  In low-temperature anti-ferromagnetic LaMnO$_{3}$, strong and localized
electronic interactions among Mn 3d electrons prevent a satisfactory
description from standard local density and generalized gradient approximations
in density functional theory calculations. Here we show that the strong on-site
electronic interactions are described well only by using direct and exchange
corrections to the intra-orbital Coulomb potential. Only DFT+U calculations
with explicit exchange corrections produce a balanced picture of electronic,
magnetic and structural observables in agreement with experiment. To understand
the reason, a rewriting of the functional form of the +U corrections is
presented that leads to a more physical and transparent understanding of the
effect of these correction terms. The approach highlights the importance of
Hund's coupling (intra-orbital exchange) in providing anisotropy across the
occupation and energy eigenvalues of the Mn d states. This intra-orbital
exchange is the key to fully activating the Jahn-Teller distortion, reproducing
the experimental band gap and stabilizing the correct magnetic ground state in
LaMnO$_{3}$. The best parameter values for LaMnO$_{3}$ within the DFT (PBEsol)
+U framework are determined to be U = 8 eV and J = 1.9 eV.
