Counting electrons in an extended structure

Three counting rules are taught in three different courses and are the same rule.

The octet rule says a main-group atom is content with eight valence electrons. Hückel’s 4n + 2 says a conjugated ring is stable with two, six, ten or fourteen π electrons. The 8 − N rule says an element of group $N$ forms $8 - N$ bonds to its neighbours, which predicts that selenium forms chains, arsenic forms sheets and silicon forms a three-dimensional network.

Every one of them says the same thing: every occupied level should be full, and none should be partly occupied. They differ only in what counts as a level.

Why a partly filled level is unstable #

The reason behind all three is one argument and it is worth stating before the counting.

A partly filled level is one where an electron could be moved somewhere else at no cost, or at very small cost. That is exactly the condition for something to happen: a distortion that splits the level and drops the occupied part, a reaction that fills or empties it, a magnetic moment that forms because the spins have no reason to pair.

A completely filled level offers none of those. Every state below is occupied, every state above costs a definite amount, and the system has nothing cheap available to it.

So “closed shell” and “stable” mean the same thing, and all three rules are ways of computing when the first has been achieved.

The counting at three scales #

One centre. A carbon atom has four valence orbitals. Filling all four takes eight electrons, and it gets them by contributing four and sharing four from neighbours. Hence four bonds, hence the octet. Hypervalency without d orbitals is about what happens when the arithmetic seems to demand more than four, and shows that the resolution does not require expanding the shell.

One ring. A conjugated ring of $n$ atoms has $n$ π levels, arranged as one lowest level and then degenerate pairs. Filling the lowest and $k$ complete pairs takes $2 + 4k$ electrons, which is the $4n + 2$ rule with the letters shuffled. It is not a rule about aromaticity so much as a rule about which counts fill the available pairs, and this site derives rather than quotes it — aromaticity as a computed shell closure fills every ring from three to ten and asks whether its highest occupied shell came out full.

An extended structure. Here the levels have become a band, and filling every level takes two electrons per orbital per site. An element with $N$ valence electrons contributes $N$ and needs $8 - N$ more, one from each of $8 - N$ neighbours.

That last step is the one with real predictive content, and it is worth checking against the periodic table.

The 8 − N rule, checked #

Group seven, one bond each: the halogens form diatomic molecules. Correct.

Group six, two bonds each: selenium and tellurium form helical chains, and sulfur forms rings of eight. Both are two-connected structures. Correct.

Group five, three bonds each: arsenic, antimony and black phosphorus form puckered sheets in which every atom has three neighbours. Correct.

Group four, four bonds each: carbon as diamond, silicon, germanium and grey tin all form the three-dimensional four-connected network. Correct.

Group three, five bonds each — and here it stops. Boron does not form five-connected structures; it forms icosahedral clusters with electron-deficient multi-centre bonding, and aluminium is a metal. The rule breaks because five bonds from four valence orbitals is impossible, which is the same impossibility where two-centre bonding stops is about.

So the rule works for exactly the range in which the required number of bonds can be made from the available orbitals, and fails outside it — which is a precise statement of its domain rather than a list of exceptions.

Why the rule is about orbitals and not about octets #

There is a piece of vocabulary worth correcting here, because the usual statement of the octet rule attaches it to a number that is a coincidence of the second row.

The rule that does the work is fill the available valence orbitals. For carbon, nitrogen, oxygen and fluorine there are four such orbitals — one $s$ and three $p$ — so filling them takes eight electrons and the number eight is right. For hydrogen there are none but the single $1s$, so the number is two, and nobody finds that mysterious.

For the third row and beyond the number of valence orbitals is still four, because the $d$ orbitals lie far too high to participate — which is hypervalency without d orbitals’s whole subject, worked through with the computation rather than assumed. So the count is eight there too, and the apparently ten-electron sulfur of sulfur hexafluoride is described without expanding anything, by three-centre bonds that put electron pairs on the ligands.

The reason to insist on the orbital form rather than the number is that only the orbital form survives into an extended structure. There is no octet in a band; there are levels, and the rule is that they should be filled. Restating the familiar rule in the form that generalises is most of what this essay is for.

What happens when the counting cannot be satisfied #

For groups one, two and three the counting has no solution. Sodium has one valence electron and would need seven neighbours each donating one; there are not enough orbitals, and no structure achieves it.

