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# Pure and Applied Chemistry

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# Toward a comprehensive definition of oxidation state (IUPAC Technical Report)

Pavel Karen
/ Patrick McArdle
/ Josef Takats
Published Online: 2014-04-12 | DOI: https://doi.org/10.1515/pac-2013-0505

## Abstract

A generic definition of oxidation state (OS) is formulated: “The OS of a bonded atom equals its charge after ionic approximation”. In the ionic approximation, the atom that contributes more to the bonding molecular orbital (MO) becomes negative. This sign can also be estimated by comparing Allen electronegativities of the two bonded atoms, but this simplification carries an exception when the more electronegative atom is bonded as a Lewis acid. Two principal algorithms are outlined for OS determination of an atom in a compound; one based on composition, the other on topology. Both provide the same generic OS because both the ionic approximation and structural formula obey rules of stable electron configurations. A sufficiently simple empirical formula yields OS via the algorithm of direct ionic approximation (DIA) by these rules. The topological algorithm works on a Lewis formula (for a molecule) or a bond graph (for an extended solid) and has two variants. One assigns bonding electrons to more electronegative bond partners, the other sums an atom’s formal charge with bond orders (or bond valences) of sign defined by the ionic approximation of each particular bond at the atom. A glossary of terms and auxiliary rules needed for determination of OS are provided, illustrated with examples, and the origins of ambiguous OS values are pointed out. An electrochemical OS is suggested with a nominal value equal to the average OS for atoms of the same element in a moiety that is charged or otherwise electrochemically relevant.

Article note: Sponsoring body: IUPAC Inorganic Chemistry Division: see more details on p. 1063.

## 1 Preamble

Like many other ideas in chemistry, the concept of oxidation state (OS; or oxidation number, ON), traces its origin prior to the development of the electronic theory of chemical bonding. As a result, and not surprisingly, a lot of ink has been devoted to this subject over the years – not only in attempts to define it clearly and uniquely but also to debate inconsistencies in terms used for it and misconceptions about its meaning. It must be admitted that IUPAC, inadvertently perhaps, may have contributed to some of the debate and polemics. Indeed, Compendium of Chemical Terminology, 2nd ed. (the “Gold Book”) offers a set of rules on how to assign OS but no definition,1 whereas ON is defined,2 but only for compounds with central atoms and ligands [1]. Nomenclature of Inorganic Chemistry, IUPAC Recommendations 2005 (the “Red Book”) [2] adopts that ON definition in its rule IR-9.1.2.8 while adhering to the semantic distinction of ON as an indicator of the OS of the central atom. The IUPAC recommendation “Nomenclature of organometallic compounds of the transition elements” [3], while recognizing the importance of OS when discussing reaction mechanism, states that due to the ambiguity of OS in many organometallic compounds “no formal oxidation numbers will be attributed to the central atoms in the following section on organometallic nomenclature”.

Latimer [4] appears to be the first to “officially introduce oxidation number or oxidation state” [5] within the context of redox half-reactions. Later, Jørgensen devoted a whole book to the concept [6], and discussed a “formal oxidation number and four different forms of oxidation states”, concluding that “The fundamental reason why one cannot define a universal oxidation number and oxidation state in a way satisfactory for all compounds is the mixture of electrocovalent and covalent bonding actually occurring”, although with current thinking this is no longer an impediment to OS assignment. However, the necessity of having a clear distinction between valence and OS is an important issue. Already in the 1940s, Glasstone [7] called attention to this problem and advocated the universal adoption of “oxidation number”, which, according to him, “provides an indication of the ‘electron demand’ of a given element, in a particular molecule or ion, in terms of a consistent basis of reference.” Yet the confusion between the two concepts and their misuse persist to this day and prompted Smith [8] and Parkin [9] to write lucid articles that detail the fundamental differences.3

There is still debate concerning what the definition of OS/ON should be, but both Steinborn [10] and Loock [11] advocate a definition already extant at the time of Pauling [12], which relies on the electronegativity (EN) of the two bonded atoms to assign OS as if the bond were ionic. While agreeing with the Pauling scheme, Jensen [13] offers clarifications and amplifications of some of the points considered by Loock. Other recent publications tackle oxidation–reduction reactions and conclude that they should be defined in terms of changes in OS [14, 15]. Calzaferri [16] comments on OS assignment in organic chemistry, a topic already discussed by Jørgensen [6]. Finally, there has been an exchange between Jansen and Wedig [17] and Zunger, Resta et al. [18–20] concerning calculated charges and OS.4 One cannot disagree with the statement of Jansen and Wedig [17] that with a heuristic concept like OS, in order to minimize the risk of applying it, “the concept needs to be defined as precisely as possible, and these definitions must always be kept in mind during applications”.

As detailed above, there have been numerous past and recent publications on OS/ON, yet none differentiates the definition of the concept from the algorithms used to derive its numerical value nor addresses the role of OS in the bond-valence approach to crystal structures and vice versa.

Thus, the time seemed ripe for an IUPAC report on OS/ON. The goal of this Technical Report is to provide an unambiguous definition of OS that covers the definitory algorithms used in textbooks and the IUPAC Gold Book to obtain its numerical value. Based on this definition, we derive OS algorithms for molecules and solids and illustrate them with examples that go from simple to complex and on to those that reach the limit of the OS concept itself. This Report is self-consistent, as the terms and concepts needed are described in its introductory sections. A primary aim of this endeavor is to satisfy the diverse audience the Report is meant for; nothing less than just about every chemist. Responses to this Report will be later worked into a Recommendation to summarize the definitory algorithms and the comprehensive definition of OS that supports them. It will also suggest in a most general way where such approaches to OS might enter chemistry teaching. These two aspects, Recommendation and teaching of OS, will be addressed in two separate articles to be published subsequently.

## 2 Introduction

Historically, the idea of OS/ON comes from the realization that some elements form a series of oxides in which the amount of bonded oxygen increases by integer multiples. Wöhler in his 1835 textbook Unorganische Chemie [21] expressed this with the term “oxydationsstufe” (an older German spelling for oxidation grade), which in many languages continues to be in use. In English, OS and ON are interchangeable [13], and their difference, if any, is subtle [22a]: Being both expressed by a numerical value, the former refers more to the state of an atom as in chemical systematics, the latter more to the numerical value as in redox-equation balancing. In this Report, we’ll accordingly use the term OS, typeset as OS when it is a variable in a mathematical context.

OS has been widely used in chemistry, for various purposes:

1. As a descriptor:

• In textbooks and comprehensive treatments of inorganic chemistry, OS conveniently sorts out the descriptive chemistry of each element.

• OS is a search parameter in property databases, for example, of inorganic crystal structures.

2. As a parameter in chemical nomenclature.

3. As a variable for tabulation or plotting of properties such as:

• Bond-valence coefficients.

• Standard reduction potentials and their graphical depiction in Latimer and Frost diagrams.

4. As a value related to d-electron configurations of transition-metal (TM) ions in compounds and obtained either from spectra (a spectroscopic OS) or from spin-order sensitive methods, given that spin is the other collectively manifested property of an electron besides its electrical charge.

5. As a formalized basis for balancing redox equations.

It is the extensive use that puts various and at times conflicting demands on the OS values, and we shall call attention to this at appropriate places in this Report.

## 3 Experimental

The comprehensive definition of OS was approached in three stages: (1) anamnesis, (2) case studies, (3) write-up. The anamnesis included drafting and discussing five internal documents: (a) current IUPAC definitions, (b) textbook definitions, (c) differences between OS and ON, (d) purpose and uses of OS, and (e) list of telling examples. In the second stage, ~100 examples were examined and discussed, and OS evaluated upon varied definitions of bond ionicity via several algorithmic and computational approaches. In the third stage, this Report was drafted and discussed by the members of the team. Two quantum-mechanical approaches were used: an extended Hückel calculation of molecular orbitals (MOs) as linear combination of atomic orbitals (MO-LCAO) [23] implemented in the program Caesar 2.0 [24] and Hartree–Fock and density functional theory methods, both as implemented in Gaussian 3.0 [25].

## 4 Notation of OS

Throughout this Report, the values of OS will be denoted by Arabic numerals with a preceding sign, such as OS = −2 or OS = +2.5. This is not the same semantic category as OS or ON in the IUPAC inorganic nomenclature in the Red Book [2], rule IR-2.8.2.

## 5 The generic definition of OS

OS of a bonded atom equals its charge after ionic approximation.

The ionic approximation is illustrated in the MO-LCAO scheme in Fig. 1. In principle, the atom with lower orbital energy becomes the negative ion and is assigned all the bond’s electrons. In a finer detail, the atom with higher orbital-mixing coefficient c is assigned all of the MO’s electrons. Bonds between two atoms of the same element are divided equally. The OS of monoatomic ions equals their nominal charge,5 as OS is a quantitative concept that operates on discrete integer values of counted electrons.

Fig. 1

The essence of the adopted ionic approximation based on how the valence orbitals participate in the bonding MO. The mixing coefficients cA and cB refer to the atomic-orbital wavefunctions ψA and ψB in an MO-LCAO approach.

As will be illustrated with numerous examples, this ionic approximation leads to reasonable OS values. The definition of OS on a similar basis, albeit stated somewhat differently, has also been advocated by others [9–11] and can be traced back to Pauling [26]. Figure 1 is also in line with recent approaches to bonding systematics, such as covalent bond classification (CBC) [27–29].

The generic definition allows determination of the OS of all atoms in any compound where bond orders are known and the ionic approximation is not obscured by metallic delocalization. This leads to two basic algorithms for the calculation of the OS value, here named the algorithm of direct ionic approximation (DIA; Section 6) and the algorithm of assigning bonds (Section 7). Both provide one and the same OS while their semantic difference is reminiscent of Jensen’s [13] distinction between compositional and topological OS. In the following, the two components of the generic OS definition, the ionic approximation and the bond order, are described in more detail.

## 5.1.1 Determination

The criterion for the negative ion in Fig. 1 is the lower energy of the atomic orbital that prevails in mixing into the bonding MO. It’s a matter of practical convenience whether the energy or the mixing coefficient is used. A criterion suggested by Haaland [30] can also be applied, which interrogates where the bonding electrons go when the bond is split thermally: If the split is heterolytic, the ionic approximation follows the direction of electrons; if the split is homolytic, the typical energy of the atom’s valence orbital applies. Haaland’s criterion can also be evaluated by quantum-chemical calculations [31], which are, however, more involved than those of orbital energy and mixing coefficient. A simple estimate is therefore desirable. Criteria for ionic approximation that are not suitable for OS are examined in Appendix A.

## 5.1.2 Estimate using electronegativities

Of the many approaches to ionicity discussed by Meister and Schwarz [32], the electronegativity (EN) difference captures best the essence of the ionic approximation that originates from the properties of the two atoms that form the bond. A suitable EN scale suggests which of the two atoms will become positive and which negative. Several EN scales have been developed, and it is clear that none of them will provide the absolute OS result in each and every chemical compound. The best known and most often used EN scales, in chronological order, are those due to: Pauling [33–35], Mulliken [36, 37], Allred–Rochow [38], and Allen [39]. These EN scales are briefly discussed in Appendix B, where the respective values are given in tabular form.

The first three of these scales conform to the commonly accepted view of EN as an in situ molecular property of an atom, dependent on the bonding environment and OS of that atom, thus yielding a somewhat circular argument for OS assignment. Allen views EN as an inherent free-atom property. This is perhaps a less conventional view, but easy to visualize with ionic approximation via atomic-orbital energies as shown in Fig. 2 in a direct simplification of the MO approach in Fig. 1. Besides being more precise, the Allen EN values are also less prone to imply unusual ionization than the other scales (see Appendix A for examples). The Allen EN scale is therefore used in this Report for determinations of OS: An atom of higher Allen EN is assigned all of the bond’s electrons while bonds between two atoms of the same element are divided equally.

Fig. 2

Ionic approximation according to relative energy of the free-atom valence orbitals conveniently derived from Allen’s EN.

Being a simplification of the MO approach in Fig. 1, even this Allen-EN criterion of ionic approximation in Fig. 2 comes with a caveat: A rare exception occurs when the atom with higher Allen EN is a net acceptor of bonding electrons. This is signaled by its low orbital-mixing coefficient in the bonding MO or via loss of electrons under a heterolytic minimum-energy bond rupture in the gas phase as defined by Haaland [30]. Both imply that the energy of the relevant orbital of this acceptor atom is actually above the donor orbital as well as above the average energy of its other valence orbitals defining its higher Allen EN. Section 10 is devoted to examples of this high-EN acceptor caveat.

Of other approaches to ionicity, quantum-chemical calculations of atomic charges as indicators of the subsequent ionic extrapolation are applicable, also with caveats. Appendix C deals with this in more detail.

## 5.2 Bond order

In simple cases, the bond order is not needed for OS determination (Section 6). When used, the bond order should be accurate as its application is no longer just a question of sign as it was with the ionic approximation. IUPAC defines two alternative approaches to bond order [1]: analysis of the electron population or counting two-electron bonds. Since the commonly available quantum-chemical software does not calculate the classical integer bond orders (see Appendix C), this Report will operate with the chemist’s bond orders defined in terms of two-electron bonds. While the term “bond order” is traditionally used for molecules, “bond valence” is in use for extended solids. As the OS algorithms of this Report assign bond electrons to atoms in both molecular and solid compounds, the terms bond order and bond valence will have the same meaning – the number of two-electron ionocovalent bonds between two directly bonded atoms. It will include an electrostatic contribution since a 1+ 1− electrostatic bond is equivalent to a two-electron bond in our electron-counting approach.

## 5.2.1 A chemist’s formula approach

For small molecules, drawing a structural chemical formula according to a set of rules is an effective approach to the bond order that is applicable in OS determination. The rules are listed in Sections 6.1 and 7.3 and exemplified throughout this Report.

## 5.2.2 Bond-valence approach

While the term bond valence is equivalent with bond order, it is also synonymous with one particular approach to calculation of bond orders in solids, the origins of which, one ionic [40] and one covalent [41], are associated with Linus Pauling. Reference [41] contains a relation that laid ground for the widely used bond valence versus bond length expression:

$BVij=exp(Rij0−dijB) (1)$(1)

where BVij and dij are the respective bond valence and distance of atoms i and j, the parameter $Rij0$ is the reference single-bond length between these two atoms, and B is a parameter that in principle is variable but is often taken as a constant equal to 0.37. The value of $Rij0$ (and eventually also that of B) is determined by a least-squares fit on a collection of known structural data where the relevant bond valence (bond order) is known. Brown and Altermatt [42] used this procedure to compile a table of $Rij0$ for solid oxides. Brese and O’Keeffe [43] extended this to solid fluorides and chlorides and made some estimates for bonds to other nonmetals. A general formula for estimating $Rij0$ of bonds between 75 elements is presented in ref. [44]. Being simple and empirical, the bond-valence method provides quite accurate estimates for bond orders when the bond is typical and not affected by serious bonding compromises.

## 6 Bondless approach to OS

Given that some of the rules for drawing a structural Lewis formula (Section 7.2.1) are the same as those for determining the OS of its constituent atoms, one may wonder whether any of these rules can be used directly to determine OS from the empirical formula of the compound, making the structural formula superfluous. Because such a simplification avoids bonds, the sought-after rules for ionic extrapolation are those of stable electron configurations, such as the Lewis rule of eight [45], the octet:

## 6.1 The rule of noble-gas configuration

Electronegative main-group (sp) atoms in a compound maintain a complement of eight electrons, bonding or nonbonding. Hydrogen maintains two electrons.

## 6.2 OS algorithm of direct ionic approximation (DIA)

A stable electron configuration is assigned to the individual atoms according to their decreasing electronegativity until all available valence electrons are used up. The charges of the atoms are then evaluated.

Typical DIA-friendly species are homoleptic binaries of at least one sp element: CO, HCl, H2 O, HF2, CCl4, NO3, NO2, NH4+, SeO32–, CrO42–, BF4, SF4, SiF62–, SnCl62–, CuCl42–, RuO4, AuI4, or solids with a homoleptic periodic bonding unit: KBr, SiC, AlCl3, SnCl2, …, or molecules with such a unit: CH for C6 H6, BH⅓– for B6 H62–, etc. Allen’s EN is the sole criterion as there are no electronegative net acceptors requiring the caveat in Section 5.1.2. Additional atoms beyond homolepticity in some cases lead to ambiguities in DIA, and this is discussed in Appendix D.

## 7 Bond approach to os

Ionic approximation of a bond must involve an assignment of bonding electrons onto the bond partner that is to form the anion (typically the one that has a higher EN). This requires a structural formula. Two variants of such a bond-assignment algorithm are dealt with in this Report: the algorithm of moving bonds (Section 7.1) and the algorithm of summing bond orders, more typical of solid-state chemistry (Section 7.6).

## 7.1 OS algorithm of moving bonds

Bond electrons are moved onto the more negative bond partner identified by ionic approximation, and atom charges are evaluated, giving the OS.

Chemical formulas that express bonding can be approached on several levels of approximation and differently for molecules and extended solids. For molecules, a Lewis formula often provides a fairly precise representation of the bond order (see Section 5.2 for the definition used in this Report). Given the energy scale of the ionic extrapolation, weak interactions such as hydrogen bonding, halogen bonding [46–48], or lone-pair–π interactions [49] need not be included in the considerations. If the Lewis formula is not sufficient, bond orders can be obtained via bond-valence relationships (Section 5.2.2). For extended solids, a bond graph (Section 7.2.2) shows the connectivity of atoms.

## 7.2 Structural representations used in this Report

The purpose of this Section is not to provide authoritative definitions but rather to explain the terms as used in this Report.

## 7.2.1 Lewis formula of a molecule

In order to visualize the ionic approximation as moving bonds, a dash (–) will depict a two-electron ionocovalent bond, in principle anything from a purely covalent to a purely ionic single bond (between charges 1+ 1−), as well as a nonbonding or lone pair. The Lewis formula will be set up by distributing available valence electrons over the actual connectivity of the atoms, with bond orders that satisfy the appropriate electron-configuration rules (Sections 6.1 and 7.3). If a non-integer bond order occurs, a dashed line will be drawn for the fractional-order bond, either standalone or in addition to the already present single-bond dashes.

