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 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. 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 13 where and d Σ = g ( z ) √ 1 + g ′ $^{2}$( z ) dzdϕ, (32) n = 1 √ 1 + g ′ $^{2}$( z )    cos ϕ sin ϕ − g $^{′}$( z )   $^{.}$ (33) M$_{lm}$$_{;}$$_{l}$ ′ m ′ = δ$_{mm}$ ′ 2 π ∫ ∫ 0 z max ∫ ∫ z min A ( z, z $^{′}$, ϕ − ϕ $^{′}$) P m l (cos θ ( z )) P m ′ l ′ (cos θ ( z $^{′}$)) e im ( ϕ − ϕ $^{′}$) G ( z, z $^{′}$) dz dz ′ dϕ dϕ ′ (34) G ( z, z $^{′}$) = √ g ( z ) g ( z $^{′}$) √ (1 + g $^{′}$( z ))(1 + g $^{′}$( z $^{′}$))− $^{1}$, (35) A ( z, z $^{′}$, φ ) = ( g ( z ) − g ( z $^{′}$)) cos φ − ( z − z $^{′}$) g $^{′}$( z ) [ g $^{2}$( z ) + g $^{2}$( z $^{′}$) − 2 g ( z ) g ( z $^{′}$) cos φ + ( z − z $^{′}$) $^{2}$] 3 / 2 (36)