Heuristic algorithms for understanding chemistry via simple quantities

2.1 Octet rule and 18-plet rule

The concept of the rule of eight appeared simultaneously in papers by Lewis [1] and Kossel [2], and the term “octet” was introduced shortly after in the paper by Langmuir [3]. Inorganic chemistry textbook [4] p. 40 defines it: “An atom obeys the octet rule when it gains, loses or shares electrons to give an outer shell containing eight electrons (an octet) with a configuration ns ² np ⁶.”

A current Wikipedia wording [5] goes beyond octet: “During the formation of a chemical bond, atoms combine together by gaining, losing or sharing electrons in such a way that they acquire nearest noble-gas configuration.” For bonded sp-elements, it implies octet (or doublet for ionic compounds of Li and Be), only the “nearest” is an adjective too strong, as boron tends to achieve octet and not the doublet. For d-elements acquiring and sharing electrons upon forming chemical bonds, we specify the noble-gas configuration as the nearest that follows; the 18-plet rule. Its history starts with Langmuir [6], as noted in Ref. [7] that deals with chemistry aspects of this rule. This somewhat weaker rule stabilizes some compounds of transition metals via electron-pair donation from ligands and by formation of metal–metal bonds. Neither is the octet rule without exceptions. So called “expansion of octet” applies to a heavier p-element atom bonded to several highly electronegative atoms.

To apply the octet rule on a simple compound, one distributes all available valence electrons upon drawing the Lewis formula of the molecule or ion, most conveniently with electron pairs as dashes. Often it is unambiguous. At times not. For N₂O of 16 valence electrons, the formula | N ≡ N − O — — | of eight electron pairs is one of two possible with octets. By the algorithm of cutting all bonds in half and counting electrons around each atom, we obtain the formal charge of the central atom N⁺ with compensating O⁻.

3.1 Generalized 8−N rule

The two approaches described below are in fact combinations of the octet rule and of the 8−N rule.

3.2 The 8−N rule expanded towards electron-precise clusters of p-element atoms

An electron-precise cluster of p-atoms is bonded by two-electron bonds and fulfils both the 8−N rule and octet. We can verify the expected cluster bonding by the 8−N rule expressed per the cluster, not per individual atom. Considering the 8−N rule at 1 atom: N _BO = 8 − VEC, where VEC is per cluster atom, we convert it to the expression valid for n cluster atoms along this line of thoughts:

1. 8 becomes 8n.

2. The valence-electron count becomes the total sum of electrons, g.

3. Each N _BO in the «at» rule has counted for two atoms, now it doesn’t; divide by 2.

This formula is used in 1988 by Owen [23] and listed in Ref. [24] on p. 254. A simple example is P₄ of 20 electrons. N _BO = (32 − 20)/2 = 6 tetrahedron edges of this electron-precise cluster. For the above discussed As₇ ³⁻ of 38 electrons, we get (56 − 38)/2 = 9 two-electron edges. Since this cluster must have its three As⁻ merely two-bonded, we insert these As⁻ as linkers into three adjoining edges of the As₄ tetrahedron (to keep high symmetry). Also for the As₄S₄ cluster of two elements, the prediction (64 − 44)/2 = 10 bonds complies with the realgar [20] and pararealgar [21] structures.

3.3 The 18−N rule expanded for a cluster of d-element atoms

The derivation of the formula is now simple. Since the 18−N rule includes electrons donated by ligands, we count to g all electrons at the cluster and use 18n instead of 8n of the previous formula:

in Ref. [23], Ref. [24] p. 340 or [25] p. 139. Since n is any positive integer, this formula covers also a simple 18-plet complex of one single central atom. How to count the electron total g with the electrons donated to the cluster? The 18-plet electron counting is described in inorganic-chemistry textbooks, such as Ref. [4] p. 815. Let’s illustrate it on the nitroprusside anion [Fe(CN)₅(NO)]²⁻ by counting via neutral atoms. We evaluate how many electrons the ligands, taken as “neutral atoms”, provide to the central atom or the central cluster of atoms. The count for each ligand is a short heuristic algorithm:

1. For the cyanide ligand, we start with neutral uncharged ^•C≡N| that has to take 1e to form the |C≡N|⁻ that donates 2e; hence 1 electron in total is donated.

2. The neutral ^• N=O| gives 1e to become |N≡O|⁺ that donates 2e to a bond along straight Fe, N, O line; hence 3 electrons are donated.

