  Exchange interaction strongly influences the long-range behaviour of
localised electron orbitals and quantum tunneling amplitudes. It violates the
oscillation theorem (creates extra nodes) and produces a power-law decay
instead of the usual exponential decrease at large distances. For inner
orbitals inside molecules decay is $r^{-2}$, for macroscopic systems $\cos{(k_f
r)} r^{-\nu}$, where $k_f$ is the Fermi momentum and $\nu=3$ for 1D, $\nu=$3.5
for 2D and $\nu=$4 for 3D crystal. Correlation corrections do not change these
conclusions. Slow decay increases the exchange interaction between localized
spins and the under-barrier tunneling amplitude. The under-barrier transmission
coefficients in solids (e.g. for point contacts) become temperature-dependent.
