Oxidation State, A Long-Standing Issue

The oxidation state is the simplest attribute of an element in a compound. It is taught early in the chemistry curriculum as a convenient electron-counting scheme for redox reactions. Its applications range from descriptive chemistry of elements to nomenclature and electrochemistry, or as an independent variable in plots and databases of bonded-atom properties (such as radius, bond-valence parameter, standard reduction potentials, spectral parameters, or spin). 
 
The history of the oxidation state goes back about 200 years when it described the stepwise increase in the amount of oxygen bound by elements that form more than one oxide. In his 1835 textbook Unorganische Chemie,[1] Wohler speaks of such an “oxydationsstufe” (an older German spelling for oxidation grade). This expression remains in use for oxidation state in several languages. The equivalent term oxidation number is also common; in English this refers more to redox balancing than to the chemical systematics of an element.[2] 
 
Under the entry for oxidation number, the IUPAC “Gold Book”[3] gives a defining algorithm for the oxidation state of a central atom as the charge it obtains after removal of its ligands along with the shared electron pairs. The entry for oxidation state in Ref. [3] complements this with a set of charge-balance rules and of postulated oxidation states for oxygen and hydrogen with exceptions. Details vary from textbook to textbook. Some list the rules according to decreasing priority to avoid the explicit exceptions; here is an example:[4] 
 
 
Atoms in an element have oxidation state 0. 
 
 
The sum of the oxidation states for atoms in a compound is 0. 
 
 
Fluorine in compounds has the oxidation state −1. 
 
 
Alkaline metals in compounds have the oxidation state +1, alkaline-earth metals +2. 
 
 
Hydrogen in compounds has the oxidation state +1. 
 
 
Oxygen in compounds has the oxidation state −2. 
 
 
 
In recent debates, Steinborn[5] and Loock[6] advocate Pauling’s[7] approach of assigning shared electron pairs to the more electronegative atom. Jensen[8] elaborates on some of the points considered by Loock. Smith[9] and Parkin[10] address the oxidation state in the context of related terms. Calzaferri[11] as well as Linford and co-workers[12] make suggestions on the oxidation state of organic compounds. Jansen and Wedig[13] point out the heuristic nature of the oxidation state and require that “concepts need to be defined as precisely as possible, and these definitions must always be kept in mind during applications”. 
 
IUPAC also realized the need to approach a connotative definition of the oxidation state. In 2009, a project was initiated “Toward Comprehensive Definition of Oxidation State”, led by the author of this Essay, and its results have recently been published in an extensive Technical Report.[14] We started with a generic definition of oxidation state in terms broad enough to ensure validity. Then we refined those terms to obtain typical values by algorithms tailored for Lewis, summary, and bond-graph formulas.


Introduction
Theoxidation state is the simplest attribute of an element in acompound. It is taught early in the chemistry curriculum as aconvenient electron-counting scheme for redox reactions. Its applications range from descriptive chemistry of elements to nomenclature and electrochemistry,o ra sa ni ndependent variable in plots and databases of bonded-atom properties (such as radius,b ond-valence parameter, standard reduction potentials,spectral parameters,ors pin).
Theh istory of the oxidation state goes back about 200 years when it described the stepwise increase in the amount of oxygen bound by elements that form more than one oxide.I nh is 1835 textbook Unorganische Chemie, [1] Wçhler speaks of such an "oxydationsstufe" (an older German spelling for oxidation grade). This expression remains in use for oxidation state in several languages.T he equivalent term oxidation number is also common;inEnglish this refers more to redox balancing than to the chemical systematics of an element. [2] Under the entry for oxidation number,the IUPAC "Gold Book" [3] gives ad efining algorithm for the oxidation state of ac entral atom as the charge it obtains after removal of its ligands along with the shared electron pairs.T he entry for oxidation state in Ref. [3] complements this with as et of charge-balance rules and of postulated oxidation states for oxygen and hydrogen with exceptions.D etails vary from textbook to textbook. Some list the rules according to decreasing priority to avoid the explicit exceptions;h ere is an example: [4] 1. Atoms in an element have oxidation state 0. 2. Thesum of the oxidation states for atoms in acompound is 0. 3. Fluorine in compounds has the oxidation state À1. 4. Alkaline metals in compounds have the oxidation state + 1, alkaline-earth metals + 2.
5. Hydrogen in compounds has the oxidation state + 1. 6. Oxygen in compounds has the oxidation state À2.
In recent debates,S teinborn [5] and Loock [6] advocate Paulings [7] approach of assigning shared electron pairs to the more electronegative atom. Jensen [8] elaborates on some of the points considered by Loock. Smith [9] and Parkin [10] address the oxidation state in the context of related terms.C alzaferri [11] as well as Linford and co-workers [12] make suggestions on the oxidation state of organic compounds.J ansen and Wedig [13] point out the heuristic nature of the oxidation state and require that "concepts need to be defined as precisely as possible,a nd these definitions must always be kept in mind during applications".
IUPAC also realized the need to approach ac onnotative definition of the oxidation state.I n2 009, ap roject was initiated "Toward Comprehensive Definition of Oxidation State", led by the author of this Essay,a nd its results have recently been published in an extensive Te chnical Report. [14] We started with ag eneric definition of oxidation state in terms broad enough to ensure validity.Then we refined those terms to obtain typical values by algorithms tailored for Lewis,summary,a nd bond-graph formulas.

