Revista Mexicana de Física 40, Suplemento 1 (1994) 108-122

The role of non-additive forces in atomic and nuclearinteractions

I.G. KAPLAN* AND O. NOVARO

Instituto de Física, Universidad Nacional Autónoma de MéxicoApartado postal 20-364, 01000 México, D.F., México

Received 31 January 1994; accepted 14 March 1994

ABSTRACT. The types of interparticle forces which are non-additive are discussed both in atomicand in nuclear systems. We present the explicit formulas for calculation of m-body forces in avariational approach. An analysis is given of the role of non-additive forces in atomic clusterformation. As follows fram our study the m-body forces are decisive in tbe metal cluster stability.The present situation and possible sources of non-additivity in nuclear interactions are discussed.

RESUMEN. Se realizan los diferentes tipos de fuerzas entre partículas, que son de tipo no-aditivo,tanto para sistemas atómicos como nucleares. Se presentan formulas explícitas para el cálculovariacional de las fuerzas de muchos cuerpos. El papel que juegan las fuerzas no-aditivas en laformación de cúmulos metálicos se discute en detalle. Esto se ejemplifica por nuestra análisis sobrela estabilidad de cúmulos metálicos donde las fuerzas de m-cuerpos demuestran ser decisivas parala estabilidad de los mismos. También se presenta una discusión del status actual, asi como de lasposibles fuentes de la no-aditividad en las interacciones nucleares.

E'ACS: 34.20.-b; 21.30.+y

1. INTRODUCTlON

Most physical laws established for many-particle systems are additive. This allows for thesimplest mathematical description. A well-known example is the Coulomb law:

v = '\' qiqj,L r ..i<j lJ

(1)

where rij is the distance between charges qi and qj. This law implies that the chargesare well describedphysically as point charges. The interaction of point charges is alwayscharacterized by the pair additivity. In general, the pontential energy of structurelessparticles can be represented as a SUIIl of pair potentials gij

(2)

irrespective of an interaction law.

*On leave from Karpov Institute of Physical Chemistry, Moscow.

TIIE ROLE OF NON-ADDITIVE FORCES IN ATOMIC. . . 109

However in real atoms and molecules the charge is spatial!y distributed. The interactingatoms have an internal structure which is modified differently in different environments.This results in deviations of the simple pairwise formula (2), introducing terms that weshal! cal! non-additive.

Let us consider, for example, the dipole-dipole interaction in an atomic system. Inorder to conserve the pair additivity one has to assume that the dipole moments ofsurrounding atoms do not induce an additional dipole moment in the interacting pair ofatoms, or change the orientation of their dipole moments. Obviously, these conditions arenot satisfied in condensed matter and even in gases if the latter are not rarefied enough.On the other hand, we know many empirical and semi-empirical pair potentials which

describe quite satisfactorily the properties of liquids and solids, see chapter 5 in Ref. [1].The point is that the parameters in these potentials are not the parameters of a true two-body interaction, their values depend upon properties of a medium. So these effective two-body potentials include non-additive interactions through their parameters. The latter cannot be directly related to the definite physical properties [2]. For instance, the coefficient ofthe term R-6 in the Buckingham or Lennard-Jones potentials is not equal to the dispersionconstant C6. The widespread method of atom-atomic potentials [1,3] gives good resultsonly because its pair potentials are effective ones. However, in some cases for obtaining agood agreement with experimental data the effective potentials must be constructed with3-body [4,5] and even 4-body [61 terms.'In nuclei, realistic two-body forces in different models ensure the binding energy of

nuclear matter which is smal!er than the empirical value by several MeV / A. Introducingmany-body forces gives an additional binding and shifts the equilibrium density to theempirical value. Non-additivity in nuclear interactions, has its origin first of al! in theconventional meson exchange forces: nucleons are interacting by the exchange of mesons.The exchange energy is non-additive both in atomic and in nuclear interactions. On theother hand, the nucleons are nowadays described as objects composed of quarks. So therecan exist non-additive quark exchange and polarization interactions.The first mention of 3-body forces in nuclear interactions appeared very long ago [7].

Afterwards, many of papers devoted to the many-body problem in nuclear physics werepublished, see Refs. [8-30]. But due to the complexity of nuclear forces this problemis much more elaborated in atomic and molecular physics, see chapter 4 in book [1],review [31] and section 2 and 3 of this report.

In the present report we discuss the type of interparticle forces which are non-additiveand the methods of theír calculations. The last two sections are devoted to a brief discus-sion of the role of non-additivity in atilmic (molecular) and nuclear interactions.

2. ENERGY DECOMPOSITION AND TIIE METIIODS OF CALCULATION OF NON-ADDlTIVITY

IN ATOMIC SYSTEMS

In this section we present formulas and a discussion of the non-additivity problem foratomic SystClllS, for which the theory is rather well elaborated. Ilut wc hope that thissystematized material will be useful for a discussion of non-additivity effects in nuclearinteractions.

------------------- -- --

110 I.G. KAPLAN AND O. NOVARa

2.1 Large distances

The term "large distances" is usually used for the distances at which the magnitudeof interaction between subsystems can be considered as a small in cornparison with thesum of energies of non-interacting subsystems. In this case different types of perturbationtheory (PT) can be applied.Thc interaction cncrgy Eint is cxprcsscd as a series aver various orders of PT

00

. _ E(l) eCl) '" [eCn) eCn)]Emt - el + cxch +L pol-exch + poIn=2

(3)

The first order of PT is the well-known Heitler-London energy. It is easily calculated if thev..'ave functions 'lto of thc isolated subSystClllS are knOWIl. For thrcc intcracting subSystCIllSA, B, and e the 1 order energies can be expressed as

(4)

(5)

where V is the interaction operator, Q denotes the perlllutations of eIectrons betweeninteracting subsystems and q is the parity of the perlllutation. The normalization factorsin Eqs. (4) ami (5) are omitted, see Subsection 1.2.6 in Ref. [1]. For the expression of theexchange energy E~~~hin the density matrix formalism, see Ref. [32].

