Multipole expansions: Magnetic and electric fields generated by electrons bound in spin–orbit eigenstates

A formalism is presented that enables the calculation of atomic charge and current densities in Russel–Saunders states of free atoms and ions in terms of scalar and vector spherical harmonics. The electric and magnetic fields generated by the multipole components of charge and current distributions are easily calculated. They are found to have the same multipolarity as their generating sources and Maxwell's equations are satisfied component by component. Calculations of these fields for hydrogen-like atoms and specific transition-metal and rare-earth ions are presented using realistic analytic radial wave functions.