Comprehensive definition of oxidation state (IUPAC Recommendations 2016)

4.1 Algorithm of assigning bonds

This algorithm works on Lewis formulas (p. 1026, Ref. [1]) of molecules and ions which show all valence electrons (Fig. 3). Its practical formulation uses electronegativity to deduce the ionic sign:

Oxidation state equals the charge of an atom after its homonuclear bonds have been divided equally and heteronuclear bonds assigned to the bond partners according to Allen electronegativity, except when the electronegative atom is bonded reversibly as a Lewis-acid ligand, in which case it does not obtain that bond’s electrons.

Fig. 3:

OS (in red) in CH₂F₂ (left) and in peroxynitrous acid (right), by assigning bonds to more electronegative partners on Lewis formula with all valence-electron pairs drawn as dashes.

The bond order of the homonuclear bonds will not affect OS, provided the isolated segment of all –AA– pair bonds, with their ionic-approximation signs, is symmetrical. Thus, while the OO bond order in Fig. 3 (right) is irrelevant, because the –OO– segment has a mirror symmetry, the NN bond order in an octet-fulfilling N₂O Lewis formula controls the OS: apply Fig. 3 to N̅=N=O̅ and |N≡N–O̅| (p. 1038, Ref. [1]).

The caveat of the reversibly-bonded Lewis-acidic electronegative ligand (Section 3) is illustrated with the Rh–S bond in O₂S–RhCl(CO){P(C₆H₅)₃}₂ [8] in Fig. 4. The electronegative S atom, here a Lewis acid, does not keep the bond pair, the allegiance of which follows from the reversibility of this adduct back into SO₂ and RhCl(CO){P(C₆H₅)₃}₂.

Fig. 4:

OS of Rh by assigning bonds according to electronegativity, but invoking the caveat of reversibly bonded SO₂ as a Lewis acid, which therefore does not keep the bond electron pair: Rh does, with OS=+1.

4.2 Algorithm of summing bond orders

This algorithm works on Lewis formulas and bond graphs. A bond graph represents the infinite periodic network of an extended solid [9], [10]. It is constructed on a stoichiometric formula of the network’s repetitive unit, with atom symbols connected by a line for each instance of the atom’s bonding connectivity. Each line carries its own specific bond order.

Heteronuclear-bond orders are summed at the atom; as positive if that atom is the electropositive partner in a particular bond, and negative if it is not: the atom’s formal charge (if any) is added to that sum, yielding the oxidation state.

In bond graphs of extended 3D structures, there are no formal charges.^[8] The obtained “ionized bond order sum”, iBOS,^[9] equals OS directly as illustrated in Fig. 5 using AuORb₃ perovskite [11], where OS and bond orders follow from the 8−N rule at O, 8+N rule at Rb, and 12−N rule at Au. For a glossary of these rules, see p. 1027 in Ref. [1].

Fig. 5:

Unit cell and coordination polyhedra of AuORb₃ perovskite, with its bond graph of ideal bond orders (values in blue) and OS (in red).

Lewis formulas may or may not have formal charges, as shown in Fig. 6. On the left, a formula with no formal charges yields OS=iBOS directly. On the right, the formal charge (the numerical variable FC) must be added to the sum at each atom; OS=iBOS+FC.

Fig. 6:

OS (in red) by summing ionized bond orders (in blue) in Lewis formulas. Left: in CH₂F₂ directly, no formal charges. Right: in CO, with formal charges (in black, with trailing sign).

Bond orders in extended solids are not always obvious and may have to be estimated from bond lengths. This is done by converting each bond length into the so-called bond valence, which is a value entirely equivalent to the bond order in terms of two-electron bonds in molecules. The origins of the bond-valence approach, one ionic [12] and one covalent [13], are associated with Linus Pauling. In Ref. [13], an expression is given that evolved into the currently used relationship between bond valence and bond length:

where BV_ij and d_ij represent the bond valence and the bond distance, respectively, between the atoms i and j. R⁰_ij is the reference single-bond length parameter of the ij pair and B is a variable parameter often fixed at 0.37 Å (37 pm). This relationship makes it possible to determine OS directly from the bond distances obtained from crystal-structure data. For details, examples and relevant references, consult Ref. [1].

5.1 Non-innocent ligands

Redox-active ligands, called “non-innocent” after Jørgensen [14], [15], render OS less obvious when combined with a redox-prone central atom. Examples include complexes of O₂, NO, and dithiolenes, discussed in Ref. [1]. Drawing their Lewis formula requires information about bond orders from diffraction and spectral data. The OS of the transition-metal center can often be ascertained from spectral and magnetic measurements.

5.2 Metallic compounds

When bonding and antibonding orbitals/bands overlap in a metal, we are no longer entitled to make the ionic extrapolation implied by Fig. 1. There are simple metallic compounds of obvious OS, such as the golden TiO (+2), blue-black RuO₂ (+4), or coppery ReO₃ (+6), but, eventually, the assignment of conducting electrons to one of the bonded atoms has its limits. An example is AuNCa₃ perovskite [16], where neglecting its metallic character suggests Au³⁻ anions for which there is no support in theory.^[10] Such an OS is merely a practical value for redox balancing, like setting OS=0 for all metals in intermetallic phases of metallic character (not the semiconducting Zintl phases), or for those inner atoms in metal clusters that are solely bonded to other metal atoms.