The natural choice of zeroth-order wave functions as an antisyrnmetrized product ofunperturbed wave functions of subsystems leads to serious difficulties for the orders of PThigher than 1 because it does not allow to apply the standard techniques of the Rayleigh-Schr6dinger or I3rillouin- \Vigner PTs. The reason is that, although the total HamiltonianII commutes with the antisymmetrizer operator .4, the zeroth-order Hamiltonian lIa andthe perturbation operator V do not comlllute with .4. So the antisymmetrized functionsof zeroth-order are not the eigenfunctious of the unperturbed I1amiltonian Ha. Severalsuccessful approaches to the symmetry-adapted pertnrbation theory (SAPT) were devel-oped, for the detailed discussion see chapter 3, in Ref. [lj. The successful elaboration ofSAPT, and its application to particular atomic systellls was performed in Refs. [33-35].

The exchange and exchange-polarization energies fall exponentially with the distance Rbetween interacting subsystems. So at large enough distances there are only electrostaticand polarization interactions. The polarization energy can be subdivided into inductionand dispcrsioll cncrgics (1]

00

E - '" [E(") E(n) ]poI - L ¡nu + (l!sp 1

n=2

(6)

which are calculated by the standard quantum mechanical PT. In this energy region g(n)11Il(

and E,\;~pcan be expanded i,üG multipole series in powers of R-1. For two interacting

TIIE ROLE OF NON-ADDITIVE FORCES IN ATOMIC... 111

subsyslems in lhe second order of PT lhe dispersion energy is expressed as

(7)

In lhe case of lhe inleraclion belween aloms in lhe S-slales lhe series (7) conlains onlyeven values of n. The firsl lerm, proporlional lo R-6, corresponds lo lhe dipole-dipoleinleraclion, lhe second (~ R-s) is due lo lhe dipole-quadrupole inleraclion an,1 so forlh.It is worlhwhile lo nole lhal lhe mullipole expansion is valid al lhe dislances al \Vhich

eleclronic c!ollds of inleracling subsyslems do not overlap. In the inlermediale distancesthe charge overlap effecls must be taken inlo accOllnt. This was done in lhe series of \Vorkshy ~Ieath and co-authors 136-381, for another approach see ReL [39].In lhe energy partition (3)-(6) only t\Vo lerms are addilive. It is él

l) and éI

2) . Thee (ISP

éI)h can be considered as additive only if we neglect in Eq. (5) all pennutations exceplexc

pair lranspositions. All other energies are non-addilive. Ir atoms don'l possess mllllipolemoments (have only c!osed e1ectronic shells) the induction energy in Eq. (6) is equal lozero and the first non-addilive lerm is E~~;.The expression for dispersion energy in the 111order of PT for lhree S-state aloms was

obtained by Axilrod and Teller [40] ami Mutto [41]. Using the dipole approximation theyoblained

é3) (ddd) = f¿~J((J (J (J) C9(abc)<lIsp a, b, e R3 R3 R3

ab ac be(8)

The dispersion constant C9(abc) can be represenled as an inlegral over atomic dipolepolarizabilities [42]. The geometrical faclor is equal to

(9)

where (Jo, (Jb amI (J, are the angles in a lriangle formed hy the interacting partic!es. Thesign of the dispersion energy in the 111 order of PT is ,Ielermined by the geometricalfactor (9). E~~;pbecomes positive if all the angles (J < 1170, and negative if, at least,

one of the angles (J > 1260• The nodal struct ure of E~~;p for different atomic trimers

was investigate" by Bruch el al. [431. The modification of the expression for E~~;p in theintermediate range of distances where charge-overlap effects mllst be taken into accountwas perfonned hy O'Shea ami Meath [37]. According to' their results, as R decreases andthe overlap of atomic wave fllnctions increases, the range of angles (J, where E~~;p < O,decreases. For the system 1l(ls)-H(ls)-Il(ls) at R ~ 3.5110 E~~;pis repulsive for allanglcs.The expressions for the higher order, of PT and subseqllent terms of the multipole

expansion were obtained hy 13ade [44]. see also Refs. [45,46].The second (after Eq. (8))

112 I.G. KAPLAN AND O. NOVARa

term in the multipole expansion of the dispersion energy of the three interacting atomshas the following form [44-46]

(3) ddq Cll (abe)Ed¡,p(ddq) = f (ea,eb,ee) R3 R4 R4 (10)

ab ac be

where the quadrupole moment is located on the atom C, and the geometrical factor isequal to

The ddd-term in the IV order of PT has the following form [44,46]

(4) (dd ) _ 45 [1 + cos2 ea 1+ cos2 eb 1+ cosee]Ed¡,p d - 64 R6 R6 + R6 R6 + R6 R6

ah ac ab be ac be(12)

At large distances the energies (8-12) rapidly fallo At intermediate distances,which aremost important for calculation of the properties of clusters and condensed matter, the for-mulas (8-12) must be corrected for the exchange and charge-overlap effects. Netherthlessthey are useful for constructing the empirical or effective potentials with fitted parame-ters [47,4]. As was shown by Hom et al. [5] the ddq-dispersion energy must be used alongwith the ddd-dispersion one in the treatment of the noble-gas-cluster spectroscopy.

