Exchange interaction and correlations radically change behaviour of a quantum particle in a classically forbidden region

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 $\cos{(k_f r)} r^{-ν}$, where $k_f$ is the Fermi momentum and $ν=3$ for 1D, $ν=$3.5 for 2D and $ν=$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.

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