A while ago I was part of a conversation about how to mark an answer about rationalising the layer structure of solid PbO. Some academics held the strong view that the only acceptable rationalisation was a second-order Jahn-Teller distortion, while others were open to the idea that the structure could be explained by a stereochemically-active lone pair in a Lewis sense.

I think this discussion gets to the heart of something quite distinctive about Inorganic Chemistry as a discipline: it has a diverse – and often contradictory – battery of theories which can be applied successfully: Inorganic Chemistry has a sort of Theoretical Pluralism. I believe the way this plays out in marking merits consideration.

So in this blog I want to set up a series of exam-like questions which I think can be answered successfully with competing theories. I have written answers which embody those competing theories. My hope is that this proves a useful resource for anyone reflecting on how they set Inorganic Chemistry exam questions. How many marks would you give each answer? Why?

Family 1: Incompatible Theories

There are some cases where the theories used to answer a question can be set up as incompatible: to write about one theory – at least within a timed exam – you must abandon the other.

Ln metallic radii

The metallic radii of selected lanthanide elements are plotted below. Explain the form of this graph. [4]

Answer 1: Secondary School

Metallic bonding can be described as metal cations surrounded by a sea of delocalised outer electrons. In most lanthanides, this can be thought of as 3+ ions surrounded by 3 electrons per metal centre, as ionisation to the +3 state is readily compensated by the bond strength.

The modest start-to-finish increase in atomisation energy can be explained by contraction of the metal cation with increasing effective nuclear charge, and the discontinuities at Eu and Yb can be explained by the ion being in the (larger) 2+ state due to the electronic stability of the f7 and f14 configurations maximising exchange energy.

Answer 2: Band Theory

Metallic bonding in the lanthanides can be described as electrons occupying the 5d band (as the 4f orbitals are considered to be too contracted to overlap effectively).

The start-to-finish increase in atomisation energy can be thought of as the improved overlap of the 5d orbitals as the atoms contract with increasing effective nuclear charge, and the discontinuities at Eu and Yb can be explained by the metal providing only 2 electrons for bonding in the 5d band, due to the electronic stability of the f7 and f14 configurations maximising exchange energy.

Crystal/Ligand Field Theory

The lattice energies of MCl2 for the 3d series are shown below. Explain the form of this graph. [5]

Answer 1: Crystal Field Theory

There is a start-to-finish increase in lattice energy as the contracted metal ion forms stronger ionic bonds with the chloride.
There is a ‘double hump’ trend due to the occupancy of the d-orbitals, which are split due to the octahedral crystal field. The z2 and x2-y2 orbitals push directly into the ligand electrons, and rise in energy relative to the spherical field; the xy, xz, and yz orbitals sit between the ligand electrons, and fall in energy relative to the spherical field.
Copper’s Jahn-Teller distortion stabilises the CuCl2 lattice, as the stabilisation of the z2 orbital (further from repulsive ligand charge) and destabilisation of the x2-y2 orbital (closer to repulsive ligand charge) leads to a net stabilisation of the d9 configuration.

Answer 2: Ligand Field Theory

There is a start-to-finish increase in lattice energy as the contracted metal ion forms stronger covalent bonds with the chloride (shorter M–L distance improves overlap).
There is a ‘double hump’ trend due to the occupancy of the d-like molecular orbitals, which are split due to the octahedral ligand field. The eg* orbitals are antibonding, and filling them destabilises the compound; in contrast, the t2g orbitals can be thought of as non-bonding in the sigma-only model.
Copper’s Jahn-Teller distortion stabilises the CuCl2 lattice, as the stabilisation of the z2 orbital (poorer overlap) and destabilisation of the x2-y2 orbital (better overlap) leads to a net stabilisation of the three antibonding eg electrons.

Tin(II) oxide

SnO adopts a layer structure, with local C4v symmetry around each Sn atom. Explain this observation. [4]

Answer 1: Lewis Bond Theory

Tin(II) has a lone pair of electrons, which is stereochemically active (like a carbene). This makes the tin five-coordinate (c/f the octahedral tin in SnS, which adopts the rock salt structure), and the space taken up by the lone pair results in a puckered layer structure of square-pyramidal tin centres.

Answer 2: Jahn-Teller Analysis

Oxygen’s p-orbitals have an energy which can substantially mix with both the s- and p-orbitals of Sn(II). This means that the s- and p-orbitals of tin can mix through oxygen’s p-orbitals in non-centrosymmetric C4v symmetry, giving the HOMO not (just) spherical s-character, but directional p-character. This s-p mixed HOMO can be described as a non-bonding pair of electrons which points into the space between the layers of SnO.

