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202 PublicacionesJ.L. Movilla, J. Planelles / Computer Physics Communications 170 (2005) 144–152 147that the step-like dielectric profile localizes all induced charge at interfaces of zero width, so that real and inducedcharges can coincide at the same place when finite confining potentials are considered.In order to avoid these divergences without giving up the employment of realistic confining potentials, we followthe self-potential calculation procedure proposed by Bolcatto and Proetto [22,24]. Basically, this procedure assumesa continuous variation of ǫ(r) across an extremely thin layer at the interfaces, yielding a finite self-potential profilefor all coordinate values. We use a layer thickness of δ = 3 Å and a cosine-like model for ǫ(r):⎧⎨ǫ(r) =⎩ǫ iǫ i +ǫ o2+ ǫ i−ǫ oǫ oif rR+ δ/2.However, as the numerical nature of the calculations requires the discretization of the continuous ǫ(r) functionyielding a multistep profile within the interfacial layer, new divergences are encountered. In order to avoid thesenumerical divergences we use the discretization scheme shown next, applied to the case of two regions 1 and 2having an infinitesimal dielectric mismatch ǫ 1 −ǫ 2 = δǫ (1 corresponds to a dot of radius R and 2 to the surroundingmatrix). Eq. (14) then becomes:φ (1)s = 12ǫ 1 Rφ (2)s = 12ǫ 2 R∞∑l=0∞∑l=0( )(l + 1)(ǫ 1 − ǫ 2 ) r 2l; rR.In our discretization we avoid calculating at the edge. To this end, the dot radius R is calculated as R = (N +1/2)h,where h is the discretization step. Thus, Eqs. (16) and (17) turn into:φ (1)s (i) = δǫ2ǫ 1 R∞∑l=0φ (2)s (i) =− δǫ2ǫ 2 R(l + 1)ǫ 2 (l + 1) + ǫ 1 ll=0[]i 2l; i N,N + 1/2∞∑[ ]l N + 1/2 2(l+1); i>N,ǫ 2 (l + 1) + ǫ 1 l iwhich are finite for all (integer) i.For the sake of accuracy, the discretization scheme of the cosine-like model is carried out over a dense grid (500points) within the thin dielectric interface region. This scheme is chosen in a way, similar to the one just described,that avoids every divergence and mimics the continuous variation of the dielectric constant.The practical implementation of the cosine-like profile, or any other continuous model for ǫ(r), by means ofthe above mentioned discretization scheme of Eq. (14), additionally requires a cut-off of the infinite summation ata finite l-value. A rather poor convergence together with numerical inaccuracy for moderately large values of l isobserved. The source of this poor convergence and numerical inaccuracy in Eq. (14) comes from the coefficientsp ml and q ml . For a moderately large value of l (but much smaller than the value l conv required to reach convergence)p ml becomes extremely small while q ml becomes extremely large. In order to bypass the computer cut-off errors,we carry out a convenient rewriting of Eq. (14). We first define a new set of coefficients ˜p ml and ˜q ml as follow,⎧⎨ l(ǫ m −ǫ m−1 )+˜q m−1,l (lǫ m +(l+1)ǫ m−1 )( R m−2R ) 2l+1m−1˜q ml =if m>1,(l+1)ǫ⎩ m +lǫ m−1 +˜q m−1,l (l+1)(ǫ m −ǫ m−1 )( R m−2R ) 2l+1m−10 ifm = 1,(15)(16)(17)(18)(19)(20)