Toward a definition of valence as a quantity (IUPAC Technical Report)

1 Introduction

The term valence has a long history in chemistry and has carried various and changing connotations. Referring to the atom’s ability to bond, valence enters the name of many chemistry concepts, some descriptive, some more quantitative. However, as a quantity, valence does not have a single and uniquely accepted definition; it is all things to all people so to speak. That is a disadvantage; differing perceptions may obscure communication. In general, valence as a quantity describes a specific bonded atom.

The current IUPAC definition of valence in the Gold Book is from the “Glossary of terms used in physical organic chemistry (IUPAC Recommendations 1994)”. ¹ Valence is “the maximum number of univalent atoms (originally hydrogen or chlorine atoms) that may combine with an atom of the element under consideration, or with a fragment, or for which an atom of this element can be substituted.” No examples are given, and the meaning of “element under consideration” or “fragment” is open to interpretation. The circular wording that defines valence with “valent” is alleviated by stating H and Cl as univalent examples.

After a period of discussions, IUPAC decided to test whether a comprehensive definition of valence could be formulated. A task group was established, with members from the sponsoring IUPAC bodies listed above. In preparation for the project, an anonymized survey was conducted among members of these IUPAC groups in 2016/7, focusing on numerical valence assignments. It confirmed that chemists’ perception of valence as a quantity corresponds to several simple, heuristic, alternative definitions. After a review of quantitative aspects of valence-containing terms and analysis of the valence history up to the current use, the possible definitions of valence as a quantity were tested on 39 chemical entities with several types of formulas. Respecting the IUPAC reflectivity principle meant looking for definitions that are with high consistency behind valence values used in chemistry papers. This Technical Report describes the results of these steps and provides details on the quantities behind the alternative definitions of valence, including their mutual relations. It analyzes the current use of the term valence as a chemical quantity in general and in selected telling examples in particular. This Technical Report will be followed by a formal IUPAC Recommendation on the use of the valence quantity and guide it to avoid ambiguities.

2 The history of the concept of valence

The etymology of valence in Chemistry stems from Latin words valentia (strength or capacity) and valor (worth or value). It expresses the ability of an atom to bond other atoms, the combining power of an element. The sole term “valence” (alternative spelling “valency”) covers the quantitative aspect noticed early on. The outline of how the use of this term in English developed over the years follows. However, we may start several decades before the valence quantity appeared. After the introduction of quantitative chemistry by Lavoisier and Proust, electrochemistry by Davy, constant volume law by Gay-Lussac, and after Dalton’s atomic theory with relative atomic mass, which all enabled the Berzelius’ summarizing law of chemical proportions. ²

Berzelius uses the chemical reaction term “oxidation” as a quantity “degree of oxidation”, even in an article title “Experiments on the Nature of Azote, of Hydrogen, and of Ammonia, and upon the Degrees of Oxidation of which Azote is susceptible”. ³ In Germany, Friedrich Wöhler calls the fixed ratios of oxygen to metal in manganese oxides “Oxydationsstufen” (oxidation grades) on p. 4 of his 1834 textbook Basics of Chemistry, Inorganic Chemistry ⁴ as well as in its 1858 twelfth edition. In 1852, Edward Frankland described in “XIX. On a new series of organic bodies containing metals” ⁵ p. 440 the “combining power” of an element in a compound, which is always satisfied by the same number of involved atoms. First in 1885, Frankland and Japp in the textbook “Inorganic Chemistry” ⁶ use the term valency for this concept when stating that “atomicity, equivalence, valency or atom-fixing power” are the terms to characterize this “combining power.” What happened in the meantime?

In the German-language sphere, August Kekulé proposed in 1857 “Ueber die s. g. gepaarten Verbindungen und die Theorie der mehratomigen Radicale” or About the so-called paired connections and the Theory of polyatomic radicals ⁷ where carbon is “vierbasisch” or “vieratomig” so that it is equivalent to four hydrogens and its atomicity equals 4. Lothar Meyer in his 1864 book “Die modernen Theorien der Chemie” ⁸ favors on p. 67 terms “ein-, zwei-, drei-und vierwerthig”^[1] (with “dreiwerthig N” already in Karl Weltzien’s^[2] paper “Systematic compilation of organic compounds” ⁹ in 1860) and states that “uni-, bi-, tri-, and quadrivalent” would be acceptable as well. In 1865, August Wilhelm von Hoffmann, in his book “Modern Chemistry, Experimental and Theoretic”, ¹⁰ suggests on pages 168–9 to replace the “vague and rather barbarous expression, atomicity” with the term quantivalence and its adjectives monovalent, bivalent, trivalent, etc. In German, the term quantivalence quickly shortens into Valenz. In 1867, Kekulé, then already at the University of Bonn, uses Valenz once in a personal statement inside an article on mesitylene. ¹¹ In 1868, Karl Hermann Wichelhaus, at the Humboldt University of Berlin, has Valenz even in the title of one version ¹² of his article on phosphorus compounds.^[3] The terms Valenz and Wertigkeit persisted, more or less as synonyms.

In 1893, Alfred Werner published his concept of complexes, ¹³ explained their bonding, and proved it with isomerism in the coordination octahedron of two different ligands around a central atom M. That atom’s Wertigkeit, such as M^III in [M(NH₃)₆]X₃ or [M(NH₃)₅X]X₂ or [M(NH₃)₄X₂]X, follows from the number of bonded “einwertiger” (monovalent) “radicals” X (single-charged monoatomic ions). His Valenz of the above [M(NH₃)₅X] “radical” (a 2+ cation) is the difference of the M “wertigkeit” and of the total of “einwertiger” X directly bonded to M, hence 3 − 1 = +2. Coordinated molecules like NH₃ or H₂O do not count. Valenz is the plain number of “einwertiger radikaler” (monoatomic single-charged ions) that can be bonded to the atom/group/ion in question. Werner therefore points out on p. 328 of “Beitrag zur Konstitution anorganischer Verbindungen” (Contribution to the constitution of inorganic compounds) ¹³ that we must distinguish between the valence number (Valenzzahl) and the coordination number. The latter is the count of atoms or groups directly bonded to a central atom. 16 years later, in the revised and expanded 2nd edition of his inorganic chemistry textbook (Newer views in the field of inorganic chemistry) ¹⁴ about the constitution of inorganic compounds, Werner introduces two terms that are still in use besides the coordination number; Hauptvalenz and Nebenvalenz, translated as primary and secondary valence in the IUPAC Red Book 2005 Recommendations ¹⁵ p. 144. Werner’s measure of primary valence is the number of strictly monovalent atoms (like hydrogen) that the atom in question would bond, denoted as its valence number on p. 18 and 62 of his book ¹⁴ . That atom’s secondary valence, p. 50 and 62, bonds molecules able to exist on their own. The coordination number introduced on p. 52 is the number of atoms directly bonded to the central atom regardless whether by primary or secondary valency.^[4]

In the first decade of the 20th century, it was realized that the English expression valence number and likewise German terms Valenz or Wertigkeit need to adopt both positive and negative values in order to simplify teaching redox reactions. This was pointed out by Richard Abegg for “valenz” in 1904 ¹⁶ (p. 344). In his 1909 edition of “Handbuch der Anorganischen Chemie” Vol. III., Part 2, ¹⁷ Abegg uses positive and negative “valenz” (p. 33), whereas the more frequent “wertigkeit” appears solely as a positive number (p. 455), even when it refers to anionic charges on p. 167. In the period 1909–1915, Nelson et al. published a series of papers on “The Electron Conception of Valence.” In the fourth paper, ¹⁸ they state: “The valence of an element may be defined as the number of corpuscles (negative electrons) of an atom of that element loses or gains to form chemical bonds.” They also assumed that every chemical bond formed between two atoms involves transfer of such a corpuscle from one atom to another.