The count for [Fe(CN)₅(NO)]²⁻ is 8 electrons of Fe, 5 from cyanide ligands, 3 from nitrosyl, and 2 from the anion charge; an 18-plet at Fe.

For a simple molecular cluster, such as Os₄(CO)₁₄, where Os atoms have 32 valence electrons, and additional 28 are supplied by the neutral ligands CO, the formula yields N _BO = (72 − 60)/2 = 6. These are edges of the Os₄ tetrahedron bonded by single bonds. The cluster is electron-precise. However, the 18-plet rule is weaker than the octet, and is not necessarily always fulfilled in transition-metal compounds, in particular those of the early transition metals.

3.4 Wade rules for boranes (heuristic algorithm of several steps)

These rules [26] concern deltahedral boron clusters of at least 5 vertices. How to count electrons in the B₆H₆ ²⁻ anion? Well, how do we count C₆H₆? It has 30 electrons, hence 15 pairs. Of these, 6 must bond H and 6 must bond C. That leaves us with 3 pairs that comply with the Hückel rule [27] of odd number of itinerant π-electron pairs in aromatic flat rings. What is the counting algorithm for the B₆H₆ ²⁻ cluster?

1. Total valence electrons: 26

2. Total electron pairs: 13

3. Subtracting 1 pair per each B (to bond hydrogen) yields the number of skeletal pairs: 7

4. Of which 1 always is the radial bond, leaving n vertices for the underlying deltahedron shape: 6

   1. If the cluster has n atoms, it is termed closo (no vertex missing in the deltahedron)

   2. If 1 atom/vertex is missing, the cluster is termed nido.

   3. If 2 atoms/vertices are missing, the cluster is termed arachno.

Let’s try B₅H₁₁ as another example: Valence electrons = 26. Electron pairs 13. Skeletal pairs 13 − 5 = 8. Underlying deltahedron vertices 8 − 1 = 7 of a pentagonal bipyramid. Actual boron vertices 5. Two vertices are missing, an arachno cluster.

We can even count the network of B₁₂ clusters of the crystalline element itself. B₁₂ has 36 valence electrons, 18 pairs per cluster. Now backwards: B₁₂ has 12 vertices hence 13 skeletal pairs. Five pairs are thus left to bond the B₁₂ clusters together into the 3D network. Of these 10 electrons per cluster, 6 are shared in 6 single bonds to another cluster. Remaining six borons of B₁₂ contribute each with 2/3 of an electron (hence with 4 in total) to form six 2-electron 3-center bonds to other clusters.

3.5 Counting in networks of extended structures

The terms extended structure/solid [28] refer in a wider context to non-molecular solids, more narrowly to those containing 1D, 2D or 3D entities formed by directional bonds. In extended structures of binary compounds, the typical covalent single chain is the sp ³ chain. The variety of plain covalent atomic sheets is higher, but still rather limited. Layers of networks have a higher variety. If we include into 3D networks also ionic solids largely controlled by sphere packing, the variety increases substantially. And so on for compounds of three and more atoms. Are there any rules for, if not predict, then to verify or at least visualize such bonding patterns in crystalline solids?

One of them is the crystal-chemical formula with three balances behind it [29]. The chemical elements appear in the formula according to the symmetry of their local environments (sites) in the crystal structure. If an element occurs in two different bonding environments, it has two entries in the formula. Let’s take a binary compound of regular coordination polyhedra; TiO₂ mineral rutile of oxygen octahedra around Ti and of titanium triangles around O. The crystal chemical formula Ti⁴⁺ ₁ ^[6]O²⁻ ₂ ^[3] with each element at own crystallographic site having its oxidation state in superscript (as if ions) followed by the coordination number in square bracket.^[1] The balances on Ti⁴⁺ ₁ ^[6]O²⁻ ₂ ^[3] are:

1. Electroneutrality 1 × 4 = 2 × 2.

2. Connectivity 1 × 6 = 2 × 3. The number of Ti–O and O–Ti connections must be the same.

3. The bond-valence balance is 4/6 = 2/3 introduced as the “strength of the electrostatic valence bond” in the Pauling’s 2nd rule [30]. The quantity 2/3 per connection is called bond valence [31] or bond order. Three such bonds at oxygen sum to 2, six at titanium to 4. That’s the octet rule in ionic approximation; oxygen achieves the neon shell, titanium obtains the argon shell.

