  The Bertaut approach associated with charge spreading so as to enhance the
rate of convergence of Coulomb series in crystals is extended to the case of an
arbitrary multiple spreading with a given initial spreading function. It is
shown that the effect of spreading may in general be treated as a uniform
transformation of space, providing that zero mean potential as a universal
spatial property is sustained. As a result, electrostatic potentials driven by
different orders of multiple spreading can be obtained from the same energy
functional in a consistent manner. It is found that the effect of multiple
spreading gives rise to more advanced forms described, for example, by simple
exponential decrease, but the functional description based on a Gaussian
spreading turns out to be invariant. In addition, the effects of a multiple
charge spreading based on simple exponential and Gaussian spreading functions
are compared as typical of molecular calculations.