In 1913, William Bray and George Branch suggested in a paper “Valence and Tautomerism” ¹⁹ splitting the term valence number into two quantities: polar number and (total) valence number, the former according to usage in inorganic chemistry and the latter in organic chemistry. For example, N in NH₄Cl would then be assigned −3 by inorganic chemists and 5 by organic chemists. In the same paper, a footnote states that the “polar number” is synonymous with the German Wertigkeit. The same year, Böttger in the 3rd edition of his book on qualitative analysis ²⁰ on pages 97–101 illustrates that wertigkeit of ions is a plain number of positive or negative ionic charges, while stating that for elements (atoms) the prevailing view is that wertigkeit is given by positive or negative integers. Two numerical examples follow later on, −1 in chloride and −3 of N in ammonium cation. His verbalization is to say that, for example, O in H₂O is negativ zweiwertig, enforcing a view that wertigkeit still is a plain number to which a + or − sign can be attached. Nernst in “Theoretische Chemie” ²¹ p. 386 keeps strictly the conservative view “…Ionen, welche mit der gleichen menge von Elektrizität geladen sind, wie das Wasserstoff-oder Chlorion, nennen wir einwertige…” (…ions charged with the same amount of electricity, such as hydrogen- or chlorine-ions, are 1-wertige…). In 1915, Fry published “The electronic conception of positive and negative valences”. ²² Gilbert Lewis also wrote about valence, in 1913 and 1916. ^{23 , 24} In “The atom and the molecule”, ²⁴ he introduced a predecessor of the octet rule, the “theory of a cubical atom” to account for what had become known as “Abegg’s law of valence and countervalence”, ¹⁶ in which the total difference between the maximum negative and positive valences (or polar numbers) of an element is frequently eight and in no case more than eight.^[5] In 1918, Joel Hildebrandt in his “Principles of Chemistry” ²⁵ states on p. 85 “…we must … distinguish between two kinds of valence … by calling one positive and the other negative” and illustrates this by many examples, like “In Al₂O₃, the valence of Al is 3, because the valence of oxygen is −2” on p. 86. On p. 98, Hildebrandt lists the adjectives n-valent with either Greek or Latin prefix for positive n (not for negative valences) in a nomenclature suggestion.

By 1920, two types of valence as a numerical variable were in use; the polar number (valence) of positive or negative sign (also termed electrochemical valence or electrovalence ²⁶ ) and the stoichiometric valence (the maximum number of hydrogen atoms that may combine with an atom of the element under consideration). Both were colliding with the term valence as a description of bonding, and that conflict afflicted the German term Valenz as well. In a book, “Chemische Valenz-und Bindungslehre”, ²⁷ Fritz Ephraim commented that “die Valenz Zahl und nicht etwa Kraft ist” (valence is a number not some force), here quoted from a series of historical sketches by Arthur Berry ²⁸ p. 177. The terms polar number or polar valence number dominated in articles around this time. In 1927, yet another valence interpretation appeared in the book “The electronic theory of valency” by Sidgwick ²⁹ with absolute valency defined on p. 182 as the number of electrons of the atom engaged in attaching the other atoms, quoting Grimm and Sommerfeld. ^{30 [6]}

In the 1930s, the (polar) valence was replaced in two papers by Arthur Noyes et al. on “argentic salts” by the “state of oxidation” ³¹ or “oxidation state”. ³² Here, it may be noted that the term “state of oxidation” apparently first appeared in a 1901 paper by Leonard Morgan and Edgar Smith on chalcopyrite. ³³ In 1938, Wendell Latimer published his groundbreaking book on redox half-reaction electrochemical potentials listed as a function of the oxidation state: “The Oxidation States of the Elements and their Potentials in Aqueous Solutions”, ³⁴ where he used “oxidation state” and “oxidation number” as established terms. He considered them synonymous with German wertigkeit that he understood to mean “valence or polar number.” In 1940, a revised edition of Latimer and Hildebrand’s “Reference Book of Inorganic Chemistry” was published. ³⁵ In its Glossary, the terms oxidation state or oxidation number are defined as “The charge on a simple ion or for a complex ion or molecule: the charge which is assumed on an atom to account for the number of electrons involved in the oxidation (or reduction) of the atom to the free element.” The same year, Joel Hildebrand’s 4th edition of “Principles of Chemistry” has oxidation number or state, stating explicitly on p. 120 that these replace the earlier term valence.

In 1941, IUPAC issued its first set of rules for naming inorganic compounds and writing their formulas ³⁶ with the electrochemical valence [sic] following the atom symbol or name as a roman numeral inside bracket. For redox aspects, the term “valency stage” emerged, with a synonym “oxidation stage” for oxygen compounds. By 1948, oxidation state (or oxidation number) is used instead of the original valency. ^{37 , 38} At the 14th IUPAC general assembly in 1947 London, Linus Pauling presided over section II of the Congress. In the same year, he published a paper “The modern theory of valency” ³⁹ advocating the need to “dissociate the concept of valency into several new concepts; ionic valency, covalency, metallic valency, oxidation number.” He compares the bonding aspect of valency with these numerical values. Thus, about ionic valency he argues that the actual ionic charges in an ionic compound are smaller than their ionic valency given by the positive and negative ionic charges (analogously for the numerical value of covalency). He then states that the positive and negative valences have a special redox nature that justifies designating them by the name oxidation number and proceeds to state four rules for its values. With such a more precise term now available, Pauling wonders “whether the word valency, without qualification, need ever be used.” His answer “I think that it may be used as a synonym for one of the more precise concepts when its meaning is clear from the context. Thus, in the classification of the compounds of iron, we may include the ferrocyanides among the compounds of bivalent iron…” is his suggestion to use it as an adjective for oxidation number if the context is clear. He concludes the paper ³⁹ by a statement on valency as a concept of computation-based bonding theory: “If scientific progress continues, the next generation may have a theory of valency that is sufficiently precise and powerful to permit chemistry to be classed along with physics as an exact science.”

From the 1950s onward, the concept of numerical valence has not changed much. As the noun valence denotes ability, its numerical adjective n-valent sounds natural for simple counts. In inorganic chemistry, it occasionally expresses the oxidation state, a term without such adjective. In physical organic chemistry, the definition in the IUPAC Recommendation ¹ entered the current IUPAC Gold Book: “The maximum number of univalent atoms (originally hydrogen or chlorine atoms) that may combine with an atom of the element under consideration, or with a fragment, or for which an atom of this element can be substituted.” Another suggestion appeared in 1995, when Green ⁴⁰ outlined the Covalent Bond Classification (CBC) for coordination compounds with the recycled term “valency number” in the Werner sense ¹³ as a number of bonding pairs formed by sharing electron from both atoms. In 2006, Parkin ⁴¹ prefers “valence” according to the 1927 Sidgwick ²⁹ suggestion that never took off; the number of an atom’s electrons in bonds. The CBC movement revised its “valence number” definition to the Parkin’s one ⁴¹ around 2010 in an array of articles. ^{42 , 43 , 44 , 45} In 2022, the “Glossary of terms used in physical organic chemistry” ⁴⁶ defined valence as the “Maximum number of single bonds that can be commonly formed by an atom or ion of the element under consideration.” The adjective “commonly formed” takes the focus from the actual atom in a given compound to the element in general and involves a degree of opinion that is arguably inconsistent with anything definitional.

As for valence-history studies, in 1952, C. A. Coulson published a book titled “Valence”. ⁴⁷ A 1965 book by W. G. Palmer “A history of the concept of valency to 1930” ⁴⁸ is an extended version of his lecture notes for a short course on valence at Cambridge University. A few years later, in 1971, Colin Russell published his four-part book “The history of valence”. ⁴⁹ The Part 1 deals with origins, nomenclature, and notation. Part 2 describes the structure applications of valence. Part 3 focuses on variations in valence and Part 4 on valence versus electric charge, up to wave mechanics. In 2005, Smith ⁵⁰ in the paper “Valence, covalence, hypervalence, oxidation state, and coordination number” dealt with historical aspects of valence and similar terms. In 2015, Nelson gave an account of terms related to valence in his article “Modern version of Lewis’s theory of valency”. ⁵¹

3 The current use of the term valence

The term valence enters some chemistry concepts, but it can also denote a quantity. The relative proportion of these two aspects of its use can only be estimated. We approached it by searching chemistry textbooks. The result shows that the use of the word valence as a quantity is minor compared with its use in chemistry-related non-quantitative concepts, and it differs between organic and inorganic chemistry.

