true” values as determined by experiments. If these are not available, semiempirical values and values obtained in ab initio calculations are sometimes to be used. In contrast to such “true” values corresponding quantities are encountered which are indeed fit parameters as coefficients of the respective power in an expansion of the potential. These are used to reproduce some measured quantity, for example, a(’) in Bottcher and Dalgarno (1975) and in Peach (1978). a ( ’ ) and a(’) in Bardsley (1974, see Table 13), or a i l ) in Diiren (1977). Such values are obviously only valid in the context of the specific model potential used and will not be considered here. The determination of polarizabilities from experiments and calculations has been reviewed by Miller and Bederson (1977) very carefully. Therefore we will not consider this aspect but we give for the systems of interest within the scope of this article in Table I a summary of presently available values for a i l ) ?a(’), and /3(’). They are collected from various sources as indicated by the respective references. Results for other atoms may be found in Dalgarno (l962), Dalgarno et al. (l968), and Miller and Bederson (1977).

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72

TABLE I PRESENTLY AVAILABLE VALUESOF a ( ’ ) ,a ( ’ ) ,AND p ( ’ ) Error Li+ Na+ K+

Rb+ Cs+ He Ne A

Kr Xe Hg

0.190 1.00 [8-51 112-81 [2 1- 151 1.3833 2.667 11.068 16.737 27.265 35

<1 <5

<1

<1 <1
u

b,c d, e d,e d,e j,o k /,m I, j I, j

p(’)

a(’)[ui]

(%)

Ref.

0.114 1.54 17.4 73 189 2.3260 [9-61 53

<5

u , f 3.53 X lo-’ h, c h, g h h a, f 0.706

I 10 20 20 <1

c,

50

p

P

1.27 8.33 14.5 29.2

Error (%)

I

Ref. i

< 20 < 20 < 20

i i i

<20

i i

< 20

n, j

The percentage errors given in most cases are conservative estimates obtained from a compilation of available data. It is seen from the table that values accurate to better than 1% are available for a ( ’ ) for Li+ and the rare gases. In some cases (heavier alkali ions), limits are given in brackets; these limits reflect lower and upper values obtained in the literature. 2. Electron Scattering

To determine the contributions veB to the model potential, electron scattering may serve as a source of information. For our special purpose the scattering data from low energy experiments are of interest. For these only a few partial waves contribute to the scattering process, which has led both experimentalists and theoreticians to represent their results in terms of the phase shifts for these few partial waves. Hence in the procedure to determine ueB (see sketch in Section 11, C) the measured quantity M is represented by a set of phase shifts 77, at some energies EJ and one has to:

(I)

obtain the phase shifts from the experiment;

DETERMINATION OF INTERATOMIC POTENTIALS

73

(2) obtain the phase shifts from their calculation for the model potential [Eq. (16)]; and (3) compare the results and vary the parameters O,(B).

u,B(TB,O,(~))

Two useful inversion procedures could be thought of, namely one to obtain the phase shift from the measured cross sections (Gerber and Shapiro, 1976) and a second one to obtain the potential from the phase shift (Bottcher, 1971; Shapiro and Gerber, 1976). The first step should not be complicated, whereas in the second step the present solutions are restricted to local potentials (see later). To the author's knowledge not even the first step has been applied in electron scattering. Experiments to determine the phase shifts have been reviewed extensively (Andrick, 1973; Golden, 1978) and a brief discussion will be sufficient. Due to the low mass of the electron target system and the low energies considered (in contrast to heavy particle scattering) from all the partial waves in the sums for the cross sections only the lowest ones, namely I < 2 are important, whereas the higher ones can be calculated in the Born approximation as corrections, indicated by the summation sign 2' . . . . . C;"=3. . . with the second term calculated in the Born = C:=, approximation. There are three basic experiments that lead to the phase shifts.

+

1. Differential cross sections for direct scattering. The measured quantity at a fixed energy for the scattering angle 19 is given in these experiments by

with

and where k is the wave number and 9, are the phase shifts. For the evaluation of such measurements usually trial values for the phase shifts q,, 0 < I < 2 (taking the above-mentioned Born corrections into account) are independently varied to obtain a best fit checked with a X2-test. If the measured cross section is sufficiently structured, a unique f i t is obtained. Such measurements and evaluations are performed at various energies yielding in summary a set of phase shifts for each energy.

2 . Differential cross sections for resonance scattering. Usually at higher energies the electron target system shows more or less well pronounced resonances which can be attributed to one of the phase shifts which as a function of the energy varies rapidly in comparison to the others. Accord-

74

R. Duren

ing to Fano (1961) this particular phase shift at I = I, can be written as a function of the energy as

with

s,,

= arccot[ 2

( ~ ,- E ) / T ]

(34)

where qpR is the nonresonant part of the phase shift, E , the resonance energy, and r the width of the resonance. Introducing Eqs. (33) and (34) into (32) and (31) yields the resonance structure of the differential cross section which again may be evaluated to yield the set of phase shifts as before. 3. Integral cross sections. Integral cross sections can also be used for the determination of phase shifts. Both the total cross section uT measured by attenuation

and the momentum transfer cross section measured in drift experiments

can be fitted by trial sets of phase shifts as a function of the energy. Complications arise concerning the uniqueness of the phase shifts. Since these problems are absent or less severe in differential cross sections, the evaluation of integral cross sections plays a minor role. In summary, from these experiments the phase shifts for the lowest /-values are obtained at various discrete energies with comparably high precision. To complete this information to be applied for comparisons with theoretical values, interpolation formulas derived from the effective range theory (O’Malley, 1963) or from corresponding ( a b initio) calculations (Nesbet, 1979, and references therein) are available. Given this set of experimental values the other step is to obtain the theoretical phase shifts. Assuming the potential to be local, this reduces to the well-known analysis of the asymptotic behavior of the radial wave function as solution of the Schrodinger equation

- , , [

d2

I(/+

1)

1

+ k 2 G,(kr,) = 0

2u,,(r,,19!~’)

(37)