3.6 Niggli formula, connectivity matrix and bond graph

The parsimony rule of Pauling [30] (the 5th rule) states that “the number of essentially different kinds of constituents in a crystal tends to be small”. Let’s take an example of the B₂O₃ glass-former that only reluctantly [32] crystallizes into an open 3D network of BO_3/2 triangles [33]. The latter formula per central atom of a coordination polyhedron is called Niggli formula and reads as “B O 3 two-connected”. Only brute force makes B₂O₃ to crystallize into a network of corner-sharing tetrahedra [34]. What’s the price? Two differently bonded oxygens. Trying to set up a BO_1.5 Niggli formula for a BO_4/n tetrahedron, we get no integer n solution. With two different O atoms, there are three alternatives: BO_1/1+3/6 (1O one-connected, 3O six-connected), BO_2/2+2/4 (2O two-connected, 2O four-connected), and BO_1/2+3/3 (1O two-connected, 3O three-connected). Parsimony rule tells us which of them has a chance. We take 2 BO_1/2+3/3 rewrite it as B₂O(1)₁O(2)₂ with the two non-equivalent oxygen atoms and derive a simplified crystal-chemical formula B₂ ^[1,3]O(1)^[2]O(2)₂ ^[3] from the connectivity balance for each boron connected to 1 O(1) and 3 O(2) in its square bracket. Complete with ionic charges, the B³⁺ ₂ ^[1,3] O(1)^2−[2]O(2)²⁻ ₂ ^[3] formula yields bond orders of these closest contacts. The O(1) atom has 2 bonds of order |−2|/2 = 1, and the O(2) atom has 3 bonds of order |−2|/3 = 2/3. The boron atom has 1 bond of bond order 1 and 3 bonds of bond order 2/3 with bond-valence sum 3 = 1 × 1 + 3 × ⅔. Drawing a bond graph [35, 36] in Fig. 1 helps visually. We distribute the atoms of the crystal-chemical formula, connect them by one line for each bonding interaction, and add bond order for each line.

Fig. 1:

Bond graph for B₂O₃ of boron-coordination tetrahedra with two types of O atoms as vertices. Bond-orders in blue.

The bond-valence balances can be formalized via a connectivity matrix [37] of “cation” rows versus “anion” columns for crystallographically different atoms.^[2] For B₂O₃, the 1 × 2 matrix of connectivities in Table 1 is appended with the “total BVS” row and column that give the bond-valence sum for the cation total and for each of the two anion totals. These “total BVS” yield one equation for each such atom, but, because of the overall electroneutrality condition, one of these equations is a linear combination of the other two. When the matrix has a single column or row, we have enough equations to solve all unknown bond valences. Otherwise, the missing equation can be replaced by an estimate based on the parsimony principle in the “equal valence rule” [31] stating [35] that “the valence sum around any loop in the bond graph is zero”. It yields an auxiliary equation along a single-connection C–A–C–A– loop where the four unknown bond valences of signs +−+− sum to 0, such as the known ones do (⅔−⅔+1−1 = 0) in Fig. 1.

4.1 Bond order

Bond order is a quantity that gives the number of two-electron bonds connecting two atoms, integer or fractional. It has to be calculated for each bond by a suitable algorithm. In this case, several easy-to-grasp algorithms exist with simple starting parameters for the two bonded elements: For binary molecules, we draw the molecular-orbital (MO) scheme where the bond order = (bonding electrons − antibonding electrons)/2. A more general formulation is in the Gold Book. Alternatively, we draw a Lewis formula according to rules (8−N rule applied by electronegativity, octet, etc.) while counting electrons to obtain bond orders as integers or simple fractions. Bond order can also be calculated as the so-called bond valence from the bond length between the two atoms. A heuristic approach is based [38] on their Allred–Rochow electronegativities, the difference of which correlates with the covalence-caused bond shortening versus the sum of the two “ionic” radii that are fit by least squares to many bond lengths of known bond order between these two atoms. The electronegativity and ionic radius for the two atoms are the parameters needed to calculate their single-bond length via equation given in Ref. [38]. That single-bond length is the only parameter in the empirical exponential relation between the bond valence (bond order) and bond length, which has a long history between [39, 40] of details recapitulated in [41].

