Variance Beats the Mean

01The evidence

• Single-atom catalysts on two-dimensional supports are now screened computationally across reaction families, and the screening consistently finds that nominally identical sites behave differently depending on local coordination and support environment.Journal of Materials Chemistry A, 2026
• Orbital-selective engineering of single-atom catalysts shows that the electronic structure of the active site is tunable through the support, which implies a distribution of site energetics rather than a single well-defined value.International Journal of Hydrogen Energy, 2026

02The tension

The d-band centre model works well for extended metal surfaces, where sites are nearly equivalent. It fails for single-atom catalysts, where measured activities scatter widely and often exceed the values predicted by scaling relations. The usual response is to invoke a new descriptor, which multiplies parameters without explaining the scatter.

03The theory

A real single-atom catalyst is not a lattice of identical sites. Atoms sit at step edges, vacancies, and defect clusters. Their local coordination differs, and so does their d-band centre. The material has a distribution, and the mean of that distribution is what conventional models use.

If the reaction rate depends exponentially on the binding energy, then the total rate is dominated by the tail of the distribution, not by its centre. Two catalysts with the same mean but different variance can differ by orders of magnitude in activity. That is precisely the kind of unexplained scatter the field reports.

This also explains why scaling relations appear to break. A scaling relation is a constraint on the mean energetics of a family of sites. If activity is generated by a minority of extreme sites, the relation constrains the wrong object.

The design implication is the opposite of conventional wisdom. Instead of making every site identical to optimise the average, make the distribution as wide as possible in the direction that puts the tail at the optimal binding energy. Heterogeneity becomes a design variable.

04What this is not

05Predictions

• P1Measured activity should correlate with d-band variance across single-atom catalysts better than with the mean.
• P2Deliberately inhomogeneous site distributions should exceed the activity ceiling predicted by scaling relations.
• P3A designed variance gradient across a support should produce an activity optimum at a predictable composition.

06What would refute it

If variance adds nothing to a model that already includes the mean and coordination number, the distributional account is redundant.

07What it would change

It would turn heterogeneity from a nuisance to be minimised into a synthesis target, and reframe scaling relations as averages rather than ceilings.