4 Possible definitions of valence as a quantity

The quantity concepts identified in the survey (Section 3.3) are candidates for the definition of valence as a quantity. Two other quantity concepts of valence, not discerned in the survey, were added; the current IUPAC definition of valence ¹ and the bond-valence (bond-order) sum at an atom. As the quantities are often determined on a Lewis formula, a complementing concept of formal charge at the atom was included, which also covers ionic charges.^[7] Fig. 1 illustrates these quantities on a Lewis formula of carbon monoxide, where each dash represents an electron pair. As we are about to test these quantities via their verbal definition or algorithm, we will keep referring to them by their explanatory names. In tables, we will use suitable shorthand to avoid needless inventing of symbols. An explanatory list of these quantities (in bold) with some verbal algorithms follows.

Fig. 1:

Quantities tested as definitions of valence (and the formal charge) illustrated on octet-obeying Lewis formula of carbon monoxide with triple-bond approximation that yields non-zero formal charges.

The number of electrons an atom uses in bonds ²⁹ was suggested ⁴¹ in 2006 as a definition of valence and later on used by the related group of authors as “valence number”. ^{42 , 45} On a Lewis formula, it is counted as the number of an atom’s valence electrons that are not in the essentially nonbonding “lone pairs”.

The number of bonding pairs at an atom or co-valence ^[8] is a formal number of predominantly covalent bonding-pair equivalents counted on the formula. In Fig. 1, it corresponds to the number of dashes denoting a two-electron bonding orbital, but in structures of other compounds it can be a sum of bond-order fractions. The co-valence does not include ionic interactions via electrons not drawn on the formula of an isolated cation or via electrons already bonding inside the formula of an isolated central-atom ion.

The formal charge, a form of electrovalence,^[9] is obtained on a Lewis formula by bisecting all bonds and counting the resulting charge at each atom. In Lewis formulas of neutral molecules, formal charges compensate each other, as in Fig. 1. Carbon monoxide indeed has an experimental dipole moment ^{80 , 81} with its negative end at the carbon atom, and the actual bond order is about 2.6. ^{82 [10]} The formal charge at an atom depends on the setup of the Lewis formula. The formula may indeed be drawn to obey the 8−N rule (Section 10.1) that originates with Hume-Rothery: ⁸³ An sp-block atom of N valence electrons tends to form 8 − N two-electron bonds with atoms of equal or lower electronegativity.^[11] While formal charges on isolated single-atom anions are trivial in this aspect, for oxo-anions, or anions with electronegative ligand atoms in general, it is more realistic to apply the 8−N rule. The anion formula is then drawn with all ligand-atoms’ ionocovalent bonds as dashes with denoted bond order. It keeps all formal charges zero on the formula of that isolated anion and its bond orders close to reality (see Fig. 6).

The number of bonding pairs sharing electrons of both atoms is the valence of the Werner ¹³ type, obtained by summing bond orders of such bonds at the atom in the given formula. It requires attention to avoid counting donated bonding pairs.

The bond-valence sum ⁷³ or bond-order sum ⁷⁴ (p. 1029) is a sum of bond valences (Section 3.2.3) or bond orders at an atom. The former is used on bond graphs (Section 4.2) of extended structures, and the latter term may better suit molecules. At times, idealized representations occur. For bond graph, see Fig. 9; for idealized Lewis formula of a molecule, see Figs. 1 and 2.

Fig. 2:

Quantities tested as definitions of valence (and the formal charge) evaluated on a Lewis formula of carbon monoxide that complies with both octet rule and 8−N rule.

The oxidation state of an atom is the charge of this atom after ionic approximation of its heteronuclear bonds. ⁸⁴ Besides direct ionic approximation limited to simple formulas ⁷⁴ (p. 1025), two general algorithms, subject to a caveat, calculate its quantity value ⁸⁴ using the Allen-electronegativity criterion. Algorithm 1: Oxidation state equals the charge of an atom after its homonuclear bonds have been divided equally and heteronuclear bonds assigned to the more electronegative bond partners. Electronegative atom bonded reversibly as a Lewis-acid ligand (electron-pair acceptor) does not obtain that bond’s electrons. Algorithm 2: The sum of heteronuclear bond orders by their polarity sign at the atom, plus any formal charge on that atom. For electronegative atom bonded reversibly as a Lewis-acid ligand (electron-pair acceptor), the formal bond polarity is inverted. Both algorithms work fine on CO of Fig. 1. After assigning bonds, C has a lone pair, the two electrons of which we subtract from the N = 4 valence electrons of C to yield the oxidation state +2. Oxygen has 8 electrons, the subtraction of which from N = 6 gives the oxidation state −2. In the second algorithm, the positively taken bond order at C is summed with the formal charge −1 to +2, and the negatively taken bond order at O is summed with +1 to −2. Direct ionic approximation on CO formula yields the same.

The absolute value of oxidation state might better resemble valence. The term valence is rarely used for negative values.

The present IUPAC definition of valence (Section 1) is tricky on the Lewis formula of Fig. 1. It asks us to find maximum number of univalent atoms that combine with an atom of the element under consideration. In our Lewis formula, that certainly is not a plain carbon as an element in general. We ask, what is the bonding state (electron configuration) at our atom under consideration in the Lewis formula? The carbon under consideration is |C⁻. It may combine with 3 H into the carbanion |CH₃ ⁻. Likewise, the |O⁺ may combine with 3 H into the oxonium cation H₃O|⁺. Hence, both valences are 3. See also Sections 5.1 and 5.3.

The number of bonded nearest neighbors is an explanatory description of the common term coordination number.

While the Lewis formula in Fig. 1 nicely illustrates all valence-type quantities considered here, it is not the only one possible. Applying the 8−N rule to CO, we limit the number of bonds at the most electronegative atom, oxygen, to two, as the oxygen atom lacks only two electrons to obtain its stable configuration. Such a formula simplifies the results obtained for the “valence” quantities on CO. The number of electrons the atom uses in bonds is “per definition” the one dictated by this 8−N rule (Fig. 2). It is also equal to the number of two-electron bonds at the atom. The formal charge is zero on both atoms. As no bonds are donated, the (formal) number of electron pairs sharing both atom’s electrons equals the number of bonding pairs at the atom; the bond-valence (bond-order) sum likewise. The oxidation state naturally remains the same. As mentioned earlier in this Section, the reality of the CO molecule is a compromise between the Lewis formulas in Figs. 1 and 2.

4.1 Applied on Lewis formulas of ions of several atoms

As illustrated above in the example of CO with formal charges, the quantity values depend on whether the chemical bonding formula respects the 8−N rule or not. Figs. 3 and 4 illustrate this fact on two limiting Lewis formulas for the NH₄ ⁺ cation and Figs. 5 and 6 for the BF₄ ⁻ anion.

Fig. 3:

Quantities tested as definitions of valence (and the formal charge) on conventional Lewis formula of NH₄ ⁺ that disrespects the 8−N rule.

Fig. 4:

Quantities tested as definitions of valence (and the formal charge) on a limiting Lewis formula of NH₄ ⁺ that obeys octet rule as well as the 8−N rule (bond orders are in blue).

Fig. 5:

Quantities tested as definitions of valence on octet-obeying Lewis formula of BF₄ ⁻.

Fig. 6:

Quantities tested as definitions of valence on Lewis formula of BF₄ ⁻ that obeys the octet rule as well as the 8−N rule (bond orders are in blue).

4.1.1 Ammonium cation

The Lewis formula of NH₄ ⁺ that respects the octet rule is drawn in Fig. 3 with four bond dashes of the available 8 valence electrons. The number of electrons nitrogen uses in bonds is 5 (nitrogen’s valence electrons minus the 0 in lone pairs). The number of bonding electron pairs at nitrogen is 4 (the co-valence, so we do not include the ionic bonding), and its formal charge is +1 (the electro-valence). Each hydrogen uses 1 electron in bonds, considering that the proton accepting the nitrogen pair has arisen from an H atom that used its electron to bond the unexpressed anion (e.g., Cl⁻). We see one bonding pair at each H and hence zero formal charge.

On the second row left, only 3 bonding pairs at N atom share electrons of both atoms; one pair is donated. As the formal ionic charge 1+ resides on N that already has sp ³ configuration, one can argue against counting this ionic bond as an ionocovalent shared pair on nitrogen. Prior forming the NH₄ ⁺, the fourth H atom became the Lewis-acid H⁺ by giving an e⁻ to form an anion (e.g., Cl⁻). One might argue that the Cl⁻ remains bonded to the + charge by hydrogen’s electron and that could justify the valence 1 at that fourth H atom in this formula that leaves the anion unexpressed. This Lewis formula is somewhat unclear on this particular definition of valence.