As an example, the parameters tabulated in Ref. [38] yield the B–O single-bond length R _BO = 1.38 Å. That is one of four boron bonds in the tetrahedral network of the high-pressure B₂O₃ crystal [34]. The length of the three bonds with v _BO = 2/3 follows from the bond-valence formula in Ref. [38] with the empirical constant 0.37 Å; d _BO = R _BO − 0.37ln(v _BO) = 1.53 Å. The experimental values are 1.37 Å and 1.51 Å [34]. An alternative bond-valence approach [40] lists directly the empirical bond-valence parameters R _ij for oxides, fluorides, etc., of elements. The R _BO = 1.371 Å in [40, 41] yields 1.52 Å for the 2/3 bond via the same equation.

4.2 Oxidation state

Oxidation state is a quantity that characterizes bonding of an atom in a compound or ion. Based on ionic approximation of bonds, oxidation state of atoms in a chemical formula is typically counted by summing known ionic charges towards electroneutrality or the given charge of the chemical formula. At times, it might be useful or pedagogical to check this by a formal process.

As a dimensionless quantity derived from unit-less counts, oxidation state needs a concept [42], a verbal explanatory definition and a heuristic algorithm to calculate its value [43]. The definition [43] states: “The oxidation state of an atom is the charge of this atom after ionic approximation of its heteronuclear bonds.” Besides the direct ionic approximation limited to simple formulas (Ref. [9] p. 1025), two general alternative algorithms [43], subject to the same caveat, give the oxidation state. One algorithm uses Lewis formula, the other works also on bond graphs [35] of extended solids:

Fig. 2:

The Mn₂(CO)₁₀ molecule and Lewis formula of all valence electrons to derive the needed quantities.

Fig. 3:

Lewis formula of (CO)₂(η ⁵-C₅H₅)Ir–W(CO)₅ with all electrons at the iridium–tungsten cluster. Formal charge in black, oxidation state in red, the cyclopentadienyl (Cp) shares 3 aromatic electron pairs formed with 1 electron from Ir.

Fig. 4:

Lewis formula of the Re₂Cl₈ ²⁻ anion. Formal charge in black, oxidation state in red.

Fig. 5:

Corrugated sheet of GeS reminiscent of black phosphorus.

Fig. 6:

The As₄S₄ molecule in pararealgar.

5.1 How to alloy a ferromagnet?

Iron, cobalt, and nickel are ferromagnetic. Their orbitals of d-parentage are stabilized by full filling in one spin direction while the s orbital is occupied partially in both spin directions. The other d-orbital spin has fewer electrons, and the excess of the majority spin is behind the ferromagnetism. Alloys of metals around this range may become ferromagnetic when the average per atom of their outer-shell d + s + p electrons is close to that of Fe, Co, or Ni.

A simple generalized picture is the Pauling’s [47] nearly equilateral triangle for saturated magnetic moments per atom (in Bohr-magneton units μ_B, about equal to the effect of 1 unpaired electron) as a function of valence-electron count per atom in binary alloys from Cr to Ni. The triangle sides rise from zero close to the chromium count of 6 valence electrons (all used in bonds or itinerant^[5]) and from zero some ½ electron above the Ni valence-electron count 10. The triangle vertex is at saturated magnetic moment of about 2 µ_B close to the valence-electron count of Fe. Co has one unpaired electron less. For the saturated ferromagnetic moment μ _sat of alloys along the Cr–Fe edge of the triangle, a simple per-atom rule μ _sat/µ_B = N − 6 is known as the Slater–Pauling rule [47, 48], where N is the average valence-electron count. It works for many binary and ternary intermetallics, as can be seen from the Pauling-triangle updates in Fig. 1 of Ref. [49], Fig. 12 of Ref. [50], and Fig. 24 of Ref. [51].

Ferromagnetism of an intermetallic compound and its saturated magnetic moment depend not only on elemental composition but also on bonding. Crystal structures and molecular orbitals extended into bands become important. The general rule may need adjustments for some structure types. Heusler alloys are a group of intermetallic compounds that are often ferromagnetic:

Compositions XYZ denote half-Heusler alloys. Those relevant for ferromagnetism contain a late transition metal X, an early transition metal Y, and a p-metalloid Z. The late transition metal X has a cube of 8 nearest YZ neighbors that each have a tetrahedron of nearest neighbors X. One choice of the unit-cell origin starts with the cubic closest packing of Z with octahedral voids occupied by the early transition metal Y as in a NaCl-type cell. The late transition metal X centers four unit-cell octants, the tetrahedron of which yields the F 4 ‾ 3m space-group symmetry. The electron counting is simple [52]: Of N _total valence electrons, Zintl approach assigns 8 to the metalloid Z. The X and Y atoms form 5 bonding and 5 antibonding MO from their 5 + 5 atomic d-orbitals. The bonding MO are fully occupied, remaining electrons in antibonding MO are unpaired. Their number per XYZ formula is N _total − 8(sp ³) − 10(d ⁵) that yields the saturated ferromagnetic moment μ _sat ^total/μ_B = N _total − 18. Per atom, the Slater–Pauling rule obtains. When XYZ has 18 valence electrons, the half-Heusler alloy is not ferromagnetic.

