Exponential orbital-tail estimate of exchange ($|t(d)|\sim P(d)e^{-\kappa d},J\sim 4|t{|}^2/U$)

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.