For some applications, it may be useful to work with a gauge-invariant potential v G . However, this does not remove the gauge-dependence on the N -1 potential or on the electronic vector potential. Thus, if one's interest is to design approximations for these local scalar and vector potentials, making a sensible choice for their gauge and adhering to it accross different approximations is of utmost importance. Since it is undeniably convenient to work with a marginal which is simply the square-root of the electron density, we believe that our definitions Eqs.( ) -( ) are bona fide definitions in which the choice of a particular gauge choice does not restrict their applicability but rather enhances it. Further advantages of the chosen gauge are to be discussed in the next section.