The bond-valence (bond-order) sum on nitrogen in this formula is 5, composed of 4 single bonds to H and 1 ionocovalent bond of the + ionic charge with any number of close contacts to unexpressed anions totaling 1− ionic charge. A hypothetical purely ionic bond of 1+ and 1− charges is a two-electron bond (physically located at the unexpressed anion in this ionic extrapolation). The oxidation state of N is −3 by both algorithms (Section 4): The 8 electrons around N of 5 valence electrons yield −3. The sum of negatively taken four single-bond orders plus the formal charge N⁺ yields −3 as well. So does the direct ionic approximation performed on the NH₄ ⁺ formula.

On the third row, the present IUPAC definition of valence (Section 1) is as if already applied in this four-bonded N⁺ combined with 4H, hence tetravalent. The last value, the number of nearest bonded neighbors of H atoms in this formula is 1, as this quantity remains uncertain on N because we consider the cation as isolated and do not know the number of neighbors.

The problems encountered in Fig. 3 vanish on the other limiting Lewis formula that respects both the octet and the 8−N rule (Fig. 4). It has fractional bond orders labeled on the dashes. We count 8 valence electrons per NH₄ ⁺ but demand that nitrogen only uses 3 electrons for bonding; hence, the bond order is ¾. That is not far from reality, as the N–H bond length 0.988 Å (98.8 pm) obtained by neutron diffraction on crystalline hydrogen-bonded ND₃ at 77 K ⁸⁵ increases to 1.042 Å (104.2 pm) at 80 K in ND₄NO₃. ⁸⁶

The number of electrons nitrogen used in bonds equals the number of atom’s valence electrons minus those in nonbonding pairs. In this formula, it is dictated by the 8−N rule on nitrogen. Hydrogen used ¾ of an electron to bond N and ¼ to bond the unexpressed anion. The number of bonding electron pairs (co-valence) at nitrogen is 3 whereas ¾ at the hydrogens where we do not include the ionic bonding. The formal charge (electro-valence) at the electronegative nitrogen atom is 0 per the 8−N rule and ¼ of the ionic charge at each hydrogen.

The formal number of bonding pairs that share electrons of both bonded atoms is 3 at nitrogen and 1 at hydrogen, composed of ¾ of an electron shared with N and ¼ shared with the unexpressed anion, say Cl⁻, as there is no such thing as a purely ionic bond. The bond-valence sum concerns all ionocovalent interactions, hence 3 at nitrogen and 1 at hydrogen.

Oxidation states obtain by either of the two algorithms. Hydrogen: First algorithm leaves H with no electrons and +1 oxidation state. Second algorithm sums the bond orders ¼ and ¾ at this electropositive atom to the oxidation state +1. Nitrogen: First algorithm takes the octet at N and obtains oxidation state as 5 − 8 = −3. Second algorithm sums the bond orders to −3. The present IUPAC definition of valence (Section 1) applies to nitrogen with one lone pair and bond-order total 3. Such |N≡ would bond 3 hydrogens. With how many Cl would hydrogen of ionic charge ¼+ bond? With ¾ Cl to (HCl_¾)^¼+. Nitrogen has 4 nearest bonded neighbors. With hydrogen we do not know.

4.1.2 Tetrafluoridoborate(1−) anion

The Lewis formula for BF₄ ⁻ of 32 electrons that respects only the octet rule (and has the 1− ionic charge unlikely placed at boron) is in Fig. 5. The evaluations are as follows:

Boron uses 3 own electrons in bonds with 3 F that each uses 1 electron. How many electrons did the fourth fluorine use? Upon formation of this anion from the BF₃ Lewis acid and F⁻ Lewis base, the latter donated 2 electrons, but one of them (yellow, Fig. 5) it obtained from the unexpressed cation-forming element (e.g., K → K⁺ + e⁻). Hence, also the fourth F atom used just 1 own electron (green). The simple way to count this quantity is as the atom’s valence electrons minus those in the atom’s nonbonding pairs.

Boron has 4 bonding electron pairs that represent its co-valence; hence, we do not include the ionic bonding via its formal charge −1 due to electrons that already bond B and F. Formal charges obtain by bisecting all bonds, summing electrons in adjacent halves to each atom, and subtracting that sum from the valence-electron number of that atom. Fluorine has 1 bonding electron pair and formal charge zero.

The number of two-electron bonds sharing electrons of both atoms is simple to check on the boron that shares all 3 of its electrons and was donated one bond. One F atom formed the Lewis-base F⁻ with electron from K that became K⁺. The F⁻ then donated one electron pair with one own (thus shared) electron to B. The K⁺ keeps its ionic bond to the 1− charge of that donated pair. With sp ³ configuration at B, one can argue against counting this ionic bond as a fifth ionocovalent shared pair on boron.

While the sum of bond valences at each F is 1 single bond, the boron has two sets of ionocovalent bonds: 4 single bonds to the F atoms and the ionocovalent contacts to the unexpressed cations (in the crystal or solution) that total to bond order 1. The oxidation state is straightforward both by assigning bonds to the electronegative fluorine or by summing the formal charge with the sum of bond orders taken positively at B and negatively at F. So is the direct ionic approximation on BF₄ ⁻ of 32 electrons forming octets at F. The present IUPAC definition of valence (Section 1) is straightforward in this Lewis formula; a single bond on F suggests 1 while the B⁻ bonding to 4 fluorine atoms gives 4. The coordination number we see only at the boron atom, and only if we neglect the “ionic” bonds to its unexpressed cation neighbors.

The problems of the Lewis formula in Fig. 5 vanish upon respecting electronegativity (fluorine atoms must carry the anion charge and ionocovalent bonds) and applying the 8−N rule. The result is in Fig. 6.

In this formula of the anion, the number of electrons the atom used in bonding ( N valence electrons of the element minus those in nonbonding pairs) and the co-valence (number of two-electron bonds) are the same. All formal charges (electro-valence) are zero because the 32 electrons of the anion include those that make the ionocovalent bonds to the unexpressed cation, and one obtains 7 − [6 + ¼ + ¾] = 0 at F and 3 − 4 × ¾ = 0 at B.

The formal number of bonding pairs sharing electrons of both bonded atoms is 1 at fluorine (¾ of an electron shared with B and ¼ shared with the unexpressed K⁺ cation as there is no such thing as a purely ionic bond) and 3 at boron (4 × ¾) = 3. As all bonds are drawn, the bond-valence sums give the same values. The oxidation state by assigning bonds is straightforwardly 3 for boron of no electrons after all bonds are assigned to fluorine that obtains −1 by subtracting electrons in bonds assigned to it from its number of valence electrons, 7 − [6 + 2 (¾ + ¼)].

The present IUPAC definition of valence (Section 1) is straightforward here as the B fragment of no nonbonding electrons would bond with 3 Cl atoms (per the definition), and fluorine with 3 lone pairs and total bond order 1 would bond 1 H. The coordination number of boron is 4 in the tetrahedron but remains uncertain at fluorine since we artificially consider an isolated anion.

4.2 Applied on bond graphs of extended solids

A bond graph is a finite representation of an extended (non-molecular) solid, ^{87 , 88} a bonding formula of the crystal. On the periodic unit of suitably distributed atoms, one draws a line for each instance of direct bonding connectivity between near-neighbor atoms in the structure. The line does not have the meaning of a two-electron bond; it carries the specific bond order defined by the chemical element and the local symmetry that controls the connectivity. Such a bond order is called bond valence (Section 3.2.3). Two examples of extended solids follow.

4.2.1 Cesium trioxidoiodate

The crystal structure of this salt has an infinite network of IO_6/2 ⁻ octahedral anions with dodecahedral voids filled by Cs⁺. Its cubic perovskite cell in Fig. 7 shows that iodine coordinates 6 oxygens of the anion, while the CsIO₃ formula requires that the bond-order sum on iodine is 5. Each Cs⁺ bonds to 12 iodate oxygens, and its bond-order total must equal 1. The bond graph on the right visualizes the bond-valence sums at each atom. The valence-type quantities derived on this bond graph are in Fig. 8. For oxidation state, the ionic approximation applies in the bond-order summing (with the + sign on Cs and I and with the − sign on O). We see that, except for the coordination number, all alternative valence definitions yield the same absolute value.