Full Heusler alloys can be regular or inverse. Regular Heusler alloys have composition X₂YZ that stems from the above XYZ structure by the late transition metal X completing the tetrahedron of X into a cube that leads to the Fm 3 ‾ m symmetry. The counting differs in one point [53]: There are additional five approximately nonbonding d-orbitals in the MO scheme, of which three (t ₂) are doubly occupied. That leaves N _total − 8(sp ³) − 10(d ⁵) − 6(t ₂) unpaired electrons. The regular X₂YZ has μ _sat ^total/µ_B = N _total − 24. Again, the Slater–Pauling rule per atom.

Keeping the symbol X for late and Y for early transition metal, inverse Heusler alloys have formula Y₂XZ, where the X tetrahedron is completed by Y into an XY cube keeping the F 4 ‾ 3m symmetry. This structure type is wider in composition [54], and the Fermi level moves as a function of that:

1. When Y = Sc, Ti in Y₂XZ, (at times also V), only one d-orbital based MO set is doubly occupied, the Fermi level is lowest. The counting is then as with the half-Heusler alloys: The μ _sat ^total/µ_B = N _total − 18.

2. When X = Cu or Zn in Y₂XZ, the entire additional d-orbital MO set would be doubly occupied (not just the three t ₂ orbitals), the Fermi level is highest. There are N _total − 8(sp ³) − 10(d ⁵) − 10(d ⁵) unpaired electrons. The μ _sat ^total/µ_B = N _total − 28 rule is valid [54, 55] for these predicted compounds.

3. Apart from the above two points, inverse Heusler alloys have μ _sat ^total/µ_B = N _total − 24.

The algorithms would be:

1. For a d-metal based half-Heusler alloy XYZ with Z = p-group atom, calculate the total number of valence electrons N _total with valence electrons for the X and Y d-group metal, and only sp-electrons for the Z atom.

   1. Estimate the total ferromagnetic moment μ _sat ^total in Bohr magnetons as N _total − 18.

   2. For N _total = 18, the alloy is not ferromagnetic.

2. For a transition-metal full-Heusler alloy with a p-group atom, calculate the total number of valence electrons N _total with d-electrons for transition metals and sp-electrons for the p-group atom.

   1. If the formula has Sc₂ or Ti₂ (rarely V₂), estimate μ _sat ^total in Bohr magnetons as N _total − 18.

   2. If the formula has Cu₁ or Zn₁ (possibly also Ag₁, Au₁, Cd₁, Hg₁; not tested) estimate μ _sat ^total in Bohr magnetons as N _total − 28.

   3. Otherwise, estimate μ _sat ^total in Bohr magnetons as N _total − 24.

These estimated values often agree with quantum-chemical calculations, yet less often with the published magnetic measurements. One of the reasons is that the actual crystal structure deviates from the expected ideal, often by a minor disorder or intermixing. In particular for half-Heusler alloys, their ab-initio calculated values [56] are often contradicted by experiment, such as for CoMnSb of an ordered superstructure [57] and saturated moment close to 4 µ_B [57], [58], [59] instead of 3 µ_B [49] obtained by calculation for the ideal structure. Table 2 lists examples of half-Heusler alloys with valence-electron counts that agree with measurements on samples of checked and determined crystal structure.

Examples of predicted and observed saturated moments of regular full-Heusler alloys are in Table 3. Data for inverse Heusler alloys appear much less frequently in the literature (Table 4), and the magnetic moments are often just calculated. For majority of cases, the estimated saturated ferromagnetic moments per atom follow the Slater–Pauling rule N _average−6, which means they coincide with the left edge of the Pauling’s triangle as in Ref. [51]. The two valence-electron count extremes towards the top and bottom of Table 4 for inverse Heusler alloys would form lines parallel with that triangle edge.