2.2 Short and intermediate distances

At these distances the most appropriate method for the calculation of the interactionenergy is the variational method. In atomic and molecular calculations it is usually theHartree-Fock method supplemented by different methods for taking into account theelectron correlation, or methods based on the density functional theory (DFT).

In variational calculations the interaction energy is found as a difference of the totalenergy of a system and the energies of constituent subsystems

Eint = Etot - ¿Ea.a

(13)

This energy can be decomposed into the energy of pair interactions, E2(n), of 3-bodyinteractions E3(n), and etc., see chapter 4 in Re£. [1]. Thus, the total energy of n particlescan be represented as a finite sum

(14)

In Eq. (14)

n

E¡(n) = ¿Ea;a=l

E2(n) =¿cab;

a<b

(15)

TIIE ROLE OF NON-ADDITIVE FORCES IN ATOMIC. . . 113

Eab= E(ab) - (Ea + Eb) == E(ab) - E1(ab), (16)

where capital letter E denotes the total energy, Greek letter E denotes the interactionenergy. Similar definitions hold for the energy of 3-body interactions, .

Eabe= E(abe) - El (abc) - E2(abe),E3(n) = L Eabe>a<b<e

and 4-body interactions,

E4(n) = L Eabeda<b<c<d

Eabed= E(abed) - E,(abed) - E2(abed) - E3(abed)

(17)

( 18)

(19)

etc. up to E,,(n). It should be emphasized that the representation of the total energy asa finite sum (14) is exacto The accuracy of the calculation of the many-body contributionis determined by the accuracy of the variational method used.

Instead of the algorithm (15)-(19) a more general and convenient for calculation ex-pression for m-body forces can be obtained. From the structure of sums (17),(18) and thedefinition of the number eL for the combinations of e particles from the set of k particlesit follows

m-l

Em(n) = L E(ab ... m) - L a~"Ek(n)a<b< ...<m k=l

(20)

(n - k)!(n - m)! (m - k)!

(21)

Aceording to Eq. (21) for m = n all a~n = 1 and for En(n) we obtain

n-I

En(n) = E(ab ... n) - L Ek(n)k=1

(22)

in full accordance with the decomposition (14).For estimating the convergence of many-body expansion it is eonvenient to express the

series (14) in relative quantities. The non-additive m-body energy is usually expressed asratio to the additive 2-body energy

Em(n)Em(2, n) = E

2(n) (23)

114 LG. KAI'LAN AND O. NOVARO

3. TIIE 1~II'OlrrANCE 01' NON-ADDlTlVE I'ORCES IN ATO~IIC ANI) MOLECULAR INTERAC-

TIONS

The most investigated systems are noble gas clusters. Sinee the first ealculation of He3by Rosen [481 and Shostak [49] large number of studies were performed in whieh ratherprecise quantum ehemiealll1ethods were used, see Rcfs. [50-53]. It was shown that many-body expansion eOliverges rather wc11. For an equilaterial triangle with side length 3aoh(3,2)1 = 0.1 for lIe3 [50] and 0.05 for Ne, [52]. The ratio of 4- to 3-body foreesE4(3, 4) ,liminishes when going from He4 to Ar4 [311. For a lelrahe,lral eonformation of He4wilh the side length 3ao 1<3(2,4)1 = 0.16,1<4(3,4)1 = 0.07 [511. As the distanees betweenlIe aloms increases non-additive forces contribntions fall rapidly and beeonle negligibleairead)' at R 2= 4.5ao [43]' i.e'l at distances which are oC importan('e in crystals.A quite diff"rent situation exists for beryllinm clusters. As was shown in Rels. [54-56],

the non-additive forces are decisive for the stability of 13e3 and 13e4 clusters. The Beatom has a closed c1"ctronic shc11. As for noble gas clusters, for I3en clusters 2-bodyforces are rqlulsive, 3-body forces are attractive and 4-body forces are again repulsive.For tll(' equilateral triangular 13e3 eonformation at ,. = 4ao, 1<,(2,3)1 = 0.67. For thetetrahedral I3e4 conformation Enon-add(4) = E3(4) + E4(4) is attractive and in the region,.= 4.0 - 4.5ao it exceeds 2-body repulsions. So if for the trimer I3e, tlle mlue of 3-bodyforces is large but not enough to stabilize the clusters, for the tetramer I3e4 3-body forcesare large enough to bind it. 1t is most interesting that this oceurs at distances which are\"('ry c10se 10 experimental nearest neighbour separations in metallie beryllium (4.321aowithin the layer, and 4.205ao between adjacent layers 157]). \Ve can conclude that 3-bodyforees are responsible for the formation of the beryllium erystal [31].If we take into aecount Eqs. (13), (14) ami (23), the expression for the interaction

energy can be written in the following form

(24)

For the I3es clnster with geometry found in Re!. 1491the expansion (24) according to thecaleulations by Novaro and Kolos [55] is equal to

Eint(I3eS) = E2(5)[1 - 1.846 + 1.155 - 0.378]

\Ve can not speak about the convergen ce of many-body series. And4-body forces are greater than 2-body ones.The same situation oecurs for lithium clusters. For the Li6 (2,2,2)

distances as in the lithium erystal the expansion (24) aecording to Re!.

Eint(Li6) = E2(6)[1 - 1.69 + 1.81 - 0.86 + 0.12]

(25)

we see that even

cluster with the[591 is equal to

(26)

The Li atoll1 has an unfilled 2s-eleetronie sh"ll. This is the reason that the sign of m-bodyforces in Li" clusters is opposite (in most cases) to the sign of m-body forces in Be"clusters: in Li" clusters the 2-body forces are attraetive and 3-body forees are repulsive.For cluster Li6 with the geoll1etry eonsidered, the 4-body attraction is decisive in theclnster stability.

