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Magic Mathematical Relationships for Nanoclusters—Errata and Addendum


We correct magic formulas for body centered cubic (bcc) structures. The logical rational for this is further corroborated by calculations of the radial distribution function (RDF) for several crystal structures. We add results for truncated cubes which may be found in nature.


We recently presented magic formulas for several crystal nanoclusters [1]. However, it is known to crystallographers that bcc structures have a bulk coordination of eight. The RDF determines the nearest neighbor peaks from a central point, and the integrated peak intensity reflects the corresponding coordination for those neighbors. We use an established method [2] to calculate the RDF for several crystals. Since ideal bcc cubes have coordination cn=1, we provide results for truncated bcc and face centered cubic (fcc) clusters.

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In reviewing the many magic formulas appearing in [1], it occurred to us that equation (1), which defines the adjacency matrix, depends on the crystal structure.

$$ \mathbf{A}(i,j)=\left\{\begin{array}{ll} 1& \text{if}\ r_{ij}< r_{c}\ \text{and}\ i\ne j\\ 0& \text{otherwise} \end{array}\right. $$

Here, rij is the Euclidean distance between atom i and atom j. While it is true that rc=1.32·rmin is necessary for the different bond lengths in the dodecahedral structure, for the bcc structure, this is not the case. We have calculated [2] the RDF for selected structures, and some of the nearest neighbors are tabulated below (Table 1). The RDF has peak locations at neighbor sites, and the integrated intensity of the corresponding peak gives the coordination. We normalize the peaks in R(r) by dividing by the first peak, thus the peak locations become dimensionless. As the table indicates, bcc structures have \(r_{c} = 2/\sqrt {3} \cdot r_{\text {min}} \approx 1.15 \cdot r_{\text {min}}\), which means the adjacency matrix must be changed, and thus the magic formulas. Note that neighbor peaks are not the same as shells, which give rise to the “magic numbers.” The dodecahedron is a complicated case, where the third neighbors appear at r2=1.31·rmin. This case is challenging, and requires more analysis, which is in progress. The corrected bcc results are shown below (Tables 2, 3, 4, 5 and 6). These results agree with those in van Hardeveld and Hartog [3] if one shifts the index by one, i.e., we use the sequence 0, 1, 2... and they use 1, 2, 3... as their sequence. While perfect cubes may be of interest mathematically, they are not likely to appear in nature, due to single bonds at the corners. We have therefore generated truncated bcc and fcc cubes with the corners removed and their results are included in (Tables 7 and 8). The magic formulas of the indices for selected clusters are summarized in Table 9.

Table 1 Neighbor peaks in the normalized RDF for several structures
Table 2 Magic formulas for the bcc cube
Table 3 Magic formulas for the bcc octahedron
Table 4 Magic formulas for the bcc truncated octahedron
Table 5 Magic formulas for the bcc rhombic dodecahedron
Table 6 Magic formulas for the bcc cuboctahedron
Table 7 Magic formulas for the bcc truncated cube
Table 8 Magic formulas for the fcc truncated cube
Table 9 Magic topological formulas for BCC and FCC clusters


We have corrected magic formulas for bcc structures and added results from the RDF and for truncated bcc and fcc cubes.

Availability of Data and Materials

The dataset(s) supporting the conclusions of this article may be obtained from the corresponding author.



Body centered cubic


Face centered cubic


Radial distribution function


  1. Kaatz FH, Bultheel A (2019) Magic mathematical relationships for nanoclusters. Nanoscale Res Lett 14(1):150.

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  3. van Hardeveld R, Hartog F (1969) The statistics of surface atoms and surface sites on metal clusters. Surf Sci 15:189–230.

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We made use of the MATLAB file, Cluster Generator, which can be found in Mathworks File Exchange Central.


The authors did not receive external funding for this work.

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FHK conceived of the project and analysis. AB wrote the code in MATLAB. Both authors contributed to writing the paper and approved the final version of the manuscript.

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Correspondence to Forrest H. Kaatz.

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Kaatz, F.H., Bultheel, A. Magic Mathematical Relationships for Nanoclusters—Errata and Addendum. Nanoscale Res Lett 14, 295 (2019).

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  • Nanoclusters
  • Topological indices
  • Coordination
  • Magic numbers