As discussed below eq. (2.41), all integrals are of short range, therefore the Coulomb functions, fi(r) and gi(r), see eq. (2.3), can be expanded in terms of the functions xv, eq. (2.5), or rv, eq. (2.15), in an appropriately chosen finite interval A. Since fi and gi are solutions to the point Coulomb Hamiltonian, A has to cover the range of the interactions folded with the size of the fragments. In addition to that, the range where gi(r) deviates from gi (r) due to the regularisation factor Ti(r), eq. (2.3), has to be considered.

The starting point is to minimize the integral

The function fL is calculated numerically according to ref. [15]. The ex­pansion functions xkL are given in eq. (2.5). Two sets of width parameters вк are given in [26]. The variational parameters Ck are determined from a system of linear equations. The weight function Wi(x) is chosen in such a way that the internal region dominates and the total interval becomes finite. A typical expression is

Wl(x) = x-(L+1> e-x (4.3)

with є w 0.01fm-2.

Thus a typical expansion interval is of the order of 20 — 50 fm. The parameter є is numerically very critical: If it is chosen too large the interval will become too small and the small width parameter ek are strongly suppressed. On the other hand if є is too small, then the expansion interval will become larger and it gets numerically very difficult to reproduce the oscillating function fL by a finite number of Gaussians centered around the origin. Increasing the number of Gaussian width parameters may lead to numerical dependencies, due to the non-orthogonality of these functions, especially as one set of parameters is used irrespective of the orbital angular momentum L.

The expansion can be improved, if in addition to the functional eq. (4.2) also the derivative of fL is included in an obvious way. Outside the interval de­termined by the weight function WL the values of the sum must not become too large. Modifications of this type are discussed in [31]. An analogous procedure is used for gi.

Depending on the kinetic energy of the fragments we found up to 100 MeV

15 to 20 width parameters sufficient to obtain a good representation of the

Coulomb functions. The choice of parameters fik is not critical to scattering calculations. We can easily omit some of them without changing the results. Changing the parameter во of the regularisation factor, eq. (2.4), in a wide range does not modify the final results either, as long as TL approaches unity outside the interaction range.