2.Generic Definition
Theo xidation state is the atoms charge after ionic approximation of its bonds.T he terms to be clarified are the "atoms charge", "its bonds", and the "ionic approximation".
Theatoms charge is the usual count of valence electrons relative to the free atom. Theoxidation state is aquantitative concept that operates on integer values of counted electrons. This may require idealizing visual representations or rounding off numerical results.
Approximating all bonds to be ionic may lead to unusual results.Ifthe N Nbond in N 2 Owere extrapolated to be ionic, the central nitrogen atom would have an oxidation state of + 5a nd the terminal one À3. To obtain less extreme values, bonds between atoms of the same element should be divided equally upon ionic approximation.
Several criteria were considered for the ionic approximation:1 )Extrapolation of the bonds polarity;a )from the electronegativity difference,b )from the dipole moment, c) from quantum-chemical calculations of charges.2)Assignment of electrons according to the atoms contribution to the molecular orbital (MO).
As discussed in Appendix Bo fR ef. [14],m ost electronegativity scales depend on the atoms bonding state,w hich makes the assignment of the oxidation state as omewhat circular argument. Some scales lead to unusual oxidation states,s uch as À6f or platinum in PtH 4 2À with Pauling or Mulliken scales.Appendix EofRef. [14] shows that aLewisbasic atom with an electronegativity lower than its Lewisacidic bond partner would lose the often weak and long bond upon ionic approximation of their adduct, thereby yielding an unusual oxidation state.A ppendix Ao fR ef. [14] points out that dipole moments of molecules such as CO and NO,which are oriented with their positive end towards oxygen, [15][16][17] would lead to abnormal oxidation states.A ppendix Co f Ref. [14] illustrates the variety of calculated quantum-chemical atomic charges.This leaves the atoms contribution to the bonding MO,t he atomic-orbital energy,a st he criterion for ionic approximation (Figure 1). Figure 1implies that while AA bonds are divided equally, in an AB compound the atom contributing more to the bonding molecular orbital receives negative charge under ionic approximation of the bond. Ref.
[14a] emphasizes that the said contribution does not concern the actual origin of the bonds electrons upon its formation, only their final allegiance.F igure 1i sn ot an instruction to use the mixing coefficients;i tm erely illustrates ac oncept. Thes ame ionic approximation is obtained when the more heuristic orbital energies are considered.

Simple Estimate of Ionic Approximation
Should complicated MO schemes make the above criterion impractical, the ionic approximation can be estimated from electronegativities.O fs everal scales discussed in Appendix Bo fR ef. [14],o nly the Allen electronegativity is truly independent of the oxidation state,a si tr elates to the average valence-electron energy of the free atom. [18][19][20] Such an ionic approximation is obtained when the bonds implied in Figure 1a re abstracted away ( Figure 2). Thee lectronegativity criterion for the ionic approximation carries an exception if the more electronegative atom is reversibly bonded as aL ewis acid (a so called Z-ligand, Appendix EofRef. [14]): Its acceptor orbital is high, and the less-electronegative Lewis-base donor atom retains the electrons because of its larger contribution to the bonding MO. An allegiance criterion by Haaland [21] identifies such an adduct:A pplied to ionic approximation, one asks where the bonding electrons go when the bond is split thermally.I ft he split is heterolytic,t he ionic approximation follows the electrons;ifhomolytic, electronegativity applies.T able 1lists the Allen scale.

Algorithm for Summary Formulas
Theoctet rule [22] concerns the most electronegative atoms in the periodic system. On as ufficiently simple summary formula involving such atoms,i ta lone dictates the oxidation states.T he algorithm is named DIA (direct ionic approximation) in Ref. [14]: Atoms are assigned octets according to their decreasing electronegativity until all the available valence electrons are used up.T he atom charges then represent the oxidation states.
Ty pical DIA-friendly species are homoleptic binaries of at least one sp element ( Figure 3) …; or solids with ah omoleptic periodic bonding unit:K Br, SiC,A lCl 3 , SnCl 2 ,etc.DIA of compounds of three or more elements may become ambiguous,w ith the limitations discussed in Appendix Do fRef. [14].