TIIE ROLE OF NON-ADDITIVE FORCES IN ATOMIC... 115

TAULE I. Non-additivc euergy contributions for the billding cllcrgy oC silvcr clusters Agn (n =4-6), in a.n., [il].

na E'nt(n) E,(n) == E.",,(n) E3(n) E,(n) E,(n) E6(n) Enon -add (n)1'3(2, n)1 1,,(2, n)l 1,,(2, n)1 1'6(2, n)1 J En,~._",t.l1

EA.M

4 -0.1409 -0.2566 0.2006 -0.0850 0.11560.i8 0.33 0.45

4' -0.0889 -0.305i 0.30i8 -0.0910 0.2168LO! 0.30 O.il

5 -0.1941 -0.350i 0.3263 -0.2598 0.0902 0.156i0.93 0.i4 0.26 0.45

5' -O.liiO -0.4502 0.5i8i -0.3913 0.0859 0.2i331.28 0.8i 0.19 0.61

6 -02660 -0.4355 0.4020 -0.3432 0.1549 -0.0442 0.16950.92 0.i9 0.36 0.10 0.39

6' -0.2606 -0.5135 0.6324 -0.6513 0.3802 -0.1084 0.25291.23 1.2i 0.i4 0.21 0.49

a Notatiolls uscd Corgcolll.etrics: 4 is rhombic, D2h; 4' is tetrahedral, Td; 5 is planar. trapezoidal.C2v; 5' is tetragonal pyramidal, C2v; ú is planar, trigonal, D3h; 6' is tripyramidal. C21) The param-eters of geollletries used are given in Ref. [i3].

The many-body analysis helps to understand why some of the possible structures arepreferable to others. Let us consider, for example, the tetrahedral structure. It is themost appropriate for optimization of 3-body interactions: this structure is built from fourtriangles. As a result for Be4 cluster it is the most stable strueture [601, while for Li4cluster the most stable is the plane rhombie strueture [61]. The reason is in the di!ferenteharaeter of3-body forees: for beryllium clusters they are attraetive, for lithilnn clustersthey are repulsive.Tbe many-body eontributions to the effeetive potentials iu clusters and erystals were

elaborated in the series of papers by Murrell and co-authors [62-681. In other stud-ies [59,69,70] the importanee of non-additive effeets in ehemisorption and catalysis wasexhibited.

In this report we present our latest ealculations of non-additive contributions in thebinding energy of silver clusters (carried out in collaboration with DI. R. Santamaría) filioPreviously such study was made only for the trimer Ag3 [ni. The calculations wereperformed by the all-eleetron spin density method with non-local corrections included(NLSD) on CRAY- YMP4/432 using DGauss programo The m-body forees were calculatedaeeording to Eqs. (20) and (21).

As follows from the data presented in Table 1 the many-body eonvergence is ratherpoor or absent entircly, as for strnetures denoted by prime. For latter structures thecOlllributions of total Iloll-additivc forces is csscntially larger than fOf lllOrc stab1e 11011-primed strúetnres. Total non-additive forces are repulsivo. It is their contribution the onethat makes the spaee conformation of Agn(n :5 6) less stable than the planar structures.

116 I.G. KAPLAN ANO O. NOVARa

4. NON-ADDlTIVE FORCES IN NUCLEAR INTERACTIONS

In the simplest model a nucleus is described as a system of point nucleons interactingthrough a two-body potential V;j. This potential is often termed "realistic". A realisticpotential is one which fits the deuteron properties and the NN scattering data well,from a threshold up to E1ab = 3S0 MeY (at present there exist several realistic 2-bodypotentials, namely: the Reid [74]' Paris [7S], Urbana [76]' Argonne [77]' Bonn [78,79], amiMoscow [80,81] potentials). The necessary requirement for any 2-body nucleon-nucleonpotential (besides fitting the scattering data and deuteron properties) is that it correctlyreproduces the saturation point of nuclear matter. The saturation point is defined by thedensity at which the absolute value of the binding energy per particle as a function ofthe nuclear density p (or, equivalently, of the Fermi momentum kF, where p = 2k~/31r2)reaches its maximum. The empirical saturation point has an energy per particle betweenIS and 17 MeY (the empirical saturation Fermi momentum is in the range 1.29-1.44 fm-1).

In the reaction matrix theory different realistic 2-body forces ensure about 10-12 MeY / A for the binding energy of nuclear matter. Now it is very well known thatwith only 2-body realistic forces the empirical binding energy ami saturation density cannot be reproduced simultaneously [17,82]. As was shown in Re£. [831, sufficient bindingenergy would be obtained at too high equilibrium density in so far as only 2-body forcesare used. A similar result was obtained by Pandharipande and Wiringa [IS] for the bestavailable at that time NN potential of Reid [74]. They showed that the saturation pointoccurs at a density much higher than the empirical one.

The two-nucleon potentials have failed in reproducing the experimental data for suchlight nuclei as three-nucleonic 3H and 3He [84]. Calculations with different techniquesami realistic 2-body potentials [84,8S] resulted in a three-nucleon binding energy aboutlMeY below the experimental value 8.48 MeY. The modification of2-body potentials cangive a triton binding energy closer to experiment, but results in a deviation from the NNscattering data [86].

All these discrepancies can be considered as the manifestation of the significance ofmany-body forces in nuclear interactions. The 3-body nucleon-nucleon forces are usu-ally attractive [87]. So the allowance for 3-body forces must diminish the gap betweentheoretical and experimental binding energies.