## 7.2.2 Bond graph of an extended solid

A bond graph is a finite representation of the infinite periodic network of an extended solid [50, 51]. In a scheme of atoms that corresponds to the stoichiometric formula of the crystalline phase, the bond graph features a line for each instance of bonding connectivity between nearest-neighbor atoms in the structure. The line does not have the meaning of a two-electron bond as in Lewis formulas; instead, it carries its own specific bond order. Polyhedral representation of atomic coordination is helpful in assigning the connectivity of the central atom. Complicated polyhedra around a central atom can be determined by the Frank–Kasper method [52]: A Wigner–Seitz cell (a Voronoi cell in periodic structure) is found first as a portion of space that is closer to the central atom than to any other atom; by connecting the central atom with all its neighbors and bisecting each connecting line with a perpendicular plane. Atoms facing the sides of the thus formed Wigner–Seitz cell are vertices of the sought polyhedron of nearest neighbors.

## 7.2.3 Notations for solid-state connectivity

A Niggli formula expresses sharing vertices of coordination polyhedra in extended structures by listing the vertex connectivity. As an example, SiO4/2 represents a silicon dioxide with four 2-connected O vertices of the polyhedron around Si, meaning that each of the four oxygen vertices is shared with another tetrahedron to form a network of corner-sharing tetrahedra. Similarly, CrO2/2+2/1 is the Niggli formula for Cr(VI) oxide with the tetrahedra joined in chains and the remaining two vertices terminal.

A crystal-chemical formula [53] lists each crystallographically nonequivalent atom in the structure and contains information about that atom’s coordination to other atoms in superscripted square brackets. The coordination numbers in the brackets follow the same array order as the atoms in the formula and are separated by a comma while eventual bonding to atoms of the same element comes after a semicolon. As an example, CrO3(s) is represented as Cr[(2,2)t;]O[2n,;]O2[1;], where superscript t means tetrahedron, n stands for nonlinear, and the absence of a number after the semicolon indicates that there is no homonuclear bond in this compound. Connectivities expressed in the crystal-chemical formula fulfil the so-called connectivity balance, a condition that requires having an equal number of AB and BA connections between two atoms A and B in a periodic solid. It states that the product of the coordination number and stoichiometric coefficient at A is equal to such a value at B.

## 7.3 Stable electron-configuration rules used in this Report

With the rule of the noble-gas configuration stated in Section 6.1, some of the remaining electron configuration rules are listed here.6

## 7.3.1 The 8−N rule

An electronegative sp atom of N valence electrons tends to form 8 − N, but no more than four, two-electron bonds with atoms of equal or lower electronegativity.

Thus, phosphorus of five valence electrons forms three two-electron bonds in the P4 tetrahedron, and nitrogen does the same in N2. In heteroatomic molecules, the 8−N rule concerns the more electronegative atom of the bonded pair as it is enforced by that atom’s EN. A few examples: In sulfur fluorides, the 8−N rule concerns fluorine. Bonds in SF2, SF4, and SF6 all have approximately the same length as a single bond [54], i.e., a two-electron interaction. In the series BF, CO, and N2, the triple bond suggested by the octet rule only occurs in N2 whereas O and F force the bond order towards 2 and 1, respectively [55]. More examples follow in the text.

## 7.3.2 The (8+)N rule

An electropositive sp atom of N valence electrons tends to form N two-electron ionocovalent bonds with more electronegative atoms.7

Examples: Sodium of 1 valence electron forms one two-electron ionocovalent bond to iodine in NaI, adopting the noble-gas configuration in the 1+ and 1− ionic limit. Beryllium of two valence electrons forms two ionocovalent two-electron bonds.

## 7.3.3 The 18-electron rule

A TM atom in a molecule tends to surround itself with 18 valence electrons.

Also called an effective atomic-number rule, this is a weaker TM variant of the noble-gas configuration rule (Section 6.1); for a recent justification, see [56].

## 7.3.4 The Hückel (aromaticity) rule

An aromatically stabilized ring has an odd number of itinerant π-electron pairs.

## 7.3.5 The 12−N (s2) rule

An atom of an element close to having 12 dsp electrons in its valence shell will tend to either lose those that exceed 12 or to gain in bonds those short of 12.

While the first tendency (to form s2 cations) is magnified by other trends in the main groups of the periodic system, the second (to form s2 anions) is specific to Pt and Au, which in such compounds are rightfully called relativistic chalcogen and halogen, respectively [57].

## 7.4 Some structure-predicting rules

This rather incomplete collection is intended as an example of additional rules that go beyond the stable electron configuration in that they also predict the connectivity of atoms. This may or may not be necessary to determine OS.

## 7.4.1 The generalized 8−N rule

In a binary nonmetallic compound of octet-obeying sp elements, the valence-electron count per the electronegative atom A (VECA) is calculated. Excess electrons over 8 remain at the electropositive atom, either as bonds or as lone pairs. Electrons short of 8 are gained by forming bonds between the electronegative atoms.

This rule originates in the Zintl concept [58] and its formal expression [59, 60] for a binary compound Cc Aa is

$VECA= 8 + CC ⋅ ca − AA (2)$(2)

where VECA is the valence-electron count per A, the variable CC is the number of electrons per “cation” C that form C–C bonds or are localized at the cation as lone pairs, and AA is the number of electrons per “anion” A that form A–A bonds. For examples, see Sections 8.3 (GaSe) and 9.5 (RbPb).

The Wade–Mingos rules [61–64] concern clusters with delocalized skeletal bonding [65]. The convex boron deltahedron of a closo borane is a good example [61]. The fundamental rule is that the number of skeletal bonding MOs is maintained even if one or two vertices are not present (nido or arachno borane, respectively). A closo borane BbHhc (b > 4, c ≤ 0) has altogether 2b + 1 pairs of valence electrons: One pair per B is in the bond to H. The remaining b + 1 pairs fill the skeletal bonding MOs, one in the radial skeletal bonding MO and one pair per B is in the tangential MOs of the deltahedral skeleton. A b-vertex nido borane has 2b + 2 pairs (is derived from deltahedron of b + 1 vertices), and a b-vertex arachno borane has 2b + 3 pairs (is derived from deltahedron of b + 2 vertices). See Section 9.2 for examples. Further generalization of the electron-count rules have been suggested for clusters, such as that by Jemmis et al. [66].

## 7.5.1 Formal charge and electroneutrality principle

Formal charge is the charge of an atom in a Lewis formula (Section 7.2.1) after ½ of each of that atom’s bonds is assigned to it as electrons. Standalone, it has a trailing sign, as in ions. Ionic extrapolation to determine OS of an atom maintains the charge (if any) of a chemical formula, and the sum of the OS over all atoms equals the sum of formal charges

$∑atomsOS=∑atomsFC (3)$(3)

where FC is the numerical value of the formal charge. The sum of formal charges also gives the ionic charge on the chemical formula considered.

## 7.5.2 Bond order sum (BOS) and ionized bond order sum (iBOS)

The BOS at an atom is analogous to the bond valence sum (BVS) used in crystal chemistry. A parameter relevant for OS is the iBOS (ionized bond order sum) at an atom. The iBOS is the sum of bond orders weighted with the ionic-approximation sign, sgn(i)

$iBOS=∑n bondssgn(i)BOn or iBVS=∑n bondssgn(i)BVn (4)$(4)

As the bond order BOn of the n-th bond is equivalent to its bond valence BVn, the sum of the (experimental) bond valences yields the iBVS exemplified later in the text.

Let’s consider two examples of iBOS. The commonly used Lewis formula of carbon monoxide, |C≡O|, has a triple bond and two lone pairs; the BOS at each atom is 3, iBOS at the carbon is +3 whereas −3 at the oxygen. Hydrogen peroxide, H–O–O–H, has a homonuclear bond the BOS of which is weighted by zero on both ends, hence iBOS = −1 at oxygens and +1 at hydrogens. The following section shows that it is not by chance that the latter two values equal the OS.

## 7.5.3 Relation between OS and iBOS in a Lewis formula

There exists a very useful relationship between OS and iBOS (or iBVS rounded off to integer) and the formal charge FC at an atom in a Lewis formula:

$OS=iBOS+FC or OS=iBVS+FC (5)$(5)

For the |C≡O| Lewis formula, it yields OSC = 3 − 1 = +2 and OSO = −3 + 1 = −2.

Equation 5 is valid for any electron count,8 whether or not the 8−N rule is obeyed or octet formed on the most electronegative or any atom, whether or not under the OS determination a bond is approximated ionic (iBOS nonzero) or divided equally (iBOS zero). If divided, however, it must be divided equally.

It may be noted that the presence of formal charges across a bond in a Lewis formula suggests an ionocovalent contribution that tends to either decrease or increase that bond’s order judged from interatomic distance. The bond will shorten when the formal charges agree with the EN of the atoms, and lengthen when they do not. An example is |C≡O| with formal charges 1− and 1+, respectively. Being against the EN, the actual CO bond length suggests a bond order < 3 and closer to 2 for which formal charges would be zero in such a Lewis formula. In other words, violation of the 8−N rule on the most electronegative atom in the |C≡O| formula decreases the actual bond order towards the value of 2 predicted by this rule. More examples follow in Section 8.3.

## 7.5.4 When OS equals iBOS

OS equals iBOS when all formal charges are zero in the Lewis formula.

A nonzero formal charge typically occurs as a consequence of an sp atom violating the 8−N rule. The formal charge of this, by definition most electronegative atom,9 then obtains a 1− for each missing bond to fulfil 8−N (or 1+ for each bond in excess) and its OS ≠ iBOS. At least one other atom in the neutral structure will then have a compensating nonzero formal charge and OS ≠ iBOS. As an example, consider the adduct of BH3 with |C≡O|. The violation of the 8−N rule at oxygen creates a compensatory formal charge 1− at the B atom of the adduct (not at C as in CO). This does not happen in NaBF4 (see Section 8.3) in which F fulfils the 8−N rule by forming a ¾ bond to B and a ¼ bond to the cation, with OS = iBOS.

## 7.6 OS algorithm of summing bond orders (bond valences)

With eq. 5, we can formulate the second version of the bond-assignment algorithm for OS. We’ll refer to it as the “bond-order sum algorithm” or “bond-valence sum algorithm” with the former typically an ideal situation and the latter an experimental value (Section 5.2.2) to be rounded off to nearest integer:

iBOS or iBVS is evaluated at each atom and summed with the atom’s formal charge (if any) to give the atom’s OS.

This algorithm is particularly suitable for bond graphs, which visualize bonds but not nonbonding electrons. It can also be used on Lewis formulas. In the following examples and figures, both methods will often be used to show that both give the same OS.

## 8 Simple examples of OS

With an occasional use of DIA (Section 6), we’ll focus on the two variants of the bond-assignment algorithm (Sections 7.1 and 7.6) applied to molecules and extended solids. Cases with zero formal charges will be considered first.

## 8.1 OS from a Lewis formula

In Fig. 3, the Lewis formula of a CrO3 molecule is used to illustrate the two bond-based approaches to OS:

Fig. 3

OS via assigning bonds to the electronegative partner and via summing ionized bond orders (8−N rule on the electronegative atom) on Lewis formula of CrO3 with 12 valence-electron pairs drawn as full dashes.

While OS in CrO3 can be obtained simply by DIA (Section 6.2), CrO(O2)2 is not a homoleptic molecule and DIA would treat it incorrectly as CrO5 (see Appendix D). OS values in CrO(O2)2 follow from assignment of bonds, the two approaches of which are illustrated in Fig. 4.

Fig. 4

OS via assigning bonds to the electronegative partner (left) and via summing ionized bond orders (right) on the CrO(O2)2 Lewis formula that fulfils the 8−N rule on O.

## 8.2 OS from a bond graph

A bond graph of an extended solid can be used to generate OS either via iBOS or iBVS. In the first case, the bond orders are predicted from 8+N and 8−N rules. In the second case, bond valences are calculated with eq. 1 from bond distances. Equation 4 is then used to obtain iBOS or iBVS, and OS is evaluated via eq. 5.

As an example, let’s consider the simple binary CrO3 of already known OS from the trivial DIA approach, and determine the OS from its crystal structure. The infinite chain that repeats itself in the crystalline CrO3 is illustrated in Fig. 5 (Niggli formula CrO2/2+2/1). The bond-order and the bond-valence approach to OS, performed on a bond graph (Section 7.2.2), are compared. The ideal bond orders are set to fulfil the 8−N rule on O. The bond lengths [67] are converted to bond valences with eq. 1 using the appropriate bond-valence parameter from ref. [43] and summed according to eq. 4 to iBVS. Given experimental circumstances, and due to bond-valence coefficients being a statistical average, the iBVS cannot be considered absolute, but rather a value that either confirms or disproves the ideal model.

Fig. 5

OS in an infinite chain of CrO3 (left), estimated on a bond graph via bond orders to fulfil 8−N rule on O (middle) and via calculated experimental bond valences (right). Note that 1 Å = 100 nm.

OS in a 3D network is determined likewise. In Fig. 6, a segment of a KMgF3 perovskite-type structure is shown with typical coordination polyhedra, and the bond-order and bond-valence approaches to OS are compared. The ideal bond orders fulfil the 8−N rule on F and the 8+N rule on both Mg and K. Bond-valence parameters from ref. [43] were used in eq. 1 to convert the bond lengths [68] to bond valences, which were summed according to eq. 4 to iBVS and rounded off to OS. The sums illustrate that bonding in extended solids is a compromise, making some atoms appear overbonded and others underbonded.10

Fig. 6

OS in solid KMgF3 (left), estimated on a bond graph via ideal bond orders to fulfil the 8−N rule on F (middle) and via calculated experimental bond valences (right).

As shown in Fig. 7, analogous ideal bond orders are obtained for the inverse perovskite Cs3 AuO [69]. The OS values for all its components are dictated by configuration rules: the 8+N rule for Cs, the 8−N rule for O and the 12−N rule for Au as a relativistic halogenide Au with the 6s2 configuration.

Fig. 7

OS in inverse perovskite Cs3 AuO, estimated on a bond graph via ideal bond orders.

Let’s consider Rb2 CuCl4 as an example of a non-d0 TM compound. Already, DIA provides the correct OS by distributing the total of 41 valence electrons, Fig. 8.

Fig. 8

OS in Rb2 CuCl4 by DIA. Dotted dash represents the odd electron.

The crystal structure consists of layers of copper coordination octahedra interleaved with nine-coordinated Rb in monocapped square antiprisms (Fig. 9 left). Given these coordinations, the bond graph and the configuration rules 8+N for Rb and 8−N for Cl provide the ideal bond orders that yield OS for all elements, including Cu, as shown in Fig. 9 right.

Fig. 9

Rb2 CuCl4 (a structural segment of the octahedral layer with 9-coordinated Rb is shown on left) and its OS estimated on a bond graph (right) via ideal bond orders.

The OS of Cu is clearly the most interesting value. Figure 10 compares two Lewis formulas for the isolated CuCl42– anion: On the left, a simplified formula is drawn with single bonds (and d electrons omitted) as often used in teaching; note the anion’s charge 2− residing on Cu, i.e., the formal charge of Cu is 2−. On the right, a formula of zero formal charges is drawn that includes the ionocovalent bonds to the unexpressed Rb cations.

Fig. 10

OS in CuCl42− via moving bonds or via summing ionized bond orders on two Lewis formulas. Left: formula with negative formal charge on Cu. Right: formula with zero formal charges due to ionocovalent bonds drawn towards an unexpressed cation. The d electrons of Cu are omitted.

TM carbonyls are interesting examples of Lewis formulas of neutral molecules that include formal charges. Figure 11 illustrates how OS in Cr(CO)6 is determined via iBOS according to eq. 5. As the Lewis formula |C≡O| of carbon monoxide has formal charges on both atoms (the 8−N rule is not obeyed), donation of the carbon’s electron pair transfers the carbon’s formal charge 1− onto Cr, while 1+ remains on the oxygen. The result of this somewhat cumbersome process of course yields the expected OS.

Fig. 11

OS in Cr(CO)6 via summing ionized bond orders on Lewis formula with formal charges; calculation of OS is shown on the right.

## 8.3 Further OS examples for molecules and solids

Let’s consider tetrafluoridoborate anion, BF4, as an sp example of a Lewis formula that may or may not involve formal charges (Fig. 12). For both variants, OS is evaluated via the algorithm of moving bonds (onto the electronegative partner) as well as via the algorithm of summing ionized bond orders and eq. 5.

Fig. 12

OS in BF4 via ionic assignment of bonds on two Lewis formulas. Left: formula with negative formal charge on B. Right: formula with zero formal charges due to ionocovalent bonds drawn towards an unexpressed cation.

In solid NaBF4, sodium is eight-coordinated by fluorine. The structure is shown in Fig. 13, left. The bond graph in the middle has ideal bond orders that fulfil the 8−N rule on F (as well as 8+N rule on B and Na) while assuming equivalence of all bonds of the same type. In the second bond graph, bond lengths [70] are converted to bond valences with eq. 1 and bond-valence parameters from ref. [43].

Fig. 13

OS (in red) in solid NaBF4 (left), estimated on a bond graph via ideal bond orders to fulfil the 8−N rule on F (middle) and via calculated experimental bond valences (in blue) and their iBVS (in grey) rounded off (right).

Also for SrBe(OH)4 (Fig. 14) there is a good agreement between the ideal OS and the iBVS obtained with structure data [71] and bond-valence parameters (ref. [43]).

Fig. 14

OS (in red) in solid SrBe(OH)4 (left), estimated on a bond graph via ideal bond orders to fulfil the 8−N rule (middle) and experimental bond valences (in blue) and iBVS (in grey, right).