Fig. 7:

Cubic unit cell of the perovskite-type CsIO₃ structure and its bond graph with bond orders (in blue) summing to the expected total (bond-valence sum, in red).

Fig. 8:

Valence-type quantities derived from the bond graph of CsIO₃ perovskite in Fig. 7.

4.2.2 Calcium dicarbide

In the densest-packed Ca²⁺ spheres and C₂ ²⁻ ellipsoids of the CaC₂ crystal, calcium is coordinated by 8 carbons at a distance of 2.83 Å and by 2 carbons at 2.55 Å ⁸⁹ (Fig. 9). Because of the two different calcium–carbon distances, there is no unique set of idealized bond orders, but a simple approximation helps: A bond-order calculation by the bond-valence method ⁹⁰ from the above bond distances suggests the longer distance to be about half the bond order of the shorter distance. With this condition and bond order 3 in the C₂ ²⁻ anion, the idealized bond graph in Fig. 9 has one Ca–C bond of order 1/3 and four of order 1/6, summing with 3 to the total 4 at C.

Fig. 9:

Unit cell of CaC₂ with added anions to see the Ca²⁺ coordination (left). Bond graph of idealized bond orders (in blue) into bond-valence sums (in red).

The quantity values derived by counting on the bond graph in Fig. 9 are in Fig. 10. Because Algorithm 2 does not count homonuclear bonds, the oxidation state of carbon now differs from the other quantities. What could make the bond-valence sum differ from the number of electrons the atom uses in bonds? Dative bonds in the network, formed by electrons of the sole donor (see overview in Table 11 of Appendix A). What could make the bond-valence sum differ from the number of bonding pairs at the atom? The single ionic bond of “conventional” formulas for NH₄ ⁺ (Fig. 3) and BF₄ ⁻ (Fig. 5) ions, which counts into the bond-valence sum at the central atom but not into its co-valence. No such features appear in the CaC₂ bond graph in Fig. 9.

Fig. 10:

Quantities tested as definitions of valence on the bond graph of CaC₂ in Fig. 9.

4.3 A case with Lewis-acid ligand

This rare case is illustrated by a complex of the Au central atom bonded to Cl and a tridentate ligand of aromatic rings and four isopropyls (Fig. 11), synthesized from AuCl-based precursor and the ligand molecule. ⁹¹ The central atom has an unusual combination of bonding; one bond it shares, one bond it donates to Lewis-acid B, and two it accepts from Lewis-base P in ClAu[ ] with bond triplets to carbons. Fig. 12 illustrates evaluation of the first three valence-type quantities that also relate to each other; on Lewis formulas, the number of electrons an atom uses in bonds equals the number of bonding pairs on it plus its formal charge.

Fig. 11:

The complex with relevant segment ClAu[ ] of two Lewis-base P donors and one Lewis-acid B acceptor (left). Lewis formula with formal charges counted upon bisecting bonds (right).

Fig. 12:

The ClAu[ ] of three bond types. First three valence-type quantities.

The other three valence-type quantities are in Fig. 13. Due to dative bonds, the number of bonding pairs via sharing electrons of both atoms is low. The bond-valence (bond-order) sum corresponds to the number of bonding pairs at each atom in Fig. 12. How to apply the current IUPAC definition of valence (Section 1) on the “element under consideration”? The four-bonded P⁺ of the Lewis formula in Fig. 11 would combine with 4 H to PH₄ ⁺. Analogously, the four-bonded B⁻ would combine with 4 F to BF₄ ⁻ and the four-bonded Au⁻ with 4 F to AuF₄ ⁻.

Fig. 13:

The ClAu[ ] of three bond types. Second three valence-type quantities.

The oxidation state is obtained by ionic extrapolation via Allen electronegativity (in the so-called Pauling units, symbol PU in Mann et al. ⁹² ) of Au, B, P, C, and Cl = 1.92, 2.051, 2.253, 2.544, and 2.869, respectively, listed also in Karen et al. ⁸⁴ The ligand-exchange synthesis from AuCl(H₃C–S–CH₃) ⁹¹ and the stable tridentate ligand of the product suggest application of the caveat of the electronegative Lewis-acid ligand (Section 4) for boron in the Au–B bond. Both algorithms yield the same oxidation states in Fig. 14.

Fig. 14:

The ClAu[ ] of three bond types. Oxidation state (red) by assigning bonds or by summing bond orders (blue) with formal charge (black), both by allen electronegativity with the Lewis-acid caveat (Section 4) applied on boron.

5 Example pairs and current use of valence in them

In many of the evaluated example compounds, the quantities calculated with the alternative valence definitions had different values. Since reflection is important principle of the IUPAC normative work, we analyzed how the alternative definitions reflect the current, even if relatively modest, use of the term valence in English. For each of such telling examples, we asked the following question: Do chemists say that an atom in this compound is k-valent or l-valent or m-valent? The answer was obtained with Google Scholar search at several geographical locations (somewhat different results) for articles in which we then analyzed the context of each adjective n-valent with common variants of the prefix n, such as di- and bi- or tetra- and quadri-.

5.3 Cr₂(CH₃COO)₄ and CrO₃

The paddlewheel molecule of dichromium(II) tetraacetate dihydrate crystal is in Fig. 15 left, without the two H₂O molecules weakly bonded to Cr along the wheel axis. The Cr–OH₂ bond-order contribution is 0.22 by bond-valence calculation of O’Keeffe and Brese ⁹⁰ from atomic coordinates in the crystal, published by Cotton et al., ⁹³ where the water bonds two other paddlewheels by a hydrogen bond of O···O distance 2.84 Å (284 pm). The paddlewheel molecule is highly symmetrical as all carbon–oxygen distances are the same and so are all chromium–oxygen ones. It is compared with another wheel, CrO₃, in Fig. 15 right.

Fig. 15:

Lewis formulas of dichromium(II) tetraacetate molecule of 104 valence electrons and of CrO₃ with 12 electron pairs. The numbers of electrons the atom uses in bonds are in red.

Table 5 lists quantity values obtained by testing the alternative definitions of valence on the two formulas of Fig. 15. The results are self-explanatory. The present IUPAC definition of valence refers to each Cr atom’s 6 two-electron bonds that would bond 6 Cl.

Table 5:

Valence-type quantities evaluated on dichromium(II) tetraacetate and CrO₃ Lewis formulas in Fig. 15.

Cr(II) acetate	#e	#bp	f.c.	#p.sh.	BVS	OS	| OS |	IUPAC	CN
Cr	6	6	0	6	6	+2	2	6	5
O	2	2	0	2	2	−2	2	2	2
C carboxyl	4	4	0	4	4	+3	3	4	3
C methyl	4	4	0	4	4	−3	3	4	4
H	1	1	0	1	1	+1	1	1	1
CrO 3	#e	#bp	f.c.	#p.sh.	BVS	OS	| OS |	IUPAC	CN
Cr	6	6	0	6	6	+6	6	6	3
O	2	2	0	2	2	−2	2	2	1

1. Shorthand symbols of quantity names: #e = number of atom’s electrons in bonds; #bp = number of bonding pairs at the atom (co-valence); f.c. = formal charge in the Lewis formula (auxiliary quantity); #p.sh. = number of bonding pairs sharing electrons of both atoms; BVS = bond-valence sum or bond-order sum; OS = oxidation state.

The relevant question was: While chromium in CrO₃ is hexavalent, do chemists say that dichromium(II) tetraacetate dihydrate has divalent or hexavalent chromium? The answer was simple. Nobody says that Cr in dichromium(II) tetraacetate is hexavalent. In five papers, it is divalent, in one of those in a similar compound. In additional 14 papers, other compounds of Cr^II are described as divalent. See also overview of these comparative examples in Table 10.