Algorithm of Assigning Bonds
These algorithms work on Lewis formulas that display all the valence electrons:B onds are assigned to the more negative bond partner identified by ionic approximation. Theresulting atom charges then represent the oxidation state ( Figure 4). As only homonuclear bonds are divided (equally), the correct bond multiplicity is essential only between those pairs of atoms of the same element that appear asymmetrical within the segment of the pairs bonds,i ncluding the sign of   their ionic approximation:W hereas the OO bond order in Figure 4would not matter so long as the -OO-segment were kept symmetrical, the NN bond order in an N 2 OL ewis formula always matters.
An example of the exception to the rule of ionic approximation according to electronegativity is [(C 5 H 5 )(CO) 2 Fe À B(C 6 H 5 ) 3 ] [23] on the right-hand side of Figure 5. Despite the higher electronegativity of B, the Lewis-basic Fe atom keeps the electrons it donated to bond triphenylborane.W henBi sr eplaced by Al [24] (Figure 5l eft), the same principle applies,n ow in line with the Fe and Al electronegativities.T he weak donor-acceptor bonds in these two adducts are the telltale sign of the reversibility criterion of Haaland, [21] suggested in Ref. [14] to identify cases of electron allegiance against electronegativity such as the one on the righthand side of Figure 5.

Algorithm of Summing Bond Orders
This algorithm is tailored to bond graphs.Abond graph represents the infinite periodic network of an extended solid. [25,26] It is constructed on astoichiometric formula of the networks repetitive unit, with atom symbols distributed such that astraight line is drawn for each instance of an atoms bonding connectivity.E ach line carries its own specific bond order.T oo btain the oxidation state,asum is calculated at each atom, of the orders of its bonds weighted by their ionic sign at that atom. Such an "ionized bond order sum", iBOS,t hen equals the atoms oxidation state.F igure 6 explains this on the AuORb 3 perovskite-type structure [27] with bond orders according to the 8 + N rule at Rb,8ÀNrule at O, and the 12ÀN rule at Au.

