A crucial step toward more realistic pseudo-potentials was given by Topp and Hopfield, who suggested that the pseudo-potential should be adjusted such that they describe the valence charge density accurately. Based on that idea, modern pseudo-potentials are obtained inverting the free-atom Schrödinger equation for a given reference electronic configuration and forcing the pseudo-wavefunctions to coincide with the true valence wavefunctions beyond a certain distance r l. The pseudo-wavefunctions are also forced to have the same norm (i.e., the so-called norm-conserving condition) as the true valence wavefunctions and can be written as