Replacement of the exact analytic two electron integrals over Gaussian type orbitals by multipole-multipole interaction terms leads to discretization errors. Such errors are important in Fast Multipole Methods as well as in a number of other tree-based algorithms. We investigate the source of these errors and estimate its dependence on the angular momentum of charge distributions. Two new range schemes suitable for use in any multipole method are presented. Our best scheme allow us to achieve one and a half to two orders of magnitude higher accuracy in the total electron-electron energy than the previously proposed range scheme while requiring the same amount of CPU time. Several benchmarks are presented to illustrate the advantages of this new approach.
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