DETERMINATION OF INTERATOMIC POTENTIALS

75

As boundary condition, regularity of the solution at the origin

is required and the phase shift is obtained by comparison of the calculatcd GI with the asymptotic behavior defined by

lim G,(kr,) = sin[ k r , -

rg+m

(39)

Each set of parameters Oi(B) in the model potential yields the phase shifts and by variation of the Oi(,) these are fitted to the experimental ones. However, Valiron et a/. (1979) have shown that the orthogonality constraint of the wave function with respect to the core may play a role. The solution to this problem may occur incidentally because the model potential is deep enough to contain bound states to which the free solution is orthogonal. Then this state induces approximate orthogonality to the core and the local model potential yields the correct result. Otherwise a special choice of the generalized pseudopotential may be used (Bardsley, 1974). To the Schrodinger equation (36) an inhomogenous term with Lagrangian multipliers A, is added

+,

where the on the right-hand side are the orbitals to which the scattering solution is required to be orthogonal. By variation of A, orthogonality is achieved and 0, is then a purely local potential. Except for the additional variation of A which must be determined for each energy, the calculation of phase shifts and the variation of the parameters is the same as discussed previously for the local potential. Experimental results from electron scattering are available from various sources. Most work has been done on He scattering and the results for this target are well established. In addition to the analysis of Andrick and Bitch (1975) from direct and resonance scattering, Steph et al. (1979) have given a uniform evaluation of several types of measurements (see references in Steph et al., 1979). These determinations for He have been confirmed recently by Williams (1979) but with a strong reduction of the error limits and an extension to Ne and Ar. Compiled in the extensive tabulation of s, p, d, and (for Ar) f-wave phase shifts for the energy range from 0.58 to 20 eV, a very accurate data set for model potential calculations is available in Williams' paper for these systems. Errors range from 1.4% to 8.4% for the

R. Diiren

76

lower I-values and are 23% for the f-wave phase shift of Ar. Measurements for heavier rare gases are rare, and results are only known with less accuracy (Kr: Weingartshofer et aI., 1974).

3. Spectroscopy Spectroscopic data enter in two ways in the determination of the parameters of the model potential: as atomic spectra to determine the contributions weA and as molecular spectra to determine the resulting interatomic potential. All the atomic spectra for the atoms considered here are well known and summaries are available in the tables of, e.g., Moore (1949) and of Bashkin and Stoner (1975). The complementary theoretical values to determine the parameter in ueA are obtained by calculating the stationary levels in the respective potential. With regard to molecular spectroscopy we have to consider within the scope of this review only that of alkali-rare gas molecules. The technique to obtain these spectra has only recently been developed and results are available for relatively few systems. Due to the weak bond induced by the van der Waals interaction the basic difficulty is to produce a sufficient amount of these molecules. If this is achieved, the well-developed technique of laser induced fluorescence eventually with dispersion of the fluorescence can be applied. The production of the alkali-rare gas molecules is achieved in a supersonic beam of a carrier gas, to which the atomic species to form the molecule are added in a small fraction. The properties of the carrier gas are given by its expansion from an oven at high pressures into the vacuum. The temperature T drops to T = To[1

+ i(y - l)M*] ‘ ~

where To is the temperature in the oven, y is the specific heat ratio c,/c,., and M is the (local) Mach number (Liepmann and Roshko, 1957). M varies with the distance X measured from the orifice along the beam in units of D , the diameter of the orifice, as M = 3.26(X/D)2’3

(42)

(Becker, 1968) for monatomic gases. The temperature is seen to decrease along the beam roughly with X . Relation (42) holds only as long as collisions in the beam are sufficiently frequent. As the rate of collisions decreases M approaches a limit M , which by Anderson and Fenn (1965)

DETERMINATION OF INTERATOMIC POTENTIALS

77

has been obtained as

where &, is the mean free path and the constants are for monatomic gases. With these relations, sufficiently accurate estimates for most cases can be obtained. For very low temperatures (high Mach numbers) Toennies and Winkelmann (1977) have given a careful study revealing the importance of the correct quantum-mechanical treatment of the collisions. This effect is particularly well visible for He where it has been verified experimentally by the same group (Brusdeylins et a/., 1977) obtaining 8.4 x K as the final temperature. In these studies it is found that M , rises more rapidly with the product p o x D (,po for the oven pressure) than suggested by Eq. (42). Returning to the formation of alkali-rare gas molecules, the carrier is loaded with a small amount of the respective atomic species of the molecules [e.g., 2% Ar and lo-’% Na in 98% He in the Na-Ar experiment by Smalley et a/. (1977)l. Once the molecule is formed, the high pressure in the oven (10-100 atm) and along the early stage of the beam guarantees such a high collision frequency that in general rotational relaxation leads to stable and “cold” molecules. Parenthetically we may remark that this technique is not restricted to these peculiar molecules. Stable molecules introduced into the carrier gas or dimers of the carrier can be rotationally cooled as well (e.g., Borkenhagen et a/., 1975; Wharton et a/., 1978; Bergmann et a/., 1976). The principle and realization of such experiments seem relatively easy and the few experiments obtained so far, namely with Na-Ar (Smalley el a/.. 1977) and Na-Ne (Ahmad-Bitar et a/., 1977), look promising. As minor problems associated with this method, competing dimer formation and successive inelastic collisions or the presence of atomic fluorescence may be mentioned. A quite serious experimental problem seems to be to achieve the necessary stability of the beam for a sufficient signal-to-noise ratio in the fluorescence signal. The evaluation of such experiments from the given spectral distributions to the interatomic potential is quite involved. In the forward procedure the ground state and the excited state term energies are written qualitatively along the general rules discussed by Herzberg (1971) and the spectroscopic constants are obtained by a f i t to the observed intensities. From there the complete potential can be obtained with a suitable model. Inversion procedures basically derived from Dunham ( 1 932) and Thakkar ( 1 975) are available. On this basis an evaluation of the above-mentioned measurements has been given by Goble and Winn (1979). In their paper a

-

78

R. Diiren

comparison with a long-range analysis (Le Roy, 1973; Stwalley, 1975) is also given. Both methods require a high computational effort, and problems arise from insufficient data in the inversion and from the model dependence in the forward procedure. 4. Heavy Particle Scattering

