Multipole decomposition of LDA+$U$ energy and its application to actinides compounds

A general reformulation of the exchange energy of $5f$-shell is applied in the analysis of the magnetic structure of various actinides compounds in the framework of LDA+U method. The calculations are performed in an efficient scheme with essentially only one free parameter, the screening length. The results are analysed in terms of different polarisation channels, due to different multipoles. Generally it is found that the spin-orbital polarisation is dominating. This can be viewed as a strong enhancement of the spin-orbit coupling in these systems. This leads to a drastic decrease in spin polarisation, in accordance with experiments. The calculations are able to correctly differentiate magnetic and non-magnetic Pu system. Finally, in all magnetic systems a new multipolar order is observed, whose polarisation energy is often larger in magnitude than that of spin polarisation.