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The authors define a quadrupole moment for a right triangle in

Lazić, Predrag, Hrvoje Štefančić, and Hrvoje Abraham. “The Robin Hood Method – A Novel Numerical Method for Electrostatic Problems Based on a Non-Local Charge Transfer.” Journal of Computational Physics 213, no. 1 (March 2006): 117–40. https://doi.org/10.1016/j.jcp.2005.08.006.

PDF on ArXiv

page 122, equation 9 and 10. I am struggeling a bit on their notation, as it seems that some indices are not defined or they are using something like Einstein notation.

The equations are:

$$ Q = \frac{2}{a_jb_j}\begin{pmatrix} \frac{a_j^3 b_j}{18} - \frac{a_j b_j^3}{36} & - \frac{a_j^2 b_j^2}{24} & 0 \\ -\frac{a_j^2 b_j^2}{24} & - \frac{a_j^3 b_j}{36} + \frac{a_j b_j^3}{18} & 0 \\ 0 & 0 & - \frac{a_j^3 b_j}{36} - \frac{a_j b_j^3}{36} \end{pmatrix} $$

and

$$ I_{ij} = k\left[\frac{1}{|\vec{x}_i-\vec{x}_j|}+\frac{1}{6} Q_{nm} \frac{(\vec{x}_i - \vec{x}_j)_m (\vec{x}_i - \vec{x}_j)_n}{|\vec{x}_i - \vec{x}_j|^5}\right] $$

In the second equation I do not get the indices $n$ and $m$. Should there be a summation over $n, m$ and $Q_{nm}$ refers to a matrix element (i.e. $\sum_{n,m} Q_{nm}\dots$) or are there $Q_{nm}$ matrices defined for all combinations of $n, m$ (triangle points or edges)? and is $(\vec{x}_i - \vec{x}_j)_m (\vec{x}_i - \vec{x}_j)_n$ a scalar product or are the $n$, $m$ indexing vector elements as well?

For Einstein notation, there should be upper/lower indices to indicate what will be multiplied with each other, but here are only lower indices used.

Note that the authors used $n$, $m$ before, but it seems they are not related. Before they used them as indices of points of minimal/maximal potential.

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    $\begingroup$ For work in Euclidean (flat) spaces it's extremely common for Einstein summations to be only over repeated dummy subscripts, e.g. $\mathbf{a}\cdot\mathbf{b} = a_n b_n$, but I admit that doesn't really help with the abuse of notation in this case. mathworld.wolfram.com/EinsteinSummation.html $\endgroup$ – origimbo May 2 at 14:43

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