We will now turn to the experiments of heavy particle scattering which as mentioned earlier may be used to determine either the contribution V , to the model potential or the interatomic potential V . Let us first consider the core-core scattering experiments to determine V , . Within the scope of this article only the alkali ion-rare gas interaction needs to be considered. There are two basic problems for experiments relevant to our purpose. The first one is that, in general, low energy data are required probing the intermediate range of internuclear distances. Second, a comparably high accuracy is required since in the range of the equilibrium distances of the interatomic potentials the core-core contributions are of the same magnitude as the well depth itself. High energy data are available ( e g . Kita et al., 1975); for moderately high energies a semiempirical set of potentials for some of the interesting systems has been given (Sondergaard and Mason, 1975). But the extrapolation of such data to larger internuclear distances leads to serious errors. Concentrating on low energies, the number of experiments and of results is found to be very small ( 2 ) ; and these are restricted to the investigation of Li+ with the rare gases. Integral cross section measurements with beams (Powers and Cross, 1973) will not be considered here due to their insensitivity to details of the potential. Another integral cross section experiment is the study of the ion mobility in gases. The theoretical treatment (Viehland and Mason, 1975) and an iterative inversion procedure have been developed (Viehland et al., 1976). On this basis specifically the potential for the Li+ -He interaction has been obtained with high accuracy (Gatland er al., 1977). In differential cross section experiments the potentials at intermediate internuclear distances have been determined in two ways. Bottner e f al. (1975) have measured the cross section in experiments with slow (3 eV < E < 9 eV) ions. The well-known rainbow phenomenon (e.g., Pauly, 1979) has been observed by them for the interaction of Li+ with Ar, Kr, and Xe. At these energies the rainbow angle 9, (in radians) is given to a good approximation by

DETERMINATION OF INTERATOMIC POTENTIALS

79

where c is the well depth of the potential, and E the collision energy. This relation (obtained in the classical approximation) indicates that a measurement of the rainbow angle for a given energy yields the well depth of the potential. This simple relation quoted here to demonstrate the main sensitivity of this experiment is obviously not sufficient for the actual evaluation. This has been performed by Bottner et al. (1975) in a quantummechanical treatment. At higher energies the rainbow phenomenon is no longer visible, but forward diffraction oscillations have been observed (Wijnaendts van Resandt et al., 1976). These oscillations reflect the dimensions of the obstacle and their period is given in a semiclassical approximation by

A8x2a/kR,

Ap

(45)

where k is the wave number of the collision, R , describes the “size” of the obstacle, and A D the difference of the reduced impact parameters leading to this particular interference. Again we have given this approximate relation to demonstrate the main sensitivity of this experiment, namely to the size R , of the interatomic potential, whereas the actual evaluation by Wijnaendts van Resandt el al. (1976) has been performed on a quantummechanical basis of elastic scattering. In any case, both experiments are seen to be complementary to each other in that the well depth (c) and the equilibrium distance ( R , ) can be obtained from both together. In both experiments Li+ has been used as the only alkali ion in collision with various rare gases. Our search for experimental data for the core-core interaction appears to be unsuccessful. However, calculated potentials are available. In particular the work of Kim and Gordon (1974) is to be mentioned, who have given potentials for Li+, N a + , K + interacting with He, Ne, Ar, and Kr. Their accuracy has been confirmed by more recent calculations (Hariharan and Staemmler, 1976), and improvements which might be helpful for heavier systems have been developed (Gianturco and Lamanna, 1978). Thus the Kim-Gordon potentials may be used as a “substitute” for experimental values. We want to try to estimate their accuracy with reference to experimental ones, in particular at intermediate internuclear distances. To do so we have compiled in Table 11 a comparison of their well depths c and the equilibrium distances R , with experimental ones as far as available. For a differentiated analysis of the errors the experimental background should be improved. The work of Wijnaendts van Resandt et a/. (1976) indicates a shift of the radial coordinate of the calculated potential by 0.75 a.u. is necessary to reproduce their measurements (see their values

-

R. Diiren

80

TABLE I1 EQUILIBRIUM DATAFOR ALKALI+ -RAREGASPOTENTIALS

Li' -He

Kim and Gordon (1974)

Bottner et al. (1975) -

4.1 t 0.3

-

-

-

4.35 t 0.4

-

-

-

-

5.0 t 0.4

-

1.11-2 t 2.-4

-

-

-

5.1 t 0.4

-

€

3.68 2.54-3 3.76 4.56-3 4.25 1.09-2 4.48 1.32-2

1.44-2 t 2.-4

-

-

R,

-

-

5.4' 0.4

-

€

-

1.87-2 t 2.-4

-

-

R, €

Li+ -Ne

R, €

Li+ -Ar

R,

Li+ -Kr

R,

€

Li+ -Xe

Wijnaendts van Resandt Gatland et al. (1976) et al. (1977) 3.65 2.75-3 -

for R,,,). The experiments of Bottner et al. (1975) require a deeper potential well (by about 7%) than the calculated one. In the ion mobility experiments of Gatland et a/. (1977) a similar deviation is obtained in the well depth. but an intriguing agreement with respect to R , is observed. Despite these differences the agreement can be said to be reasonably good. We dare to make this statement particularly in the light of the usually much more coarse approximations for the determination of V , in model potential calculations. The experiments described in the remaining part of this section are used to determine the total interaction potential. In principle, scattering experiments of ground-state atoms and of excited atoms and collisional excitation experiments can be used for this purpose. Of these the excitation experiments are usually less sensitive to our particular interest in the intermediate range of internuclear distances. For instance the alkali-rare gas interactions have such high thresholds for the excitation (several 10 eV) that all detailed information about, for example, the well depth of the excited state is lost (Duren et al., 1974; Mecklenbrauck et a/., 1977; Ostgaard Olsen et a/., 1977). With one exception we will therefore discuss experiments with thermal collision energies. As the first type of experiments we will consider the line shape experiments, which can yield both the interaction potential in the ground state and the excited state. This topic has been reviewed recently (Gallagher, 1975; Behmenburg, 1978); therefore a brief summary confined to our specific interest will be sufficient.

