## 2 Methods
### Lda + U Methodology  
The use of the LDA + U methodology is not specific to UO\({}_{2}\), but is a standard approach to dealing with oxides in general. In particular, for UO\({}_{2}\), it is used to achieve f-electron localization which prevents the appearance of delocalized f-electrons which would have resulted in non-integer uranium oxidation states, a nonphysical result considering recent experimental observations [18].  
For our theoretical study the electronic structure of U was represented by [Xe 4f14 5d10] 5f63 6s2 6p6 d1 7s2 (14 valence electrons), that of Pu by [Xe 4f14 5d10] 5f6 6s2 6p6 7s2 (16 valence electrons), and that of O by [He] 2s2 2p4 (6 valence electrons). The LDA + U approach was implemented with the Dudarev scheme [19] for which only the difference in the values of Hubbard coefficients \(U\) and \(J\) matters rather than each of them individually. We used (in eV) \(U=4.0\) and \(J=0\) for both U and Pu which seems to be the widely adopted choice. See, e.g., [17] or [20], in which the following sets of \((U,J)\) are chosen for U and Pu, respectively: \((4.5,0.5)\) and \((4.7,0.7)\), so that in either case \(U-J=4.0\) eV.  
The proper choice of \(U\) is also important for the system under consideration to have the correct physical properties: the FM, AFM, or NM ground state, the values of the magnetic moment per atom and the band gap in agreement with experiment, the correct lattice constant (or density), etc. We used the occupation matrix control (OCM) method which allows one to study the physical properties of systems of either of the three magnetic states as a function of \(U\). OCM for VASP was suggested by [PERSON] and [PERSON] [21] and has become available in VASP starting with the version vasp5.3.5. Its use for both UO\({}_{2}\) and PuO\({}_{2}\) was very recently demonstrated by [PERSON] and [PERSON] [22]. Using their approach, we studied the properties of both UO\({}_{2}\) and PuO\({}_{2}\) as a function of \(U\). In the case of UO\({}_{2}\), imposing [22] (i) the AF ground state, (ii) magnetic moment of 1.8 \(\mu_{B}\), (iii) the ground state lattice constant of 5.470 A, and (iv) the band gap of \(2.25\pm 0.25\) eV leads to (in eV) (i) \(U\stackrel{{>}}{{\sim}}3\), (ii) \(2.5\stackrel{{<}}{{\sim}}U\stackrel{{<}}{{\sim}}3\) (\(U=4\) gives \(\sim 1.9\ \mu_{B}\)), (iii) \(3\stackrel{{<}}{{\sim}}U\stackrel{{<}}{{\sim}}4\), and (iv) \(4\stackrel{{<}}{{\sim}}U\stackrel{{<}}{{\sim}}5\), so that, optimizing between the four criteria gives 3.5 and 4 as equally best overall choice of \(U\) for UO\({}_{2}\). Similarly, in the case of PuO\({}_{2}\), imposing [22] (i) the NM ground state, (ii) the ground state lattice constant of 5.395 A, and (iii) the band gap of 1.8 eV leads to (in eV) (i) \(3.5\stackrel{{<}}{{\sim}}U\stackrel{{<}}{{\sim}}4\), (ii) \(4\stackrel{{<}}{{\sim}}U\stackrel{{<}}{{\sim}}5\), and (iii) \(4.5\stackrel{{<}}{{\sim}}U\stackrel{{<}}{{\sim}}5\), so that, optimizing between the three criteria gives 4 and 4.5 as equally best overall choice of \(U\) for PuO\({}_{2}\). In our study, we take \(U=4\) eV for both UO\({}_{2}\) and PuO\({}_{2}\); besides being one of the best overall choices, in view of the above considerations, it will ensure both the transferability of the \"effective\" inter-atomic interaction potentials within the UO\({}_{2}\) and PuO\({}_{2}\) components of MOX of a given composition and the ease of the reproducibility of our results by other researchers.