Another example having formal charges is the addition product of borane and ammonia, BH3 NH3. Its typical Lewis formula in Fig. 15 (left) has a negative formal charge 1− on boron and positive 1+ on nitrogen. Equation 5 relates these formal charges to OS and iBOS at these two atoms. Since these formal charges are contrary to the EN of B and N, we expect the bond to be weaker than the single bond drawn (see Section 7.5.3). Indeed, the bond length 1.672 Å (167.2 pm) [72] in the gaseous molecule corresponds to BV = 0.6 as obtained with eq. 1 and bond-valence parameter $RBN0 = 1.488 Å$ (148.8 pm) calculated from data in ref. [44]. The EN principle suggests that these negative and positive charges would partially relocate to the hydrogens in the respective boron and nitrogen portions of the molecule. This leads to intermolecular attractive forces that manifest themselves in that BH3 NH3 is a solid stable up to about 90 °C [73]. If the charge relocation is prevented by a full substitution of the hydrogens, the formal charges contrary to EN may contribute to the lability of the B–N bond and to interesting reactivity [74].

Fig. 15

OS in BH3 NH3 (g) via ionic assignment of bonds on a typical Lewis formula (left) and on two extreme distributions of the formal charge generated by the calculated B–N bond valence of 0.6: either immediately on B and N while FC = 0 on H (middle) or on H while FC = 0 on B and N (right). Note that all OS remain the same throughout (not shown in the middle and right formulas).

Trimethylamine oxide is not suitable for DIA, and the two algorithms of ionic assignment of bonds are used here to determine OS, Fig. 16. Its Lewis formula is typically drawn with an octet configuration at the nitrogen, which thus acquires a positive formal charge compensated by negatively charged oxygen, thereby violating the 8−N rule. As the 1+ and 1− formal charges comply with EN of the bonded atoms, the N–O single bond strengthens to become a “sort of double bond” [34a]. If this ionocovalent contribution is drawn with a full dash, the formula loses formal charges, and OS equals iBOS (iBVS).

Fig. 16

OS in trimethylaminoxide via ionic assignment of bonds. Left: Lewis formula with formal charges on N and O due to 8−N rule violation on oxygen. Right: Formula where the ionocovalent contribution of these formal charges is drawn as a full single bond yielding zero formal charges on N and O.

The nonmetallic GaSe (Fig. 17) is an example of an extended structure with the electropositive atom forming pairs as implied by the generalized 8−N rule (Section 7.4.1), which requires the excess electrons over 8 per electronegative atom (9 − 8 = 1) to remain at the electropositive atom. This leads to a two-electron bond between the two Ga. The structure is straightforward in that every Ga has three equal bonds to one Se. Because each Ga has the same bonding pattern, we may remind the reader that already the trivial DIA is sufficient to provide OS: Eight of the total nine valence electrons of this periodic bonding unit (which by chance is identical with the formula of the solid compound) are assigned to Se, leaving one electron on Ga, which provides OSSe = −2 and OSGa = +2.

Fig. 17

GaSe structure with one full Ga coordination shown (two of its three Se are outside the unit cell) and the OS values determined from its bond graph (right). The bond order dictated by the 8−N rule is listed above the three connectivity lines. Below them is the bond valence determined from the bond length.

The same result is obtained in the bond graph in Fig. 17, where the 8−N rule dictates bond order of ⅔ for each of the three GaSe bonds, and iBOS yields OS (eq. 5 in the absence of formal charges) for both Se and Ga, as the Ga–Ga bond is weighted with zero. An identical result is obtained with bond valences calculated from bond lengths [75] with eq. 1 and the appropriate bond-valence parameter from ref. [43].

## 9 OS in species with homonuclear bonds

As the OS definition in Section 5 operates with equal partitioning of homonuclear bonds in Lewis formulas, uncertain bond order remains a possible obstacle in OS determination. Some important cases are exemplified in this Section.

## 9.1 Chains of atoms of the same element

In this Section, we visit two simple examples of homonuclear chains, one neutral and one charged, before cases of increasing ambiguity.

## 9.1.1 Polysulfanes

Polysulfanes are straightforward examples with single bonds along the chain. Their Lewis formulas obey the 8−N rule, formal charges are therefore zero, and OS = iBOS. The following OS are obtained for the two examples (Fig. 18):

Fig. 18

OS in trisulfane and pentasulfane via assigning bonds to the electronegative partner and via summing ionized bond orders.

## 9.1.2 Triiodide anion

Although a Lewis formula of triiodide I3 may appear simplest if drawn with single bonds, it would violate the 8−N rule on terminal iodines and the octet on the central iodine. Therefore, only the Lewis formula with iodine–iodine bond order ½ (the formula includes bonds to the unexpressed cation) is considered in Fig. 19. As an added benefit, this avoids formal charges. The bond order ½ is justified from comparison of the bond distance of 2.90 Å (290 pm) for I3 in a soft cationic environment [76] with the distance of 2.68 Å (268 pm) in I2 gas [77]. The difference of 0.22 Å (22 pm) is close to an estimate with eq. 1 for the bond order decreasing from 1 to ½; −0.37ln(1/2) = 0.256 Å (25.6 pm). As in the previous example, OS = 0 for the central iodine. For electrochemical purposes, the average OS = −⅓ per iodine is the value to use.

Fig. 19

OS in triiodide anion via ionic assignment of bonds.

## 9.1.3 Azide anion

In this example, we’ll see that the azide anion, N3, is similar to triiodide. This time we’ll consider two Lewis formulas: Fig. 20, left, with an octet on all nitrogens and violation of the 8−N rule (which leads to formal charges). Figure 20, right, with ionocovalent bonds to the unexpressed cation and the octet and 8−N rules obeyed on the flank nitrogens. The formulas differ in their NN bond order.

Fig. 20

OS via ionic assignment of bonds in two Lewis formulas of the azide anion: one with all octets (left), the other with all formal charges zero in which the 1e ionocovalent bond to an unexpressed cation is drawn from both terminal atoms (right).

What is the actual NN bond order in azide? A least-squares fit with eq. 1 of bond valences versus NN distances for integer bond orders is shown in Fig. 21. Considering the NN distance in ionic azides of about 1.16–1.17 Å (116–117 pm) [78], we see that the bond order in N3 is ~2.5, as seen in the formula on the right (the formula on left indicates this via the ionic interactions of the formal charges). The formula on right provides OS values that are analogous to the triiodide I3 in the previous Section. Also here the average OS = −⅓ per N is suitable for electrochemical purposes.

Fig. 21

Fit with eq. 1 of the NN bond lengths: triple (1.097 Å in N2 [79]), double (1.247 Å in N2 H2 [80] and dimethyldiazene [81], 1.226 Å of N22− in SrN2 [82] or calculated 1.236 Å in dimethyldiazene [83]), single (1.447 or 1.453 Å in hydrazine [84, 85]) (1 Å=100 pm).

## 9.1.4 Dinitrogen monoxide

No octet-obeying Lewis formula of N2 O exists which has all formal charges zero. With octets and formal charges, two Lewis formulas of N2 O can be drawn, as shown in Fig. 22. They differ in bond orders, leaving the OS of nitrogen ambiguous:

Fig. 22

OS of N2 O via ionic assignment of bonds in two octet obeying Lewis formulas.

Can this bond-order ambiguity be resolved? A look at Fig. 21 shows that the NN bond of 1.131 Å (113.1 pm) in N2 O at 293 K [86] is much closer to the triple-bond length than to double-bond length. This suggests that the nitrogen OS values approach those on the right-hand side of Fig. 22.11 A typical simplification is to consider the average OS = +1 on each N (the electrochemical OS). That is obtained from OSO= −2 dictated by the 8−N rule (Section 7.3.1) and the electroneutrality condition (Section 7.5.1).

## 9.1.5 Pentanitrogen cation (1+)

The Lewis formula of N5+ requires several resonance forms of various weights. We will not enumerate these but, instead, look at bond distances [87] to estimate the bond orders. The bond lengths are shown in Fig. 23. Comparisons with Fig. 21 suggest a bond order of nearly 3 between the wing nitrogens and about 1.5 on both sides of the central nitrogen. These approximate bond orders provide the OS values shown in Fig. 23.

Fig. 23

Bond distances in N5+ [87] and bond orders/valences deduced from Fig. 21 (left). A round-off of these bond orders to obtain OS = FC given that iBOS = 0 for atoms of the same element (right).

The bond orders obtained suggest that the two mid-wing nitrogens are using more than 4 electrons for bonding. This is caused by the ionocovalent interaction (dashed) due to formal charges 1+, 1− and 1+ that would be present on the central three nitrogens if they were singly bonded; this interaction would shorten (strenghten) the single bond as its charges agree with the actual NN bond polarity (estimated by the calculated Mulliken charge being negative at the central N and positive at the mid-wing N). The implied near absence of the lone pair on the two mid-wing nitrogens is in agreement with the near linearity of the bond. The angle at the central nitrogen, approaching the tetrahedral value, suggests that nearly two lone pairs are present on this N, hence the Lewis formula in Fig. 23 is still somewhat off, and the obtained OS values are an approximation.12 Clearly, for electrochemistry and redox balancing, the average OS = +1/5 per nitrogen is the value to use.

## 9.1.6 Thiosulfate

Structural properties of sodium thiosulfate [88] suggest that all its terminal atoms carry some of the anion charge. In other words, all form ionocovalent bonds to a cation. The calculated [24] Mulliken charges (−1.4 for O and −1.1 for terminal S) confirm this but, being different, they suggest unequal SS and SO bond orders. The SS bond distance of 2.025 Å (202.5 pm) [88] is shorter than the single bond of 2.055 Å (205.5 pm) in crystalline S8 [89] or 2.056 Å (205.6 pm) in the H2 S2 gas [90], but substantially longer than the double bond of 1.883 Å (188.3 pm) in S2 O [91, 92] or 1.889 Å (188.9 pm) in S2 [93]. Given this uncertain situation, the approach chosen here is to draw in Fig. 24 two limiting Lewis formulas.

Fig. 24

OS in thiosulfate via ionic assignment of bonds in two limiting Lewis formulas having bonds to unexpressed cation to avoid formal charges: SS bond single (left), double (right).

The result in Fig. 24 means that the individual OS of the two sulfur atoms in thiosulfate are ambiguous, and no formula in-between these two limits yields reasonable values. The formula on left provides an unusual yet not entirely illogical OS of −1 for the terminal sulfur, reminiscent of the oxygen OS in peroxides. The formula on right, while having an SS bond order that is too high, suggests OS values that at times are used, making the S + SO32− = S2 O32− synthesis a non-redox process. The latter fact is not necessarily an advantage as that synthesis in an aqueous environment is well described with standard reduction potentials for both its half-reactions when an average OS = +2 for the thiosulfate sulfur is considered. That average is clearly the preferred OS for sulfur in thiosulfate in Latimer and Frost diagrams of the electrochemical standard potentials (see also Section 14).

The only route to an unambiguous OS for both S atoms in thiosulfate would be to resolve the SS bond polarity. The resulting set of OS is often seen in textbooks: the terminal sulfur has OS = −2, the central sulfur +6, independent of their bond order.

## 9.2 Boranes

The Wade–Mingos rules in Section 7.4.2 predict the skeletal shape of BbHhc boranes (b > 4, c ≤ 0), so that OS can be evaluated not only with DIA on a repeat unit if all B atoms are equivalent, but also from the structural connectivity, provided bond orders are known. Considering OSH = −1, an average OS of boron can be calculated as OSB = (h + c)/b. This average represents the generic OS only for closo boranes that have all vertices equivalent. Let’s illustrate the relationships with two examples.

## 9.2.1 The closo-borane B6 H62– anion

All B are equivalent, and DIA can be performed on the repeat unit BH(1/3) with a total 4⅓ electrons. After assigning the stable doublet configuration to the most electronegative atom, H, we are left with 2⅓ electrons per B, which means that OSB = +2/3. The same result is obtained with OSB = (h+ c)/b.

The structure is shown in Fig. 25. The anion has in total 26 valence electrons or 13 electron pairs. Of these, 6 pairs are in the six BH single bonds and the remaining 7 are the skeletal pairs. Of these 7, one pair is the radial-skeletal MO. The remaining 6 pairs, one from each of the 6 boron atoms, are in skeletal B–B bonds. Given that there are 12 such bonds and considering the symmetry of the molecule, each is a one-electron bond. The algorithm of moving bonds onto the electronegative bond partner assigns bond electrons as shown by sectioning with dashes in Fig. 25: Hydrogen obtains 2 electrons and OSH = −1. Boron obtains 4/2 electrons from BB bonds divided in half and 1/3 electron from the radial-skeletal MO. As with DIA above, 2 and 1/3 electrons on B give OSB = +2/3.

Fig. 25

The B6 H62 anion and its bond sectioning under the OS algorithm of moving bonds onto the electronegative partner. The number of electrons in bonds and the resulting OS are denoted.

## 9.2.2 The nido borane B6 H10

If all borons were equivalent in B6 H10, OSB = (h+ c)/b = +5/3 would be valid for this BbHhc borane. The electron counting by Wade–Mingos rules is as follows: B6 H10 has 14 valence-electron pairs in total, of which 6 are in B–H bonds, 1 is the radial-skeletal MO, and the remaining 7, in the tangential-skeletal MOs, identify a 7-vertex parent deltahedron (pentagonal bipyramid), of which one vertex is missing in B6 H10. B6 H10 is a nido borane, Fig. 26. Its B atoms are not all equivalent. The five basal borons are bonded to nine hydrogens that maintain the 2−N rule. Three of these borons have iBOS = OS = +2 and two +3/2 (+9/5 on average). The apical boron has iBOS = OS = +1 from one bond to hydrogen. The weighted average of these values is +5/3.

Fig. 26

The nido borane B6 H10.

## 9.3 Carboranes

Wade–Mingos rules in Section 7.4.2 are valid also for carboranes, clusters in which some boron is replaced with carbon. Replacing B1− with C will consume some or all of the anionic charge. Because of this substitution, the vertices are no longer equivalent and generic OS cannot be evaluated from DIA. An example is the icosahedral ortho-carborane B10 C2 H12 in Fig. 27, which is derived from B12 H122 by formally replacing two B1– with two C atoms. Because of this isoelectronic relation, we will assume that all bonds in the icosahedron are equivalent. This allows evaluation of OS by moving bonds.

Fig. 27

OS determined by the moving-bond algorithm on ortho-carborane B10 C2 H12 with each skeletal bond of 4/5 electrons, a radial-skeletal MO of 1/6 electrons per vertex and each vertex bonded to H with a two-electron bond.

The evaluated OS are shown in Fig. 27. The molecule has 50 valence electrons or 25 electron pairs. Of these, 12 are in the single CH and BH bonds. Of the remaining 13 pairs, one is the radial-skeletal MO (hence, 1/6 electron per each B/C) and 12 pairs bond the icosahedron of borons and carbons with 30 edges, hence, 4/5 electrons per edge. To obtain OS, the bonds are moved according to EN. Two H atoms on C have OS = +1 and 10 H atoms on B −1. Moving bonds around the two C atoms: 1/6 + 4(4/5) + (1/2)(4/5) + 2 = 173/30, which yields OSC = −53/30. Around the two B atoms that have 2C neighbors: 1/6 + (3/2)(4/5) = 41/30 yields OSB = +49/30. Around the four B that have 1C neighbor: 1/6 + (4/2)(4/5) = 53/30 yields OSB = +37/30. Around the four B that have no C neighbors: 1/6 + (5/2)(4/5) = 13/6 yields OSB = +5/6 for this ortho isomer. Note that the OS values will be different for the meta and para isomers of B10 C2 H12.

If the above result appears complicated, the obvious simplification is to consider all vertices alone by neglecting the EN difference between B and C. The two C atoms will then have OS = −1 because their hydrogens have +1. The 10 B atoms will have OS = +1 because there are 10 hydrogens with OS = −1 bonded to them. Besides simplicity, the advantage of this assignment is that it is valid for all isomers of this carborane: ortho, meta, and para.

## 9.4 Metallaboranes

Clusters with partially or fully integrated metal M cations into anionic borane cages are known as metallaboranes. While the parent deltahedra of smaller borane anions are readily recognized to define the OS of the metal via the anionic charge that satisfies Wade–Mingos rules, the distinction is harder for the large icosahedron or isocahedron parent cages of the borane subunits in, e.g., MB10 H12. Here the anion can either be a nido B10 H122− of octadecahedral parentage or an arachno B10 H124− of icosahedral parentage, with subtle differences in shape [94], Fig. 28. When the boron subunit is arachno B10 H124−, M adopts the position of a regular vertex, making MB10 H12 a true nido heteroborane of 11 vertices. However, when the subunit is a nido B10 H122−, the M atom is shifted away from the missing octadecahedron vertex and does not bond to the front B atoms in Fig. 28; it occupies a μ4-bridge over the back boron trapezoid [95]. As a result, these two metallaboranes of two different borane-anion charges and OSM have similar shapes that can only be distinguished by careful evaluation of distances or via computerized comparisons [94]. The latter reveal bonding situations in between these two extremes, depending on how many orbitals the M atom can supply [94]. If one (e.g., Au+), it yields the 2− anion, if three (e.g., Co3+), it yields the 4− anion. The OS of M needs to be confirmed from physical measurements (e.g., Mössbauer spectra in ref. [95]).

Fig. 28

Similar deltahedral shapes of metallaboranes (on right) may lead to ambiguous OS of the metal M (vs. the subunit anions in center) if their parent deltahedra (left) are misidentified.

## 9.5 Other main-group clusters

Benzene is an aromatic cluster, and the OS is straightforwardly −1 for carbon and +1 for hydrogen. In naphthalene, C10 H8, two of its carbons are bonded solely to other carbons. These two carbons obtain OS = 0 via the bond-assignment algorithms (iBOS weighted by zeros), with the remainder of the molecule having the same OS as benzene.

RbPb is a nonmetallic solid for which the generalized 8−N rule (Section 7.4.1) predicts tetrahedral cluster anions Pb44− isoelectronic with white phosphorus. The crystal structure indeed contains lead tetrahedra [96] with OSPb = −1.

As4 S4 forms two different cluster molecules in which both elements maintain the 8−N rule. This information is sufficient to obtain the OS in Fig. 29 as iBOS, and similarly so for other electron-precise main-group clusters.