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Essays 4718 www.angewandte.org line-earth metals.T he 8inits name symbolizes the preceding noble-gas shell.
The8ÀNrule:Anelectronegative sp atom with N valence electrons tends to form 8ÀN but not more than four twoelectron bonds with atoms of equal or lower electronegativity. As an example,p hosphorus with 5valence electrons forms 3 two-electron bonds in the P 4 tetrahedron, and nitrogen does the same in N 2 .I nh eteroatomic molecules,t he 8ÀN rule is enforced by higher electronegativity.F or example:I ns ulfur fluorides,t he 8ÀN rule concerns fluorine.B onds in SF 2 ,S F 4 , and SF 6 have all approximately the length of asingle bond. [28] In the series BF,CO, and N 2 ,the full triple bond suggested by the octet rule only occurs in N 2 ,w hereas Oa nd Ff orce the bond order towards 2a nd 1, respectively. [29] The8 À Nrule is not the same as the octet rule,a se ach can be violated independently:T he Lewis formula of N 2 Oc an be drawn as j N N-O j ,w hich has octets yet violates the 8ÀN rule on oxygen. Hydrogen obeys an analogous 2ÀN rule. The1 2 À Nrule:A ne lement having close to 12 dsp electrons in its outermost shells tends to lose those that exceed 12 or to gain in bonds those less than 12. While the first tendency (to form s 2 cations) is magnified by other trends in the main groups of the periodic system, the second (to form s 2 anions) is specific to Pt and Au,which in such compounds are called relativistic chalcogens and halogens,r espectively. [30] Besides bond graphs,t he algorithm works on Lewis formulas that display bond orders.I tw orks directly if the formula does not carry formal charges.I fi td oes,t he atoms formal charge FC is added to the atoms positive or negative sum of bond orders iBOS to yield its oxidation state [Eq. (1)]. [14b] This relationship is illustrated for CO and [Cr(CO) 6 ]i nF igure 7.
Thebond orders in extended solids are not always obvious and may have to be estimated from bond lengths.This is done by converting each bond length into the so-called bond valence,which is avalue entirely equivalent to the bond order in terms of two-electron bonds in molecules.T he origins of the bond-valence approach-one [31] ionic and one [32] covalent-are associated with Linus Pauling. In Ref. [32],a n expression is given that morphed into the current relation for bond valence versus bond length [Eq. (2)].
In this expression, BV ij and d ij are the respective bond valence and distance of the atoms i and j, R 0 ij is the single-bond length between them, and B is av ariable parameter often fixed to 0.37. Although for the best accuracy R 0 ij is afunction of the coordination number and oxidation state of the "cation" for ag iven "anion" (fitted [33,34] to as et of such structures), ag eneral approach [35] lists two parameters for each atom, related to the size and electronegativity,f rom which R 0 ij is calculated for any atom pair i and j.A st he oxidation state operates on integer electrons,around off is required on the obtained bond valences/orders or on their ionized sums at an atom.
We will now analyze WCl 4 in Figure 8. Thebond graph of its infinite chain has aW -W group and eight Cl atoms,w ith each Wc oordinating six Cl atoms by two single,t wo 3 = 4 ,a nd two 1 = 2 bonds,asestimated from experimental bond lengths [36] in Ref. [14c].P articipation of more than one porbital at the Cl bridge above and below W-Wm akes its bond-order sum (1.5 at Cl) exceed the 8ÀN rule.Achain edge resembles locally am olecule,a nd such aC la tom in aL ewis formula would carry aformal charge of 0.5 + ,compensated by 0.25À at each of the two Watoms it bridges.This feature can be seen in the bond graph in Figure 8.
Theb ond-valence approach to bond orders can also be used for finite species.A ne xample is Cu 5 I 7 2À [37] in Figure 9. One of the five Cu atoms bonds to four iodine atoms,w hile the remaining four Cu atoms bond three.T wo of the seven iodine atoms bond to three Cu atoms and the remaining five only to two.T he oxidation state is evaluated as ar ound off value of the positive and negative sums of bond valences calculated with Equation (2)] from bond distances d CuI in Ref. [37] and R 0 CuI = 2.188 o btained from parameters in Ref. [35].A st he CuÀCu bonds are not approximated to be ionic, only the CuI bonds are relevant besides the 2 = 7 bonds from each iodine atom to the cations omitted in Figure 9, which yields ionized bond valence sums of À1.05(4) per iodine and + 1.07(2) per Cu atom. Around off yields asingle oxidation state for each element in an ice demonstration of the Paulings [31] parsimony rule.
An example of an sp molecule shows another kind of bonding compromise:( C 6 H 5 ) 3 P = C = P(C 6 H 5 ) 3 .T his extreme Lewis formula emphasizes the high order of the phosphorus-

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Chemie to-carbon bond because of the 8ÀN rule working for the more electronegative carbon atom. Ab ond order of 1.9 is calculated as above from the distance [38,39] of 1.63 . As the PCP angle is not 1808 8 but only 1348 8, [39] the 8ÀN rule is somewhat violated due to the only small difference in the electronegativity of Pand C, and it is clear that this bond has as trong ionic contribution from the formal charges 1 + ,2 À , and 1 + that would appear on the PCP segment if s bonded. As these charges comply with the electronegativity,the bond strengthens and becomes a [40] "sort of double bond". When this ionocovalent interaction is drawn with two full dashes as above,the formula loses its formal charges,and the oxidation states equal directly to the sums of the ionized bond order: À4 at the carbon and + 5a tt he phosphorus atom. That makes sense redox-wise within the molecule [41] as well as in its full [42,43] synthesis.G iven the bond strength, the Haaland criterion is unlikely to apply in this case.
An example of an sp cluster is As 4 S 4 .I to ccurs as two different molecules,w here both elements maintain the 8ÀN rule (an electron-precise cluster). This information is sufficient to obtain their oxidation states by sums of the ionized bond orders (Figure 10). Ac luster where the 8ÀN rule is weakened due to steric compromise is S 4 N 4 .I ti st he same as on the left side of Figure 10, except that Nr eplaces Sw hile Sr eplaces As. Neither atom complies with the 8ÀN rule,w hich would require formation of three short two-electron bonds from the small Ntothe bulky Satom. As the bond lengths and angles in solid S 4 N 4 [44] are irregular, data for gaseous S 4 N 4 [45] are considered here:T he molecule has as ulfur tetrahedron with a42mpoint symmetry and bond lengths of 2.666 and 2.725 , far longer than as ingle bond of 2.055 . [46] One approach to sensible oxidation states is to neglect these weak S-S interactions and consider S 4 N 4 ac yclic tetramer with an SN summary formula, to which DIA applies to yield + 3and À3 for the oxidation states.Bond-based algorithms give the same result ( Figure 11) on as ymmetrical Lewis formula with 22 electron pairs,byconsidering the SNS bond angle of 105.3(7)8 8 is tetrahedral and complemented by two lone pairs of electrons at N, with each Sa tom forming one single bond to another Satom (with a1Àformal charge at Nand 1 + charge at S). Ther eality is somewhere in between these two simplifications:T he S À Nb ond valence calculated with parameters from Ref. [35] with aS À Nb ond length of 1.623(4) [45] is 1.40(2)-more than the single bond of the Lewis formula but not quite the 1.50 required by the 8ÀN rule.T he similarly calculated sum of the homonuclear bond valence at each Satom of the tetrahedron is about 0.5, half of the 1.00 generated by the single bond in the Lewis formula.
In general, the bonding in nonmetallic sp binary compounds C c A a is rationalized with the Zintl concept [47] and its formalization with the generalized 8ÀN rule [48,49] [Eq. (3)].In this relation, VEC A is the valence-electron count per "anion" A, CC is the number of electrons per "cation" Cthat form CÀ Cbonds or are localized at the cation as lone pairs,and AA is the number of electrons per At hat form A À Ab onds.T he VEC A value lends itself to av erbal approach to the rule: Whereas electrons in excess of 8r emain at the less-electronegative atom as bonds or lone pairs,electrons short of 8are gained by forming bonds between the more electronegative atoms.
Foro ur S 4 N 4 example above,w eo btain VEC A = 11. The three electrons in excess of eight remain at the Satom as one SÀSbond and one lone pair per Satom in our Lewis formula; in reality this is somewhat violated, since electronegativity   . .