Because of its long-range nature, the two-pion exchange three-nucleon potential is ex-pected to be the most important among other proposed three-nucleon forces. It was firstproposed by Fujita and Miyazawa [U]. The process considered is the following. Particle Aemits a pion which is scattered by particle B and then absorbed by particle C. Sorne partof this process can be represented as exchanges of pions between particles A and B and,independently, between particle B and C giving a contribution to the 2-body forces. Theother part, which can not be separated in this way, gives rise to the 3-body interactionhctw('cn partic!es A, 13and C.

Other 3-body potentials were also introduced [88]. The 3-body forces arising fromp-meson-pion exchanges and two-p-meson exchanges were considered along with two-pion-exchanges in Re£. [13]. Osman [24] studied their contribution to the properties ofthree-nucleon systems. According to the results obtained [24]' all considered 3-body forcesmust be taken into account, a considerable contribution to the binding energy is given

TIIE ROLE OF NON-ADDITIVE FORCES IN ATOMIC. . . 117

by their tensor parts. The importance of taking into account the nucleon resonances (~degrees of freedom) for calculating 2- and 3-body forces was shown in Refs. [26-28].Two main methods can be distinguished in the nuclear matter calculations: the Brueck-

ner-I3ethe method [89-91) and the variational method [17,29,92]. In the Brueckner-I3ethemethod simultaneous correlations among n particles can be taken into account only bysolving n-body Faddeev equations 193]. This is up to now possible for n ~ 3. The three-nucleon forces are incorporated into the Faddeev equation and the latter is sol ved inthe first-order perturbation theory concerning the triton wave function [20,22]. In theabove-mentioned Ref. [24) the Faddeev equation with different types of 3-body forces wassolved by the expansion of an unknown wave function on partial S-, P- and D-waves.Including the D-waves enables us to study tensor forces.The variational method has the aclvantage that n-body correlations can be taken into

account without solving an n-body equation. But in this method a very complicated situa-tion occurs for the tensor forces. There are two approachs for solving this problem [17]. Thefirst is based on the Van Kampen cluster expansion [94). The 2- and 3-body interactionsare evaluated for successive terms in the cluster expansion, see [13,14]. In the secondapproach, which was begun by Pandharipande and Wiringa [15,16], the most importanthigher-order terms in the correlation operator are summed using integral equations inanalogy with the Fermi hypernetted chain (FHNC) method. We suppose that the approachdescribed in section 2.2 of this report, in particular Eqs. (20) and (21), can also be usefulfor many-body decomposition in nuclei.The numerical results for the contribution of the 3-body forces in the tri ton binding

energy are rather dispersed beca use of the complexity of nuclear forces and the differ-ent approximations for their calculation. This contribution ranges from 0.158 MeY [221,0.5 MeY [20], and 0.6 MeY [30] till (1.3-1.6) MeY in Ref. [24]. The latest calculations byPandharipande [30] gave for the ground state energy of 3H the value -8.47 :l::0.02 MeYwhich is in an excellent agreement with the experimental value 8.48 MeY. In this calcula-tions, an Argonne 2-body potential [771and model 3-body potential obtained in Ref. [29)were used.

In larger nuclei as in the dense nuclear medium the influence of many-body forces mustbe even more pronounced. It results in medium modification of the properties of mesonsand nucleons. As was shown in Refs. [95,96], the masses of vector mesons (p, w and 4»as a function of nuclear density decrease almost linearly. This leads to an increase in themeson-exchange interactions in the medium, for further elaboration see Refs. [97,98).

Another aspect of the non-additivity in nuclear interactions, in some respects sim-ilar to the atom-molecular case, arises in the so-called alpha- particle model of lightnuclei [99-101,9,10]. In the original model alpha particles were considered as stable well-defined clusters with additive attractive forces between theIll. This model did not givesatisfactory results [99]. Later on, the idea of the resonating group structure [102] wasapplied with all possible grouping of nucleons into "alpha clusters [lOO]. This modelcontains two types of the non-additivity described in Section 2.1 of this reporto Theinteraction between alpha clusters can polarize (excite) the.clusters. This effect leads tothe polarization non-additive alpha-alpha interaction [9,10J. The antisymmetrization ofthe total nuclear wave function [lOO] results in the exchange non-additive alpha-alphainteraction.

118 I.G. KAPLAN ANO O. NOVARa

It is now well known that the nuc1eon itseIr has a finite size and an internal structure.The proton has a r.m.s. radius of about 0.S3 fm which is comparable to the average inter-nucleon distance of I.S fm. Each nucleon is composed of three colored quarks interactingby gluon exchange forces [103,1041. In the qnark cluster modcl the NN system is describedas a six-quark system using the resonating group mcthod in which the rauli principie isfully taken into account [1051. Robson [ISI investigated in a phenomenological model the3- and 4-body forces between nucleons arising from quark-exchange interactions inducedby the antisymmetrization of the total wave fnnctiou. In ReC. [106] the influence of therauli principie for quarks on some properties of the three-nuc1eon system was investigated.The antisymmetrization of the 9-quark system was performed with using of the doublecoset expansion [107, lOS], which reduces the number of permutations from 9!j(3,)3 = 16S0to 55 of 5 basic types. The calculatious of the 3-particle point density p(r) showed thatpure Pauli e[fects, associated with quarks, resulted in very small changes in the centralvalue of p(r), far from the experimental depressiou (of ~ 30-40% below the maximumvalue of the density). In the general case the influence of the quark exchauge, arising fromantisymmetrization of a nuclear wave function at quark level, on nuclear properties wasstudied recentIy in ReC. [109).In the last part of this section we briefly discuss the situation with the long-range forces

in nuclear interactions, which are the color analog of the van der Waals forces in atomicinteractions. If they exist they must be non-additive (in perturbation treatment: in ordershigher than two).It is well known that the transverse-photon exchange bctween electrons yields long-

range van der Waals forces proportional to R-7 [110]. The long-range character of thevan der Waals potential is a consequence of zero rest mass of the photon. Gluons arealso massless particles. I3ut because the hadrons (nucleons) themselves are color-neutralthey can not exchange a single color gluon. 1I0wever, the exchange of a pair of gluons isnot forbidden since two gluons can he in a color-singlet state. So the two-gluon exchangebetween hadrons can give rise to a long-range forces between hadrons. The expressionsfor these forces were derived by a numher of authors [111-118J. An interhadronic 2-bodypotential, corresponding to these forces can he written in the following form [116]