5.4 ReF₇ and ReCl₃

Halogenides of second- and third-row transition metals form often molecular or network solids. Two such examples are compared here. While ReF₇ is a molecule of a shape close to trigonal bipyramid, ⁹⁴ ReCl₃ is a crystalline solid of Re₃ clusters bonded to Cl, bridged by Cl, and joined with other clusters by Cl into a 3D network. ⁹⁵ The network repeat unit in Fig. 16 also lists the bond lengths. Bond-valence calculation with parameters of O’Keeffe and Brese ⁹⁰ from the actual bond lengths yields the following bond-order estimates for the chlorine bridges (for the nomenclature μ convention, see Connely et al. ¹⁵ p. 208). The sole Cl–Re “µ₁” bond has bond order 1.137, each Cl–Re bond of µ₂ bridges on Re–Re edges has 0.800, and the µ₂ bridges between repeat units have one Cl–Re bond order 0.785 and the other 0.442. The >1 bond-order sum of bridging halogens justifies the standard electron counting by two-electron bonds. Counting electrons at the Re₃ cluster of the Re₃Cl₆Cl_6/2 repeat unit via “neutral atoms” (Housecroft and Sharpe ⁹⁶ p. 815) yields 21 electrons from three Re, 3 electrons from the three single-bonded Cl, 3 (2 + 1) = 9 electrons from the three µ₂ bridges on the Re₃ edges, and 6 (2 + 1)/2 = 9 electrons from the six inter-unit bridges. In total, 21 + 3 + 9 + 9 = 42 electrons. With g = 42 for the n = 3 Re atoms, the generalized 18-plet formula for clusters, N _BO = (18 n − g )/2 derived in Section 10.6, yields six orbitals that bond the Re₃ cluster of 18-plet at each Re. The Re₃ cluster has double bonds. All other bonds are single bonds in this approximation.

Fig. 16:

Repeat units of solid ReF₇ on left and ReCl₃ on right, the latter with rounded off bond distances (1 Å = 100 pm) and identification of single bonds as via electrons shared with or via both electrons donated by the bridging ligand. The cluster triangle is in blue. Numbers of atom’s electrons used in bonds are in red.

With the Fig. 16 of single Re–Cl bonds and double Re=Re bonds, it is not too difficult to derive the set of valence-type quantities (Table 6) as done in the first paragraphs of Section 4. Let’s take a few:

Table 6:

Valence-type quantities for ReCl₃ (repeat unit Re₃Cl₉) and ReF₇ in Fig. 16.

ReCl3	#e	#bp	f.c.	#p.sh.	BVS	OS	| OS |	IUPAC	CN
Re (of 3)	7	9	−2	7	9	+3	3	9	7
µ1 Cl (of 3)	1	1	0	1	1	−1	1	1	1
µ2 Cl (of 6)	3	2	+1	1	2	−1	1	2	2
ReF7	#e	#bp	f.c.	#p.sh.	BVS	OS	| OS |	IUPAC	CN
Re	7	7	0	7	7	+7	7	7	7
F (of 7)	1	1	0	1	1	−1	1	1	1

1. Shorthand symbols of quantity names: #e = number of atom’s electrons in bonds; #bp = number of bonding pairs at the atom (co-valence); f.c. = formal charge in the Lewis formula (auxiliary quantity); #p.sh. = number of bonding pairs sharing electrons of both atoms; BVS = bond-valence sum or bond-order sum; OS = oxidation state.

The formal charge: The single-bonded chlorine has formal charge 0. The two single bonds of bridging Cl we cut in half giving the atom two electrons and formal charge 7 − (4 + 2) = +1, where N = 7, and four electrons are in nonbonding pairs on Cl. Bisecting bonds around Re gives it 4 + 5 = 9 electrons and formal charge 7 − 9 = −2.

The oxidation state by Algorithm 1 is the charge at the atom after ionic approximation by assigning bond electrons to the partner of higher Allen electronegativity and dividing homonuclear bonds equally. In the Fig. 16 approximation, Re loses the 3 electrons it shares with Cl but keeps the 4 shared with the two other Re, and OS _Re = 7 − 4 = +3. The Cl gains all electrons in bonds, and OS _Cl = 7 − 8 = −1. The oxidation state by Algorithm 2 is the sum of the atom’s heteronuclear bond orders by their polarity sign at the atom, summed with any formal charge on that atom. Hence, the five heteronuclear bonds sum as +5 on Re of formal charge −2, and the OS _Re = 5 − 2 = +3. Single-bonded chlorine of formal charge 0 has OS _Cl = −1 + 0 = −1, and the bridging chlorine of formal charge +1 sums up its two single bonds as −2, and OS _Cl = −2 + 1 = −1. Direct ionic approximation on ReF₃ is straightforward.

With the present IUPAC definition of valence (Section 1), we ask: What is the state (electron configuration) at our atom under consideration and how would it bond with F or H? Rhenium of formal charge −2 and 9 two-electron bonds would be in ReF₉ ²⁻. The single-bond counting on Cl is straightforward with 1 as in HCl and 2 on the bridging Cl, as the – Cl ‾ ‾ – corresponds to ClF₂.

The question about the current use was: The rhenium in ReF₇ is heptavalent; do chemists say that ReCl₃ has trivalent, heptavalent, or nonavalent rhenium? Nobody says that solid ReCl₃ has a heptavalent or nonavalent rhenium. In seven papers, Re is stated as trivalent in ReCl₃. See also overview of these comparisons in Table 10.

5.5 MoCl₆ and MoCl₂

Likewise, we compare the MoCl₆ ⁹⁷ molecule with an extended solid MoCl₂ ⁹⁸ Counting electrons at the Mo₆ cluster of the Mo₆Cl₁₀Cl_4/2 repeat unit via “neutral atoms” starts with 36 electrons from six Mo and 2 electrons from the two single-bonded Cl atoms. Then, 6 electrons are added from the four inter-unit µ₂ bridges counted per the repeat unit as 4/2 = 2 Cl atoms sharing 1 and donating 2 electrons. The µ₃ bridges on the eight faces of the Mo₈ octahedron add 40 electrons (5 electrons from each Cl, 1 shared and 4 donated). In total, 36 + 2 + 6 + 40 = 84 electrons. With g = 84 for the n = 6 Mo atoms, the formula N _BO = (18 n − g )/2 derived in Section 10.6 yields 12 orbitals bonding the 12 edges of the Mo₈ cluster having 18-plet on each Mo. All bonds in the repeat unit are therefore single bonds.

With all bonds single in this approximation, the valence-type quantities in Table 7 are straightforward. In MoCl₂ of Fig. 17, the numbers of electrons that the atom uses in bonds verbalize as follows: The “μ₁” Cl shares 1 and donates 0; the μ₂ Cl shares 1 and donates 2; the μ₃ Cl shares 1 and donates 4. Molybdenum uses all its valence electrons.

Table 7:

Valence-type quantities for MoCl₂ (repeat unit Mo₆Cl₁₂) and MoCl₆ in Fig. 17.

MoCl2	#e	#bp	f.c.	#p.sh.	BVS	OS	| OS |	IUPAC	CN
Mo (of 6)	6	9	−3	6	9	+2	2	9	9
µ1 Cl (of 2)	1	1	0	1	1	−1	1	1	1
µ2 Cl (of 2)	3	2	+1	1	2	−1	1	2	2
µ3 Cl (of 8)	5	3	+2	1	3	−1	1	3	3
MoCl6	#e	#bp	f.c.	#p.sh.	BVS	OS	| OS |	IUPAC	CN
Mo	6	6	0	6	6	+6	6	6	6
Cl (of 6)	1	1	0	1	1	−1	1	1	1

1. Shorthand symbols of quantity names: #e = number of atom’s electrons in bonds; #bp = number of bonding pairs at the atom (co-valence); f.c. = formal charge in the Lewis formula (auxiliary quantity); #p.sh. = number of bonding pairs sharing electrons of both atoms; BVS = bond-valence sum or bond-order sum; OS = oxidation state.

Fig. 17:

Repeat units of solid MoCl₆ on left and MoCl₂ on right. Numbers of atom’s electrons used in bonds are in red. The cluster octahedron is in blue.

The formal charge is zero for the two single-bonded Cl atoms. The μ₂ Cl atom has two lone pairs of 4 electrons and obtains 2 electrons from bisecting its two bridging bonds, hence 7 − (4 + 2) = +1. The μ₃ Cl atom has one lone pair of 2 electrons and obtains 3 electrons from bisecting its 3 bridging bonds, 7 − (2 + 3) = +2. Bisecting all nine bonds around Mo gives it 9 electrons, and 6 − 9 = −3 for its formal charge. These charges per neutral formula sum to zero: The 12 Cl atoms per repeat unit have total formal charge 0 × 2 and + 1 × 2 and + 2 × 8 equal +18, and the six Mo atoms have −3 × 6 equal −18.