Angewandte
Essays makes sulfur enforce the 8ÀN rule almost as strongly as nitrogen.
When there is sufficient difference in electronegativity, bonding predictions with the generalized 8ÀN rule are precise.C onsider GaSe with VEC A = 9: Thes ingle electron in excess of 8w ould remain on Ga, thereby forming singlebonded GaÀGa dumbbells.T hat is indeed the case,a nd the oxidation states in GaSe are evaluated by summing bond orders in Figure 12.  [51][52][53][54] are applied first:As an example,B 6 H 10 has 14 valence-electron pairs of which 6are in BÀHsingle bonds,1always is in aradial-skeletal MO,and the remaining 7, in the tangential skeletal MOs,f orm a7vertex parent deltahedron (pentagonal bipyramid), of which one vertex is "missing" in B 6 H 10 (a nido-borane;F igure 13). Its Batoms are not all equivalent. Thefive basal boron atoms are bonded to nine hydrogen atoms that maintain the 2ÀN rule.Summation of the bond orders with apositive sign yields an oxidation state of + 2f or three of these Batoms (those in front) and + 3 = 2 for the remaining two.The apical Batom has an oxidation state of + 1a rising from one single bond to H.

What if the Compound is Metallic?
When bonding and antibonding orbitals/bands overlap in am etal, we are no longer entitled to make the ionic extrapolations as in Figure 1. However,t here are simple metallic compounds with obvious oxidation states,such as the golden TiO, dark RuO 2 ,o rs ilvery ReO 3 .S ome sp elements also form stoichiometric metallic compounds:B a 3 Si 4 obeys the Zintl concept [47] in that it forms butterfly-shaped Si 4 6À anions,i nw hich two Si atoms have two bonds and two Si atoms three bonds to each other according to the generalized 8ÀN rule,but the compound is weakly metallic. [55] Ultimately,the assignment of conducting electrons to one of the two bonded atoms has its limits.A ni ndication of the problem is an unexpected electron configuration or an unexpected bonding pattern. Thef ormer is exemplified by the AuNCa 3 perovskite [56] (Figure 14), where neglecting its metallic character suggests Au 3 À anions,for which there is no support in theory.T he latter may be illustrated on two platinides:red transparent Cs 2 Pt [57] and [58] black BaPt. In line with the 12ÀN rule,C s 2 Pt contains isolated Pt 2À anionsar elativistic sulfide.H owever,B aPt does not have such anions;ithas chains of Pt as if there were adeficit of electrons at the Pt atom. This means that some electrons left Pt 2À to make BaPt metallic,and it is this deficit that is compensated by forming Pt À Pt bonds.I ft hese Pt À Pt bonds were single bonds,t heir chain would be an eutral relativistic sulfur and BaPt would be built of Ba 2+ ,2e À ,and Pt in infinite chains.As BaPt appears formally stoichiometric,the + 2oxidation state for Ba leaves Pt with À2, which does not comply with the actual bonding.
Unsatisfactory oxidation states are also obtained when DIA is applied to ordered alloys with compositions and structures dictated largely by the size,such as LiPb or Cu 3 Au, where the 8ÀN or 12ÀN rules for the most electronegative element are not valid. If an oxidation state is needed to balance redox equations,i ti sb est considered zero for all elements.

