Formal valence, charge distribution, and chemical bond in a compound with a high oxidation state: KMnO$_4$

KMnO$_4$ has unusual formal manganese oxidation state Mn$^{+7}$ that seems puzzling: the energy of creating such ion ($119$ eV) is much greater than the energy of chemical bonds (up to $\sim10$ eV). We have used the Wannier functions formalism to analyze the distribution of Mn-$3d$ electrons and O-$2p$ electrons for empty electronic states in the MnO$_4^-$ complex and have found that, while formally one has $d^0$ configuration for manganese ion in this compound, in reality only about one-half of the hole density described by these Wannier functions corresponding to this configuration belongs to $d$-electrons, while the other half is spread over surrounding oxygen atoms. This corresponds much more to Mn$^{+7}$ state than to Mn$^{+2}$, because the calculated total number of $d$-electrons is equal to $5.25$. Our analysis has also sown nearly perfect covalent type of chemical bond within the MnO$_4^-$ complex with negligible contribution of the ionic part.