Direct Summation of the Madelung Constant Using Axial Multipoles
We present an absolutely convergent real-space summation method for electrostatic potentials in ionic lattices. The method constructs the lattice from translated axial multipole units whose low-order moments are systematically eliminated, producing a prescribed rapid far-field decay. For the RU-13 construction, the leading contribution decays as r^(-13). In three-dimensional NaCl, this gives the Madelung constant to 13 decimal places using summation radii of only a few tens of nearest-neighbor spacings, and spherical and cubical growth geometries converge to the same limiting value. The same real-space construction is applied to nonpolar surface and edge geometries, off-lattice points, and interstitial positions. The Madelung constant of bulk CsCl was also determined with the same accuracy and convergence speed. The method is validated in higher dimensions (d<=6 in practical calculations).