Magnetic phase transitions that involve multipolar degrees of freedom have been widely studied during the last couple of decades, challenging the common approximation which assumes that the physical properties of a magnetic material could be effectively described by purely dipolar degrees of freedom. Due to the complexity of the problem and to the large number of competing interactions involved, the simple (fcc) crystal structure of the actinide dioxides made them the ideal playground system for such theoretical and experimental studies. In the present paper, we summarize our recent attempts to provide an ab initio description of the ordered phases of UO2, NpO2, and AmO2 by means of state-of-the-art LDA+U first-principles calculations. This systematic analysis of the electronic structures is here naturally connected to the local crystalline fields of the 5f states in the actinide dioxide series. Related to these we find that the mechanisms which lead to the experimentally observed insulating ground states work in distinctly different ways for each compound.
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