Nominal Oxidation States
Theapplications of oxidation state in chemistry are wide, and one value does not always fit all. In systematic descriptive Figure 12. Unit cell of GaSe with one full Ga coordination shown (left, two of the three coordinated Se atoms are outside the unit cell) and the oxidation states determined from its bond graph (right). The bond order dictated by the 8ÀN rule is listed above the three connectivity lines. Below them, the bond valence is listed, calculated from the bond length determined [50] by X-ray diffraction.

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Chemie chemistry,t he oxidation state sorts out compounds of an element;i ne lectrochemistry,i tr epresents the electrochemically relevant compound or ion in Latimer diagrams and Frost diagrams of standard (reduction) potentials.S uch purposeoriented oxidation states that differ from those by definition may be termed nominal, here "systematic" and "electrochemical".
An example of both is thiosulfate.I ts structural properties [59] suggest that all its terminal atoms carry some of the anion charge,e ven if the S À Sa nd S À Ob ond orders are not entirely equal. TheS À Sbond distance of 2.025 [59] is shorter than the single bond of 2.055 [46] in crystalline S 8 or 2.056 [60] in H 2 S 2 gas,b ut substantially longer than the double bond of 1.883 [61,62] in S 2 Oo r1 .889 [63] in S 2 . Although the single S À Sb ond is the closest approximation, two limiting Lewis formulas are considered in Figure 15.
Theformula on the left provides oxidation state À1atthe terminal sulfur atom, reminiscent of the value in peroxides. Thef ormula on right suggests oxidation states that at times are used for aLewis acid-base interpretation of the synthesis reaction S + SO 3 2À = S 2 O 3 2À ,m aking it an onredox process. This is not necessarily an advantage,a st his reaction in an aqueous environment is well-described with half-reaction standard potentials that utilize the average sulfur oxidation state + 2, which represents thiosulfate in Latimer and Frost diagrams-an electrochemical oxidation state.The only route to unambiguous oxidation states for both Sa toms in thiosulfate would be to resolve the S À Sb ond polarity,a si n some textbooks:the terminal sulfur has À2, the central sulfur + 6, independent of their bond order and emphasizing the similarity of the Oa nd Sl igands-a systematic oxidation state.

Non-Innocent Ligands:H 2
J ø rgensen [64,65] coined the adjective "non-innocent" for redox-active ligands that render the oxidations tate of the central atom less obvious.A dditional information from diffraction, spectra, or magnetic measurements is needed. Of the many examples, [66][67][68] the simplest non-innocent ligand is molecular H 2 .
Complexes with molecular H 2 resemble hapto complexes of olefins or aromatic hydrocarbons,e xcept that H 2 only has a s bond. Despite this,H 2 attaches to metal cations even in the gas phase,free of solvents,substrates,and intervening atoms, as elaborated in arecent review [69] by Bieske and co-workers. An intriguing ambiguity arises:w ith some metal ions,t he atoms of the H 2 molecule remain bonded to each other, while with others they form ad ihydride.Crabtree [70] attributes this to the central transition-metal atom having both empty and filled do rbitals:T he former participate in the three-center bonding MO that binds the H 2 moiety while the latter sabotage this by back donation into the empty antibonding MO of H 2 .T he two extreme outcomes are presented schematically in Figure 16.
Theoctahedral complex in Figure 17 is stabilized by the d 6 electronic configuration and the 18-plet on Ru. While the two cis-hydride anions are 2.13 a part, the HÀHd istance in the H 2 ligand is 0.83 ( 0.09 l onger than in H 2 gas). Its bond order is 0.78, alittle more than 2 = 3 for at hree-center bond of equal partners,a nd this can be attributed to the 2ÀN rule working for the more electronegative hydrogen atom.
Areview by Morris [76] of iron-group H 2 complexes shows that the HÀHbond distances vary,uptoabout 1.60 for the largest Os, [77] and also depend on the ligand trans to H 2 :When that ligand is an electron-rich sp atom, such as oxygen or ahalogen [78] capable of strong p donation to the central atom, the H À Hdistance is long. When that ligand is p-acidic,such as CO,orwhen it is electron-poor, the H À Hdistance is short. [76] Thedistance increases with the extent of back donation from the central atom into the s*M Oo fH 2 ,a sc ontrolled by the metals size and by the trans ligand. By our oxidation-state definition, the back-bonding metal atom gets its electrons    [75] obtained by assigning bonds ontothe electronegative partner (left) and by summing ionized bond orders (right).

