The spatial part of our wave function eq. (2.36) consists of Gaussian func­tions and products of solid spherical harmonics. To keep the notation as simple as possible, we disregard in this section the coupling of the various angular momenta. Therefore a single term on the right-hand side of the matrix elements (marked by the index r) is of the structure

N-1

L, a>= П П У, m (SN-n, r +j)

i =

where we have converted the single particle coordinates ri in eq. (2.34) into Jacobian coordination si via an orthogonal matrix. The numbering of the Jacobians starts with the internal ones, see fig. 2. The number of clusters on the right-hand side is denoted by nCr. Note that sN, proportional to the coordinate of the center-of-mass, is absent in eq. (3.2) due to the translational invariance of our wave function. The index Lra > is just a reminder of the fragmentation and the angular momentum structure of the

ket wave function. The function on the left-hand side of the matrix element Lia’ > can be expressed in an analogous way by the Jacobian coordinates si on the left-hand side, which differ in general from those on the right-hand side. So the general spatial matrix element is of the form

JLla’Lra(P) = < Lla PwijLra > (3.3)

The orbital operators w®® contain the coordinates in the form of eq. (3.2), but in addition to that also differential operators may occur, like in the spin — orbit force. We have put the permutation P to the left of the symmetric interaction for convenience, see the discussion below. Since the evaluation of all interactions can be reduced to the calculation of certain overlap matrix elements [16], [17], [4], see also section 3.4, we restrict our considerations to the norm, because there all the essential steps become apparent. We can express the norm matrix element of eq. (3.3) by choosing the Jacobian coordinates on the left-hand side as independent variables in the form

z

Уі„т„ (Qn) = Г1іті…lzmz

Since we used an orthogonal transformation from the single particle coor­dinates r to the Jacobian coordinates s, whose index l we suppressed for simplicity, no determinant appears in the integral. The matrix p in the ex­ponent results from applying the permutation P to the coordinates on the right-hand side and then expressing these coordinates by those on the left — hand side. The vectors Qn are the intercluster coordinates pk, see eq. (2.35) and fig. 1, on the left — and right-hand sides, after applying the permutation P onto the latter. Again these can be expressed as linear combinations of Jacobi coordinates s, the former are just some of these coordinates.

In case of a genuine interaction its radial dependence in Gaussian form is also included into p and its angular dependence is then an additional spherical harmonic in eq. (3.4). For treating explicitly other radial dependencies, e. g. the Coulomb interaction, see ref. [4] and the discussion below. The number of angular momenta z is thus the sum of relative coordinates on the left-hand side and those on the right-hand side plus possibly one from the interaction.

Except for the solid spherical harmonics the matrix element eq. (3.4) is just a multi-dimensional Gaussian integral, which can be evaluated by bringing the matrix p into diagonal form. For treating the solid spherical harmonics we use their generating function [18]

L

(b • r)L = bL ^2 CLm b-m YLm (r) (3.5)

m=-L

with the vector b = (1 — b2, г(1 + b2), —2b) being a null vector with respect to a real scalar product, i. e. b • b = 0 and the scalar product of two vectors given by

bn • bn = 4bnbn’ — 2(ЬП + b2n) (3.6)

The coefficients CLm are given by [18]

CLm = (“2)LL! y^ + 1)(L — m)(L + m) ‘ (3J)

To evaluate the matrix element eq. (3.4) we now consider the generating integral

(3.8)

Expanding the expression expQ^ anbn ■ Qn) into a power series in an and bn we find

taking into account eq. (3.5).

Since the generating integral eq. (3.8) is just a Gaussian integral it can be done explicitly and the result can again be expanded in a power series in an and bn to find the desired integrals via eq. (3.9). The transformation

N-i

sn = ^ T^tA » = 1,…,^ — 1 (3.10)

A=^

with TAA = 1 and Ty = 0 for X> X’ brings the matrix p into diagonal form. This transformation results in

I (aibi… az bz) = J dti… dtN-i exp ^- ^ ^ PnaJon ■ tAj j

where we expressed the vector Qn as

N-i

Qn — 52 Pnntn

n=i

and used the properties of the transformation eq. (3.10). Employing the method of completing squares the integral yields

and taking into account eq. (3.6). Expanding the exponentials into a power series and ordering the terms in the form of eq. (3.9) yields the final re­sult [16], [4]

The sums over gnn>, hnn’ ,knn> run over all possible combinations of nonneg­ative integers, which fulfill the following relations for n1 = 1,…,z

/ t (gnn’ + hnn’ + knn’ + gn’n + hn’n + kn’n) ln’

Eq. (3.16 a) results from comparing the exponent of an in eq. (3.13) and eq. (3.9), whereas eq. (3.16 b) results from that of bn>. In these relations gnn’ = hnn’ = knn’ = 0 if n < П. If there are more than two angular mo­menta different from zero, e. g. more than 2 clusters, eqs. (3.16) allow many solutions, which have to be found by trial and error. Starting from group theoretical considerations, Stowe [19] developed a general scheme to find all solutions, which is realized in a very efficient Fortran program. Since the solution of eqs. (3.16) is independent of the permutation and the width pa­rameters used to describe cluster internal and cluster relative wave functions, it need only be done once.