The consequence is that the levels cannot all be filled, whatever the structure. Something is left partly occupied, and by the argument at the top of this essay that is a system with cheap excitations available — which is to say a metal. What a metal actually is makes that definition precise and measures it.

This is a better account of why the left of the periodic table is metallic than the usual one about electrons being loosely held. Caesium’s outer electron is indeed loosely held, and that is a fact about the atom; the metallic state is a fact about the structure, and it arises because no arrangement of caesium atoms closes a shell. An element with too few electrons is stuck with partly filled levels, and being stuck with them is what being a metal consists of.

The counting also explains the diagonal. Elements near group four have just enough electrons for a closed-shell structure to be possible, and whether they take it depends on fine energetics — which is why tin has a semiconducting grey form and a metallic white one differing only in structure, and why the metalloids sit exactly where the counting becomes marginal.

What failing to close a shell leaves is a cheapest excitation falling without limit as the system grows. That is what an element with too few valence electrons is left with, and it is the extended structure’s version of a half-filled degenerate pair.

The compounds, where the counting still works #

The rule survives into compounds in a form that is worth knowing, because it is how a great deal of solid-state chemistry is actually done.

Take the average number of valence electrons per atom rather than the number belonging to one element. Gallium arsenide averages $(3 + 5)/2 = 4$ per atom, and it adopts the same four-connected network as silicon and is a semiconductor. Zinc selenide averages $(2 + 6)/2 = 4$ and does the same. Copper gallium selenide averages four and does the same again.

The generalisation is that a compound with four valence electrons per atom on average has a closed-shell structure available, and takes it. That single arithmetic step accounts for most of the useful semiconductors, and it explains their family resemblance: they are all the same electron count wearing different elements.

The Zintl concept extends this further, to compounds in which electrons transfer from an electropositive element to an electronegative one and the latter then obeys the 8 − N rule for its new count. Sodium tetrelide is a case: sodium gives up its electron, silicon becomes formally like phosphorus with five valence electrons, and it forms three-connected clusters exactly as the rule for group five requires.

The transition metals, where a different count applies #

Everything above is about the main group, and the transition metals need a paragraph because the counting is genuinely different there rather than merely harder.

A transition metal has five $d$ orbitals in addition to the $s$ and $p$, so there are nine valence orbitals and the closed-shell count is eighteen rather than eight. That is the eighteen-electron rule, and it works for organometallic complexes about as well as the octet works for organic molecules.

In the extended metals themselves it does not work at all, and the reason is instructive. The $d$ orbitals are compact and overlap their neighbours weakly, so the band they form is narrow; the $s$ and $p$ orbitals are diffuse and form a wide band. The two overlap in energy, so there is no arrangement in which one is full and the other empty, and the count has no closed solution — which is why every transition metal is a metal, across the whole series, with no exceptions and no diagonal.

The consequences are the ones that make the transition metals useful. A narrow band with states at the Fermi level is exactly the case where electrons repel one another enough to matter, which is where magnetism comes from; and partly filled $d$ levels on a surface atom are what lets a molecule bind to it and be activated, which is where catalysis comes from. Neither is describable in this model, and both are downstream of a count that has no solution.

The ring rule is checked rather than assumed at every size from three to ten — eigenvalues against the closed form, both trace relations, and pairing in both directions — so the shell closures the counting rules are read off are computed results rather than a restatement of the rule.

What the counting cannot do #

Three limitations, and they are the same three that limit every counting argument.

It does not say how large the gap is. Closing a shell says there is a finite cheapest excitation and says nothing about its size. Diamond’s gap is $5.5$ electronvolts and grey tin’s is essentially zero, and both are four-connected closed-shell structures. Whether a material is a transparent insulator or a semiconductor is decided by the interaction strength, not by the count. The gap is not the band width separates the quantities.

It does not say which of several satisfying structures is taken. Sulfur has rings of eight, chains, and several other allotropes, all two-connected and all satisfying the count. Choosing between them needs energies.

It assumes the electrons do not repel each other. The whole argument treats levels as things to be filled, with the cost of putting two electrons in the same place ignored. That is a half-filled band is not always a metal’s subject, and it is where a closed-shell count can be right about the arithmetic and wrong about the material.

Filling a chain one electron per site stops halfway up its column, where the spacing is smallest; filling it two per site fills every level. The two cases behave quite differently as the chain grows, and the counting rules of molecular chemistry are all about the second.

