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for the surface Σ: (A) for clusters of two particles we use spherical coordinates { x, y, z } = { r ( θ ) sin θ cos φ, r ( θ ) sin θ sin φ, r ( θ ) cos θ } , and (B) for clusters with more than two particles we specify the surface in terms of a function g ( z ) as { x, y, z } = { g ( z ) cos φ, g ( z ) sin φ, z } . 

In the case B, the surface element is 

and the normal to surface Σ is 

In the basis of generalized hyperspherical harmonics the operator M has matrix elements given by 

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where 

and 

After the angle integration in equation (34) and by using the elliptic integrals given in the Appendix, the matrix elements are obtained by numerical evaluation of the resulting ( z , z $^{′}$) double integral using an NQ -point Gauss-Legendre quadrature [29, 30]. Because of the integrable singularity apparent in the kernel of the operator M in equation (6), the mesh of z must be shifted from the mesh of z ′ by a transformation which insures that there is no overlap between the two meshes. 

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<img src='content_image/340766.jpg'>

The delta symbols δ$_{mm}$ ′ in equation (34) reflects the fact that we consider only objects with rotational symmetry in this paper. Moreover, for fields parallel with the cluster axis m = 0, while m = 1 for perpendicular fields. 

The convergence of the results is obtained in two steps. First, the number NQ of quadra- ture points is increased until the matrix elements of M converge, and then the size NB of 

