Kohn’s nearsightedness principle, that a local electronic property depends only on the effective potential within a finite decay length, recast as a Bayes-optimal conditional expectation whose agreement across overlapping neighbourhoods is exactly nearsightedness at a given radius.
Kohn’s nearsightedness principle states that a local electronic property at a point depends, beyond a finite decay length, only on the effective potential in a bounded neighbourhood. Distant perturbations at fixed chemical potential change local observables by an amount that falls off with distance. Equivalently the one- and two-particle density matrices decay away from the diagonal with a density-matrix decay length, short for gapped, near-equilibrium systems and long (or absent) for metals, delocalised
It is a physical hypothesis about a system, not a theorem about the formalism. It holds for organic molecules near equilibrium, gapped insulators, and enzyme active sites, and fails for the cases above.
Proposition in conditional-expectation form
Let a local target
be predicted from a neighbourhood of radius . Under a data distribution the Bayes-optimal prediction is the conditional expectation . Nearsightedness at scale on an overlap holds exactly when is (almost surely) determined by the common refinement ; then the two neighbours’ conditional expectations agree,
and a non-zero difference exhibits dependence on information private to one side, so nearsightedness fails at that radius.
References
- E. Prodan and W. Kohn, Nearsightedness of Electronic Matter (PNAS 102, 11635, 2005)