Fig. 29

OS via iBOS in two sigma-bonded As4 S4 clusters.

## 9.6 TM clusters

Let’s consider a ring of five copper atoms in the anionic cluster Cu5 I72– (Fig. 30) in a diffusely charged environment of solvated tetrabutylphosphonium cations [97]. One Cu atom bonds to four iodines, whereas the remaining four Cu bond only to three iodine atoms. Two of seven iodine atoms bond to three Cu and the remaining five only to two. Such an irregular connectivity might better suit a bond graph of an extended solid, but here it actually is a single cluster ion. We will treat the structure in Fig. 30 as if it were a bond graph and evaluate OS via bond valences calculated with eq. 1 from bond distances given in ref. [97], using $RCuI0 = 2.188 Å$ (218.8 pm) calculated from parameters in ref. [44]. Since all CuCu bonds are divided equally, iBVS applies only to the Cu to I bonds and bonds of I to the unexpressed cation, yielding iBVS = −1.05(4) per I and +1.07(2) per Cu, which after round-off give the expected integer OS for Cu and I. Despite different bonding patterns, this is a remarkably homogeneous set of iBVS for atoms of the same element, making this a nice example of Pauling’s Parsimony Rule, which states that “the number of essentially different kinds of constituents in a crystal tends to be small” [40].

Fig. 30

Bond valences in Cu5 I72– and their summing into iBVS on the atoms, from which the expected integer OS values follow by a round-off. Bonds to the unexpressed cation are counted with their ideal bond order of 2/7.

Whereas the Cu5 I72– cluster exemplified sigma bonding at iodine similar to CuI, halogenides of early TM clusters often feature π contributions from halogen’s second p-orbital. Let’s consider WCl4 with alternate W–W bonds along a chain of edge-sharing coordination octahedra (Fig. 31) [98]. The repeat unit WCl4 is suitable for DIA: OSW = +4 and OSCl = −1. It is more interesting to see how this transcribes into the bond-order/valence approach to OS. The short segment of this chain in Fig. 31 has several bonding features. There are two terminal W–Cl bonds of 2.28 Å (228 pm), which is close enough to the single-bond length of 2.24 Å (224 pm) in WCl5 [99] and WCl6 [100] to be also considered single in agreement with the 8−N rule. There are two chlorines joining the two nonbonded W atoms with Cl–W distance of 2.50 Å (250 pm), a length that corresponds well to the increase by −0.37ln(½) = 0.26 Å (26 pm) required by eq. 1 when bond order decreases from 1 to ½ for maintenance of the 8−N rule. Finally, there is a W–W bond (2.69 Å) (269 pm), which is bridged by two Cl atoms, each at a distance of 2.36 Å (236 pm) from both W. This corresponds to a W−Cl bond order of about ¾, making BOS ≈ 1.5 at Cl due to involvement of more than one p-orbital in this three-center configuration. Note that if a Lewis formula were drawn to reflect such an involvement, the bridging Cl would obtain a positive would obtain a positive formal charge, compensated by a negative one at W. Equation 5 would then provide the correct OS with the algorithm of summing bond orders, as in Fig. 31 on the bond graph.13

Fig. 31

A segment of the infinite WCl4 chain with cluster-like alternate W–W bonds (top). Bond graph with idealized or rounded off bond orders (see text) and OS from their sums according to eq. 5 involving formal charges created on the WW cluster by the Cl bridges (bottom).

OS in TM binary clusters can be obtained by DIA on their repeat unit, for example WCl3 for the W6 Cl18 cluster [101] or MoCl2 for a Cl linked network of Mo6 Cl12 [102] of a similar cluster. Equally well, the 8−N rule (Section 7.3.1) and the electroneutrality condition (Section 7.5.1) can be used on the summary formula [MmXx]c of the cluster: OSM = [x(8 − N)+c]/m, where c is the ionic charge of the cluster (a positive number for a cation, negative for an anion) and N is the amount of valence electrons of the halogen or chalcogen atom X. If the cluster does not have a repeat unit, that is, if all metal atoms do not have equivalent bonding environments, such an OS is merely nominal.

If two TMs form a cluster, the OS might become ambiguous. An example is the icosahedral WAu12 predicted on the basis of the 18-electron rule [103] and later detected [104]. There is no rule that convincingly predicts the WAu bond order, and hence OS cannot be determined à priori. In such a case, assignment of zero OS for both atoms is a convenient solution that both conforms to the picture of 12 6s-electrons of Au(0) being donated to fulfil the 18-plet on tungsten, and follows the suggestion for metallic alloys at the very end of Section 13.

## 9.7 Haptocomplexes

TM π-complexes of aromatic rings or conjugated chains14 require some additional rules for Lewis formula construction and provide some additional challenge to determining OS from it. Some cases are trivial, such as the bis(η6-benzene)chromium where two neutral aromatic ligands are bonded to Cr, which, being less electronegative than carbon, obtains OS = 0 upon ionic approximation. The full hapticity is not an absolute condition for arene complexes, and η4 and η2 examples are well established [105–107]. Such a “dearomatization” has an impact on reactivity [108], but the ligands usually remain neutral and contribute no charge to the OS of the TM center. On the other hand, even arenes of full hapticity may acquire some anionic charge [109–112], the exact value of which is not easy to establish and suggestions vary up to the full 4− charge of the aromatic 10 π-electron C6 H5 CH34− anion ligand in an inverse-sandwich complex of uranium [113]. In such cases, the determination of OS for the central atom relies on physical-property measurements. Once the anionic charge is established, the OS is straightforward as shown in the following.

## 9.7.1 Five-membered rings

The prime example is the cyclopentadienyl anion C5 H5 in its sodium salt or in neutral complexes with TMs, such as ferrocene shown in Fig. 32. The strong aromatic stabilization15 is the OS-defining rule once three aromatic π-electron pairs are accounted for16 on this five-membered ring, producing a charge of 1− per ring and hence OSFe = +2 in Fe(C5 H5)2 upon ionic approximation. In the cyclopentadienyl anion itself, three π-electron pairs are used in bonds to the central atom, making all five carbons equivalent with an OS of −1.2, while OS of the hydrogens is +1. In this case, the aromatic stabilization controls the bonding17 and there is also an 18-plet on Fe. Neither of these two circumstances is compulsory, and examples of η1 and even η3 complexes are known [114]. The cyclopentadienyl behaves as a 1− charged ligand per ring, whether bonded to one TM center or several [115, 116] or whether it occurs with one or several hapticities as in Ti(C5 H5)4 [117] or Zr(C5 H5)4 [118]. OS is then straightforward.

Fig. 32

Left: schematic formula of Fe(C5 H5)2 and formal balance of valence electrons to determine the OS of Fe. Right: OS in Fe(C5 H5)2 via two formally single ionocovalent bonds and 18-plet by counting 6 electrons per ring contributing to the Fe environment. Given Fe–C distances [119], the total bond order of the five Fe–C bonds is likely to be higher than 1. Thus, a Lewis formula with multiple hapto-bonds would not change the OS but produce a formal charge on Fe.

## 9.7.2 Seven-membered rings

A seven-membered hydrocarbon ring must have formal charge of 1+ in order to be aromatic with 6 π-electrons, as in tropylium bromide (C7 H7)Br. Its aromatic stabilization is weaker than in benzene and a full (aromatic) η7 hapticity in its TM complexes is more easily violated. The 18-electron rule often controls the hapticity of the cycloheptatrienyl ring from seven to five to three to one, depending on the TM and its valence electron count. The central atom supplies or accepts one electron to enable the appropriate conjugation in the ring or its segment.

Three examples of such a series along 3d metals are shown in Fig. 33, all with three additional carbonyl ligands: a complex of V [120] with the aromatic tropylium cation and hapticity of 7, of Mn [121] with a hapticity of 5, and a Co complex [122] with a hapticity of 3. In the η5 complex, the π-electron donor is a pentadienyl-anion segment of the ring, a conjugated 6-electron donor, while the rest of the ring stashes two excess electrons in a CC double bond so that Mn maintains its 18-plet. In the η3 complex, the π-electron donor is an allyl-anion segment of the ring, which is a conjugated 4-electron donor, and the four excess electrons over the 18-plet at Co are in two double bonds. These three hapticity alternatives have a total of 70, 72, and 74 electrons, respectively, and isoelectronic cationic or anionic variants exist, such as the Cr(0)-based cation [123] analogous to the V complex, an Fe(2+)-based cation [124, 125] analogous to the Mn complex, or an Fe(0)-based anion [126, 127] analogous to the Co complex of Fig. 33. Analogies are found also with several 4d and 5d metals [123, 128, 129].

Fig. 33

Lewis formulas of V(CO)37-C7 H7), Mn(CO)35-C7 H7), and Co(CO)33-C7 H7) and the formal balance of valence electrons to obtain OS at the TM. This OS is compensated by an opposing ionic charge of the conjugated donor portion of the ring (see text). The central atom supplies or takes the electron, and the 18-electron rule for such an atom of N valence electrons then determines the hapticity of the ring as η = 18 − N − 6 + c, where c would be the eventual ionic charge of the entire complex and the numeral 6 expresses electrons supplied by the three carbonyl ligands.

Although a full balance of valence electrons is shown in Fig. 33 in order to evaluate how many remain at the TM and what its OS is, a quick determination of OS follows from recognition of the ionic approximation for the π-electron donor segments. In the Lewis formulas in Fig. 33, the aromatic tropylium is a 1+ cation18 and the five- or three-membered segments are conjugated carbanion(1−) species.

Obeying the 18-electron rule does not guarantee a unique OS assignement. As an example, consider the 18-plet (η5-C5 H5)Mo(η7-C7 H7) compound where the presence of a cyclopentadienyl anion and tropylium cation implies Mo with OS = 0. However, a spectroscopic and computational study [130] suggests that the cycloheptatrienyl ring acts more like an aromatic 10 π-electron (C7 H7)3− ligand, and consequently OS of Mo is +4. Analogous mixed sandwich compounds with TM from groups five and four also exist, featuring a deficient 18-plet [131, 132] and ambiguous OS. There are also examples of cycloheptatrienyl charge ambiguities due to metal-dependent electron donation to the ligand, which may in some cases render the (C7 H7)+ or (C7 H7)3− charge description inappropriate [133]. Such difficulties with OS assignment and subsequent ambiguities in classification of covalent organometallic compounds led Green [27] to devise the CBC. Notwithstanding the increased acceptance and use of this scheme, assignment of OS will continue to be necessary, especially in the study of reactions of organometallics.

## 9.8 Dihydrogen- and hydrido complexes

Compounds with molecular H2 as a ligand resemble haptocomplexes of olefins or aromatic hydrocarbons, yet are different in terms of bonding. Whereas organic hapto-ligands provide π-electrons to bond the TM atom and keep their σ-electrons for themselves, H2 has only one single σ bond. Despite this, H2 attaches to metal cations even in the gas phase, free of solvents, substrates and intervening atoms, as elaborated in a recent review by Bieske [134]. An intriguing ambiguity arises; with some metal ions, the atoms of the H2 molecule remain bonded to each other, with others, the ion is inserted into the H2 molecule to form a dihydride. As argued by Crabtree [135], the reason for this variability is that the typically involved TM atom features both empty and filled d-orbitals, of which the former participate in the three-center bonding MO that binds the H2 moiety, whereas the latter try to sabotage this by back donation into the empty antibonding MO of H2. The two extreme products of the H2 reaction with the TM ion are shown in Fig. 34.

Fig. 34

Two extremes of an adduct of H2 with a generic metal atom M and their differing OS.

The formation of H2 adducts is not limited to the gas phase. In 1984, Kubas et al. [136] synthesized the first TM complex with an H2 ligand, the yellow W(CO)32-H2){P(C6 H11)3}2, by precipitating it from a toluene solution of W(CO)3{P(C6 H11)3}2 with H2 gas. An example of hydride formation is Ir(CO)Cl(H)2{P(C6 H5)3}2 [137, 138], obtained from H2 and the square-planar Ir(CO)Cl{P(C6 H5)3}2, aka Vaska’s complex [139]. Only the latter reaction is an oxidative addition. An example that combines both types of bonded hydrogen is Ru(H)22-H2)2{P(C5 H9)3}2 [140]. The OS of hydrogen and ruthenium in the last example are evaluated on its Lewis formula in Fig. 35.

Fig. 35

OS of hydrogen and ruthenium from the Lewis formula of RuH22-H2)2{P(C5 H9)3}2 [140], obtained via moving bonds onto the electronegative partner (left) and by summing ionized bond orders (right).

The octahedral complex in Fig. 35 is stabilized by the d6 electronic configuration and 18-plet on Ru. While the two cis hydride anions are 2.13 Å (213 pm) apart, the H–H distance in the H2 ligand is 0.83 Å (83 pm), 0.09 Å (9 pm) longer than in H2 gas. The corresponding bond order is 0.78, a little more than the ideal 2/3 for a three-center bond of equal partners, and this can be attributed to the 2−N rule working for the somewhat more electronegative hydrogen.

Morris [141] reviewed H2 complexes of the iron group and showed that the H–H bond distances are variable. Distances up to about 1.60 Å (160 pm) occur, in particular for osmium as the central atom [142], and depending also on the ligand trans to H2: When that ligand is an electron-rich sp atom, such as oxygen or halogen [143], capable of strong π donation to the central atom, the H–H distance is long. When that ligand is π acidic, such as CO, or when it is electron-poor, the H–H distance is short [141]. Thus, the distance is controlled by the extent of back donation from the central atom into the σ* MO of H2. In line with the OS definition in Section 5, the back-bonding metal atom gets its electrons back because it is the main contributor to this additional metal–ligand bonding interaction, which is antibonding with respect to the H–H bond. This avoids ambiguity in OS for this class of complexes that are important [144, 145] in catalysis and reversible hydrogen binding. In OS determinations, back donations have typically been disregarded, and only the primary σ bonding is considered. Given that hydrogen has a higher Allen EN than any TM, the OS of all such η2 hydrogens then also comes out as zero if the EN approach to ionic extrapolation is applied.

## 10 OS in general for Lewis acid (LA) Lewis base (LB) adducts

We have already considered OS in typical LA–LB adducts CuCl42− or H3 B–NH3 and followed the usual rules to obtain unambiguous OS. However, is this always the case or can complications arise?

Consider the formation of HBr in Fig. 36 under the guise of the general LA–LB concept and from two LA–LB reactions; one real and one imaginary. Everyone agrees on OS in the HBr product. We then see that one of these LA–LB additions proceeds with a change in OS. That addition (Fig. 36, right) has the LA (acceptor) more electronegative than the LB (donor). Should that electronegative LA acceptor qualify for the caveat in Section 5.1.2 and keep its OS under the addition? Clearly not.

Fig. 36

Formation of hydrogen bromide in two LA–LB reactions. The LA–LB addition leads to change in OS only if LA is more electronegative than LB.

It is not the deliberately chosen LA–LB addition but the actual orbital mixing coefficient in the bonding MO of the adduct, or the energies of that MO’s contributing atomic orbitals, or the minimum-energy bond-rupture path of Haaland [30], which suggest the appropriate ionic approximation. What matters in the latter criterion is not where the bond electrons come from but where they go. Thermal splitting of a bond identifies the relative energies of the two immediate atomic orbitals participating in the MO formation in Fig. 1: If the split is homolytic, the average valence-electron energy by Allen EN (Fig. 2) is sufficient to decide the two atomic-orbital energy levels that control the ionic approximation. This is the case of HBr with OSH = +1 and OSBr = −1. If the split is heterolytic, the atom that keeps the bond electrons, keeps them also for OS counting, as this implies a higher mixing coefficient in the original bonding MO.

Only some bonds will split heterolytically as the Haaland criterion requires, and, in almost all such cases, the LB will keep its donated electron pair by virtue of its higher EN and pose no problems for OS determination by EN alone. However, there is a group of unusual LA (Z) ligands, and those of them that have the acceptor atom with EN higher than the LB donor require further consideration.

## 10.1 Example of the electronegative-acceptor caveat

Fig. 37

Sulfur dioxide as LA ligand in Rh(CO)Cl{P(C6 H5)3}2–SO2 [146].

As noted in Section 5.1.2, using EN as a sole gauge for ionic approximation may lead to incorrect OS when the acceptor of bonding electrons is more electronegative than the donor. An example is the addition product in Fig. 37 of Rh(CO)Cl{P(C6 H5)3}2 with SO2 [146]. Its geometry suggests the electronegative S as LA (acceptor), hence the electropositive Rh is the LB (donor). The generic definition (Fig. 1) then assigns the donated electrons to the electropositive Rh on the grounds that the Rh–S bonding MO is mainly formed by the orbital wavefunctions of the Rh donor. The Rh–S bond order is 0.49 when calculated from the Rh–S distance 2.45 Å (245 pm) with eq. 1 and $RRhS0 = 2.19 Å$ (219 pm) obtained from data in [44]. Indeed, the adduct breaks up easily into its uncharged components, heterolytically, complying with the Haaland criterion [30]. For the EN-based OS to agree with this generic OS, the electronegative-acceptor caveat (Section 5.1.2) needs to be applied so that SO2 does not end up with the electron pair donated by Rh. Sulfur dioxide is a classical example of an LA ligand, see Appendix E for more details.

The electronegative LA has one intriguing consequence: the TM donor maintains its OS yet uses one of its originally nonbonding pairs for bonding. Does this OS still imply the correct electron configuration dn even after the nonbonding pair became bonding? A fruitful debate on this topic has been initiated by Hill [147] and Parkin [148].

## 10.2 OS and the electron configuration of TM

In TM chemistry, OS is used to calculate the electron configuration dn at an atom of N valence electrons

$n = N−OS (6)$(6)

Conversely, if n is determined via magnetic or spectral properties, a “spectroscopic” OS can be calculated from it [6]. Clearly, eq. 6 is a desirable feature of the OS, and it applies similarly to sp-block elements. Let’s therefore consider LA–LB adducts with electronegative LA and see whether n in dn should comprise valence electrons in all weakly bonding/nonbonding/antibonding orbitals (complying with eq. 6) or just those in orbitals that aren’t bonding (not complying with eq. 6). Let’s simultaneously look at how the ionic approximation in Fig. 1, based on the LA and LB orbital mixing coefficient in the bonding MO, resolves the OS and how this is handled with the ionic approximation in Fig. 2 via Allen EN of the LA and LB atoms.