R»Ro (27)

where Ro = 1 fm, the value of N ::; 7 and depends upon the assumed quark-quarkpotential, and AN is the dimensionless coupling constant that characterizes the long-rangepotentia1.The detailed analysis of the existing experimental data was performed by Feinberg and

Sucher [116]. They carne to the conclusion that the values of N ::;5 in Eq. (27) are ruledout by the experimental data, whilc /{-6_ and n-7-<1cpcndcnt potentials are Bot, unlessthe length para meter Ro is several Fermi.On the other hand, in [lefs. [114-116, 119,120] it was shown that the additive two-body

confinement potential for quarks leads to van der Waals forces which are in strong contra-diction with cxpcrimcnts GIl the Illlclcon-nuclcon scattcring and somc other experimental

TIIE ROLE OF NON-ADDlTIVE FORCES IN ATOMIC... 119

data. So the majority of particle physicists do not believe in the existence of color vander Waals forces. Nevertheless, there are no rigorous theoretical prohibitions to theirexistence. Barut amI Raczka [1211 snggested to introduce the color degree of freedom inthe confinement potential as a remedy against unacceptably large van der Waals forces.In Refs. 1122,1231 possiblc experiments sensitive to the color van der Waals forces areproposed. In a recent high precision experiment on 208Ph +208Pb Mott scattering 11241 acomparison with the angular shift produced hy the color van der Waals forces includingnuclear polarizabiblity, vaccllm polarization, relativistic efrects, and c1ectronic screeningprovided an upper limit for the strength of these forces '\7 = 10:J: 1.

ACKNOWLEDGEMENTS

We are grateful to Drs. A. Mondragón. K.W. McVoy, YU.F. Smirnov, T. Seligman, A.Shirokov, and E. Stephenson for helpful discussions on the nuclear interaction problems.One of the authors (I.G.K.) acknowledges CONACYT for financial support under thecontract No. 920100.

REFERENCES

1. LC. Kaplan, Theory o/ Molecular Intemetions, Elsevier, Amsterdam-Oxford-New York-Tokyo(1986).

2. LC. Kaplan and o.n. Rodimova, Dokl. AN SSSR 265 (1982) 1179.3. A.I. Kitaigorodskii, Ghem. Soe. Reviews 7 (1978) 133.4. W.J. Meath and R.A. Aziz, Mol. Phys. 52 (1984) 225.5. T.R. Horn, R.n. Cerber, J.J. Valentini and M.A. Ratner, J. Ghem. Phys. 94 (1991) 6728.6. A.D. Mistriotis, N. Flytzanis and S.C. Farantos, Phys. Rev. 839 (1989) 1212.7. H. Primakoff and T. Holstein, Phys. Rev. 55 (1938) 1218.8. S.D. Drell and K. Huang, Phys. Rev. 91 (1953) 1527.9. A. Herzenberg, Nuov. Gim. 1 (1955) 986, 1008; Nud. Phys 3 (1957) 1.10. A. Herzenberg and A.S. Roberts, Nud. Phys. 3 (1957) 314.11. J. Fujita and H. Miyazawa, P1'Ogr.Theoret. Phys. 17 (1957) 360, 366.12. J.W. Clark and P. Westhaus, J. Math. Phys 9 (1968) 131.13. M.L. Ristig, W.Y. Ter Louw and J.W. Ciar k, Phys. Rev. C3 (1971) 1504; C5 (1972) 695.14. K.E. Kürten and M.L. Ristig, Phys. Lett. 866 (1977) 113.15. V.R. Pandharipande and R.n. Wiringa, Nud. Phys. A226 (1976) 269.1G. R.n. Wiringa and V.R. Pandharipande, Nud. Phys. A299 (1978) 1.17. n.o. Day, Revs. Mod. Phys. 50 (1978) 495.18. D. Robson, Nud. Phys. A308 (1978) 381.19. N. Yamaguchi, T. Kasahara, S. Nagata and Y. Akaishi, P1'Ogr.Theoret. Phys. 62 (1979) 1018.20. Muslim, Y.E. Kim and T. Ueda, Nud. Phys. A393 (1983) 399.21. Ch. Hajdllk, P.U. Sauer, and W. Strueve, Nud. l'hys. A405 (1983) 581.22. A. nomelburgh and W. Clockle, Phys. Rev. C28 (1983) 2149.23. R.C. ElIis, S.A. eoon and n.H.J. McKeller, Nud. Phys. A438 (1985) 631.24. A. Osman, Hadronic JOU1'1l. 12 (1989) 283.25. M.n. Johnson, J. Haidenbauer and K. Holinde, l'hys. Rev. C42 (1990) 1878.26. M.T. Peña, H. lIennillg and P.A. Sauer, Phys. Rev. C42 (1990) 855