5.2 Suggest a high-T _c superconductor

High-T _c superconductivity in a periodic network solid is associated with a central-atom behind a largely nonbonding HOMO that is half-occupied and separated in energy by suitable ligand field (approximates the narrow nearly half-filled band advocated by physicists [76]). A diluted hole pair can be imagined if this orbital is slightly less than half filled, and diluted electron pair if slightly more than half filled. These become the superconducting pairs, perhaps by condensation [77] of the Bose–Einstein type.

An example is YBa₂Cu₃O_7−x of two Cu-coordination square pyramids joined by square-planar copper coordination that accommodates the x oxygen vacancies per unit cell. Already the YBa₂Cu₃O_6.5 composition of average Cu²⁺ (d ⁹) is superconducting because the square-pyramidal coordination of 5 oxygens keeps Cu^>2+ with a single nearly half-filled orbital d _{x ² −y ²} [78], while the x-vacant square-planar coordination has Cu^<2+ on average. Removing the x-vacancies stabilizes the superconductivity. Under aliovalent substitutions, the critical temperature correlates with the bond-valence sum at Cu in the coordination pyramids where the hole pairs form, not with an integer oxygen content or disappearing distortions [79, 80]. Another example of hole-doped d ⁹ configuration is the Nd_0.8Sr_0.2NiO₂ superconductor [81]. An example of electron doping is Nd_2−x Ce _x CuO₄ superconducting below 24 K when x = 0.15 [82].

A similar case is the nearly half-occupied relativistic-stabilized 6s ² orbital. It leads to superconductivity up to 13 K in BaPb_1−x Bi _x O₃ [83], increased to 30 K in Ba_0.6K_0.4BiO₃ [84]. In the Tl–Ba–Cu–O system, zero resistance at 81 K [85] was raised to 125 K [86] and 127 K [87]. HgBa₂CuO_4+δ becomes superconductor below 94 K [88], and HgBa₂Ca₂Cu₃O_8+δ [89] holds the record of zero resistivity at 134 K.

In 2001, it was discovered that MgB₂ is a superconductor below 39 K [90]. Taken as a Zintl phase of VEC _A = 4, its boron forms 8−4 = 4 bonds in graphene-type sheets separated by sheets of Mg²⁺ that is placed above and below each hexagon. However, the charge transfer from the cation is not entirely perfect; MgB₂ is black, B–B distances are longer than expected, and calculations suggest an s→p charge transfer with superconductivity driven by s-band holes [91]. Similar cases are potassium [92], rubidium [93], or cesium [94] alkali-metal fullerides A₃C₆₀ with cations A filling all octahedral and tetrahedral holes in the cubic closest packing of the anion balls [95]. Cs₃C₆₀ is superconducting up to 33 K.

In 2006, superconductivity below 4 K in a LaOFeP layered network was reported [96]. The critical temperature increased to 26 K [97] upon electron-doping the ∞ 2 [ F e As − ] anionic-network layer by the LaO_1−x F _x ^(1+x)+ cation to ∞ 2 [ F e As ( 1 + x ) − ] . Hole doping via Ba_0.5−x K _x ^(1−x)+ to ∞ 2 [ F e A s ( 1 − x ) − ] enhanced it to 38 K [98]. Even the simplest undoped variant, LiFeAs with Li in square-pyramids of As, is superconducting up to 18 K [99]. The chemist wonders; why should Li⁺ Fe²⁺As³⁻ be a superconductor? The ∞ 2 [ F e As − ] layer network is a stack of three square-planar nets, As 2Fe As, with Fe in coordination tetrahedra of As that share 4 edges upon forming the layer. With double the occupancy of As nets, the Fe–Fe net distance 2.68 Å is within the range for Fe–Fe single bonds [100]. As–As bonds are absent. Bond-valence analysis [38] on LiFeAs structure [99] yields bond-valence sum 3.36 of the four Fe to As bonds at both atoms. The Li–As bond-valence sum is 0.95. The infinite planar Fe cluster can be counted on its repeat unit Fe₂As₂ ²⁻ in Fig. 7 via neutral atoms. The µ₄-bridge ligand is counted as a neutral As that has to obtain 3 electrons to donate 4 bonds, hence is a donor of 8−3 = 5 electrons. The electron count g = 28 at the repeat unit of the Fe cluster therefore consists of 10 from two As, 2 from the anion charge, and 16 from two Fe. With 2 cluster atoms per repeat unit, the 18-plet cluster formula N B O = ( 18 n − g ) / 2 gives (36 − 28)/2 = 4 cluster-bonding orbitals at each iron atom. Figure 7 indeed has one single bond + six halves of it at the two Fe atoms. The 28 electrons at the cluster repeat-unit make 14 pairs, of which 8 count as octets on As leaving 6 pairs to Fe, of which 4 are in Fe–Fe bonds and 2 make a lone pair at each Fe. However, the Li–As bond is not a full transfer of electrons for As to freely bond Fe with them. As is not able to fully form 4 two-electron bonds to Fe, the bond-valence estimate is mere 3.36. The Fe lone-pair orbital is therefore only partially occupied. The T _c of iron pnictides correlates with the nearly tetrahedral bond angle that controls the pnictogen-atom height above the Fe plane [101] and with that height’s homogeneity across the structure [102]. Transitions to weak antiferromagnetism (<1 μ _B) [103] are suppressed in favor of superconductivity already upon minor doping [104].