DETERMINATION OF INTERATOMIC POTENTIALS

81

The principle of these experiments is to measure the dispersed fluorescence from a cell which contains the two atomic species, where the exciting light is “tuned” to the atomic line of one of the atoms. To a good approximation the light emitted at a certain frequency o off from the line center is given by the energy difference between the upper and the lower state of the transition both distorted by the interaction, that is,

where V , and V/ are the interatomic potentials for the interaction in the upper and the lower state, respectively, and R is a certain constant internuclear distance corresponding to this w. For such an experiment the apparatus is in principle comparably simple: A gas cell contains mainly the perturber gas and a small fraction of the species to be excited. Excitation is achieved with a discharge lamp for the latter material or a laser tuned to a particular transition thereof. Fluorescence light is observed perpendicular to the light beam and dispersed by appropriate means. The requirement with respect to the wavelength resolution of this element is not very high. But special care is needed to guarantee sufficient suppression of the central line. This is crucial since the intensity of the spectrum in the interesting range is 10-6-10-8 of the central line intensity. For the same reason the parameters of the cell must be chosen to avoid radiation trapping as well as possible. As parameters of the measurements, the temperature of the cell or the density of the perturber gas is varied. In the observed spectral distribution (of the fluorescence) one can according to the basic equation (46) distinguish three regions: 1. The impact region which typically stretches by AA x -+ 1 A around the line center: According to our relation above, far this region the interaction is governed by small distortions, which means that the interaction potential at large internuclear distances is involved. I t can be related to the width and the shift of the Lorentzian profile near the center, but we will not discuss this further (see, e.g., Lewis er at., 1971; Roueff, 1972, and references therein). 2. The far wing region with lAAl x 1-50 A: I n this region the distortion is larger but large internuclear distances are still involved. In this domain no temperature and density dependence of the intensity other than direct proportionality to the density is observed, which makes i t less attractive for our purpose. 3. The extreme wing region [MI2 50 A: Here the distortion reflects a comparably strong interaction (within the thermal limit) and it is this region in which we are mainly interested. We will see later how the

82

R. Diiren

temperature and pressure dependence of the intensity can be used to determine the interatomic potential here. The theory of line shapes is well developed especially for the binary collisions which we are interested in (see, e.g., Cooper, 1967; Reck, 1969; Szudy and Baylis, 1975; Herman and Sando, 1978; West and Gallagher, 1978). We will discuss the evaluation of these experiments by means of the quasi-static theory, of which we have anticipated the central idea earlier by Eq. (46). This states that transitions of the perturbed oscillators occur with unchanged internuclear distance for the upper and lower state. I t applies quantitatively in the region of our interest (region 3) and it allows a proper discussion of the experiments and their relation to the interatomic potential. To determine the spectral distribution we first consider the number N, of transitions which lead to an emission with a frequency between w and w + dw. By the correspondence between w and R established by Eq. (46) this is given by N,dw = NO4rR2 dR P ( R ) d o do

(47)

where No is the average density of the perturber, and P ( R ) the probability for the occurrence of internuclear distances between R and R dR. The usual expansion of the density yields for this probability the pair distribution

+

g,

P ( R ) = - exp[ V,(R)/KT] ga

(48)

where V , denotes the initial state of the experiment which is the lower one in an absorption experiment and the upper one in an emission experiment [the asymptotic value V,(co) being always subtracted]. g,/g, describes the ratio of the statistical weights for the molecular and the corresponding asymptotic atomic states. In this expression for P , equilibrium for all possible states is implied. To convert the number of transitions into intensity we have

Z(O) d w = N, . hwA (R ) d w

(49)

where A ( R ) is the transition probability, which can be written as Z(w)do= N o 4 r R d2w~ h 4 w 4 P ( RD ) .( R ) d w

(50)

where D ( R ) is the local dipole moment of the transition. In the following we will assume this to be independent of R and equal to its asymptotic value Do. This assumption should hold within 10% for the internuclear

DETERMINATION OF INTERATOMIC POTENTIALS

83

distances considered here, in particular for the alkali-rare gas interaction. Normalizing this expression to the total intensity, one obtains " "

where w, is the central frequency. For simplicity we will normalize the intensity ratio above with respect to the trivial factors on the r.h.s. of Eq. ( 5 I ) and define as normalized measured intensity

I*(o)dw = 4 a R 2 dR P ( R ) d w dw

As we have defined the far wing region (region 2) by small distortions V , is in general negligible there. With P ( R ) 2~ 1 the expression (52) for I * becomes independent of the temperature and the density. In contrast to this the influence of P ( R ) becomes visible in the extreme wings. If the spectral distribution is measured as a function of the temperature, one obtains the potentials in the following way (Hedges et a / . , 1972).

1. The plot of log Z*(o) versus 1 / T at different values w yields the initial potential, say, V , for emission versus w and since Aw = V , - V! the lower potential V,(w). By using a value for V/ determined from other sources (e.g., scattering experiments) on the w scale for one point, say wo. the corresponding R value R, can be fixed. 2. To obtain the scaling of the w-axis in R the spectral intensities I: obtained from extrapolation of I * to 1/ T = 0 are integrated to yield

s::

I*(o)do = (4a/3)[R 3

( ~ )-

R '(wg)]

(53)

By inversion of this (measured and integrated) function the scaling in R of the potentials V , and V/ obtained in step 1 is fully established. If the spectral distributions are measured for a fixed temperature at a high and at a low density of the perturber, the evaluation to obtain the potential is similar. I t is complicated by the fact that the distribution as given by Eq. (48) is valid in the high density limit (where all states are populated), whereas in the zero density limit contributions from inside the rotational barrier, that is, all bound states, must be excluded (York et a/., 1975). Clearly the calculation of this latter distribution requires the knowledge of the initial potential. In the above mentioned publication, however, this difficulty has been solved by an iterative procedure which obtains its starting potential from the assumed classical distribution.