120 I.G. KAPLAN AND O. NOVARa

27. A. Picklesimer, R.A. Rice and R. Brandenburg, Phys. Rev. C44 (1991) 1359; C45 (1992) 547,2045,2624.

28. A. Picklesimer, R.A. Rice and R. Brandenburg, Phys. Rev. Lett. 68 (1992) 1484.29. R.B. Wiringa, Phys. Rev. C43 (1991) 1585.30. V.R. Pandharipande, Nud. Phys. A553 (1993) 191c.31. O. Novara, Kinarn 2 (1980) 175.32. B. Jeziorski, M. Bulski and L. Piela, Int. J. Quant. Chern. 10 (1976) 281.33. G. Chalalinski and B. Jeziorski, Mol. Phys. 32 (1976) 82; Theor. Chern. Acta (Berl.) 46

(1977) 277.34. K. Szalewicz and B. Jeziorski, Mol. Phys. 38 (1979) 191.35. G. Chalalinski, M.M. Szczesniak and S.M. Cybulski, J. Chern. Phys. 92 (1990) 2481.36. M. Kreek and W.J. Mealh, J. Chern. Phys. 50 (1969) 2289.37. S.F. O'Shea and W.J. Mealh, Mol. Phys. 28 (1974) 1431; 31 (1976) 515.38. P.J. Knowles and W.J. Mealh, Mol. Phys. 59 (1986) 965; 60 (1987) 1143.39. G: Chalasinski and B. Jeziorski, Mol. Phys. 27 (1974) 649.40. B.M. Axilrad and E. Teller, J. Chern. Phys. 11 (1943) 299.41. J. Mutlo, Proc. Phys. Math. Soco Japan 17 (1943) 629.42. M.R. Aub and S. Zienau, Proc. Roy. Soco London, A257 (1960) 464.43. L.W. Bruch, O. Novara and A. Flores, J. Chern. Phys. 67 (1977) 2371.44. W.L. Bade, J. Chern. Phys. 27 (1957) 1280; 28 (1958) 282.45. R.J. Bell, J. Phys. 83 (1970) 751.46. M.B. Doran and !.J. Zucker, J. Phys. C4 (1971) 307.47. L.W. Bruch, E. Blaislen-Barajas and O. Novara, J. Chern. Phys. 67 (1977) 4701.48. P. Rosen, J. Chern. Phys. 21 (1953) 1007.49. A. Shoslak, J. Chern. Phys. 23 (1955) 1808.50. O.A. Novara and V. Bellrán-López, J. Chern. Phys. 56 (1972) 815.51. G.E. Musso, V. Magnasco and M.P. Giardina, J. Chern. Phys. 60 (1974) 3749.52. O. Novara and F. Yáñez, Chern. Phys. Lett. 30 (1975) 60; Int. J. Quant. Chern. Suppl. 9

(1975) 349.53. J.P. Daudey, O. Novara and M. Berrondo, Chern. Phys. Lett. 62 (1979) 26.54. W. Kolos, F. Nieves and O. Novara, Chern. Phys. Lett. 41 (1976) 431.55. O. Novara and W. Kolos J. Chern. Phys. 67 (1977) 5066.56. J.P. Daudey, O. Novara, W. Kolos and M. Berrando, J. Chern. Phys. 71 (1979) 4297.57. J. Donohne, The slmclures of elemenls, Wiley, New York, 1974.58. C.W. Bauschilicher, D.M. Liskow, C.F. Bender and M.F. Schaefer IlI, J. Chern. Phys. 62

(1975) 4816.59. J. García-Prielo, W.L. Feng and O. Novara, Surf. Sci. 147 (1984) 555.60. R.J. Harrison and N.C. Handy, Chern. Phys. Lett. 123 (1986) 321.61. 1. Bouslani, W. Peweslorf, P. Fanlucci, V. Bonacié-Kouleckyand J. Koulecky, Phys. Rev.

835 (1987) 9437.62. J.N. Murrell, J. Tennyson and M. Kamel, Mol. Phys. 42 (1981) 747.63. J.N. Murrell, Int. J. Quant. Chern. 37 (1990) 95.64. J.N. Murrell and J.A. Rodríguez-Ruíz, Mol. Phys. 71 (1990) 823.65. A.R. AI-Derzi, R.L. Johnslon, J.N. Murrell and J.A. Rodríguez-Ruíz, Mol. Phys. 73 (1991)

265.66. B.R. Eggen, R.L. Johsnlon and J.N. Murrell, Mol. Phys. 76 (1992) 619.67. J.Y. Fang, R.L. Johnslon and J.N. Murrell, Mol. Phys. 78 (1993) 1405.68. F. Gao, R.L. Johnslon and J.N. Murrell, J. Phys. Chern. 97 (1993) 12073.69. M.E. Ruíz, J. García-Prielo, W.L. Feng and O. Novara, J. Mol. Catal. 33 (1985) 311.70. M.E. Ruíz, O. Novara and J. García-Prielo, J. Mol. Catal. 40 (1987) 37.71. I.G. Kaplan, R. Sanlamaría and O. Novara, Mol. Phys. (in press).