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Essays back because it is the main contributor to this additional metal-ligand bonding interaction, which is antibonding with respect to the H À Hbond. Allen electronegativities also yield zero for the oxidation state of all such h 2 hydrogen atoms.

Non-Innocent Ligands:N itrosyl
Perhaps the best known nitrosyl complex is nitroprusside. While CN in [Fe(CN) 5 NO] 2À is easy,N Oo ffers three alternatives for the nitrogen oxidation state:N O + ,N O, and NO À .T hey differ in bond order:either j NO j + with OS N = + 3, or j N=O j À with OS N =+1( by DIA), or the nitrogen monoxide of OS N =+2inb etween.
We adopt the bond-valence approach:S ingle-crystal neutron diffraction of Ba[Fe(CN) 5 (NO)]·3 H 2 O [79] yields an NO bond length of 1.12 , shorter than the 1.15 [80] in NO gas.C onsidering that the 8ÀN rule for oxygen will tend to decrease the actual bond order towards two,t he observed bond length suggests j NO j + ,hence Fe 2+ .The diamagnetism of nitroprusside confirms this:T he electron configuration at Fe is low-spin d 6 and OS Fe =+2. Theoctahedral field of strong splitters keeps the low-spin configuration even upon reduction to [Fe(CN) 5 NO] 3À ;itisthe NO + ligand that is reduced to NO not iron. At ruly non-innocent ligand! Many nitrosyl complexes are not as straightforward. The MNO segment should be linear for j N O j + but bent for j N = O j À . [81,82] Thes nag is that the MNO angles vary,i ndicating fractional NO bond orders and problematic oxidation-state assignments. [83][84][85] Enemark and Feltham [86] avoided oxidation states in nitrosyl complexes altogether by adopting a{ MNO} n notation, where n is the number of valence electrons on the metal when the ligand is formally NO + .
Ar ecent study [87] reinvestigated [Fe(CO) 3 (NO)] À ,w hich seems isoelectronic with [Fe(CO) 4 ] 2À in which iron has an oxidation state of À2-a mere replacement of CO with NO + . Something was not right though:the FeNO angle is linear, but the NO bond distance of 1.21 s uggests ad ouble bond j N=O j À .S pectroscopic and quantum-chemical considerations in Ref. [87] brought an explanation:T he Lewis-basic N atom of the j N = O j À anion donates both electron pairs as two p bonds to the Fe central atom (no s bond), thereby linearizing the FeNO angle and validating the double bond within NO.T he resulting oxidation state of 0f or iron is corroborated in Ref. [87] by the diamagnetism of the complex, caused mainly by antiferromagnetic coupling of two unpaired electrons at the tetrahedrally coordinated d 8 iron with two unpaired electrons at the NO À ligand, isoelectronic with O 2 .R ef. [87] therefore also lists these dc onfiguration electrons in an expanded Enemark-Feltham notation: {Fe 8.2 (NO)} 10 (8.2 was calculated). Figure 18 illustrates the stabilizing 18-plet at Fe.

Oxidation-State Tautomerism
Oxidation-state tautomerism, also known as valence tautomerism, concerns thermally induced oxidation-state changes involving redox-active ligands and redox-prone central atoms.M anganese catecholate is an example.A tl ow temperatures,m agnetic measurements suggest [Mn-(C 6 H 4 O 2 ) 3 ]h as one catecholate and two semiquinonate ligands around ac entral Mn atom of oxidation state + 4. [88] At high temperatures,t he magnetic moment suggests reduction to high-spin Mn 3+ upon oxidation of the catecholate to semiquinonate.Lewis formulas for the two ligand alternatives are drawn in Figure 19, where the transition is illustrated and relevant oxidation states evaluated by both algorithms.More examples have been surveyed. [89][90][91] An oxidation-state tautomerism among solely central atoms is exemplified in Ref. [14d].