84

R. Diiren

In some cases satellites are observed in the far wings of the spectral distribution. Usually they are found as a shoulder or a maximum followed in some cases by small regular oscillations in the distribution. They occur at frequencies w where d w / dR vanishes leading in the quasi-static theory to a singularity in the spectral distribution [see Eq. (52)]. For a quantitative treatment of such observations the quasi-static theory is therefore insufficient. The theoretical work on this subject is well advanced (Sando, 1974; Szudy and Baylis, 1975, and references therein). Since such calculations are complicated, especially in relation to the information about the potential to be expected, we need not consider this special feature here. As mentioned previously these experiments yield the interatomic potential. Therefore we will postpone the discussion of results to the following section. The second type of experiments to be considered here are beam experiments again at low energies. We will first consider experiments to determine the differential cross sections. A special feature of the alkali-rare gas interaction which we have already mentioned is a separation between the energy of the excited states and the ground state which is large compared with thermal (- 100 mecollision energies. As a consequence such experiments and their evaluation can be performed independently for the excited states and for the ground state. Of course, the ground state potentials play a central role in our comparison of experimental and theoretical results. On the other hand, techniques and the evaluation of such experiments are well known and reviewed (Pauly and Toennies, 1965, 1968; Fluendy and Lawley, 1973). In particular results for the alkali-rare gas interaction have been obtained by two independent investigations (Buck and Pauly, 1968; Duren el af.,1968) for most alkali-rare gas combinations which agree within a few percents in the minimum of the potential. For the excited states one expects similar accuracy from similar experiments, but it is only in recent years that such results have become available. We have reported recently the experimental aspects and the evaluation of such experiments in detail (Duren and Hoppe, 1978), so a brief summary will be sufficient. Basically in such an experiment the differential cross section for the excited state is measured as usual in a crossed beam experiment where the excitation of the primary beam is achieved by a laser tuned to a specific hyperfine structure level of this atomic species. The difference A of scattered particles measured for laser on and off reflects the difference between the respective cross sections:

A

= (da*/du) - (da/dw)

(54)

DETERMINATION OF INTERATOMIC POTENTIALS

85

where du*/dw stands for the effective cross section in the excited state, and d u / d o for the one in the ground state. d u * / d w is the total differential cross section; that is, it is the sum over the cross sections for the various exit channels of the collision. In the examples investigated so far the initial state is one of the doublet components of the resonance state of Na and the sum for du*/dw is over the fine structure components. In many cases th.: well depth for the ground state and .rr-components of the excited states can be expected to be quite different from each other. In particular the cross section for the ground state may be monotonic in the range of large angles. Then a structure in the difference A can be attributed to the excited state and can be evaluated accordingly. The use of the laser for the excited process will produce a selective preparation of the initial state. Only a particular fine structure level ( j ) is excited, which by the polarization of the exciting laser light has a welldefined distribution of the projection quantum number (m,).From the stationary distribution in the total angular momentum ( F , M ) (Hertel and Stoll, 1974; Macek and Hertel, 1974) one obtains with linearly polarized light for the j = 3/2 component the weights W ( j ,m,) as j=3/2,

mJ= -3/2

W(j,m,)=

1/12

-1/2 5/12

+1/2 5/12

+3/2 1/12

(55)

The knowledge of this distribution is desirable since it enters the evaluation of the data. The theoretical background for this evaluation derived from the general quantum-mechanical scattering theory has been given by Mies (1973) and Reid (1973). Compared with the general case it is simplified by the above mentioned separation of ground state and excited states such that only the excited states must be taken into account. The scattering amplitude for transitionsj'mJ' to j , m, is given as

Aa) = 2 / ( k ' k ) ' / 2 C i ' - ' ' Y : P ( r i ) G ~ ; Y ~ ' ( a ) (56) where i? is the beam direction, and a is the direction of the detector. Y are j(j'mJ',j,m, 1

the spherical harmonics and the sum is over all angular momenta of the relative motion. The matrices G are obtained from the usual T matrices by GF;p=

2 (jlm, I J M ) T J ( j ' l ' j l ) ( j ' l ' m ~I ~J M' )

JM

(57)

where the ( ) are the Clebsch-Gordon coefficients. To obtain the T matrices the coupled Schrodinger equations for each J

86

R. Duren

are integrated

to the asymptotic limit. The potential matrix V for the coupled equations can be expressed by the adiabatic potentials involved. The differential cross sections for individual j‘m; +jmj transitions are then given as the squares of the scattering amplitude. Averaging of these cross sections with respect to the initial distribution of mi described previously and summing over the final state quantum numbers yields the measured cross section as

No inversion procedure is available but the above mentioned steps must be calculated for each iteration in the determination of the potential parameters. Since these experiments are moderately complicated, only a few results are available at the present time: Carter et al. (1975), Hertel et al. (1976, 1977), Diiren et al. (1976), Diiren and Hoppe (1978), and Diiren and Groger (1979). We will discuss of these the alkali-rare gas systems later in the section on the comparison between experiment and theory. The most recent development of these experiments has been the use of Doppler fluorescence for the resolution of the fine structure of Na in the exit channel (Phillips et al., 1978). Results have been obtained for the differential cross section for fine structure transitions from Na (32P,,2) in collisions with Ar. The angular resolution of these measurements is rather low. But the future development may see an interesting source of information emerge from this type of experiment. In addition to the differential cross section experiments described so far, a large number of integral cross section experiments of this type have been reported. In particular, cross sections for fine structure transitions and depolarizations have to be mentioned (for recent reviews, see Baylis, 1979; Elbel, 1979). Such experiments are usually performed in cells, but beam experiments have also been reported (e.g., Anderson et al., 1976; Phillips et al., 1977). Clearly the cross sections measured in such experiments depend mainly on large internuclear distances and not that much on the behavior at intermediate distance. In reference to a very accurately measured energy dependence of a fine structure transition cross section (Phillips et al., 1977), potentials differing in the well depth by as much as a factor of two can hardly be distinguished (Saxon et al., 1977). Thus for our special

DETERMINATION OF INTERATOMIC POTENTIALS

87

interest such experiments usually will not be sufficiently differentiated to determine the interatomic potential. C. COMPARISON OF INTERATOMIC POTENTIALS FROM DIFFERENT EXPERIMENTS