A telling example is the Au–B bond in the [Au{B(PC6 H4)2(C6 H5)}Cl] complex synthesized by Bourissou’s group [149], featuring B as the LA ligand of higher Allen EN. In Fig. 38, the OS and n are each evaluated via the two approaches specified in the previous paragraph. We see that eq. 6 is satisfied for both LB and LA atoms (Au and B) in only one of four combinations of the two definitions, namely, with the generic ionic approximation (Fig. 1; Au populates the Au–B bonding MO) and including in dn all antibonding, nonbonding, and weakly bonding orbitals of Au (its bond to B). Bourissou’s [149] Mössbauer spectrum of the 197Au nucleus sees exactly these n electrons (d10) and OSAu = +1 is suggested in compliance with eq. 6, even though the square-planar coordination implies Au+3 with d8 since the donated Au pair became the Au–B bond itself (Fig. 38). This MO sinks in energy as illustrated by other examples [150, 151] and loses the stereochemical ligand-field effect of the slightly antibonding MO or the “lone” pair. The square-planar geometry is then controlled by the ligand-field behavior of the remaining four d pairs of Au (8 electrons). The electric-field gradient seen by the 197Au Mössbauer nucleus then originates from these 8 electrons plus the donated 2 in the weakly bonding MO close to Au. To avoid ambiguity of the d10 spectroscopic and the d8 ligand-field configuration, the electron configuration might best be described after Parkin [148] as n = 10 in dn−2; a notation in which n fulfils eq. 6.

Fig. 38

Adduct of a lower-EN TM as a LB: Au–B [149], with OS evaluated via generic (Fig. 1) and purely EN-based (Fig. 2) ionic approximations and dn via two alternative definitions of the electron configuration. States in bold comply with eq. 6.

The same combination of electron-configuration definition and ionic approximation as above fulfils eq. 6 for other adducts of metal LB with more electronegative LA, regardless of the LA–LB bond order. Two examples of Fe complexes illustrate this in Figs. 39 and 40 with MO energies and occupations as reported in [152].

Fig. 39

Adduct of a lower-EN TM as a LB: Fe–B [152], with OS via generic (Fig. 1) and EN-based (Fig. 2) ionic approximations and the electron configuration dn via two alternative definitions. States in bold comply with eq. 6. The LA–LB bond order is 0.25 when calculated from the Fe–B bond distance of 2.46 Å with eq. 1 and $RFeB0 = 1.95 Å$ obtained from [44]. (1 Å=100 pm).

Fig. 40

Adduct of a lower-EN TM as a LB: Fe–B [152], with OS via generic (Fig. 1) and EN-based (Fig. 2) ionic approximations and the electron configuration dn via two definitions. Bold states comply with eq. 6. The LA–LB bond order is 0.18 when calculated from the Fe–B bond distance 2.59 Å [152] with eq. 1 and $RFeB0 = 1.95 Å$ obtained from [44]. (1Å=100 pm).

On going from Figs. 39 to 40, we see the LA–LB bond weakening from a bond valence of 0.25 to 0.18, which corresponds to the increasing energy of the Fe–B bond orbital as drawn in [152]. That particular bond order is low yet illustrative of the span of bond orders occurring with LB TMs, which extends from these low values up to about unity.

As seen in the previous examples, the first-row TM atoms are weak LBs even if their donor strength can be somewhat increased by supplying electrons. An example of the latter is a series of three copper boratranes with Cu reduced from cationic(1+) to neutral to anionic(1−) [153]. The strength of a TM LB interaction increases more profoundly down a group [154]. An example [150] of a practically single-bond order is in Fig. 41. The same combination of definitions works in Fig. 42 for the SO2 adduct discussed in Section 10.1.

Fig. 41

Adduct of a lower-EN TM as a LB: Ir–B [150], with OS via generic (Fig. 1) and EN-based (Fig. 2) ionic approximations and electron configuration dn via two definitions. Bold states comply with eq. 6. The LA–LB bond order is 0.99 when calculated from the Ir–B distance of 2.165 Å [150] with eq. 1 from $RIrB0 = 2.16 Å$ obtained from data in [44]. (1Å=100 pm).

Fig. 42

Adduct of a lower-EN TM as a LB: Rh–SO2 [146], with OS via generic (Fig. 1) and EN-based (Fig. 2) ionic approximations and the electron configuration dn via two definitions. Bold states comply with eq. 6. The LA–LB bond order is 0.49 (Section 10.1). In all cases there is an s2 lone pair on S because the Rh–S bond does not utilize sulfur’s electrons.

Let’s now consider (C5 H5)(CO)2 Rh−Pt(CO)(C6 F5)2 [155]. Since the Rh(CO)2(C5 H5) component is a neutral 18-plet, it must be a Rh-based LB, and this is an example of a donor–acceptor TM–TM bond. Once this is clear, we see in Fig. 43 that the same combination of definitions works also for the more electronegative LA atom being another TM. While the electropositive Rh LB atom has a deformed tetrahedral bonding environment which conforms better to the dn−2 = d6 than the spectroscopic d8 configuration, eq. 6 is fulfiled with n = 8. The Pt coordination is square planar as expected from its own d8 configuration.

Fig. 43

Adduct of a lower-EN TM as a LB: Rh–Pt [155], with OS evaluated via generic (Fig. 1) and EN-based (Fig. 2) ionic approximations and the electron configuration dn via two definitions. Bold states comply with eq. 6. There is always d8 on Pt because the Rh–Pt bond is formally not being filled with Pt’s electrons.

When the TM LB is more electronegative than the LA acceptor, the ionic approximation via Allen EN (Fig. 2) gives the same result as the generic one (Fig. 1). This is illustrated in Fig. 44 for [(C5 H5)(CO)2 Fe–Al(C6 H5)3]. As long as LB remains the more electronegative partner, the same is valid for an LA–LB adduct of two TMs, as shown for [(C5 H5)(CO)2 Ir–W(CO)5] in Fig. 45.

Fig. 44

Adduct of an electronegative TM as a LB: Fe–Al [156], with OS evaluated via generic (Fig. 1) and EN-based (Fig. 2) ionic approximations and the electron configuration dn via two definitions. Bold states comply with eq. 6. LA–LB bond order 0.55 is calculated from the Fe–Al distance of 2.51 Å [156] with eq. 1 and $RFeAl0 = 2.29 Å$ obtained from data in [44]. (1Å=100 pm).

Fig. 45

Adduct of an electronegative TM as a LB: Ir–W [157], with OS evaluated via generic (Fig. 1) and EN-based (Fig. 2) ionic approximations and the electron configuration dn via two definitions. Bold states comply with eq. 6. LA–LB bond order 0.45 is calculated from the Ir–W distance of 3.05 Å [157] with eq. 1 and $RIrW0 = 2.75 Å$ obtained from data in [44]. (1Å=100 pm).

The last combination examined here is the adduct of an electropositive TM LA and an electronegative main-group LB. One such example, CuCl42−, was treated in Fig. 9 and the outcome, as indicated at the start of this section, is too trivial to deal with in a separate figure: all four definition alternatives considered yield a d9 Cu2+ and octets on chloride ions.

All examples from Figs. 38 to 45 have one single combination of the ionic-approximation and electron-configuration definitions that fulfils eq. 6: It is the generic ionic approximation and the electron-configuration that includes all weakly bonding, nonbonding, or antibonding orbitals filled with an atom’s own electrons. The suggested notation for this TM electron-configuration follows Parkin [148] by stating that n = … in dn−2. In that notation, the “spectroscopic” configuration n fulfils eq. 6 in compliance with OS whereas the “ligand-field” configuration n−2 is the number of d electrons interacting with the ligand field of the coordination geometry adopted.

## 11 Ambiguous OS

Ambiguous OS with several choices or intermediate values has already been encountered in some examples in Sections 9.1.6 (the SS bond order in thiosulfate), 9.4 and 9.7 (the anionic charge of borane, arene, or C7 H7 clusters) and in 9.8 (the anionic charge of H2-related ligands). In this Section, examples are selected of sole ambiguous ionicity, sole ambiguous bond order and combinations of these. Either a choice has to be made for OS, depending on the intended purpose, or the ambiguity needs to be resolved with physical measurements that go beyond simple Lewis formulas.

## 11.1 Ambiguous ionic approximation; the case of H3 PO3

The ENs of hydrogen and phosphorus are close (Appendix B), and ionic approximation of the P–H bond to obtain OS may be considered ambiguous. Two alternative assignments of OS in H3 PO3 are used in textbooks and are shown in Fig. 46.

Fig. 46

Alternative OS in H3 PO3.

Formation of H3 PO3 by hydrolysis of PCl3 suggests that OS of phosphorus remains +3 as the reaction ends with protonation of the lone pair on P. The electrochemical OS of H3 PO3 also needs to be +3 so that the species defined by the identity of its central atom is distinguished from H3 PO4. The same is valid for the systematic description of phosphorus chemistry where oxidation of H3 PO3 to H3 PO4 is conveniently assigned to the increase of the OS of phosphorus. However, a Mulliken population analysis, carried out by us, shows phosphorus to be more positive by about 2 elementary-charge units than the phosphanoic hydrogen. This admittedly less common assignment, P(+5) and H(−1), also follows from Allen EN.

## 11.2 Ambiguous bond order; the case of pentaazole

As we have seen in Section 9.1.5, nitrogen can be a rich source of bonding conflicts, compromises, and exceptions that extend to OS. One more example of hard-to-resolve bond orders is 1H-pentaazole, HN5. With the help of Fig. 21, the almost equal NN bond distances along the ring shown in Fig. 47 suggest that the NN bond orders have a span of about 0.1. This is due to aromatic contribution in this ring. As this contribution is not 100 %, it is impossible to determine OS precisely in this molecule except for the +1 for the hydrogen. Two limiting Lewis formulas are drawn in Fig. 48, one with localized bonds and one with full aromatic conjugation, leading to two different OS sets. The OS set on the localized formula agrees with a rule that an atom bonded solely to atoms of the same element has OS = 0 if the 8−N rule is obeyed. The OS set on the aromatic formula would be the one to use for electrochemistry.

Fig. 47

Bond lengths (in Å) in pentaazole; experimental [158] (left; phenyl-substituted), calculated [159] (right).

Fig. 48

OS in two limiting Lewis formulas of pentaazole, localized and aromatic.

## 11.3 Non-innocent ligands

Jørgensen [160, 161] coined the term “non-innocent” or “suspect” for ligands that exhibit somewhat hidden or unexpected redox properties, thereby rendering assignment of OS non-obvious. The redox behavior may seem to involve changes in the metal’s OS while actually it is the ligand that undergoes electron transfer. Assignment of OS requires detailed spectroscopic, magnetic, and/or structural studies. Given that the redox behavior of non-innocent ligands is based on bond-order changes in the ligand framework, the simplest examples are the nitrosyl and dioxygen ligands, but there are many more [162–165].

## 11.3.1 Nitroprusside

While the cyanide ligands in [Fe(CN)5 NO]2− obey the 8−N rule and pose no problem for OS determination, NO allows several chemically feasible alternatives in which the OS of nitrogen differs: NO+, NO, and NO. This is due to variable bond order within the ligand, either as an |N≡O|+ cation with OSN = +3, or an |N=O| anion with OSN = +1 (both via DIA), or as the octet-violating yet neutral nitrogen-monoxide in between, of OSN = +2.

As the generic definition of OS requires knowledge of this very NO bond order, the bond-valence approach is attempted here: Single-crystal neutron diffraction of BaFe(CN)5(NO)·3H2 O [165] yields NO bond lengths between 1.12 and 1.15 Å (112 and 115 pm), which, being shorter than 1.15 Å (115 pm) in NO gas, suggest that the actual bond order leans somewhat towards the |N≡O|+ formula with a triple bond. Since the 8−N rule for oxygen will tend to limit the real bond order to two for that cation, these observed bond lengths may well correspond to true NO+. With NO+ and five CN as ligands, the electroneutrality condition in eq. 3 suggests Fe2+ as the central atom.

Considering the ambiguous NO identity, a reliable approach to OS in nitroprusside is via the magnetic properties of the central atom. Nitroprusside is diamagnetic, hence the electron configuration at Fe must be a low-spin d6 and OSFe = +2.19

## 11.3.2 Nitrosyl complexes in general

Not all NO complexes are like nitroprusside. As apparent from the Lewis formula, the MNO bond angle will be linear with |N≡O|+ but bent for |N=O| [166, 167]. The snag is that the MNO bond angles in various NO complexes vary in between those two extremes, suggesting fractional and variable NO bond orders that can complicate OS assignments [168–170]. In fact, two different sets of such angles can occur within a single molecule; Fe(NO)4 has two tetrahedral angles and two ~160° [171]. As a result, some authors find the assignment of OS difficult or even undesirable, such as Enemark and Feltham [172] in their {MNO}n classification of nitrosyl complexes, where the superscript n is the number of valence electrons on the metal when the ligand is formally NO+ (n then includes the unpaired electron in case of NO or the donated electron pair in case of NO) and where no assumption is made about electron distribution between metal and NO ligand, hence OS. An example is {Fe(NO)4}13 for Fe(NO)4. Recent sophisticated analyses [173, 174] aim past the Enemark and Feltham classification, although the ambiguous nature of metal and nitrogen OS remains.

## 11.3.3 Hemoglobin

Another example is one of the biologically most important molecular segments, the diamagnetic adduct of the otherwise paramagnetic hemoglobin’s heme and O2. The problem is that O2 in its ground state is also paramagnetic (a triplet state). Several alternatives with compensating spins can be considered with the mental picture of the MO diagram for O2 and of a strong ligand field due to the four planar nitrogens: (1) the singlet–singlet form of Fe2+(S=0)–O2(S=0) [175] justifiable by a low-spin iron and the excited state of singlet oxygen, (2) the idea of compensating triplets of Fe2+(S=1)–O2(S=1) [176] in which one might see an intermediate-spin iron in a tetragonal ligand field, (3) the idea of Fe oxidation to Fe3+(S=½)–O2(S=½) [177] where the remaining spin of the superoxide anion O2 is compensated with an opposite spin of a low-spin Fe. Let’s say we omit the fourth variant with Fe4+ and the peroxide anion O22− on the basis of its paramagnetism. In this situation, bond lengths might be of guidance. The OO distance in the oxygenated hemoglobin is difficult to ascertain due to disorder, but a value of 1.26(8) Å is available from X-ray diffraction [178], which seems to indicate an increase from 1.22 Å (122 pm) in O2 gas. Calculation in a model system [179] yields 1.263 Å (126.3 pm) in agreement with an earlier estimate of 1.27 Å (127 pm) [180]. The difference of ~0.04 Å (4 pm) from O2 gas is less than −0.37ln(3/4) = 0.106 Å (10.6 pm) estimated with eq. 1 for the OO bond order decreasing from 2 in dioxygen to 1½ in the superoxide anion formed when heme’s iron in the adduct is oxidized to OSFe = +3. Accordingly, the OO bond order is somewhere in between 2 and 1½, with iron between +2 and +3; like a “mixed-valence” compound according to the Day and Robin [181] designation. The OS definition in Section 5 then suggests OSO = 0 and OSFe = +2. The concealed redox variation of O2 in heme mimics a non-innocent ligand.

## 11.3.4 Dithiolenes

Consider the Ni dithiolate complex, Ni(S2 C2 H2)22−. Its formation via salt-metathesis, NiCl2 + 2K2 S2 C2 H2 = K2 Ni(S2 C2 H2)2 + 2KCl, maintains Ni2+. However, the anion easily oxidizes in an oxygen atmosphere. That reaction, Ni(S2 C2 H2)22− + ½O2 + 2H+ = Ni(S2 C2 H2)2 + H2 O, then looks as though Ni2+ were oxidized to Ni4+ (Fig. 49).

Fig. 49

Ni(S2 C2 H2)22− − 2e = Ni(S2 C2 H2)2; is Ni being oxidized?

The snag is that Ni4+ is an extremely strong oxidant, very unlikely to be present in this molecule. If there is Ni2+ instead, which atom in the ligand is oxidized? Figure 50 provides two possible answers to that question.

Fig. 50

Two alternatives for the oxidation of the dithiolate ligand in Ni(S2 C2 H2)22− that remove two of the “anionic” 1e bonds, while the remaining 1e per ligand either forms SS bonds such that S is oxidized in the end (top), or conjugates with the double CC bond such that C is oxidized in the end (bottom).

The SS bonding is supported by the shortening of both inter- and intraligand SS distances upon oxidation to the neutral complex [182]. The 8−N rule is obeyed on S as the most electronegative atom in this complex. Both C and S have octets.

The SCCS conjugation is supported by the shortening of the CS bonds, expansion of CC bond and a corresponding change in vibration frequencies upon oxidation to the neutral complex [183]. The 8−N rule is obeyed on S as the most electronegative atom, and both C and S have octets.

The OS +2 of nickel in both Ni(S2 C2 H2)22− and Ni(S2 C2 H2)2 is proved by the diamagnetism of the complexes [184], corresponding to Ni(II) with d8 configuration in a square-planar field. The determination of OS of S and C requires knowledge of precise bond distances in the oxidized and reduced forms, unaffected by bulky ligand substituents and cation charges in the crystal packing. A possible outcome of such an analysis is that the conjugation also affects the S atoms, yielding fractional OS on both C and S. Interestingly, an intermediate Ni(S2 C2 H2)2 anion can be obtained with a suitable electron donor, such as tetrathiafulvalene, 2,2′-bis(1,3-dithiolylidene) [185].

## 12.1 OS tautomerism in a molecule

This is often called valence tautomerism in the literature, but actually concerns a thermally induced OS ambiguity involving a redox-active ligand with a redox-prone central atom of dichotomous OS of close stability. The internal redox reaction, causing the OS switch, is typically triggered by a spin crossover occurring as a function of temperature.