VSEPR/MO Theory

Why is XeF2 is a linear molecule, rather than bent? [3]

Answer 1: VSEPR

XeF2 has five pairs of electrons around the central Xe, leading to a VSEPR prediction of a trigonal bipyramidal parent shape. The F atoms occupy the sterically-crowded axial sites and the lone pairs occupy the sterically-open equatorial sites. This gives the molecule a linear atomic geometry.

Answer 2: MO Theory

The SALCs of the F2 fragment maximise their overlap with Xenon’s p- (and arguably d-) orbitals in a linear arrangement. The u-symmetry SALC perfectly fits the u-symmetry pz orbital of Xe, and the g-symmetry SALC overlaps well with the vacant d-orbital (which is admittedly of high energy, so the bond would be weak). The linear geometry therefore permits 3c,4e bonding, allowing hypervalency through delocalising the electrons over several atomic centres.

Family 2: Partial Picture

There are also interesting cases where theories are generally seen to coexist. Here, applying several different models is something which could be seen as building up a fuller picture of the phenomenon (and perhaps developing only one model might attract partial credit in a student’s answer).

Sterics/Electronics

[NiCl4]2- is tetrahedral, but [Ni(CN)4]2- is square planar. Explain these observations. [3]

Answer 1: sterics

The big chloride ligands (larger Tolman Cone Angle) favour a larger bond angle than the small cyanide ligands.

Answer 2: electronics

The strong-field cyanide ligands favour the geometry which best stabilised the d8 configuration of Ni(II). The net stabilisation of electrons in the square planar geometry is larger than that in the tetrahedral geometry.

Is one of these answers enough? Each of them completely rationalises the observation without needing to appeal to the other.

Expanding octet with(out) d-orbitals

Explain how sulphur ‘expands its octet’ in the hypervalent compound SF6. [4]

Answer 1: involving d-orbitals

Sulphur has vacant d-orbitals, which – while not a very good energy match with the fluorine orbitals – permit it to develop six bonds with the small F atoms (whose SALCs span a1g and t1u [match for sulphur s- and p-orbitals] and also eg [which match sulphur’s d-orbitals]).

Answer 2: Not involving d-orbitals

The SALCs of the F6 fragment can match with the a1g s-orbital of sulphur and the t1u p-orbitals of sulphur, letting the SF6 molecule fill four bonding MOs and giving a bond order of ⅔ per S–F interaction. The non-bonding eg set can be thought of as bearing a 2- charge and developing ionic bonding with the octet-satisfying covalent [SF4]2+ system.

Conclusion

Acknowledging the Theoretical Pluralism of Inorganic Chemistry is a really important part of writing and marking high-quality exam questions. The pluralism of the subject changes what we reward, what we teach, how we teach it, and how students engage with Inorganic Chemistry because it is Inorganic Chemistry. The purpose of this blog has been to try and provide various examples, in the hope that one or two cases resonate with Inorganic examiners.

To return to the initial anecdote: I’m sure a student who describes the PbO structure as relating to the second-order Jahn-Teller distortion deserves marks. But I also think we should scrutinise the basis for withholding marks from the student who describes a stereochemically-active lone pair at the lead atom. In my view – and I recognise that other people feel very differently – this approach to marking would produce false negative results: everyone who gets high marks deserves high marks, but not everyone who fails deserves a fail. This threatens the validity of our assessments in ways which might not be obvious to people writing or scrutinising exam questions.

The solution is to write clearer exam questions. That student who wrote “stereochemically-active lone pair” might well have had a deep grasp of the MO theory which reconciles the lone pair idea with s-p mixing (“the s-p mixed HOMO can be thought of as a stereochemically-active lone pair”). With a better-written question, they could have been judged on their understanding of the subject rather than their understanding of the examiners’ subtext. They have answered the question, they just haven’t answered the question you really wanted to ask.

It’s not too fanciful to frame this as a disciplinary example of the hidden curriculum. There is a tacit understanding that Lewis Theory is less sophisticated than MO Theory, and that Inorganic Chemists should reach for MO arguments where possible. But this is not normally taught! And in a subject railroaded by the RSC into ‘Problem Solving’ as an outcome, it’s awkwardly true that you can solve a problem perfectly satisfactorily with an inferior theory: a stereochemically-active lone pair explains the structure of SnO completely adequately!

Exam marking in a theory-diverse subject