Before we enter the comparison of experimental and theoretical results we want to compare the results from different experiments yielding the interatomic potential. This comparison can be given in detail only for one particular system, namely Na-Ar. For this system all the experiments mentioned earlier have been performed: the spectroscopy of the Na-Ar molecule (Smalley et al., 1977), line shape measurements in the extreme wings (York et al., 1975), and various molecular beam experiments (von Busch et al., 1967; Diiren and Groger, 1975, 1979). In some cases evaluations of the same data set have been given, which will be also quoted later. We want to use this comparative study to establish error bounds for the following comparison with theoretical results. The comparison of the methods which is given simultaneously with this study should be seen as definitely restricted to results and their accuracy. We will not compare the methods in terms of feasibility and the like. We do not have a preference for any one of these methods. To the contrary we have to stress the point that the available results show the different experiments to yield complementary information. Let us first consider the ground state of Na-Ar for which we have compiled the available equilibrium data in Table 111. The inspection of this table reveals by the evaluations of the same experimental data set by different groups the dependence on the model (see entries 1 , 2 and 3,4). In this respect the spectroscopic data and molecular beam data show no difference. (Note that we have chosen from the multiple choice results of the spectroscopic values the one closest to differential scattering experiment.) For these reasons it seems to us appropriate to give the unweighted average value as a recommended value with error as given by the standard deviation. Figure 1 gives a plot of the potential obtained from the differential cross section experiment mentioned earlier together with the equilibrium points as given in Table 111. The shape of this function has been given previously (Diiren and Groger, 1978, and references therein), and it compares well with the spectroscopic result in a fairly large range of internuclear distances around the minimum. A similar compilation for the h,,: state is given in Table IV. We will not discuss this table in detail because similar arguments apply as before. With

R. Duren

88

TABLE 111 EXPERIMENTAI. EQUII.IRRIUM DATAFOR

THE

Na-Ar

(GROUND STATE)

INTERACTION

Evaluation

Experiment Type"

Ref.

Year

MBI MBI SP SP MBD

h h e e g

1967 1967 1977 1977 1978

a.u.)

c (

R , (a.u.) 9.5 9.1 9.4 9.4 9.5

0.204 0.188 0. I76 0.190 0.195 0.190 t 2%

Average value"

9.4

Ref.

Year

c

1968 1968 1977 1979 1978

d e

1

g

* 2%

"The type of experiment is denoted by: MBI. molecular beam experiments, integral cross sections (glory undulations): SP, spectroscopy of van der Waals molecule; MBD, molecular beam experiments, differential cross sections. hvon Busch ef al. (1967). 'Buck and Pauly (1968). dDuren er al. (1968). 'Smalley et al. ( 1977). 'Goble and Winn (1979) (their potential TI). KDuren and Groger (1978). "Unweighted average with percentage standard deviation.

0:

-

0'

T

O

3 0

0

I

>

-0 1

-0

: I

8

1

I

12

10

R

1

11

1

16

1

18

[a u.1

FIG. I . Interatomic potential for Na-Ar (ground state). Equilibrium data: 0, Buck and Pauly (1968); 0 , Duren e? al. (1968); +, Smalley e? al. (1977); +, Goble and Winn (1979); solid line, Duren and Groger (1978).

89

DETERMINATION OF INTERATOMIC POTENTIALS TABLE IV EXPERIMENTAL EQUILIBRIUM DATAFOR

THE

?TI/*

STATE OF THE Na-Ar SYSTEM

Experiment

Evaluation

Type*

Ref.

Year

LS SP SP MBD

b

1915 1977 1977 1979

Average valueg

c

c e

c

(lo-’ a.u.)

R , (a.u.)

Ref.

Year 1975 1977 1979 1979

2.56 2.58 2.6 1 2.55

5.99 5.49 5.49

b

(5.75v

e

2.57 2 1%

5.7 & 2%

c

d

“The type of experiment is denoted by: LS, line shape measurements; SP, spectroscopy of van der Waals molecule; MBD, molecular beam experiment, differential cross sections. bYork et al. (1975). ‘Smalley et al. (1977). dGoble and Winn (1979). ‘Duren and Groger (1979) (the experiment is primarily sensitive to c). /The particular evaluation in reference (e) yields this value from a calculated potential. g Unweighted average with percentage standard deviation.

respect to the average value we note that a similar error limit as before has been obtained. A figure of the potential will be presented later together with theoretical results. In summary then for a system studied as well as Na-Ar. the equilibrium data can be obtained with an accuracy of approximately 2%. But unfortunately there are not more systems studied that well. Similar tables can be compiled for Na-Ne [Diiren et al., 1972 (ground state MBI only); Carter er a/.. 1975; Ahmad-Bitar et al., 19771. But for this system due to various reasons, molecular beam experiments provide data with errors up to 30%. In this case the most reliable results are certainly obtained from spectroscopy of the van der Waals molecule with quoted error limits of about 2% for the equilibrium data of the ground state and the excited state (’T~,’) (Ahmed-Bitar er a/., 1977). In the light of an independent evaluation of the same data (Goble and Winn, 1979), we are rather inclined to estimate a (still sufficiently impressive) limit of 5%. To close this section on the experimental potentials we want to give a direct comparison between the measured quantities and the ones calculated from the potentials, namely the differential cross sections for Na-Ar in the ground state (Fig. 2 ) and the excited state (Fig. 3) (Diiren and Groger, 1978, 1979).

0

200

100

300

400

500

600

700

a,,,,

800

9 0 0 1000

1400 1 2 0 0

00

1300 1400

[degl

FIG. 2.

n'

Y!OO

'

0.

no

1I.d

1b.d

2b.

00'

24. u d

2b

JcM

od

3'2

od

[degl

FIG. 3.

3b.00

h o d

Yq

on

Y A . 00

82.

no

56 00

I

nu

DETERMINATION OF INTERATOMIC POTENTIALS

91

In the ground state experiment both rapid oscillations and one rainbow angle are resolved, which leads to an accurate determination of both E and R,. In the experiment for the excited state, rapid oscillations are not resolved but three rainbow maxima are visible.

