Magnetic orbitals and mechanisms of exchange I.: Direct exchange

The problem of direct exchange is examined in the simplest cases (formal coupling of two electrons initially isolated and hydrogen molecule). Then we consider the general case of a solid characterized by weak overlaps between the atomic orbitals of first-nearest neighbours so that the involved electrons are quasi-localized. We recall the physical origin of spin-orbit coupling. Its influence on direct exchange allows one to revisit Hund's rules and to express phenomenological Hamiltonians describing anisotropic couplings as well as the antisymmetrical Dzialoshinski-Moriya coupling.