Multipole expansion truncation errors

This applies to explicit charge-charge interactions, while the error in multipoles remains controlled by the order at which the multipole expansions are truncated. The explicit pairwise interactions between Gaussian charges in the leaf boxes are still computed explicitly using the erf(f,yr,y)/r,y kernel. 

The absolute error caused by truncating the multipole expansion is also proportional to 1/R. The error is therefore larger for close pairs of boxes of the same size and smaller for remote pairs of the same size. In our example, we could thus combine all four boxes Ci to C4 into a four times larger box C [Figure 9(b)], and the errors in the interaction energies of the close boxes (Uab) and of the remote boxes (Uac) would still be on the same order of magnitude. The computational cost, however, is reduced drastically by increasing the size of remote boxes. 

The accuracy of a truncated multipole expansion for reproducing V(r) can be quantified by the root mean squared error (RMSE),... 

Thanks to efficient recurrence formulae, multipole moments and multipole moment derivatives can be calculated at very high order with a low computational cost. The calculation of reaction field factors, however, may become computationally expensive at high order due to the increasing number of linear equations to be solved. Thus, in practice, the multipole moment expansion is cut off at a maximum value of f (/max), usually taken around 6. In order to get an order of magnitude of the error introduced by the truncation, let us consider Kirkwood s equations [5] for the free energy of a charge distribution of charges q, and r, in a spherical cavity of radius a ... 

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