IV. Interatomic Potentials Determined with Model Potentials In this final section we will discuss the comparison of experimental and theoretical results. Having established in the last section error limits of experiments, this will lead to an estimate for the reliability of the theory in its present status. Discrepancies will predominate even if the most recent development of the theory is considered, as we will demonstrate with few examples. We will begin our discussion with the previously mentioned Na-Ar interaction. For this system as for many others the potentials have been calculated by Baylis (1969) and Pascale and Vandeplanque (1974). Their procedure for determining the parameters of the model potential has been described previously. We recall in particular that one parameter is varied to be determined from the experimental interaction potentials in the ground state, represented by c, R , , the equilibrium data. However, agreement of both these values with the experimental ones is not obvious. We note that in the theoretical results both values E and R , are correctly correlated, such that a fit of both quantities can be achieved with the variation of only one parameter in the model potential. This leads us to an interesting feature of this model potential, namely that it reproduces the shape of the ground state potential quite well. This determination of the correct shape of the alkali-rare gas interaction from experiments has been a longstanding problem. Finally it has been resolved by the simultaneous evaluation of many independent measurements for the same system (Buck and Paul, 1968; Duren et al., 1968). The claim of these papers was a universal shape function, practically deduced from the fit to few systems with particularly many measurements and generalized to the others. Recently we have confirmed its use for Na-Ar (Duren and Groger, 1978), and recent spectroscopic work seems to support it (Goble and Winn, 1979). In Fig. 4 this shape function is given together with the LennardFIG. 2. Differential cross section for Na-Ar (ground state). Measured points and fit with interatomic potential of Fig. 1.

FIG. 3. Differential cross section for Na(3’P,,+Ar. interatomic potential of Fig. 5.

Measured points and fit with

92

R. Duren Reduced Distance X = R / R m

FIG. 4. Reduced potentials for the ground state alkali-rare gas interaction: Lennard-Jones (12.6) potential; . . . ., model potential calculations; , Duren ~

-el

-, a/.

(1968).

Jones (12,6) potential, the one deduced from Baylis (l969), and the one from Pascale and Vandeplanque (l974), the latter ones differ within the resolution of the drawing only for internuclear distances smaller than R , . (The same result is obtained from the reduction of heavier alkali-rare gas pairs.) In the critical region of R / R , x 1.2, V ( R ) / V ( R,) x 0.5 the calculated potential lies midway between the Lennard-Jones (12.6) potential and the modified one. This agreement (in absolute units, values of about 2x a.u. are involved) excellently supports the model potential used in the calculation. Before we discuss the results for the excited states we note again that the ground state data are the only ones used in the fitting procedure of Baylis (1969) and Pascale and Vandeplanque (1974) and they were the only ones available at that time. The inherent assumption was that this fit would determine the model potential sufficiently to yield the potentials for the excited states simultaneously. The experiments to test the excited state and this assumption have been discussed in detail previously and we compare in Table V the calculated results with the average values. The results of a recent CI calculation (Saxon et al., 1977) (deviating by 10%in 6 ) are also given. Figure 5 shows these data together with the potential determined from differential cross section measurements (see Fig. 3). Inspection of this table yields discrepancies by roughly a factor 2 for the well depth and a deviation of 10% for

93

DETERMINATION OF INTERATOMIC POTENTIALS TABLE V COMPARISON OF CALCULATED AND EXPERIMENTAL VALUESFOR THE f , pSTATE OF Na-Ar

a.u.)

c

Experimentalu Model potential* C I calculation‘

2.57 2 I% 1.30 2.24

R, (a.u.1

5.7 2 2% 6.35 5.75

~~

~~

“See Table IV. hPascale and Vandeplanque (1974) (see Note Added Proof. p. 100). Saxon el a/. ( 1977).

5.0

7.5

10.0

R

[J.u]

in

12.5

15.0

+,

FIG. 5. Interatomic potentials for Na(32P,/3-Ar. Equilibrium data: Pascale and Vandeplanque (1974); 0 , Saxon et u/. (1977); solid line, Duren and Groger (1979).

94

R. Diiren

R , . Unfortunately more recent model potential calculations are not available for Na-Ar (see Note Added in Proof, p. 100). From the study of this particular case we think that two general conclusions can be drawn. First one finds that the model potential discussed previously quite accurately reproduces the shape of the ground state potential for the alkali-rare gas interaction. It badly fails for the excited states. The second conclusion refers to the use of experimental data: I t is seen that the reliability of a model cannot be established with ground state data only. As a second example we would like to discuss the Na-Hg interaction where we have used the model and algorithm of Baylis (1969). Our starting point however was quite different in that we had the data for the ground state and for the excited state available. (A second difference was our more phenomenologically oriented interest, to obtain the interaction potential.) Using the model as it stands, it turned out to be impossible to obtain a simultaneous fit to the results for the ground state and the excited state by the variation of the free parameter r o . So we used the dipole polarizability a(1 ) as a second free parameter. The fit to both data sets was then obtained by finding in the r o ,a(')-plane the common overlap of the areas compatible with the individual data sets. The results have been given elsewhere (Diiren and Hoppe, 1978) and a brief summary may be sufficient. Concerning the ground state interaction one pair of parameters ro, a(') was found to fit the ground state interaction for the systems Na, K, Cs interacting with Hg measured by Buck et al. (1972). Good agreement was found for the shape of these potentials determined by the inversion of differential cross sections (Buck, 1974). (See Fig. 1 of Diiren, 1977.) Comparison with differential cross sections for the excited state, at the present time only possible for Na-Hg, is also satisfactory. The *?r states exhibit a deep potential well (- 11.1 X l o w 3 a.u.) leading to orbiting collisions at low energies. Due to this effect a note of precaution has been given in Diiren and Hoppe (1978) due to some uncertainty in the indexing of the maxima which could be verified only partly in this work. Unfortunately this problem has not yet been completely settled, even though we have a new confirming piece of evidence from the direct observation of the orbiting backward peak (Diiren et af., 1979). We are well aware of the limitations imposed by the phenomenological basis of this model, which has been seen to work well for one particular system only. So we are not inclined to draw generalizing conclusions concerning this model. However, concerning the method of the determination of the potentials in general the example demonstrates that a simultaneous evaluation of ground state and excited state data is urgently required.

