Calculation of Multipolar Exchange Interactions in Spin-Orbital Coupled Systems

A new method of computing multipolar exchange interaction in spin-orbit coupled systems is developed using multipolar tensor expansion of the density matrix in local density approximation+U electronic structure calculation. Within the mean field approximation, exchange constants can be mapped into a series of total energy calculations by the pair-flip approximation technique. The application to uranium dioxide shows an antiferromagnetic superexchange coupling in dipoles but a ferromagnetic one in quadrupoles which is very different from past studies. Further calculation of the spin-lattice interaction indicates it is of the same order with the superexchange and characterizes the overall behavior of the quadrupolar part as a competition between them.