The case of Mn catecholates is used here as a rare example of a tautomeric complex that has the simplifying feature of being homoleptic. The low-temperature X-ray determined structure of Mn(C6 H4 O2)3 is interpreted as containing two semiquinonate and one catecholate ligands around a central Mn atom with OSMn = +4. Magnetic measurements support this conclusion [186]. However, the magnetic moment at high temperatures indicates a high-spin Mn3+ with three semiquinonate ligands,20 and the change in OS from +4 to +3 occurs by oxidation of the catecholate ligand into semiquinonate via an internal electron-transfer process. This transition appears to be continuous, not abrupt.

A Lewis formula with two-electron and one-electron bonds (obeying octets and fulfilling the 8−N rule on oxygens) is drawn in Fig. 51 to depict the two limiting OS alternatives. OS can be determined on each of them either by assigning bonds to the electronegative partner or via summing ionized bond orders, with the same result. For other examples and further discussion on this topic, see [187, 188].

Fig. 51

Temperature induced OS tautomerism in Mn catecholates evaluated via summing ionized bond orders on Lewis formula of Mn(C6 H4 O2)3.

## 12.2 OS tautomerism in an extended solid

OS transitions of the tautomeric type occur also in extended solids, and are known as “charge ordering” (upon cooling) and “valence mixing” (upon heating). An example in Fig. 52 is the perovskite-related YBaFe2 O5 [189]. At low temperatures, smaller and larger coordination square pyramids around Fe alternate as shown in Fig. 52 (left). Their iBVS values, calculated from bond distances in ref. [189] and parameters in ref. [43] with eqs. 1 and 4, show that the OS of the two differing iron atoms approaches +2 and +3 upon the bonding compromise of having to distort the Y Ba mesh. Upon heating, one single type of square pyramid is formed, and not even Mössbauer spectroscopy detects two different Fe in this high-temperature phase [190].

Fig. 52

Thermally induced mixing of OS in YBaFe2 O5.

## 13 What if the structure becomes metallic?

As bonding and antibonding orbitals/bands overlap in the metallic state of a compound, we are, strictly speaking, no longer entitled to make the ionic extrapolations suggested in Fig. 1. Nevertheless, there are many compounds with luster and electrical conductivity that maintain a simple stoichiometric formula. Consider a short series of TM oxides: the golden TiO, the black RuO2, and the silvery ReO3. One applies DIA to obtain the OS of +2, +4, and +6, respectively, which in fact confines the delocalized electrons to the TM. Some sp elements may also form stoichiometric metallic compounds. Ba3 Si4 obeys the Zintl concept [58] in that it forms butterfly-shaped Si46– anions, in which two Si have two bonds and two Si have three bonds to each other according to the generalized 8−N rule (Section 7.4.1), yet the compound is weakly metallic [191].

Ultimately, however, the assignment of the free electrons to one of the two bonded atoms has its limits. A sign of the problem is an unexpected electron configuration or an unexpected bonding pattern.

The example of an unexpected electron configuration is shown in Fig. 53, where two inverse perovskites Rb3 AuO [192] and Ca3 AuN [193] are compared. Unlike Cs3 AuO in Fig. 7, both are metallic. While weakly metallic Rb3 AuO extrapolates to the same OS as Cs3 AuO, the Ca3 AuN metal does not. Neglecting the pronounced metallic character in Ca3 AuN suggests Au3− anions for which there is no support in theory. This phase may better be understood as a metal with an electride character of Au + 2e.

Fig. 53

OS in the slightly metallic Rb3 AuO (top) and a full metal Ca3 AuN (bottom) estimated on a bond graph of the perovskite-type structure. The Ca3 AuN estimation is based only on the 8+N rule on Ca and 8−N rule on O, violating the 12−N rule on Au.

The example of an unexpected bonding pattern concerns two platinides: red transparent Cs2 Pt [194] and black BaPt [195]. In line with the 12−N rule, Cs2 Pt contains isolated Pt2− anions, a relativistic sulfide. Yet BaPt does not have such anions; it has chains of Pt as though there were a deficit of electrons at Pt. This means that some electrons have abandoned Pt2− to make BaPt metallic, and this deficit is compensated by forming Pt–Pt bonds. If these Pt–Pt bonds were single bonds, the chains would be a neutral relativistic sulfur, and BaPt would be built of Ba2+, 2e and $∞1[Pt].$ As BaPt is still formally stoichiometric, the 8+N rule gives the OS of Ba while assigning Pt an OS of −2 under the implied understanding that the metallic electrons are counted with the anion and the OS does not comply with the actual bonding adopted by Pt.

Unsatisfactory values of OS are obtained when DIA is applied to ordered alloys the composition and structure of which are dictated by size factors, such as LiPb, Cu3 Au, and many others. In such alloys, the bonding rules for the most electronegative element (such as the 8−N or 12−N rules) are not fulfilled and the covalent bonding pattern predicted by the generalized 8−N rule is lacking.

OS in metallic compounds should be evaluated with great caution. If OS is needed for alloys in order to balance redox equations, it is best considered zero for all components of the alloy.

## 14 Electrochemical OS

Electrochemical OS is an OS value that has been nominally adjusted to suit the electrochemical purpose of representing a specific charged or thermodynamically relevant entity. The adjustment may concern adopting an average OS for atoms of the same element in such an ion or molecule, or altering the OS in order to include that ion or molecule in a specific electrochemical series. A practical example of the first type is the average OS used for thiosulfate and dithionate (+2 and +5, respectively) in the Latimer diagram for sulfur in acid solution [22b], shown in Fig. 54. This must be done because the experimental electrochemical data refer to such an ion or molecule as a whole.

Fig. 54

Latimer diagram for sulfur in acidic solution.

A practical example of the second type is keeping OSP = +3 in H3 PO3 instead of the EN-compliant +5, as discussed in Section 11.1. Such a convenience stretched even further is the Latimer diagram for boron that would includes tetrahydridoborate (Fig. 55). In order to set up such a diagram, one keeps expediently OSH = +1 (despite EN telling otherwise) and has OSB = −5 in BH4.

Fig. 55

Latimer diagram for boron in acidic solution that would uses an electrochemically expedient yet unusual oxidation state for B in order to include tetrahydridoborate(1−) anion.

In view of the widespread use and importance of Latimer and Frost diagrams for predicting redox-reaction outcomes and evaluation of redox systems, the use of electrochemical OS will continue. It illustrates that OS is a flexible and purpose oriented concept.

## 15 Cautionary note on OS and its uses

• OS is not the charge on an atom. OS is not the charge on an atom/ion represented by electron density. Being a formal ionic extrapolation, the absolute value of OS always exceeds that of the “observable” charge on the atom (except with gas-phase ions). As noted in ref. [17], it is in fact impossible to objectively state the true charge of an atom embedded within a particular chemical environment.

• OS and the bond-valence sum BVS. BVS at a TM atom is often evaluated for newly reported crystal structures and compared with the OS deduced from other methods. On Lewis formulas, however, eq. 5 shows that OS is not always or not necessarily equal to BVS or BOS, and this is demonstrated by several examples in this Report (Figs. 11, 15, 16, 20, 22, 31, 35).

• OS and redox reactions. Redox reactions can and should be defined as suggested by Vitz [14] as those that can be decomposed into two half reactions in which OS changes. Given that ∆G° = –nFE°, the standard reduction potential E° is available whenever a changing OS is found or defined in a half equation. However, the definition of oxidation and reduction per se via OS is impractical if not impossible. As shown in the example of thiosulfate in Section 9.1.6, OS does not necessarily define oxidation and reduction: Consider the formula with OS = 0 at the terminal S. Protonation of this salt’s anion leads to sulfite and elementary S, with no change in OS, yet the reaction is perfectly describable in terms of E° as a disproportionation of the electrochemical (average) OS = +2 of sulfur in thiosulfate into +4 and 0. As a rule, if an oxidation takes place with an added oxidant, an OS should increase in the species that is being oxidized; if there is a reduction (with a reducing agent), it should decrease an OS in the species that is being reduced.

• Ambiguity in OS. Although there is one sole generic OS for each element in every compound, its true value may not be easily ascertained (Section 11), in particular when the compound has several resonance forms or when the EN values of the bonded atoms are very close. Practicality may dictate which OS is most convenient for the intended use, but application of this practicality should be clearly specified.

## 16 Summary: the OS definition and algorithms

Although the concept of OS is a formalism (ionic approximation), it is a very useful electron-counting method that assigns whole electrons to atoms in molecules or extended solids. That count matters not only in redox reactions and electrochemistry, but also in spectroscopy and physical properties such as those related to the electron spin of an atom.

## 16.1 Generic OS

There is a generic definition: OS of a bonded atom equals its charge after ionic approximation. (Implicit in this definition, OS of a monoatomic ion equals that ion’s nominal charge.)

## 16.1.1 Ionic approximation

The ionic approximation considers the contribution that the atomic orbitals make to the MO of the bond:

• The atom that contributes more to the bonding MO becomes the more negative ion/atom (its participating atomic orbital has lower energy and a higher mixing coefficient). It keeps all that MO’s electrons.

• Bonds between atoms of the same element are divided equally between the two partners.

• In general, the atom that contributes more to a given MO or MO* keeps all that MO’s electrons.

OS of a bonded atom then equals that atom’s charge after the atom that contributes more to a given MO takes that MO’s electrons.

## 16.1.2 Ionic approximation to OS via EN

Comparison of two EN values is the most convenient and easiest way to arrive at the ionic approximation of a bond. In this process, we suggest using the Allen EN scale because it follows the relative energy of the free-atom valence electrons, and hence in the majority of cases such ionic approximation will provide an OS identical with the generic one. With one caveat: If in a particular bond the more electronegative atom is a net acceptor of bonding electrons (manifested by a heterolytic thermal split in which that atom does not keep the bond’s electrons), this atom will not be assigned the bond electrons (algorithm 1 below) or not receive that bond’s order with a negative sign (algorithm 2 below).

In homoleptic compounds of two chemical elements, the use of EN to evaluate OS does not require knowledge of the bond order, and the ionic approximation can be performed on the summary formula: A stable electron configuration is assigned to the individual atoms according to their decreasing Allen EN until all available valence electrons are used up. The charges of the atoms are then evaluated to give the OS.

## 16.1.3 OS algorithms

The OS of every atom can be evaluated when orders are known for all the bonds that atom forms. There are two variants of the bond-assignment algorithm in a Lewis formula:

• The OS algorithm of moving bonds: Bond electrons are moved onto the atom that is to become negative and the charge is then evaluated at each atom giving its OS.

• The OS algorithm of summing bond orders (suitable also for bond graphs of solids): The iBOS (or a rounded off iBVS in a bond graph) is evaluated at each atom and summed with the atom’s formal charge (if any) to give the atom’s OS.

## 16.2 Nominal OS

A nominal OS is one that has been nominally adjusted against the generic OS definition in order to suit a specific purpose. That purpose may be the electrochemical identity of the ion, maintained via the nominal electrochemical OS (Section 14). Or it may be the chemical systematics brought about by the inversed ionic approximation of the P–H bond in H3 PO3 in order to obtain OS = +3 on phosphorus (Section 11.1). Or polarizing a homonuclear bond in thiosulfate to OSS = +6 and −2 in order to emphasize the uniqueness of the central atom and similarity of all four ligands (Section 9.1.6).

## 16.3 OS definition versus postulatory rules for OS

The ionic approximation described in Section 5.1.1 and advocated by this Report is at the root of the IUPAC Gold Book rules/algorithms for OS/ON. Even simpler rules are used in teaching chemistry, where OS enters at various levels. When coupled with nomenclature, the approach to OS is even language specific. OS is often introduced with redox reactions before chemical bonds are discussed in detail. There are many approaches to these simple initial rules but most start by stating that H is usually +1 and O −2 with a few exceptions. Later, a set of hierarchical algorithmic steps is introduced in some curricula. This Report does not seek to abolish these simple rules, but it attempts to provide a comprehensive definition of OS that puts the rules on a firmer basis. Of course, the bonding approach to OS requires at least a rudimentary understanding of bonding and ability to draw Lewis formulas of molecules and bond graphs of solids.

## 17 Acronyms and variables used throughout the text

The following list contains acronyms (not in italics) and variables (in italics), except those of single instance which are explained immediately in the text.

BO

bond order

BOS and BOS

bond order sum at an atom (Section 7.5.2)

iBOS and iBOS

“ionized bond order sum” (Section 7.5.2)

BV

bond valence (Section 5.2.2)

BVS and BVS

bond valence sum at an atom (Section 7.5.2)

iBVS and iBVS

“ionized bond valence sum” (Section 7.5.2)

CBC

covalent bond classification (Section 5)

DIA

direct ionic approximation (Section 6.2)

EN

electronegativity

FC

formal charge

LA

Lewis acid

LB

Lewis base

MO

molecular orbital

N

number of valence electrons of a neutral atom

NBO

natural bond orbitals (a method, Appendix C)

LCAO

linear combination of atomic orbitals

ON and ON

oxidation number

OS and OS

oxidation state

$Rij0$

bond-valence parameter (single-bond length in Å for atoms i and j; Section 5.2.2)

TM

transition metal.

## Membership of the sponsoring body

Membership of the IUPAC Inorganic Chemistry Division for the period 2012−2013 is as follows:

President: R.D. Loss (Australia); Vice President: J. Reedijk (Netherlands); Secretary: M. Leskelä (Finland); Titular Members: M. Drábik (Slovakia), N.E. Holden (USA), P. Karen (Norway), S. Mathur (Germany), L.R. Öhrström (Sweden), K. Sakai (Japan), E.Y. Tsuva; Associate Members: J. Buchweishaija (Tanzania), T. Ding (China), J. García-Martinez (Spain), A. Kiliç (Turkey), D. Rabinovich (USA), R.-N. Vannier (France); National Representatives: Y.F. Abdul Aziz (Malaysia), V. Chandrasekhar (India), B. Prugovečki (Croatia), A. Saqib (Pakistan), H.E. Toma (Brasil), N. Trendafilova (Bulgaria), S. Youngme (Thailand).

## Appendix A: Ionic-approximation criteria not suited for OS

The reader of this Appendix should be familiar with the introductory simple examples in Section 8. Here we present three ionic-approximation criteria that in some cases yield an unusual OS, different from the OS obtained via the generic definition: the dipole moment, EN scales other than Allen’s and Brønsted–Lowry acidity.

## A-1 Dipole moment and exceeding the 8−N rule

Exceeding the 8−N rule for the number of bonds formed by the most electronegative element is rare, but occurs in the CO and NO molecules, to name those that are stable. It has interesting consequences, even if not for the OS itself. As shown on the carbon-monoxide example in Fig. 56, OS remains defined solely by EN and does not even require the bond order when DIA is used to distribute the five valence-electron pairs.

Fig. 56

OS of carbon monoxide via DIA (left) and via ionic assignment of bonds (right).

However, violating the 8−N rule at oxygen in CO means that the oxygen’s bonding-electron fraction that exceeds bond order of 2 manifests itself as being partly in the vicinity of carbon. The consequence is that the CO molecule has an experimental dipole moment [196, 197] oriented with its negative end at the carbon atom. The same situation has only recently been confirmed for NO [198], as the polarity is not easy to ascertain experimentally. If, instead of ionic approximation based on EN, the dipole polarity itself were extrapolated to determine the OS in these molecules, CO would be an oxygen carbide, O(+4)C(−4) and NO an oxygen nitride O(+3)N(−3).

Sophisticated quantum-chemical calculations are able to reproduce the experimental CO dipole moment [199, 200] and provide an explanation for this unusual situation [201–203]. The dipole moment is a vector quantity, and the shape of the electron distribution may be more important in the determination of its sign than the scalar atomic charges. Thus, according to ref. [203], it is the highly asymmetrical partitioning of the electron cloud around carbon, and in particular the placement of the lone pair, that results in carbon being the negative end of the dipole moment while, at the same time, carrying a positive partial charge compliant with EN.21

Calculations show a similar situation for BF [201] and NF [202], which, however, are unstable [204, 205]. Indeed, Huzinaga et al. [201] cautioned against using EN to assign the direction of bond dipole moment in heteronuclear diatomics with lone pairs and a potential for multiple bonding.

## A-2 Different OS from different EN scales

OS values determined with different EN scales may differ. While the Allen scale yields an OS value of +4 for lead in PbH4, the Pauling scale suggests −4. This becomes more serious for TM elements, where central atoms such as Au or Pt have a higher Pauling EN than ligating atoms H, P, or Te. Clearly, OS = −6 for platinum in PtH42− according to the Pauling scale is unusual, and the +2 value obtained with the Allen scale is more in line with systematic descriptive chemistry.22

## A-3 Ionic approximation against Brønsted–Lowry acidity

TM hydrides can act as hydride, hydrogen, or proton donors [206]. Given the EN difference at the metal−hydrogen bond, one may find it surprising that Co, Fe, and Mn hydridocarbonyls are Brønsted–Lowry acids in aqueous solution: HCo(CO)4 is a strong acid [207], H2 Fe(CO)4 is comparable to acetic acid [208, 209], and HMn(CO)5 to “carbonic acid” [210]. This would seem to suggest that the hydrogen is protic. EN values tell the opposite, and so do quantum-chemical calculations in the literature [211, 212]. Also Mulliken-charge calculations with the programs Caesar [24] and Gaussian [25] for HCo(CO)4 of structure determined in ref. [213] yield H more negative than Co, by 0.2 to 1.2 elementary charge units.23 The respective H and Mn charges in HMn(CO)5 [214] are −0.1 and +1.3 by Gaussian and −0.35 and +0.35 by Caesar. The acidity is apparently not caused by polarity but by a weak metal–hydrogen bond (weakest for 3d TM elements [206]) aided by the thermodynamics of the subsequent hydration in solution. This means that despite being proton acids in aqueous solutions, these hydridocarbonyls do have OSH= −1 in agreement with EN considerations. For related aspects see refs. [215, 216].24

Thus, the HCo(CO)4 gas of our example has OSCo = +1. As it dissolves in water and dissociates as a protic acid, OSCo in the anions becomes −1 and remains so in the ionic solid KCo(CO)4 that is formed when the acid is neutralized with KOH.