The oxidation state via Algorithm 1: By ionic approximation, Mo loses 2 electrons it shares with Cl yet keeps the 4 it shared with four other Mo; OS _Mo = 6 − 4 = +2. Single-bonded Cl gains all electrons in bonds and OS _Cl = 7 − 8 = −1. Algorithm 2: The polarity-weighted sum of the atom’s heteronuclear bond orders plus any formal charge gives +5 for Mo of formal charge −3, and OS _Mo = 5 − 3 = +2. Single-bonded chlorine of formal charge 0 has OS _Cl = −1 + 0 = −1. The μ₂ Cl atom of formal charge +1 sums up the two bridging single-bond orders 1 as −2 and its OS _Cl = −2 + 1 = −1. The μ₃ Cl atom of formal charge +2 sums up the three bridging single bonds as −3 and its OS _Cl = −3 + 2 = −1. Direct ionic approximation on MoCl₂ is straightforward.

What is the state (electron configuration) at one Mo atom “under consideration” and how would it bond with F or H? Such an Mo of formal charge −3 with 9 two-electron bonds would be in MoF₉ ³⁻. That gives valence 9 for Mo by the present IUPAC definition of valence. The single-bonded Cl yields 1, the μ₂ Cl yields 2 as the – Cl ‾ ‾ – configuration corresponds to ClF₂, and, likewise, the μ₃ Cl configuration corresponds to ClF₃.

The number 9 appearing among the quantities for Mo may appear extreme. Is it an artifact of the 18-plet counting or is there a reality behind the 9? An estimate of bond orders from interatomic distances in the crystal by Schäfer et al., ⁹⁸ calculated according to O’Keeffe and Brese, ⁹⁰ gives 8.62 for the bond-order sum at the two apical Mo atoms, 8.39 at the four equatorial Mo atoms, 0.76 at the single-bonded Cl, 1.07 at the inter-cluster μ₂ Cl, and 1.85 for the μ₃ Cl. That largely justifies the cluster bonding idealized with single bonds, while indicating that the inter-cluster Cl bridges might be just two one-electron bonds.

We asked the following question: With Mo in MoCl₆ always referred to as hexavalent, do chemists say that MoCl₂ has divalent, hexavalent, or nonavalent molybdenum? The answer was simple; nobody says MoCl₂ has hexavalent or nonavalent molybdenum. In eight papers, Mo is divalent in this chloride. In additional four papers, similar Mo-cluster complexes are described as of divalent Mo. See also overview of such comparisons in Table 10.

5.6 Ni(C≡O)₄ versus Ni(C=O)₄

Tetracarbonylnickel forms from Ni and CO. Table 8 illustrates the subtle difference the 8−N rule makes for the valence-type quantities evaluated on two alternative Lewis formulas in Fig. 18, each of 25 electron pairs. Nickel uses zero electrons in bonds and has 4 bonding pairs. With 10 electrons in nonbonding pairs, the Ni formal charge is 10 − (4 + 10) = −4. There are 4 bonding pairs at Ni, and the sum of bond valences is also 4.

Table 8:

Valence-type quantities evaluated on alternative Lewis formulas of Ni(CO)₄ in Fig. 18.

Ni(C≡O|)4	#e	#bp	f.c.	#p.sh.	BVS	OS	| OS |	IUPAC	CN
Ni	0	4	−4	0	4	0	0	4	4
C	4	4	0	2	4	+2	2	4	2
O	4	3	+1	2	3	−2	2	3	1
Ni(C= O ‾ ‾ )4	#e	#bp	f.c.	#p.sh.	BVS	OS	| OS |	IUPAC	CN
Ni	0	4	−4	0	4	0	0	4	4
C	4	3	+1	2	3	+2	2	3	2
O	2	2	0	2	2	−2	2	2	1

1. Shorthand symbols of quantity names: #e = number of atom’s electrons in bonds; #bp = number of bonding pairs at the atom (co-valence); f.c. = formal charge in the Lewis formula (auxiliary quantity); #p.sh. = number of bonding pairs sharing electrons of both atoms; BVS = bond-valence sum or bond-order sum; OS = oxidation state.

Fig. 18:

Alternative Lewis formulas of tetracarbonylnickel with formal charges marked as counted within the green-line cuts.

Fig. 19:

Molecule of Mn₂(CO)₁₀ with lone pairs marked at Mn atoms as black dashes.

The oxidation state of Ni is zero by Algorithm 1 that assigns bonds to atom of higher Allen electronegativity and leaves Ni with 10 electrons in lone pairs. Or, by the second algorithm, the polarity-weighted sum of Ni heteronuclear bond orders gives +4 on Ni of formal charge −4, and 4 − 4 yields oxidation state zero. Table 8 shows that the alternative formulas for the ligands give the same valence-tested quantities for the central atom Ni. The present IUPAC definition of valence for Ni atom “under consideration” follows from its state that tells how many F or H it would bond. A d ¹⁰ Ni, with four two-electron bonds and formal charge −4, would bond as in NiH₄ ⁴⁻, hence valence 4. The C and O values would differ on the two formulas in Fig. 18.

We asked the following question: Do chemists say that tetracarbonylnickel has zero-valent or tetravalent Ni? The answer is that nobody says Ni(CO)₄ has tetravalent Ni. In 12 papers, Ni is zerovalent in this carbonyl. In additional five papers, Ni is considered zerovalent in this carbonyl, even with a small portion of the carbonyl ligands replaced with another electrically uncharged donor. In additional six papers, Ni is zerovalent in complexes where it bonds with three or four donor atoms of a neutral ligand into an uncharged complex. See also the overview of comparative examples in Table 10.

6 Relations among the tested quantities

All valence-type quantities tested in this study stand on their own as parameters that characterize some aspects of bonding. In this section, we list the main general relationships among the quantity values obtained on our 39 example entities and several types of chemical formulas or bonding schemes:

7 Summary of the current use of adjective n-valent in tested quantities

In Section 3, we concluded that the quantity term valence and its adjective n-valent are not used very often. Does their context imply that one of the investigated valence-type quantities prevails? In Section 5, we illustrated such current-use analysis with Google Scholar searches on eight examples of two compounds of the same element, where we could compare the preferred context of the adjective n-valent for chemists. Section 9 presents five more examples of a compound that yields high variation of the alternative valence quantities. The overview of the current use of n-valent on these telling examples is in Table 10.

The results show that inorganic chemists working with metals use n-valent as an adjective for positive oxidation states. Nobody uses any other alternative for n-valent metals in cases 1, 3, 5, 11, 12, and 13 listed in Table 10. On the other hand, organic chemists count n-valent as the number of bonding electron pairs at the atom. This follows from comparison of CH₄ and C₂H₂ (Section 5.2 and case 2 in Table 10) and from many more collateral statements in searched papers.

A conflicting use of n-valent appears on sp-block atoms. In clear-cut cases 8 and 9 in Table 10, every user of n-valent takes N in NH₄ ⁺ as tetravalent and O in H₃O⁺ as trivalent. Opinion is mixed about H₂S₂, S₂X₂, CO, N₂, and P₄ (respective cases 6, 7, 10, 14, and 15 in Table 10). Organic chemists still use n-valent here for the number of bonding pairs at the atom, and those few inorganic chemists who use n-valent in such molecular examples often take it as the oxidation-state in absolute value, but not all of them. In the rare papers that use n-valent for N₂, both zerovalent and trivalent N appear, while the heavier P₄ is mostly stated as zerovalent.

The diversity in use of n-valent has a cause. In organic chemistry, the current use corresponds to the current Gold-Book IUPAC definition of valence formulated in “Glossary of terms used in physical organic chemistry (IUPAC Recommendation 1994)”. ¹ It fits how organic chemists find the carbon valence 4 in CH₄ and C₂H₂ (Table 10 and Section 5.2) if the triple bond is taken as three single bonds. Likewise, chemists who work with sp-block nonmetals are often counting the actual number of single bonds, such as in the sulfur hydrides and bromides in Section 5.7 or in the NH₄ ⁺ and H₃O⁺ formulas (Table 10). Inorganic chemists working with metals follow the Pauling’s suggestion for use of valence (see Section 2) as an adjective for oxidation state. As this feels natural only for positive oxidation states, terms “negative n-valent” are nearly absent in the literature.

