In order to determine the reaction matrix amn from eq. (2.24) and eq. (2.31) we need the matrix elements of H between regular and irregular Coulomb functions. Hence, we need matrix elements of H, eq. (2.18 — 2.19) between Fl, Gl and rv, respectively. Since FL and GL are not square-integrable functions, some care is necessary. In the discussion below eq. (2.41), FL and Gl, but obviously not GL, had to be solutions to the point-Coulomb Hamil­tonian, or to the total Hamiltonian for large separation of the fragments so that the identity operator from the antisymmetrizer between fragments could not lead to infinite contributions. Hence, one has to correct for the fact that Gl is not a solution in just this case.

All matrix elements of FL, containing Fl in the ket can be calculated using the expansion coefficients determined from eq. (4.2). Using eq. (2.29) the only critical matrix elements are < GHG > and < Г&HG >. Operating with the r. h.s. of eq. (2.41) onto GL we find that the regularisation factor TL can be factored out from all terms except the kinetic energy of the relative motion. Hence, it suffices to consider this term in detail.

Since there are no permutations across fragment boundaries we can restrict the discussion to just the relative coordinate. Omitting all unnecessary fac­tors we arrive at

[g(r)T (r)]

dr [g(r)T2(r)g//(r) — (g(r)T'(r))2] (4.4)

where the first term is already taken care of by g being a solution to the point-Coulomb Hamiltonian and the ’ denotes derivation with respect to r.

Taking into account eq. (4.2) we can write in the obvious notation < GHG > = 9mgn < XmHxn > —

mn

гж

-C (GT’fdr (4.5)

J 0

here gm denote the expansion coefficients of G and the constant C contains essentially the internal norm of the fragments.

In an analogous way we find

<XmHG > = gn <XmHXn > —

n

г ж

— C Xm(2G’T’ + GT") dr (4.6)

0

Note that the point Coulomb contribution has to be taken out of < xm H xn >. The correction terms of eqs. (4.5) and (4.6) sometimes exceed the expansion terms appreciably.

Now we have all the necessary ingredients to calculate the reactance ma­trix amn according to eq. (2.31). Since the Hamiltonian is symmetric, the eigenvalues ev, eq. (2.16), are real and therefore depending on the number of expansion functions Xv there are certain energies E for which the denomina­tor in eq. (2.21) or (2.31) vanishes. It is easy to convince oneself, that this factor is cancelled against a corresponding one in the numerator, see also ref. [2]. There could be a slight numerical problem, if the energy E is too close to one of the eigenvalues ev, due to division by a very small number or even zero. This difficulty can, however, be overcome quite easily by omitting the corresponding eigenvector rv, or by reducing the number of expansion functions just by one, so changing the eigenvalues ev slightly.

It can, however, not be excluded that accidentally the denominator in eqs. (2.21) or (2.31) becomes zero, without the existence of a physical resonance. Therefore it is argued [32, 33] that using a variational principle for e. g. the S — matrix, i. e. using complex scattering functions, this problem can be avoided. This might be true in practice, but there exist counter examples [34].

Since the position of the pole depends also on the regularization parameter в0, these accidental poles can always be avoided by changing f30, a procedure which needs only a small amount of computing time, compared to the calcu-

lation of the matrix elements as described in chapter 3.1. Also the measures taken to avoid division by a small number, discussed above, can be used.

Following along the general lines discussed in [5] it is possible to construct various variational principles, out of which the described K-matrix, K-1- matrix or the S-matrix, are just certain limiting cases. One can show that for all these different cases the matrix elements calculated so far suffice and it is only necessary to form the proper linear combinations [35]. So one could just use different methods to avoid spurious resonances. How to calculate the S-matrix and search for the complex energy poles of the S-matrix is described in [8].