## Appendix B: Electronegativity (EN)

The concept of electronegativity, well known to every chemist and student of chemistry, has a long history but, as fascinatingly chronicled by Jensen [217–219], many of the early inputs and contributors have seemingly been relegated to the annals of history. Already in the early 1800s Avogadro and Berzelius provided tables that listed elements in order of their relative electropositive/electronegative character. After the advent of the electron and the Bohr atom model 100 years later, EN was recognized as the ability of an atom to hold its valence electrons, and it was suggested by Rodebush [220] that “we might represent electronegativity as a function of N/S where N is the number of valence electrons and S the number of shells”. This calculation method, although not actually used, predates Pauling’s landmark paper [33]. Thus, the stage was set for Pauling to observe in his famous book [34b] that the “qualitative property that a chemist calls electronegativity” is “the power of an atom in a molecule to attract electrons to itself”.

The first empirical EN scale was established by Pauling based on bond energies [33], and this was followed by a plethora of publications on the subject to this day. The major aims of the numerous works were: to put EN on a firm theoretical or experimental basis, to relate EN to more directly measurable and precise atomic properties, and, later, to obtain EN values via purely computational means, with the ultimate goal of providing EN values for use in various applications for prediction and rationalization of molecular properties and chemical reactivity.

It is not the objective of this Appendix to provide even a short overview of the vast field of EN, or to resolve ongoing discussions concerning what exactly EN is and how best to obtain it. Interested readers are directed to relevant reviews [221–224]. In addition, a lucid treatment by Huheey [225] and two recent publications [226, 227] provide good summaries of and references to the various approaches, while two commentaries on EN scales by Pearson [228] and Allen [229], albeit now somewhat dated, give some of the flavor of past debates on the subject.

## B-1 EN of an atom

The prevalent, if not universally accepted view of EN, and one in accord with the above definition provided by Pauling, is that it is an in situ atom-in-molecule property, hence not an invariant property of the atom but a property dependent on the atom’s environment in the molecule, on the number and types of atoms attached to it and on the atom’s OS. Although Pauling also stated in his book [34c] that “The property of electronegativity… represents the attraction of a neutral atom (our emphasis) in a stable molecule for electrons”, he went on to assign different EN values to an element, depending on its OS. Other approaches, espousing the view that EN is a property of atoms in molecules, also give multiple EN values for an atom, taking into account its molecular environment and OS.

This variability of EN presents a dilemma for OS assignment. If an EN scale depends on OS, it could be construed that using EN to assign OS is but a circular argument. Fortunately, for OS assignment, the EN grid need not be so fine since only the relative signs of the residual atomic charges are needed. And for this, EN values based on the most common OS of the elements as well as those of the “free atom” provide satisfactory answers. Keeping this in mind, we’ll now review the four most often used EN scales, those of Pauling, Mulliken, Allred−Rochow, and Allen, together with a short outline of the less common approach by Sanderson and its specialized use. Outside the scope of the present discussion are the chemical-potential approach to EN [230–232] and the related concept of chemical hardness, Pearson’s well-known hard/soft acid/base (HSAB) principle [233–236]. These approaches, in Pearson’s own words [228], are more appropriate “as a measure of the chemical reactivity of an atom, radical, ion, or molecule.”

## B-2.1 Pauling EN

Pauling [33, 34d] based his EN scale on his observation that the bond energy E(AB) of the bond between two dissimilar atoms A and B was almost always greater than the measured bond energies of the homonuclear species, AA and BB. He assumed that the “ideal” covalent bond energy of A–B should equal either the arithmetic mean [33] or the geometric mean [34e] of the homonuclear bond energies, and that the “excess” bond energy of AB is caused by electrostatic attraction between atoms A and B, partially charged due to their differing EN. Pauling famously postulated this “excess” bond energy to be proportional to the square of the EN difference, and this was confirmed experimentally.

Pauling found the largest EN difference between Cs and F. The value for χF has been set arbitrarily to 3.98, and this gives a scale in which EN for all elements is < 4 and positive. The values listed in the Table 1 below are those updated by Allred [35]25 (except those with one decimal place, which are from Pauling [34f]).

Table 1

Pauling EN scale.a

aUp to the carbon group, the EN are valid for OS = N (the number of valence electrons), up to halogens for OS = 8 – N. For comparison, EN of Tl(I), Sn(II) and Pb(II) are listed in [35] as 1.62, 1.80 and 1.87, resp. For TM, the values concern OS = +2 for the Ti to Ni triads and +1 for the Cu triad and increase for higher OS, for example Fe(III) has 1.96 [35]. For variation with OS, Pauling [34f] refers to the review by Gordy and Thomas [237]. Values for lanthanoids (Ln) are not listed above but for Ln(III) they increase towards 1.27 for Lu [35].

## B-2.2 Mulliken EN

Shortly after Pauling, R. S. Mulliken [36, 37] proposed an EN scale composed of the ionization energy, IEb, the ability of the bonded atom to hold its own electron, and the electron affinity, EAb, the ability to attract an extra electron. These two properties conspire to justify the notion of EN as the power of the atom to attract the bonding electron pair to itself. The Mulliken EN, χM, is given by χM = (IEb + EAb)/2, where the subscript “b” denotes the bond state of the atom (usually referred to as a “valence state” or “orbital hybridization”).

With the availability of more accurate and reliable EA data, the Mulliken approach was developed to a full-grown EN scale over many years: In a series of articles, Jaffe and co-workers [238–241] tabulated Mulliken EN values for most elements, including the first-row TMs. Since EN was associated with the type of orbitals used by the atom in molecules, the term “orbital electronegativity” was suggested. Bratsch [242] has revised the Jaffe values based on more accurate EA data and used a method to calculate valence-state promotion energies that is said to be better suited for teaching purposes. According to Bratsch, the values for first-row TMs by Jaffe [241] are not sufficiently reliable for proper use. In fact, evaluation of reliable EN values for TMs appears to be a remaining challenge for this method. The most recent revision is due to Bergmann and Hinze [223], who used the most recent spectroscopic data and claimed that their Mulliken EN values “supersede similar earlier compilations”, including that of Bratsch. They also provide EN values for the TMs, but only with one-decimal point precision.

Mulliken ENs have a sound theoretical basis, but their value depends on the bond state (orbital hybridization) of the atom and its OS. The values listed in Table 2 (in Pauling units, from Bergman and Hinze [223]) are valid for the “typical” hybridization that reflects best the chemical environment of the atom as in the Pauling scale.

Table 2

Mulliken EN scale.

a For noble gases, the EN values are for doubly occupied orbitals.

b The values for various hybridizations of carbon are: sp3 2.45, sp22.69, sp 3.17.

## B-2.3 Allred and Rochow EN

Allred and Rochow [38] introduced a scale of EN based on the electrostatic force of attraction between the nucleus and the electron from a bonded atom, FZeff/r2, where Zeff is the effective nuclear charge and r is the atom’s covalent radius. To convert the Allred–Rochow EN (χAR) to the Pauling scale the following expression was used, with r in picometers: χAR = 3590 Zeff/r2 + 0.744, in which the effective nuclear charges were obtained from Slater’s rules while counting all electrons as shielding the electron coming from the other atom.

It is of historical interest to note that because the covalent radius (hence χAR) depends on the atom’s hybridization, Allred and Rochow opined in their contribution [38], prior to Jaffe [238–241], that “it is more accurate to speak of orbital electronegativity and to assign a range of electronegativity values to an element than to consider electronegativity as an invariant atomic property.” Thus, the χAR values are calculated using the most common covalent radius of the element. The values listed in Table 3 are from Allred and Rochow [38], and from Little and Jones [243] for the second-row TMs and for elements 58 onward.

Table 3

Allred–Rochow EN scale.

aThe Ln(III) EN remain close, with a minor increase towards 1.14 for Lu [243].

This method has gained popularity due to its simplicity and the ease with which it can be applied to heavier elements as well as d- and f-metals, and hence it is widely used in TM chemistry. The method reproduces some, but not all, of the anticipated exceptions to the general trend in EN values. Thus, the higher EN of Ga than Al or Ge than Si is due to the intervening poorly shielding d-electrons, while the higher EN values for most of the third-row TMs compared to their second-row congeners can be attributed to the “lanthanide contraction”. However, the expectations of EN Au>Ag or Hg>Cd are not met. There are other questionable anomalies. For instance, the EN values of Pd and Pt are lower than those of Rh and Ir, respectively, contrary to the normal trend of increasing EN in a period. This reversal can be traced to the anomalous ground-state electronic configurations, Rh [Kr]4d85s1but Pd [Kr]4d10 and Ir [Xe]4f145d76s2but Pt [Xe]4f145d96s1, and to the attendant decrease in Zeff in Pd and Pt, which is also contrary to expectations. Thus, the anomaly in these EN values may be inherent to the method used by Allred and Rochow for EN calculation, and may not be real. It would be interesting to see whether an eventual update of the Allred–Rochow EN values with the more sophisticated Clementi–Raimondi [244] approach to Zeff or the use of more recent covalent radii would resolve the anomalies and produce more reliable χAR values.

## B-2.4 Sanderson EN

Sanderson based his EN scale on his idea that electronegativity of an atom or ion can be evaluated from the relative compactness of its electronic cloud [245–247]. He argues that as a more compact atom holds more tightly onto its own electrons, it has a higher EN. In a spherical approximation of covalent/ionic radius, the atom/ion’s average electron density (ED) is calculated and related to a value for a hypothetical isoelectronic inert atom, (EDi), interpolated from ED for noble gases [248]. The ratio of ED/EDi (“stability ratio”) then allows comparison between the relative compactness of the electronic shells of different atoms/ions and the setting up the EN scale.

While experimental data carry uncertainties associated with covalent/ionic radii as well as a dependence on atomic charge [249], these variables together with electron densities become self-consistent within computational methods of the density functional theory and are used there as parameters [250]. It should also be noted that Sanderson’s approach to EN is a direct follow up of his idea of EN equalization [248] in which EN values of two atoms making a bond become equalized upon forming partial charges. The EN equalization continues to be a useful concept [232, 251].

Although successfully applied by Sanderson for calculation of bond energies and rationalization of molecular properties [249, 252], this EN scale never gained the following of the other scales. We therefore do not list any of its values beyond noting that for the main-group elements, after conversion [253, 254], they are very close to those of Pauling with which they correlate strongly.

## B-2.5 Allen EN

A fundamentally different view of EN was presented by L. C. Allen [39, 255, 256] who argues that EN is an intrinsic free atom property and equates it to the average one-electron energy of the valence-shell electrons of the free atom in its ground state; the configuration energy, CE. For the s- and p-block elements, it is given by CE = (s + p)/(n + m), where n and m are the number of s- and p-electrons, and εs and εp are the experimentally determined, multiplet averaged, one-electron ionization energies [39, 255].

The Allen EN values are sometimes given the symbol χspec because they are calculated from accurate spectroscopic data and hence possess greater intrinsic precision than the other EN scales. The method has been extended to the d metals, although taking account of their valence electrons is not as straightforward as with the main-group elements [256].

Politzer et al. [227], in a very recent and purely computational work, use the property known as “average local ionization energy” to obtain free-atom EN values. Although the authors note that Allen’s approach carries some ambiguity in enumerating valence d-electrons and it does not take into account the interpenetration of valence- and core orbitals, it is too early to tell how widely these most recent EN values will be accepted and hence Table 4 lists the Allen EN (χspec values) converted to Pauling units [39, 255, 256].

Table 4

Allen (spectroscopic) EN scale.

Allen’s EN values correlate well with the Pauling and Allred–Rochow scales yet avoid their anomalies, perhaps not surprisingly, as χspec tracks periodic variation in atomic ionization energies. Indeed, Allen proposed some guidelines to test EN scales [257] and finds the Pauling scale wanting in this regard. It is not a question of subscribing to all such self-described rules, but some are useful, such as the “Si rule: All metals must have EN values less than or equal to that of Si” for assignment of OS to TMs, and we noted a similar caveat when applying the Pauling scale to these metals.

Spencer et al. [258] advocate the Allen scale of EN as being the most pedagogically sound for introductory chemistry students as it relies on ionization energies, a concept that will have been introduced early in the course to help understand the structure of the atom. To allow the student to do their own calculations, a simpler Average Valence Electron Energy (AVEE) formula can be used instead of Allen’s configuration energy, and the simplified approach yields values that are close to Allen’s EN. However, the authors opine that since “these values measure the average force with which an atom holds its valence electrons” they are “only an approximation to the electronegativity that applies to an atom in a molecule.” That is, they hold the traditional view that EN is an in situ molecular property of an atom.

## B-3 Conclusions on EN scales

While the approaches of Pauling, Mulliken, and Allred–Rochow are valid for the most common bonding environment of the given atom, their EN are in situ, atom-in-molecule values that depend, among other, on the OS. Only the Allen EN scale is a free-atom view that is appropriate for the ionic approximation suggested in Section 5. Its values track the other scales and provide a reliable ionicity sign for OS according to Fig. 2.

In this Report, we have used the Allen scale and suggest this scale be used henceforth for ionic approximation of bonds that leads to the OS assignment. Allen’s free-atom valence-electron view of EN is perhaps less conventional, but the scale avoids the apparent anomalies that are present in some of the EN values in the other scales, and hence it is less prone to produce unusual ionic approximations of bonds.

## C-1 Calculation of atomic charges

Net charges calculated on two mutually bonded atoms by a quantum-chemical method never equal the OS of those atoms. Still, in most cases, their sign is indicative of the direction of the ionic approximation. It might even be tempting to obtain both the ionic sign and bond order component of the OS from such a calculation. However, chemists and physicists alike are aware [17, 259–261] of pitfalls associated with the partitioning of real space into regions attributable to individual atoms. The quantum-chemical process employed is called population analysis, and its procedures depend on what is being partitioned, usually either orbitals or electron densities.

Of the orbital-based population analyses, the Mulliken charge [262] and the Löwdin charge methods [263, 264] should be mentioned. The Mulliken charge is obtained by subtracting the number of an atom’s valence electrons from the sum of the calculated net populations of its atomic orbitals and one-half of its calculated overlap populations. The Löwdin method avoids this equal partitioning by transformation to an orthogonalized basis prior the population analysis. Various schemes have been suggested to alleviate the sensitivity of Mulliken populations to basis-set choice (the starting atomic parameters) [265–268] and typically adopt some form of orthogonalization that avoids overlap [269]. The original Löwdin population analysis has also been developed further [270, 271]. Thus, schemes can be applied with various implementations of various quantum-chemical methods, each time yielding a somewhat different charge.

Of the electron-density-based population analyses, the Bader method [272] of AIMs (atoms in molecules) partitions a molecule [273] or a solid [274] into atomic basins with boundaries defined by zero electron-density gradient (or flux). A related method uses the electron-localization function [275, 276] to partition structures for the generation of net atomic charges [277]. A rudimentary topological version is the method of Voronoi, used, for example, by Guerra et al. [260], which partitions the electron-density space by bisecting each nearest-neighbor connection line by a perpendicular plane (an analogy to Voronoi polygons). An alternative to AIM is the so called “stockholder partitioning” of Hirshfeld [278], in which the electron density at each point is divided among the contributing atoms in proportion to their contribution to a hypothetical promolecule [279–281] as a reference electron-density model of spherical ground-state atoms prior to the molecule formation. Charges can also be calculated via electron-density-dependent properties instead of just electron densities. In one such approach, the total Madelung potential of the calculated atomic charges is required to reproduce the electrostatic potential in an ionic structure [282]. In another case, atomic charges are calculated from the trace of the atomic polar tensors, that is, from the gradients of the dipole moment with respect to Cartesian displacements of the atoms (these also contain information on infrared vibrational intensities) [283, 284]. The atomic polar tensors scheme is relatively insensitive to the choice of basis set used in the calculations. However, as was the case for orbital population analysis, different net atom charges still may result as a consequence of choices to be made during the calculation.

## C-2 Calculations of bond order

Mulliken analyses [262] provide overlap populations, which, while of physical significance [285], are not the integer numbers expected for the total bond orders that sum to integer OS. Even for homonuclear bonds an integer covalent bond order is not obtained. Various approaches have appeared in the literature to overcome this. One is to use Löwdin orthonormalization [263] to avoid overlaps altogether. Mayer [264, 286, 287] refers to Wiberg’s understanding [288] of the quantum-chemical bond order and has gained a substantial following [289–291]. A similar approach was adopted by Jug et al. [267, 292, 293]. A different line of reasoning is the natural-resonance method, aka natural bond orbitals (NBO) [294, 295] that are obtained from averaged weighted resonance forms. This method yields “chemist” bond orders composed of ionic and covalent contributions, and the results are similar [291, 295] to those obtained with the Mayer method. The NBO methods are used across the periodic table [296, 297] and have been developed further [298, 299]. Of other approaches, the Hirshfeld stockholder partitioning has been modified to produce bond orders [300]. In a brief summary, one can say that besides the basis-set dependence, the overlap populations calculated with nonspecialized quantum-chemical software may deviate substantially from the integral bond orders chemists expect for OS determination and are therefore not suitable for bond-order estimation.

## C-3 Direct quantum-chemical calculation of OS

Several approaches have been adopted. Besides bond orders, Mayer [264] and others [267, 295] calculate a value termed valence or valence number, which in a homoleptic molecule is the absolute value of OS for an atom. An explicit quantum-chemical calculation of OS is suggested by Zhan et al. [301] and shown to work for sp molecules below a certain level of complexity. OS in TM complexes has been approached via a localized-orbital analysis [302]. A method for the calculation of OS for any TM compound has been recently suggested [303]. It simulates an ionic approximation by projecting pairs of bonding and antibonding orbitals back onto the original d-orbitals, and assigns a unity contribution to OS when a particular d-orbital is fully occupied and zero when it is less than fully occupied.