8 Conclusions

The manifold context of valence as a quantity in the current use interferes with launching a single definition of valence. The situation is not much different from what Pauling describes in 1948. ³⁹ The majority of the current and relatively infrequent use of “valence” as a quantity splits into two contexts of the adjective n-valent: (a) the number of two-electron bonds at the atom, typically of elements relevant for organic chemistry, and (b) the positive oxidation states of semimetals and metals. Already in 1913, Bray and Branch ¹⁹ suggested splitting the term “valence number” into “total valence number” in organic- and “polar valence” in inorganic chemistry. In addition to the two prevailing current uses, the CBC approach^[14] has a “valence number” defined as the number of electrons an atom uses in bonds, following Parkin ⁴¹ who revived the 1927 suggestion by Sidgwick of absolute valency. ²⁹ Unfortunately, this deviates from the above two current common understandings of valence as a quantity.

As these uses are bound to continue, clarity in communication is essential. It means stating what one means by valence via verbalizing the first use of valence quantity in organic chemistry or sp-block chemistry papers, or with an early-on specification that, say, divalent chromium is Cr²⁺ or chromium(II).

It would be worthwhile to alleviate the three problems of the current Gold-Book IUPAC definition of valence “The maximum number of univalent atoms (originally hydrogen or chlorine atoms) that may combine with an atom of the element under consideration, or with a fragment, or for which an atom of this element can be substituted”: (1) Valence is defined with “valent.” (2) The terms “element under consideration”, “fragment”, and “substituted” are not entirely clear. (3) The definition covers only a portion of the current use of valence as a quantity.

The problem (1) is easy to solve; the mentioned hydrogen atom clarifies that “univalent atom” has one two-electron bond. Naming fluorine instead of chlorine would strengthen this context if the “originally” were omitted. As for the problem (2) of the “element under consideration”, “fragment”, and “substituted”, it is the atom’s bonding environment that must be “considered” because valence is a quantity that always describes a bonded atom in a specific compound. The present IUPAC definition of valence then applies to that atom’s bond-order total and its charge in Lewis formula, both of which we keep upon our thought bonding with “univalent” H or F. That yields values identical to the often-used (Section 7) number of two-electron bonds or their equivalents at the atom ^[15] (see Table 11). The problem (3) would be amended by stating the two different uses of valence as a quantity and emphasizing clarity in communications.

Table 11:

Overview of important tested examples (co-valence = number of bonding pairs at the atom; electro-valence = formal charge on the Lewis formula; BVS = bond-valence or bond-order sum; OS = oxidation state; IUPAC = present IUPAC definition of valence; CN = coordination number of bonded atoms).

Unless specified otherwise, central atom or any atom of central cluster in	Atom’s electrons in bonds	Co-valence	Electro-valence	Number of 2e-bonds by sharing	BVS	OS	| OS |	IUPAC	CN
H2O	2	2	0	2	2	−2	2	2	2
H2O2	2	2	0	2	2	−1	1	2	2
HgCl2	2	2	0	2	2	+2	2	2	2
Hg2Cl2	2	2	0	2	2	+1	1	2	2
SBr2	2	2	0	2	2	+2	2	2	2
S2Br2	2	2	0	2	2	+1	1	2	2
H2S	2	2	0	2	2	−2	2	2	2
H2S2	2	2	0	2	2	−1	1	2	2
CH4	4	4	0	4	4	−4	4	4	4
C2H2	4	4	0	4	4	−1	1	4	2
CrO3 Fig. 15	6	6	0	6	6	+6	6	6	3
Cr(II) acetate Fig. 15	6	6	0	6	6	+2	2	6	5
MoCl6 Fig. 17	6	6	0	6	6	+6	6	6	6
MoCl2 Fig. 17	6	9	−3	6	9	+2	2	9	9
ReF7 Fig. 16	7	7	0	7	7	+7	7	7	7
ReCl3 Fig. 16	7	9	−2	7	9	+3	3	9	7
Ni(CO)4 Fig. 18	0	4	−4	0	4	0	0	4	4
Mn2(CO)10 Fig. 19	1	6	−5	1	6	0	0	6	6
Os3(CO)12 Fig. 21	2	6	−4	2	6	0	0	6	6
H3O+ conventional	4	3	+1	2	4	−2	2	3	?
H3O+ by 8−N rule	2	2	0	2	2	−2	2	2	3
NH3	3	3	0	3	3	−3	3	3	3
NH4 + conventional Fig. 3	5	4	+1	3	5	−3	3	4	?
NH4 + by 8−N rule Fig. 4	3	3	0	3	3	−3	3	3	4
BF4 − conventional Fig. 5	3	4	−1	3	5	+3	3	4	?
BF4 − 8−N rule Fig. 6	3	3	0	3	3	+3	3	3	4
CO3 2− conventional	4	4	0	4	4	+4	4	4	3
CO3 2− 8−N rule	4	4	0	4	4	+4	4	4	3
C in |C≡O| Fig. 1	2	3	−1	2	3	+2	2	3	1
C in |C= O ‾ ‾ 8−N rule Fig. 2	2	2	0	2	2	+2	2	2	1
O in |C≡O| Fig. 1	4	3	+1	2	3	−2	2	3	1
O in |C= O ‾ ‾ 8−N rule Fig. 2	2	2	0	2	2	−2	2	2	1
C in Ni(C≡O|)4	4	4	0	2	4	+2	2	4	2
C in Ni(C= O ‾ ‾ )4 8−N rule	4	3	+1	2	3	+2	2	3	2
O in Ni(C≡O|)4	4	3	+1	2	3	−2	2	3	1
O in Ni(C= O ‾ ‾ )4 8−N rule	2	2	0	2	2	−2	2	2	1
KI3 expand. octet Fig. 22	3	3	0	3	3	−1	1	3	3
KI3 oct., 8−N rule Fig. 22	1	1	0	1	1	0	0	1	2
N2	3	3	0	3	3	0	0	3	1
P4	3	3	0	3	3	0	0	3	3
| O ‾ ‾ −O= O ‾ ‾ |	4	3	+1	2	3	+1	1	3	2
O3 ↑ average reasonance formula	4	3	+1	2	3	+1	1	3	2
O ‾ ‾ =O= O ‾ ‾ a	4	4	0	4	4	0	0	4	2
⁞ O ‾ ‾ – O ‾ ‾ – O ‾ ‾ ⁞ b	2	2	0	2	2	0	0	2	2
KO3 paramagnetic, octets	3	2.5	+½	2	2.5	+½	½	?	2
KO3 paramagnetic c	3	3	0	3	3	0	0	3	?
I in CsIO3 Fig. 7 , 8	5	5	0	5	5	+5	5	5	6
O in AuORb3	2	2	0	2	2	−2	2	2	6
N ‾ ‾ =N= O ‾ ‾ octets	5	4	+1	3	4	+3	3	4	2
|N≡N− O ‾ ‾ | octets	5	4	+1	3	4	+2	2	4	2
N2O ↑ average formula	5	4	+1	3	4	2.5	2.5	4	2
|N≡N= O ‾ ‾ after Pauling (Ref. 57 p. 9)	5	5	0	5	5	+2	2	5	2
Au in ClAu[ ] Fig. 11	3	4	−1	1	4	+1	1	4	4
B in ClAu[ ] Fig. 11	3	4	−1	3	4	+3	3	4	4
Cl in ClAu[ ] Fig. 11	1	1	0	1	1	−1	1	1	1
P in ClAu[ ] Fig. 11	5	4	+1	3	4	+3	3	4	4

1. ^aNo formal charge, a nonet at central atom. ^bNo formal charge, septet at terminal atoms. ^cNonet at central atom.

The 2022 “Glossary of terms used in physical organic chemistry” (IUPAC Recommendations 2021) ⁴⁶ defines valence as the “maximum number of single bonds that can be commonly formed by an atom or ion of the element.” Would it be 1 for halogens, 2 for chalcogens, 3 for pnictogens, and 4 for tetrels, as enforced by the 8−N rule when the element is the most electronegative bond partner? Or, is it 4 for all these, up to the octet limit? Or, could it be the saturated valence of the atom? What would the interpretation of “commonly formed” be for d-metals combining ionic and covalent bonding aspects?

There are two valence definitions in use over the main fields of chemistry, and they both concern the actual bonding of the atom in question. Only with that strict focus, the valence as a quantity, a chemical variable, becomes useful.