= Exponential orbital-tail estimate of exchange
{title2=$|t(d)|\sim P(d)e^{-\kappa d},\quad J\sim4|t|^2/U$}
For localized orbitals with an exponential tail $e^{-\kappa r}$, overlap-mediated hopping over separation $d$ inherits an exponentially small factor $e^{-\kappa d}$, multiplied by geometry-dependent polynomial factors. A kinetic-exchange scale then falls approximately as $e^{-2\kappa d}$. Comparisons must use actual nearest-neighbour distances, not interchange <lattice> constants between different crystal structures. The estimate is not a replacement for a screened solid-state orbital calculation.
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