Oxidation State and d n Configuration
Thec onfiguration d n is ac entral-atom descriptor for transition-metal complexes.Itbecomes tricky when the ligand is bonded by the more electronegative atom as aLewis acid. One of the examples discussed in Ref.
[14e] is [Au{B-(PC 6 H 4 ) 2 (C 6 H 5 )}Cl]. [92] In this adduct, Au populates the Au À Bw eakly bonding MO so that the Mçssbauer spectrum still sees this MO together with the rest of the delectrons at Au as d 10 ,thus suggesting an oxidation state of + 1f or gold, despite the square-planar coordination at Au that is typical of Au 3 + with d 8 (Figure 20). Thes quare-planar Au appears because the donated Au pair became the AuÀBb ond itself,l ost its ligand-field effect, and the coordination geometry is now controlled by the 8e lectrons of Au remaining in the weakly antibonding MOs.Our generic definition also suggests + 1for gold. Fort his oxidation state to maintain the important formula n = NÀOS valid for d n at at ransition-metal atom with N valence electrons, n must also include the weakly

Angewandte
Chemie bonding pair donated by the central atom. To fulfill the equation and avoid the emerging ambiguity exemplified above by the d 10 "spectroscopic" versus the d 8 "ligand-field" or "magnetic" configurations,w ef ollow Parkin [93] by noting the configuration as n = 10 in d nÀ2 ,w here "2" symbolizes aw eakly bonding "donated" pair.

Choices, Estimates, and Round Offs
Fori ntermetallic compounds,t he ultimate choice of the oxidation state zero at all atoms is best if needed in redox chemistry.U sage-related choices also define the nominal oxidation states (see Section 8).
Subtler estimates and round offs are required for compounds with electrons delocalized over non-equivalent atoms, as expressed by several resonance formulas with weights in arbitrarily long decimal numbers.Without round offs of bond orders in Lewis (2). Their decimal values are inherent to the statistical distribution of bonding compromises when the length of agiven bond is compared to an average length of as elected group of reference bonds.Inaddition, an empirical function is used for the bond-length to bond-order conversion.

Outlook on Computational Approaches
Theg eneric definition in Ref. [14] states:" The oxidation state of ab onded atom equals its charge after ionic approximation". Only heteronuclear bonds are extrapolated to be ionic, and the atom to become negative is the one that contributes more to the bonding MO.T he heuristic MO diagram in Figure 1d oes suggest that quantum-chemical calculations might be used to evaluate oxidation states.A s discussed in Appendix CofRef. [14],this carries an inherent degree of ambiguity because of the variety of computational methods available and of the basis-set data to choose from. Within this limitation, ap ossible MO approach might use ag eneralization stating that an atom reversibly contributing more to ag iven MO or MO* of ah eteronuclear bond keeps that MOs electrons, [14k] with ab uilt-in condition that homonuclear bonds are split evenly.T his is illustrated with nitrogen monoxide in Figure 21. Somewhat obscured by the sp interaction, we see that aMOiscloser in energy to one of its two contributing AOs. That AO then receives the MOs electrons upon ionic approximation. When repeated over all the MOs,t he expected oxidation states are obtained (Figure 21).
Am olecule with ah omonuclear bond, N 2 O, is treated similarly in Figure 22. During such a"manual" approach, one has to identify atoms that are actually or predominantly Figure 20. Square-planar Au in [Au{B(PC 6 H 4 ) 2 (C 6 H 5 )}Cl]. [92] Figure 21. Oxidation states in nitrogen monoxide by assigning the MO's electrons to the energetically closest AOs in aM Os et deduced from the orbital energies estimated with an extended-Hückel program. [94] Figure 22. Oxidation states in N 2 Ob yassigning the MO's electrons to the energetically closest AO in aM Oset from an extended-Hückel program [94] calculation. Four unoccupied MOs are omitted.

Angewandte
Essays bonded together by each particular MO.A lthough merely illustrative of ac onceptual suggestion, the oxidation-state approach in Figure 22 circumvents the dilemma encountered in Ref.
[14f] over two alternative Lewis formulas of N 2 O.

ASummary of the Algorithms
As implified approach to the ionic approximation and oxidation states identifies the negative atom by comparing Allen electronegativities with the exception of the more electronegative atom being bonded as aL ewis acid. This approach comes in three algorithms for three different types of chemical formulas (summary formula, Lewis formula, bond graph) covering molecules,i ons,a nd 1D (chains), 2D (planes), or 3D infinite networks of solids.A no verview of the inputs and validity is given in Table 2.

Conclusion
Thes uggested oxidation-state definition justifies both IUPAC algorithms in Ref. [3],while removing exceptions and including some defiant cases,such as those of ligand acceptor atoms with an electronegativity higher than the donor.Being based on chemical bonding,o ur definition does not replace the algorithms needed early on in the chemistry curriculum. It might be helpful at ahigher level. [a] The simplifying use of electronegativity as ac riterion for the ionic approximation (Figure 1v ersus Figure 2) carries an exception:I nreversible adducts of aLewis-acidic atom with an electronegativity higher than its Lewis-basic counterpart, the base keeps the donated pair.
[b] In 1D and 2D networks, possible formal charges of atoms at molecule-likee dges or faces need to be identified on the bond graph (see example in Figure 8).