in order of increasing values of the function This formula correctly predicts both the first and second parts of the Madelung rule (the second part being that for two subshells with the same value of n + , the one with the smaller value of n fills first). Wiswesser argued for this formula based on both the pattern of nodal and radial nodes, the concept now known as orbital penetration, and the influence of the core electrons on the valence orbitals. In 1961 the Russian agricultural chemist V.M. Klechkowski proposed a theoretical explanation for the importance of the sum n + , based on the Thomas–Fermi model of the atom. Many French- and Russian-language sources therefore refer to the Klechkowski rule. The full Madelung rule was derived from a similar potential in 1971 by Yury N. Demkov and Valentin N. Ostrovsky. They consider the potential where and are constant parameters; this approaches a Coulomb potential for small . When satisfies the condition , where , the zero-energy solutions to the Schrödinger equation for this potential can be described analytically with Gegenbauer polynomials. As passes through each of these values, a manifold containing all states with that value of arises at zero energy and then becomes bound, recovering the Madelung order. Perturbation-theory considerations show that states with smaller have lower energy, and that the s-orbitals (with ) have their energies approaching the next group. In recent years it has been noted that the order of filling subshells in neutral atoms does not always correspond to the order of adding or removing electrons for a given atom. For example, in the fourth row of the periodic table, the Madelung rule indicates that the 4s subshell is occupied before the 3d. Therefore, the neutral atom ground state configuration for K is , Ca is , Sc is and so on. However, if a scandium atom is ionized by removing electrons (only), the configurations differ: Sc is , Sc+ is , and Sc2+ is . The subshell energies and their order depend on the nuclear charge; 4s is lower than 3d as per the Madelung rule in K with 19 protons, but 3d is lower in Sc2+ with 21 protons. In addition to there being ample experimental evidence to support this view, it makes the explanation of the order of ionization of electrons in this and other transition metals more intelligible, given that 4s