Fig. 7:

Bonds in the LiFeAs unit-cell repeat unit of the infinite iron cluster (in green) with magenta bonds to As ligands (large white) that are charged by lithium cations (small white). Cluster counting (see text) suggests ideally one lone pair per Fe.

The idea of nearly half-filled orbital/band, with implicit use behind hundreds of tried superconductor compositions, lends some heuristic quality to the rules considered in this section. An old empirical rule for superconductivity is worth mentioning; the 1955 Matthias rule [105] that there is a dip in d-metal alloy superconductivity around 6 valence electrons per atom (that are all unpaired in an isolated atom) with maxima around d ⁵ and d ⁷. The rule covers a wide range of d-metal alloys that may also contain pre- and post-transitional elements and form a wide range of structure types including Heusler alloys [106] as well as high-entropy alloys [107]. Some studies [108] see common denominators behind this rule and the high-T _c superconductivity in theories that go beyond heuristic approach.

6.1 Redox reactions in aqueous solutions

Chemists balance redox equations by counting exchanged electrons in a simple algorithm: The per-formula decrease in oxidation state of the oxidant-element atoms (the electron total they accept upon reduction) becomes the reaction coefficient in front of the reductant. The inverse for electrons the reductant formula gives away. Ionic charges are then balanced via a protic acid or base if needed. How to find useful reductant for a given oxidant, and vice versa? By considering oxidation and reduction separately. Every redox reaction is a sum of two half-reactions, one for oxidation upon losing electrons and one for reduction upon accepting them. In 1938, Wendell Latimer published “The Oxidation States of the Elements and their Potentials in Aqueous Solutions” [42]. Since then, we have tables of standard electrochemical reduction potentials E° for half-reactions of inorganic compounds or ions in their 1M (1 mol/L) aqueous solutions.

The redox-prediction algorithm is simple: Take the two half-reactions, invert the oxidation half-reaction so that it too refers to reduction, and look up the two standard reduction potentials. The half-reaction with higher standard reduction potential provides the oxidant to be reduced. The half-reaction with lower standard reduction potential provides the reductant to be oxidized. The oxidant and reductant then react with each other.

Reduction potentials concern aqueous solutions and tell us something about the chemistry of the element. Even the solid-state redox behavior may roughly follow these aqueous redox estimates. Standard reduction potentials are typically listed for pH 0 ( E 0 ° ) and pH 14 of unity activity (often approximated as concentration, as done hereafter) of the respective water ions H⁺ and OH⁻ (in simplified notation). As can be seen from the mass-action consideration of any balanced reduction half-reaction that involves water ions, increasing pH will decrease the half-reaction’s reduction potential. When the associated protolysis takes place at a pH relevant for the redox reaction in the 1M solution, it changes the pH dependence of the half-reaction E°.

Let’s illustrate it on the HClO and Cl₂ pair. At pH 0, their half-reaction equation includes H⁺ ions that balance the electron charge, HClO + e⁻ + H⁺ = ½Cl₂ + H₂O. Its E° under varying pH beyond the standard pH 0 is expressed from the simplified 298 K Nernst equation while keeping the unity standard pressure of Cl₂:

The amount concentration (molar) of H⁺ ions equals 10^−pH, and the concentration of HClO molecules changes with pH as well. How it changes, we find out via the equilibrium constant of the protolysis reaction:

Since [HClO] + [ClO⁻] = 1 in the standard 1M solution, the unknown [ClO⁻] = 1 − [HClO] is plugged into the K _a expression, yielding the [HClO] = [H⁺]/(K _a + [H⁺]) needed in our adopted form of Nernst equation:

where we take [H⁺] = 10^−pH. Fig. 8 shows the plot, calculated with E 0 °  = 1.63 V [109] p. 791. Notice the kink at pH equal to pK _a of HClO. The mass action applied to half-reactions in Fig. 8 illustrates why increasing pH decreases their reduction potential.

