We develop a new scheme for determining molecular partial atomic charges (PACs) with external electrostatic potential (ESP) closely mimicking that of the molecule. The PACs are the "minimal corrections" to a reference-set of PACs necessary for reproducing exactly the tensor components of the Cartesian zero- first- and second- molecular electrostatic multipoles. We evaluate the quality of ESP reproduction when "minimally correcting" (MC) Mulliken, Hirshfeld or iterated-Hirshfeld reference PACs. In all these cases the MC-PACs significantly improve the ESP while preserving the reference PACs' invariance under the molecular symmetry operations. When iterative-Hirshfeld PACs are used as reference the MC-PACs yield ESPs of comparable quality to those of the ChElPG charge fitting method.