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In the manual, there is a nice example how Born effective charges of salts can be calculated within the Berry phase formalism. As a newbie to that method, I'd like to know if one can also find out the the effective charge of an ion in single component bulk metals in this or a similar way. For instance, can one calculate that 1 electron of each sodium ion is free while the rest are bound to the ionic core?

The Born effective charge tensor gives the change of polarization of an atom (along i) if it is slightly displaced along direction j. For your purpose, it should be suffucient to calculate the partial local charges, or to take the charge density difference between the electronically converged system and the cell which just has the atoms superposed (i.e. without accounting for interaction in the lattice)