Fig. 8:

Standard potential E° (in V) for reduction of HClO or ClO⁻ to Cl₂ as a function of pH ([HClO]+[ClO⁻] = 1 mol/L, p _Cl2  = 1 bar). Notice the reaction equation suggesting the double slope in alkaline solutions.

Let’s now bubble Cl₂ gas into water of varied pH (by a redox-prone acid/base). One product will certainly be chloride ion. The reduction half-reaction Cl₂ + 2e⁻ = 2Cl⁻ does not involve water ions; hence its standard potential E 0 ° = 1.36 V is independent of pH. Draw the horizontal line in Fig. 8. For the mutual reaction, the upper-line E° half-reaction provides its oxidized specie, the lower-line E° half-reaction its reduced specie. Above pH 4.5, both are Cl₂ that will disproportionate into Cl⁻ and HClO/ClO⁻. Below pH 4.5, the inverse reaction prevails. This approach can be generalized further to stability fields versus pH and E° taken both as free variables. In our drawing, we add also a vertical line above the kink on the curve, in order to separate HClO and ClO⁻. Since any Cl⁻ or Cl₂ is oxidized above the highest E° lines of this plot, we mark the top right ClO⁻ and top left HClO. In the triangle on left below the HClO field, Cl₂ is stable. Further right, Cl⁻ is stable together with HClO prevailing up to pH 7.5 and ClO⁻ above that. Below the lowest E° lines of our plot, any ClO⁻ or Cl₂ is reduced. We mark this stability field Cl⁻ and obtain a simple Pourbaix diagram [110] for this limited system. Several redox reactions are covered in Pourbaix diagrams for compounds of elements forming an array of oxidation states, and may include formation of water-insoluble components along the pH range via their solubility products.

6.2 Products of carbide hydrolysis

Carbides contain C₁ or C₂ or C₃ carbon groups. Some carbides hydrolyze in water or protic acids to the corresponding hydrocarbon; some form hydrocarbon mixtures. Salt-like carbides of electropositive sp-metals contain C⁴⁻ or C₂ ²⁻ or C₃ ⁴⁻ anions and hydrolyze into the respective weak acid CH₄ or C₂H₂ or C₃H₄. Some carbides of d- or f-transition metals are hydrolyzable, such as Sc₂OC and Mn₇C₃ of single carbon atoms, or YC₂ and other rare-earth dicarbides of C₂ groups. Whereas Sc₂OC and its rare-earth analogues hydrolyze to methane [111], Mn₇C₃ yields a hydrocarbon array of concentration decreasing with the chain length [112]. So does YC₂, except that odd-numbered chains are missing.

To predict the outcome, we identify the carbon groups of the carbide and then use standard reduction potentials to predict the metal ion stable in the acidic aqueous environment. Balancing the primary hydrolysis-reaction gives the answer:

1. Sc₂OC would form Sc³⁺ as Sc₂OC + 6 H⁺ = 2 Sc³⁺ + CH₄ + H₂O.

2. Mn₇C₃ would form Mn²⁺ as Mn₇C₃ + 14 H⁺ = 7 Mn²⁺ + 3 CH₄ + H₂.

3. YC₂ would form Y³⁺ as YC₂ + 3 H⁺ = Y³⁺ + C₂H₂ + ½ H₂.

The algorithm is simple: Those carbides that yield hydrocarbon series in the real reaction, release also hydrogen. Correspondingly, Mn₇C₃ is silvery and YC₂ golden metallic.

How does hydrogen polymerize the carbon groups? In the reaction on the solid surface, one hydrated H⁺ ion protonates the carbon group towards the corresponding hydrocarbon while the metal reduces another H⁺ into the “nascent hydrogen” ^•H that interferes with the protonation in the vicinity by initializing polymerization of hydrocarbon radicals in n steps [112] into hydrocarbon chains of C_1n or C_2n . Hence, if the carbide composition suggests formation of hydrogen upon hydrolysis, an array of hydrocarbons forms as if by polymerizing the structural carbon groups present in the solid [113].