In a quantum-mechanical context, this can also be written as where −e and me are the charge and mass of the electron, Ψ is the ground-state wave function, and L is the angular momentum operator.
For a crystal of volume V composed of isolated entities (e.g., molecules) labelled by an index j having magnetic moments morb, j, this is However, real crystals are made up out of atomic or molecular constituents whose charge clouds overlap, so that the above formula cannot be taken as a fundamental definition of orbital magnetization.
However, because of the factor of r in the integrand, the integral has contributions from surface currents that cannot be neglected, and as a result the above equation does not lead to a bulk definition of orbital magnetization.
Nevertheless, it is often a good approximation because the orbital currents associated with partially filled d and f shells are typically strongly localized inside these atomic spheres.
A general and exact formulation of the theory of orbital magnetization was developed in the mid-2000s by several authors, first based on a semiclassical approach,[5] then on a derivation from the Wannier representation,[6][7] and finally from a long-wavelength expansion.