5.3 Nominal OS

In systematic descriptive chemistry, OS is used to sort out compounds of an element; in electrochemistry, it represents the electrochemically relevant compound or ion in Latimer and Frost diagrams of standard (reduction) potentials. Such purpose-oriented OS may differ from the OS per definition and are termed nominal, or more specifically “systematic” or “electrochemical”. Consider thiosulfate, the S₂O₃²⁻ anion: The sulfur–sulfur bond is practically a single bond, which yields OS=+5 for the central S and −1 for the terminal S atom by both algorithms (p. 1039, Ref. [1]). The electrochemical OS of sulfur in thiosulfate is their average, +2 (p. 1060, Ref. [1]). On the other hand, a systematic OS of +6 may be chosen for the central S atom, as if the sulfur–sulfur bond were approximated ionic, to emphasize the similarity of the peripheral O and S atoms obtaining OS=−2 as in sulfate.

5.4 Nominal OS and redox reactions

As discussed by Vitz [17], redox reactions can be decomposed into two half reactions in which OS changes. Since ΔG°=–nFE°, the standard reduction potential E° is available whenever we set up an OS to change in a half reaction. Consider the synthesis of S₂O₃²⁻ from SO₃²⁻ and sulfur. We may see this Lewis acid-base reaction as a non-redox process by assigning a nominal OS=0 to the thiosulphate terminal sulfur atom. Yet the same reaction can also be described as a redox one in terms of E° and the electrochemical OS=+2 of sulfur in thiosulfate. As a rule, if an oxidation takes place with an added oxidant, the OS should increase in the species that is being oxidized; if there is a reduction (with a reducing agent), it should decrease the OS in the species that is being reduced.

5.5 Need for choices, estimates, and round offs

Usage-related choices define the nominal OS, vide supra. Options arise also when the AB ionicity in Fig. 1 approaches zero and the distinction between positive and negative OS wanes. In Ref. [1] (p. 1054) this is illustrated with H₃PO₃; in textbooks the P–H bond electron pair is assigned to either H or P, yielding two OS alternatives. A similar example is CSe₂ where OS of −2 for Se is obtained via MO considerations or by analogy with CO₂.^[11] The same OS also suits ···C–Se–C··· compounds. For intermetallics of metallic character, the ultimate choice of OS=0 is best if needed for redox chemistry.

Subtler estimates or round offs are required for compounds with electrons delocalized over nonequivalent atoms, manifested by a set of weighted resonance formulas; as an example in Ref. [1] for 1H-pentaazole on p. 1054, N₅⁺ on p. 1038, or thiosulfate on p. 1039. Round offs are needed also in compounds that enter bonding compromises, as exemplified with S₄N₄ in Ref. [18].

Rounding is also necessary for bond-valence sums after the bond-length to bond-valence conversions with eq. 1. Their decimal values stem from the statistical distribution of bonding compromises in the set of compounds used to obtain the single-bond length parameter R⁰_ij in eq. 1.

On the other hand, reasonable fractions of small integers are obtained for OS in compounds such as B₆H₁₀ or B₁₀C₂H₁₂ (p. 1040, Ref. [1]), or when two vicinal OS are indistinguishably mixed, such as for Fe in YBaFe₂O₅ (p. 1058, Ref. [1]), or when the ionic charge is distributed over several equivalent atoms, such as in B₆H₆²⁻ (p. 1039, Ref. [1]) or I₃⁻ and N₃⁻ (p. 1037, Ref. [1]). OS in non-stoichiometric solids with randomly distributed defects is decimal: NiO_1.031 with interstitial oxygens has Ni of OS=+2.062, as does Ni_0.970O with nickel vacancies.

6.1 Allen electronegativity scale

6.2 OS and dⁿ configuration

The configuration dⁿ is a central-atom descriptor in transition-metal complexes. For a transition metal of N valence electrons, dⁿ yields OS as:

A problem arises when that central atom acts as a Lewis base and donates one of its dⁿ non- or anti-bonding electron pairs to an acceptor ligand (Lewis acid, a Z-type ligand [6]): Whereas OS remains the same (since the ionic approximation assigns the donated electron pair back to the Lewis base), the coordination geometry and magnetism at the central atom may no longer refer to the original dⁿ before that donation.

One of the examples discussed in Ref. [1] is the Au–B bond in Fig. 7 [19], where two Au d-electrons populate the weakly bonding MO so that Mössbauer spectroscopy still sees this MO’s two electrons together with the rest of the d electrons on Au as d¹⁰, suggesting OS=+1 for gold, in accord with the OS definition in Section 2. Yet the coordination at Au is square planar, typical of d⁸ Au³⁺. The square-planar Au appears because the donated Au d-electron pair became the Au–B bond itself, and the geometry is now controlled by the energy minimum for the remaining 8 Au d electrons in the weakly antibonding MOs. To avoid the emerging ambiguity of the d¹⁰ “spectroscopic” versus the d⁸ “ligand-field” or “magnetic” configurations, a notation used by Parkin [6] may describe the central atom: Au of n=10 in dⁿ⁻², where “2” symbolizes the weakly bonding “donated” d-electron pair, n enters eq. 2 for OS determination, and n−2 are the electrons in d orbitals subject to ligand field.

Fig. 7:

Square-planar Au bonded to more electronegative B acceptor in a Z-type ligand [19].

In the same way for similar adducts of Z-type ligands and transition-metal Lewis bases: Rhodium in Fig. 4 has configuration dⁿ⁻² of n=8 that yields OS=+1 from eq. 2 (Rh is assigned the adduct-bond electron pair it donated) while the n−2=6 electrons at the low-spin 4d atom comply with the square-pyramidal coordination. More examples are found in Ref. [1].