Oxidation Number: Pragmatically speaking, the ON is defined as a counting rule that students of chemistry learn, for example, to balance redox reactions. In contemporary f-element chemistry research the ON is often meant to indicate the (formal or calculated) nf-shell electron count [4f (n = 4) for Ln, 5f (n = 5) for An] relative to the number of valence electrons in the neutral atom, irrespective of the extent of donation into the (n+1)d and (n+2)s metal shells. As far as shell occupations or charges of atoms in molecules are concerned, one or more of the available chemically motivated QM tools must be used to extract this kind of information from the calculated groundstate electronic structure. Alternatively, shell occupations and atomic charges can be deduced from spectroscopic measurements, for example, but they are not principal observables. It is important to keep in mind that QM calculation-based measures of an ion charge or its ON depend on the unavoidable approximations made in the electronic structure calculations, and also on the specific definitions used for these quantities. Different measures of charges of atoms in molecules, for example, vary considerably depending on their definition, even though they probe the same underlying physics for the most part. Insights from Crystal Field and Ligand Field Theories: Crystal field (CF) and ligand field (LF) theories are the basis of our understanding of the electronic structure and properties of the ground and excited states of ionic compounds. For the application of these theories to ionic compounds in general, there are several excellent texts; see, for example, References 54-56. The standard assignment of ligand and crystal field theory for oxides with the composition M n O m is to assume the nominal ionicity of oxygen is O 2-and that the nominal ionicity of the metal cations M is (2m/n) where, for compounds where CF and LF theory are used, m and n are such that 2m/n is integer. Thus, for a few specific examples, M in MO is assigned the nominal oxidation state M +2 or M(II), M in M 2 O 3 is assigned as M +3 or M(III), and M in MO 2 is assigned as M +4 or M(IV). It is worth stressing the open-shell occupations corresponding to these oxidation states for Pu: Pu(II) is 5f , Pu(III) is 5f , and Pu(IV)-as for PuO 2 -is 5f . Furthermore, in an MO-theoretical context, including LF theory, it is understood that the filled, empty, and open 5f shell orbitals are not purely atomic. Instead, these orbitals are predominantly Pu(5f), with secondary O(2p) admixtures being related to donation (and back-donation, when possible). For example, when an occupied O(2p) combination donates into a formally empty 5f orbital, this results in an occupied bonding combination that is mostly ligand-centered, along with an empty, mostly metal-centered 5f-O(2p) antibonding combination. The latter is the LF analog of an empty 5f orbital in CF theory. We stress that the properties derived from LF and CF assignments of oxidation states cover a broad range from magnetic properties to excitation energies to core-level spectroscopy including X-Ray Photoemission (XPS, see, for example, Ref. 57) and XANES (see, for example, Ref. 38).