DETERMINATION OF INTERATOMIC POTENTIALS

95

TABLE V1 EQUILIBRIUM DATAFOR THE GROUND STATEINTERACTION OF Na-Ne a.u.1

t

Experimento Model potentialb

R, (am)

0.037 0.035

10.0 10.0

‘?Ahmad-Bitaref a/. (1977). ’Philippe ef a/. (1979).

Recent results for our final example, namely Na-Ne, obtained by Philippe er al. (1979) can be interpreted to support our conclusions. Unfortunately we have not obtained sufficiently detailed data on these results to go through the same analysis as with Na-Ar and are restricted to the published values. Again the experimental data from scattering (Carter el a/., 1975) and from spectroscopy (Ahmad-Bitar er a/., 1977) demonstrated that neither the ground state nor the excited state could be interpreted by various model potentials. By use of the model potential of Bottcher and Dalgarno (1974) with parameters fitted to e--scattering data the above mentioned calculation of Philippe et a/. (1979) obtained potentials without reference to the measured interatomic potential. For the ground state this leads to excellent agreement with the experiment as shown in Table V1. For the excited state there is the problem that the currently available core-core potential ( N a + -Ne) is not sufficiently well established. Each of the various available choices leads to values of the equilibrium data for the A27i-state outside the error limits of the experiments. Out of these choices the spectroscopy of .the van der Waals molecule is able to select two which lead to results close to the experimental findings as compiled in Table VII. It should be noted however that this particular choice of the core-core repulsion is not supported by other reasons.

TABLE VII EQUILIBRIUM DATAFOR THE %-STATE INTERACTION c

Experimento Model potentialb

a.u.)

0.638 2 2% 0.601

“Ahmad-Bitar el al. (1977). Philippe ef al. (1979).

OF

Na-Ne

R , (a.u.)

5.1 ? 2% 5.0

96

R. Diiren

In contrast to the philosophy advocated in this article, namely to use the experimental interaction potential to determine the parameters of the model, Philippe et a/. (1979) have used the determination of subsets of parameters only. Because of this their work is especially valuable as a study of the limitations in accuracy imposed by that procedure. Of course it is difficult to assess the validity of consequences for general use from this work, but a few suggestions may be allowed. Concerning the construction of the model potential the expansion of Bottcher and Dalgarno (1974) [see Eq. (29)] is confirmed as a most valuable tool. Concerning the use of experimental data a serious lack of accurate values concerning the core-core interaction is made visible. We want to close the comparison of experimental and theoretical results here being aware that many valuable contributions had to be left out. We hope that the main roots for future work could be described by these examples.

V. Conclusions Great advances in experimental techniques concerning the atomic interaction at intermediate and moderately large internuclear distances for excited states in addition to the ground state have been obtained in recent years. Various different experiments-heavy particle scattering with laser excited atoms, molecular spectroscopy with laser induced fluorescence, line shape broadening, and electron scattering4an contribute to the knowledge of the interatomic potentials involved in such interactions. To determine the potentials quantitatively from the original experimental data with an accuracy comparable to the high accuracy of the original experimental data some a priori knowledge must be introduced. Ab initio calculations are usually too expensive and they are, on the other hand, not as accurate as experiments can be. The model potential calculations present an alternative. First, they provide some qualitative insight quite easily but much more importantly they are from the beginning ready to be combined with experimental results in a natural way. Compared with the traditional phenomenological determination of the interatomic potential, this combination has the advantage that the modeling is shifted from the eventually many interatomic potential functions involved to one function, namely the model potential, and that the additional knowledge of the electronic wave functions is included. In this way the interatomic potentials for various states and eventually for various interacting systems of a series are obtained with one model potential. An additional less obvious

DETERMINATION O F INTERATOMIC POTENTIALS

97

advantage is found in the analysis of the individual contributions to the model potential. From there one finds a rational way to combine experiments as diversified as electron scattering from the target and atomic spectra of the projectile and investigations of the molecule (by scattering or by molecular spectroscopy) in a unified evaluation. As a matter of experience then the interaction contribution is observed to be described with fewer parameters than needed in the phenomenological approach, provided of course that the model potential is designed with care. However. this combined evaluation has until now been demonstrated with only a few examples. These examples lead to an accuracy of the order of lop5a.u. for the well depths (corresponding to some c m - ’ or better) and of 0.1 a.u. for the equilibrium distances. It seems fair to identify this as a satisfying achievement. Of course severe problems have been brought to light with these more recent experimental results, concerning the flexibility and reliability of currently available models. More research in this direction will be necessary to elaborate the present status of their development. To this end the combination of experimental data and model potential calculations proposed in this article may present a helpful concept.

ACKNOWLEDGMENTS

I wish to thank W. E. Baylis, F. Gianturco, and H. Pauly for fruitful discussions concerning this article and Mrs. Ch. Schneider-Rogier and Ms. B. Jung for their help in the preparation of the manuscript. Special thanks for the many discussions in the course of the experimental work with scattering experiments with laser excited beams are due to my co-workers W. Groger, H.-0. Hoppe, R. Liedtke, and H. Tischer.

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NOTE ADDED IN PROOF Some investigations related to the subject of this article that have been published recently need to be mentioned. Concerning experimental results. a new evaluation of the spectroscopic data for Na-Ar has been published (Tellinghuisen et a/., 1979). It confirms the equilibrium data used in this article and gives a refined shape function for the respective potentials. A central question concerning the various model potentials proposed. namely the accuracy of their numerical treatment, has been investigated for the Baylis model in two papers (Czuchaj and Sienkiewicz, 1979: Duren and Moritz, 1980). I n these more accurate treatments the model yields good results concerning the excited states (with the free parameter fitted to the ground state). Specifically, for the A2v,,, state of Na-Ar, c = 1.85. lo-’ a.u. and R , = 5.55 a.u. (Duren and Moritz, 1980) are obtained, which are to be compared with the values of Table V.