With this formalism we can now calculate each individual overlap matrix element from eq. (3.15) using eqs. (3.7, 3.10 — 3.12, 3.14 and 3.16). Straight­forward extensions cover potentials, whose spatial form is a linear combina­tion of products of Gaussian functions, solid spherical harmonics and powers of r2. In case of a Gaussian radial dependence, this dependence has to be put into the construction of the matrix p^> in eq. (3.4), which then modifies the transformation T^ in eq. (3.10) and hence also the diagonal elements fix (eq. (3.11)) and the final coefficients ann> (eq. (3.14)). Forces containing ex­plicitly a solid spherical harmonic, like the tensor force, are taken care of by
increasing the number of angular momenta by one by adding an additional vector Qn. With the relation [20]

2 i

C = 2ІТТ £ (-irJWr№-„.(r) (3.17)

m=-l

We can treat even powers of r2 by just increasing the number of angular momenta by two. Obviously all three radial dependencies can be treated simultaneously by modifying the matrix p and the number of angular mo­menta accordingly. The results are always of the form of eq. (3.14), i. e. of the type norm integral.

The Coulomb potential ZZ2/r is apparently not of the above form, but it can be written as

і Гв f°°

-=20— dkexp(—k2{3 r2) (3.18)

r VO0

yielding again a Gaussian radial dependency and an additional integration over k. The explicit reduction of the matrix elements of the Coulomb inte­gration in terms of overlap integrals is given in ref. [4]. Other negative powers of r are easily obtained by modifying the integral in eq. (3.18) by the appro­priate monomial in k2. For the tensor potential in the non-relativistic quark model, which contains an ^ term, an explicit expression is given in [21].

For potentials containing differential operators, like the spin orbit interac­tion, the differential operators are applied to the wave function on the right — hand side, resulting in additional polynomials in vectors r — rj, which can be treated by the method described above. For an explicit calculation of the kinetic energy, see [16]. All the potential terms are finally reduced to linear combinations of certain norm-type matrix elements. Each of these integrals can be evaluated explicitly by hand, the sheer number of matrix elements, however, necessitates the use of a computer program. As will be shown below, we need between 15 and 20 different Gaussian width parameters in order to facilitate the expansion of the Coulomb functions in the interaction region. So for one permutation we have typically 200 — 400 different ma­trix elements per structure taken into account (neglecting the symmetry of entrance and exit channel). On the other side the number of terms in the antisymmetriser grows rapidly with the increasing number of particles and furthermore in general the number of important configurations grows too.

Therefore one is forced to avoid multiple calculations of the same matrix elements. For this purpose the symmetry of the ansatz for the internal wave function, eq. (2.36), can be utilized. Permutations inside a cluster will not modify the wave function eq. (2.36). Furthermore, interactions between different pairs of particles might yield identical spatial matrix elements.

Let us consider 12C as an illustrative example. Without any symmetry the 12-nucleon antisymmetrizer contains 12! = 479 001 600 terms. Since protons with spin up are distinguishable from protons with spin down, we can reduce the huge number to (3!)4 = 1296 terms. If we group the nucleons in 12C together into three alpha-particles, then only 120 terms survive. Allowing for a two-body interaction, each of the above numbers has to be multiplied by 12 • 13/2 = 78, the number of possible interactions. Realizing that each of the two interacting particles can be in one of the 3 clusters initially and after interaction (and permutation) in some other one yields 9 • 10/2 = 45 possibilities, where we assumed a spatially symmetric interaction and did not check if this combination is possible for the permutation considered, thus arriving at 120 • 45 = 5400 possible interaction terms. The actual calculation yields 2601 different terms under the assumption that the 3 clusters have different width parameters.

What was shown here by an example is a quite general decomposition of a finite group into double cosets of appropriate subgroups [22]. Here we comply with the cluster symmetry of the wave function eq. (2.36) by considering subgroups Sni x Sn2 x… of the symmetric group SN with ^щ = N. Each double coset can be characterized uniquely by one permutation. It is easy to convince one-self [22] that the overlap matrix elements eq. (3.15) for all permutations belonging to one double coset are identical because the matrix p^i is unchanged. In the above example we consider the subgroup S4 x S4 x S4, yielding 120 double cosets. Also the interacting particles can be easily marked within the double coset expansion, which is described in some detail in the following section.