A worked case where two rules disagree #

The rules are the same rule, so a case where two of them give different answers is worth examining, and one is available in the same one-orbital chain.

Cyclobutadiene is a four-membered ring with four π electrons. The octet is satisfied at every carbon — each has four bonds and eight valence electrons. The $4n + 2$ rule is not satisfied, since four is neither two nor six.

The ring rule is the one that is right, and the diagonalisation says why. Cyclobutadiene’s four π levels come out as one low, two degenerate at exactly the non-bonding energy, and one high; four electrons fill the lowest and put one in each of the degenerate pair. That is a partly filled level, with all the instability that implies, and the molecule duly distorts to a rectangle to remove it — which delocalisation is stabilising, and other things that are false in general computes.

So the octet was satisfied and the molecule was unstable anyway, because the octet counts electrons at a centre and the instability lives in a level shared between four of them. A counting rule sees only the levels it is a count of, and a rule that counts locally cannot see a partly filled level that belongs to the whole system.

That is precisely the situation in an extended structure, where every level belongs to the whole system, and it is why the local rules have to be replaced rather than extended.

The rule dissolving, one element at a time #

Each stops being reliable at exactly the point where closing the shell becomes impossible is the essay’s claim, and the elements of one group make it visible as a sequence rather than as a boundary — the rule does not fail, it fades.

Group 15 has five valence electrons, so the rule demands three bonds. The structures oblige and then stop obliging by degrees.

Phosphorus in its white form is a tetrahedron of four atoms, each bonded to exactly three others, with nothing else nearby. The rule is met and the molecule is discrete.

Arsenic is a puckered layer in which each atom has three close neighbours at 2.52 ångström and three more at 3.12 — a fourth of a nanometre further, and clearly a different kind of contact. Three bonds, plus three neighbours that are not bonds.

Antimony has the same arrangement with the two distances at 2.91 and 3.36. The gap has narrowed.

Bismuth has it again, and the two sets are closer still. Push bismuth with pressure and they become equal: the three long contacts shorten into bonds, the atom has six equivalent neighbours, and the element becomes a straightforward metal.

So the rule’s failure is a continuous approach of the second shell of neighbours, and the “3 + 3” structure is a rule half kept.

Group 16 supplies the endpoint. The rule demands two bonds, and sulfur, selenium and tellurium duly form rings and chains — two close neighbours each, with the rest of the coordination at longer range. Polonium abandons it entirely. It is simple cubic: six neighbours, all at the same distance, no short set and no long one. It is the only element that adopts that structure at ordinary conditions, and it is a metal.

That progression is the essay’s sentence run down a column. Where the shell can be closed, it is, and the structure is whatever closes it — a molecule, a chain, a layer. Where the atoms are large enough and the bonding weak enough that closing it gains little, the distinction between a bond and a contact stops being sharp, the two sets of neighbours merge, and the count has nothing left to count.

It also says why the metalloids sit where they do. The diagonal of the periodic table where the classification becomes unhelpful is exactly the diagonal along which the short and long contacts are converging, and metalloid names a structure with a rule half obeyed rather than a substance with an intermediate property.

And it gives the rule a measurable form that a count cannot have. The quantity that says how far the rule is being obeyed is the ratio of the long contact to the short one — 1.24 for arsenic, 1.15 for antimony, converging on one for a metal — which is read off a diffraction pattern, varies continuously, and does not require anybody to decide whether a contact counts as a bond. A count of neighbours has to make that decision; a ratio of distances reports the same thing and leaves the decision out.

The rule as a single sentence #

Everything above compresses to one statement, and it is worth carrying because it applies at every scale this site works at.

A structure will adopt whatever arrangement lets it fill its levels completely, and where no arrangement does, what is left is a metal.

The octet is that sentence for one centre, $4n + 2$ for a ring, $8 - N$ for an extended structure, and the average-of-four rule for a compound. Four rules that are usually met in four different contexts, taught by four different arguments, and derivable from one.

The unification is worth more than the tidiness. A reader who holds the four separately has four things to remember and no way to tell which applies when they conflict; a reader who holds the one sentence can derive each of them, knows that the level being counted is what distinguishes them, and knows to trust the count over the largest system involved. Cyclobutadiene above is the smallest demonstration of that last point, and an element with too few electrons to close any shell is the largest.