TIIE ROLE OF NON-ADDITIVE FORCES IN ATOMIC. . . 121

72. A. Ramírez-Solis, J.P. Daudey, O. Novaro and M.E. RuÍz, Z. Phys. D-Atoms, Molecules andClusters 15 (1990) 71.

73. R. SantamarÍa, I.G. Kaplan and O. Novaro, Chem. Phys. Lett. 218 (1994) 395.74. R.V. Reid, Ann. Phys. (N.Y.) 50 (1968) 411.75. M. Lacombe, B. Loiseau, J.M. Richard, R. Vinh Mau, J. Coté, P. Pirés and R. de Tourreil,

Phys. Rev. C21 (1980) 861.76. I.E. Lagaris and V.R. Pandharipande, Nucl. Phys. A359 (1981) 331.77. R.B. Wiringa, R.A. Smith and T.G. Ainsworth, Phys. Rev. C29 (1984) 1207.78. K. Holinde and R. Machleidt, Nucl. Phys. A247 (1975) 495.79. R. Mahchleidt, Adv. Nucl. Phys. 19 (1989) 189.80. V.G. Neudatchin, I.T. Obukhovsky, V.1. Kukulin and N.F. Golovanova, Phys. Rev. CH (1975)

128.81. V.1. Kukulin and V.N. Pomerantsev, Progr. Theor. Phys. 88 (1992) 159.82. H.C. Kiihler, Phys. Let!. C18 (1975) 217.83. F. Coester, S. Cohen, B. Day and C.M. Vincent, Phys. Rev. C1 (1970) 769.84. J.L. Friar, B.F. Gibson and G.L. Payne, Ann. Rev. Nucl. Par!. Sci. 34 (1984) 403.85. B.H.J. McKeller and W. Gliickle,Nucl. Phys. A416 (1984) 435C.86. R.A. Brandenburg, G.S. Chuliek, R. Maehleidt, A. Pieklesimer, ami R.M. Thaler, Phys. Rev.

C37 (1988) 1245.87. R.B. Wiringa, J.L. Friar, B.F. Gibson, G.L. Payne and C.R. Chen, Phys. Lett. 143B (1984)

273.88. T. Veda, T. Sawada and S. Tagaki, Nucl. Phys. A285 (1977) 429.89. H.A. Bethe, Ann. Rev. Nucl. Sci. 21 (1971) 93.90. J.-P. Jeukenne, A. Lejeune and C. Mahaux Phys. Rep. C25 (1976) 84.91. B.D. Day and R.B. Wiringa, Phys. Rev. C32 (1985) 1057.92. S.C. Pieper, R.B. Wiringa and V.R. Pandharipande, Phys. Rev. Let!. 64 (1990) 364.93. L.D. Faddeev, Mathematical Aspeets o/ the Three-Body Problem in Quantum Scat!ering

Theory, Davey, New York (1965).94. N.G. Van Kampen, Physica 27 (1961) 783.95. G.E. Brown and M. Rho, Phys. Let!. B237 (1990) 3; Phys. Rev. Let!. 66 (1991) 2720.96. T. Hatsuada and S.H. Lce, Phys. Rev. C46 (1992) R34.97. M. Asakawa, C.M. Ko, P. Lévai and X.J. Qui, Phys. Rev. C46 (1992) IU159.98. M. Asakawa and C.M. Ko, Phys. Rev. C48 (1993) R526.99. L. Rosenfeld, Nuclear Forces, North Holland, Amsterdam (1948), Ch. 13.100. S.J. Biel, Proc. Phys. Soco A (1957) 866.101. V.G. Neudatehin and YU.F. Smirnov, Nucleon Clusters in Light Nuclei, Nauka, Moseow (1968),

in Russian.102.J.A. Whceler, Phys. Rev. 52 (1937) 1083,1107.103. Quarks and Nudei, (N. Weise, ed.) World Seientifie, Singapore-Philadelphia (1984).104. H.J. Lipkin, Nucl. Phys. A446 (1985) 409C.105. A. Faessler, A. Buehmann and Y. Yamauchi, Intern. J. Mod. Phys. 2 (1993) 39.106. Y. Suzuki, K.T. Heeht and H. Toki, Kinam 4 (1982) 99.107. P. Kramer and T. Seligman, Nucl. Phys. A136 (1969) 545; A186 (1972) 49.108. P. Kramer, M. Moshinsky and T. Seligman, in Group Theory and its Applications, VIlI (E.M.

Loebl., ed. ) Aead. Press, New York (1975).109. W.F.M. Spit, V. Vento, A.G.M. Hces, P.J. Brussard and P.J. Mulders, Nucl. Phys. A553

(1993) 689.110. H.B. Casimir and D. Polder, Phys. Rev. 73 (1948) 360.111. T. Sawada, Progr. Theor. Phys. 59 (1978) 149.112. Y. Fujii and K. Mima, Phys. Let!. 79B (1978) 138.113. T. Appelquist ami W. Fisehler, Phys. Lett. 77B (1978) 405.

122 I.G. KA PLAN AND O. NOVAllO

114. S. Matsuyama and H. Miyaza",a, Progr. Theor. Phys. 61 (1979) 942.115. ~1.B. Gavela, A. Le Yaouanc, L. Oliver, O. Pene, J.C. Raynal and S. Sood, Phys. Let!. 82B

(1979) 43I.116. G. Feinberg and J. Sucher, Phys. Rev. D20 (1979) 1717.117. K-F. Liu, Phys. Let!. 131B (1983) 195118. G. Feinberg, Comments Nud. Port. Phys. 19 (1989) 51.119. O.W. Greenberg and H.Y. Lipkin, Nud. Phys. A370 (1981) 349.120. Th. Pfenninger and A. Faessler, Nud. Phys. A484 (1988) 476.121. A.O. Barut and R. Raczka, lntem. J. Mod. Phys. 2 (1987) 265.122. C.V.K. Baba, V.M. Datar and A. Roy, Pramana-J. Phys. 29 (1987) 143.123. M.S. Hussein, C.L. Lima, M.P. Pato and C.A. Bertulani, Phys. Rev. Let!. 65 (1990) 839.124. A.C.C. Villari, W. Mittig, A. Lépine-Szily et al., Phys. Rev. Let!. 71 (1993) 2551.