## Appendix D: When dia becomes ambiguous

As shown in Section 6.2 and examples that followed it, DIA is a simple approach to calculation of OS on summary formulas of binary homoleptic compounds. Upon adding more atoms, however, DIA will become ambiguous if the summary formula allows for two or more OS isomers of either redox or connectivity type.

## D-1 Redox isomerism

Redox isomerism invalidates DIA when two or more atoms are able to vary their OS in a compensatory manner.

This applies to ternary and higher compounds. An example from the sp elements is a Tl, O, H composition with a total of 5 valence-electron pairs to be distributed, which leads to ambiguity in an electron-pair location on Tl or H. The DIA result for the thallium(3+) hydride oxide is invalidated by the existence of thallium(1+) hydroxide as an alternative. Among p-group compounds, SnSeO4 (17 pairs) is ambiguous with the electron pair either on Sn or Se. One of many examples among TM compounds is FeTiO3(Fe+2/Ti+4 or Fe+3/Ti+3). A combined case is VSbO4 (V+3/Sb+5 or V+4/Sb+4 or V+5/Sb+3), in which Mössbauer spectra helped to determine OS as V+3/Sb+5 [304].

## D-2 Connectivity isomerism

Connectivity isomerism invalidates DIA for a sequence of three or more bonded atoms extrapolable to anions. For such a sequence, DIA provides OS only for the isomer with a central atom at a local minimum in EN and flanks at maximum (further referred to as V condition). If there is another atom attached to the flank, this added atom must be at an absolute minimum in EN (like an M) not allowing it to extrapolate to an octet or doublet.

Examples: A correct OS is obtained with DIA for [STeSe]2− with 20 valence electrons but not for [SSeTe]2− and [SeSTe]2−. The correct OS is obtained with DIA for cyanate [NCO] but not for fulminate [CNO]. A correct OS is obtained for the carbamic acid NH2 COOH but not for nitromethane CH3 NO2. Both H3 C–BO and H2 C=BOH have the same OS as given by DIA, but boranecarbonyl H3 B–CO does not. Figure 57 illustrates that OS in BrNF2 cannot be determined by DIA. While the F–N–F segment causes no problems, the Br–N–F segment of the molecule violates the V condition for EN of the central atom. OS must be obtained from bond-assignment algorithms.

Fig. 57

DIA on NBrF2 (left) does not give the correct OS (right).

If the V condition is violated by atoms of the same element bonded to each other, a correct OS can be obtained if the compound has a sole repetitive bonding unit that contains one single atom of that particular element, and DIA is performed on that repeat unit. Examples are the empirical formulas HgCl for Hg2 Cl2, HO for H2 O2, CH for C2 H2 or C6 H6 or C8 H8, CCl3 for C2 Cl6, KGe for the K4 Ge4 solid with Ge44− cages, BH for the B12 H12 closoborane (Section 9.2), or GaSe for Ga2 Se2 with gallium dumbbells (Section 8.3). DIA will, for example, not work on CF10/4 for the not entirely periodic C4 F10. Neither will DIA provide correct OS for a binary compound that contains bonds between ligand atoms (which violates the V condition), such as CrO(O2)2 exemplified in Section 8.1.

## Appendix E: LA Ligands

The term ligand immediately conjures up images of electron-donating LB bonded (ligare, “to bind”) to electron-acceptor LA metal center. However, TMs by virtue of their incompletely filled d-orbitals are inherently ambivalent and can act also as electron donors, especially late TMs of low-OS that are electron-rich. The concept of TM basicity has long been recognized [216, 305, 306], and indeed it is of crucial importance in several important organometallic reactions, such as oxidative addition [307] and C–H activation [308, 309]. The ability of a TM to donate electrons manifests itself in the classical back-bonding to typical π-acid ligands, but this secondary bonding component is, by convention, ignored under OS assignment. On the other hand, TMs can also act as pure σ-donors to σ-acceptor ligands (so-called Z ligands [310]), and it is such neutral Z-type ligands that are the object of this Appendix.

Fig. 58

Three types of sulfur bonding in SO2 ligand. Left: LB sulfur atom (L-type SO2 ligand) [311]. Middle: LA sulfur atom (Z-type SO2 ligand) [146]. Right: π-complex (M-type SO2 ligand) [312].

The first structurally authenticated examples were the pyramidal SO2 adducts of Vaska’s complex and its Rh analog in ref. [313] and ref. [146], respectively. In view of the SO2 relevance in the environment, numerous TM–SO2 complexes were prepared and many structurally characterized. Three types of bonding modes of the SO2 ligands were identified in mononuclear complexes, Fig. 58. The structural similarities between the κ1 planar/pyramidal modes and the linear and bent M–NO geometries have been noted, and the bonding in the MSO2 fragment interpreted in a similar fashion as in MNO [314–316], including the suggestion that the fragment should be denoted according to the Enemark and Feltham notation [172] as {MSO2}n. Translating this into the OS of M and S was not unambiguous. On the basis of M–S and S–O distances in Rh(CO)Cl{P(C6 H5)3}–SO2 [146] with a {RhSO2}8fragment, Ibers [146] noted that the SO2 ligand is on the way toward the geometry in dithionate, S2 O42–, Fig. 59. However, the attendant OSS = +3 and OSRh = +2 in the complex would be difficult to reconcile with the diamagnetism of the compound, which is more in line with a Rh(I) (d8) donating a pair of electrons to a neutral SO2 ligand (OSS= +4). This is by no means unique, and SO2 adducts with, for example, Pt(PR3)2(R = cyclohexyl, tert-butyl) [317] or [PtX(C6 H2{CH2 N(CH3)2}2-2,6-R-4)] (R = OH, X = Cl; R = H, X = Cl, Br, I) [318] are considered to contain respectively Pt(0) and Pt(II). Their reversible binding of SO2 is more in accord with these OS than with ionic approximation towards Pt+2 or Pt+4 and [SO2]–2.

Fig. 59

Bond lengths and OSO angle in Rh(CO)Cl{P(C6 H5)3}–SO2 [146] to be compared with 1.43 Å and 117° in solid SO2 [319] vs. 1.50 Å and 108° in S2 O42− [320]. (1Å=100 pm).

The first structural identification of a TM → B donor–acceptor bond in the [Ru(CO)(P(C6 H5)3){B(mt)3}] (B(mt)3 = tris(methimazolyl)borane) metallaboratrane by Hill [321] in 1999 has rekindled interest in the field, which has seen spectacular progress in the intervening years when similar complexes of Co [322], Ni [323], and others have been synthesized. The TM → Z-type ligand interactions can be divided into two main classes: supported and unsupported by donor buttresses, Fig. 60.

Fig. 60

TM complexes with LA ligand: Center: unsupported, of Pt more Allen-electronegative than Be (top) [328], of Pt less Allen-electronegative than Ga (bottom) [332]. Left: supported, with octahedrally coordinated Ru [321]. Right: supported, with square-planar Au [149].

The supported TM → Z complexes include bis- or tris-(methimazolyl)boranes and phosphine boranes, with main contributions from the groups of Hill [324], Parkin [325], Bourissou [154], and more recently of Mösch-Zanetti [326] and Nakazawa [327]. Contemporary examples of unsupported structurally characterized TM → Z complexes are chronicled in a remarkable series of articles by Braunschweig and co-workers utilizing the electron-rich, two-coordinate Pt(PCy3)2 (Cy = cyclohexyl) or Pt(PCy3)(IMes) [IMes = 1,3-bis(2,4,6-trimethylphenyl)imidazole-2-ylidene] fragments. The Z-type ligands in the so-formed T-shape, tri-coordinate complexes include BeCl2 [328] (Fig. 60, center top), AlCl3 [329–331], GaCl3 [332] (Fig. 60, center bottom), and ZrCl4 [333]. The richness and the vibrancy of the field are eloquently illustrated in numerous recent reviews which focus not only on the structures of these unusual complexes but also on their intriguing reactivities [151, 334–338].

The bonding in these unusual compounds is conceptually simple; donation of the TM σ-electron pair into σ-acceptor orbital on the Z ligand. However, the extent of this transfer is dependent both on the TM and the Z ligand, and the effect that this has on the assignment of the OS and dn electron configuration of the TM center has generated lively debate. To circumvent the ambiguity on dn and OS created by the extent of this interaction, Hill [147] suggests notation akin to one advocated by Enemark and Feltham [172] for TM nitrosyl. In such a {M → Z}n notation, the superscript “n” would denote the number of electrons associated with the TM d-orbitals and the M → Z bond. Since neutral Z-type ligands contribute no electrons or charge, within the confines of OS the superscript “n” also identifies dn and vice versa. A somewhat different view was presented by Parkin [148] who based his arguments on the CBC method of Green [27, 29] and a strict interpretation of the electron configuration as “the number of electrons that are housed in metal d-orbitals that are either primarily nonbonding or have a metal–ligand antibonding character” [148]. Since, as shown by a simple MO diagram [148, 339, 340], the M → Z interaction is bonding in nature, it effectively removes two nonbonding metal d-electrons and hence the metal center should be considered as having a dn–2 configuration (note that this n remains the same for Hill and Parkin). We devoted Section 10.2 to this important question that relates to OS.

Computational studies and physical measurements on TM → LA complexes confirm that an electronegative LA ligand does not influence the OS of the TM center, whereas the coordination geometry is that of a dn–2 TM central atom supplying two bonding electrons. For instance, in [{κ3-(o-(iPr2 P)C6 H4)2 B(C6 H5)}RhCl(dmap)] (iPr = isopropyl, dmap = N,N-dimethyl-aminopyridine), an NBO analysis suggests that 79.3 % of the Rh−B bonding comes from the Rh dz2-orbital [341]. We therefore denote rhodium as being Rh(I, n = 8 in dn−2) where I is the Stock notation for the OS, n = 8 complies with that OS via eq. 6 and n − 2 = 6 complies with the square-pyramidal coordination geometry. Similarly, in the square planar [{κ3-(o-(iPr2 P)C6 H4)2 B(C6 H5)}AuCl] [149], gold would be denoted as Au(I, n = 10 in dn−2) where n = 10 corresponds to the generic OSAu = +1, also seen by Mössbauer spectroscopy, whereas n − 2 = 8 to the square-planar Au coordination. Braunschweig in his series of [(Cy3 P)2 Pt–Z] complexes analogously observes that the 31P NMR chemical shifts are in line with a Pt(0, d10) and not a Pt(II, d8) center [331]. We denote this as Pt(0, n = 10 in dn−2) where n = 10 complies with the generic OS of Pt via eq. 6 and n − 2 = 8 with the trigonal-planar coordination. Similarly, the authors of the (silox)3 Ta(BH3) complex of silox = {(CH3)3 C}3 SiO conclude [342] that Ta is best considered having an OS = +3, yet concur with Parkin that it is a d0 center. We denote this as Ta(III, n = 2 in dn−2) where n = 2 complies with the OS of Ta via eq. 6.

## Acknowledgments

We are indebted to the following colleagues: Prof. Reinhard Nesper (ETH Zurich, for an initial discussion and for scans of OS definitions in German textbooks), Prof. Karlheinz Schwarz (TU Vienna) and Prof. Trygve Helgaker (University of Oslo) for reading Appendix C, and to Prof. Arne Haaland for fruitful discussions.

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Note: Republication or reproduction of this report or its storage and/or dissemination by electronic means is permitted without the need for formal IUPAC permission on condition that an acknowledgment, with full reference to the source, along with use of the copyright symbol ©, the name IUPAC, and the year of publication, are prominently visible. Publication of a translation into another language is subject to the additional condition of prior approval from the relevant IUPAC National Adhering Organization.

Corresponding author: Pavel Karen, Department of Chemistry, University of Oslo, P.O.B. 1033 Blindern, 0315 Oslo, Norway, e-mail:

Published Online: 2014-04-12

Published in Print: 2014-06-18

Oxidation state: a measure of the degree of oxidation of an atom in a substance. It is defined as the charge an atom might be imagined to have when electrons are counted according to an agreed-upon set of rules: (l) the oxidation state of a free element (uncombined element) is zero; (2) for a simple (monatomic) ion, the oxidation state is equal to the net charge on the ion; (3) hydrogen has an oxidation state of 1 and oxygen has an oxidation state of −2 when they are present in most compounds. (Exceptions to this are that hydrogen has an oxidation state of −1 in hydrides of active metals, e.g., LiH, and oxygen has an oxidation state of −1 in peroxides, e.g., H2 O2; (4) the algebraic sum of oxidation states of all atoms in a neutral molecule must be zero, while in ions the algebraic sum of the oxidation states of the constituent atoms must be equal to the charge on the ion. For example, the oxidation states of sulfur in H2 S, S8 (elementary sulfur), SO2, SO3, and H2 SO4 are, respectively: −2, 0, +4, +6, and +6. The higher the oxidation state of a given atom, the greater is its degree of oxidation; the lower the oxidation state, the greater is its degree of reduction. (Ref. [1] online version as of 2013-12-22).

Oxidation number of a central atom in a coordination entity: the charge it would bear if all the ligands were removed along with the electron pairs that were shared with the central atom. It is represented by a Roman numeral. (Ref. [1] online version as of 2013-12-22).

Discussion of valence is not the subject of this self-consistent Report, but the usage of “low/high valent” to indicate “low/high OS” is strongly discouraged.

There is still a lingering misconception in some circles that OS represents “real charges”. It must be stated emphatically that this is not the case.

An alternative ionic approximation is one according to bond polarity. Its disadvantage is that it yields unusual OS values when polarizing a homonuclear bond of two almost equivalent atoms into ions of high and opposing charges, such as the NN bond in N2 O into N3− of the terminal and N5+ of the central nitrogen of the molecule (Section 9.1.4). Furthermore, certain exceptional dipole moments might suggest unusual ionic extrapolations (such as CO and NO in Appendix A, Section A-1).

Although not usually stated, the 8−N and (8+)N rules introduced in this Section derive from elementary bonding considerations. They are not necessarily identical with the rule of octet since each can be violated independently. A simple try shows that N2 O with eight valence-electron pairs can have octets on all atoms and yet violate the 8−N rule on O (the most electronegative element): $|N≡N−O¯_|.$

The 8 in the shorthand symbol merely implies a noble gas shell, and the parentheses around it are dropped in subsequent sections. The rule also covers the second-period electropositive atoms.

Any ionization would change OS together with the charge; a hypothetical [C=O|]4+ has OSC= +2 + 2 = +4 and OSO= −2 + 2 = 0. Obviously, the ionic charge is the sum of OS or the sum of FC of the atoms as the iBOS cancel on summation over the bond.

The rule is formulated such that it focuses on the most electronegative atom.

Relative to the average of structures on which the particular bond-valence coefficient has been statistically determined.

The only route to unambiguous OS for all three atoms in N2 O would be to resolve the NN bond polarity. Making the terminal nitrogen more negative gives it OS = −3 with +5 for the central nitrogen and O remaining at −2; not a very realistic assignment for two nitrogen atoms that are not that dissimilar.

The only route to an unambiguous OS for all nitrogens in N5+ would be to resolve the bond polarity of each NN bond and assign the bonding electrons to its more negative nitrogen. As Mulliken charges suggest that the central nitrogen and both terminal nitrogens are negative, they would obtain OS = −3 while the mid-wing nitrogens would have OS = +5, again not a very realistic OS assignment.

A bond graph can be constructed also for an infinite chain. It maintains the chemical composition, but it is not identical with the Lewis formula, and the connections are not necessarily two-electron bonds. The bond graph of WCl4 has 2W and 8Cl and tells us that, besides one WW bond, each W atom in the chain forms two single bonds, two 3/4 bonds, and two 1/2 bonds to chlorine atoms.

The number of conjugated carbons that are bonded to the central atom is called hapticity, n, and symbolized as ηn in a formula or name.

Immediately after the anion’s conjugate acid, cyclopentadiene, is deprotonated, an anionic lone pair on carbon plus two double bonds can be imagined on the ring.

The delocalized symmetrical cyclopentadienyl anion does not obey octets unless those three π-pairs are localized and the symmetry violated.

It remains a cation also under ionic approximation to OS. Even though carbon is more electronegative than any TM, it leaves the ionic electron on the central atom. This is because the aromatic stabilization makes the C7 H7 radical analogous to an electropositive “atom” (imagine Na) that easily gives up one electron to become a cation (Na+) hence the “tropylium atom” is at the higher energy level in Fig. 1.

The octahedral field of strong splitters will favor pairing of six electrons in the three stabilized t2g orbitals. The stable low-spin configuration and OS of Fe is retained even upon reduction to [Fe(CN)5 NO]3−, when it is the NO+ ligand that is reduced to NO, not iron.

The high-temperature form has not been structurally characterized.

Positive Mulliken-population charges on carbon in CO are typically obtained with commonly available software. This is, however, just an artifact of a small overlap population being calculated (<2).

The difference between two such alternative OS values equals the number of electrons in the bonds between the atoms whose EN is in question, here 4 two-electron bonds.

The scatter refers to results with different basis sets available in the programs used and with the Hartree–Fock versus Density Functional Theory approach in Gaussian.

As the preparation of TM hydridocarbonyls from carbonyls usually involves protonation of a reduced, intermediate anion, M(CO)xn, the change in OS from H(+1) to H(−1) in the product can be viewed as a result of the carbonylate anion’s strongly reducing nature. However, the conceptual, heuristic nature of OS can be appreciated from the fact that protonation of neutral metal carbonyls is also possible. Thus, protonation of Fe(CO)3{(C6 H5)2 PCH2 P(C6 H5)2}, basic but not strongly reducing, yields according to ref. [215] the cationic [Fe(CO)3{(C6 H5)2 PCH2 P(C6 H5)2}H]+, in which OS of the hydrogen directly bonded to Fe is also −1 by comparison of Allen EN. There are more such examples [216].

The arithmetic mean was used in the Allred updates [35] of Pauling EN.

Citation Information: Pure and Applied Chemistry, Volume 86, Issue 6, Pages 1017–1081, ISSN (Online) 1365-3075, ISSN (Print) 0033-4545,

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