The +2 oxidation state of Cr incorporated into the crystal lattice of UO2

Doping by Cr is used to improve the performance of uranium dioxide (UO2)-based nuclear fuel. However, the mechanism of structural incorporation of Cr remains unclear. Here, in order to understand this process on the atomic scale and the redox state of Cr in UO2-based nuclear fuel, we performed intensive ab initio atomistic simulations of the Cr doped UO2 matrix. We unexpectedly found that Cr in UO2 exists as Cr2+ species and not as the widely claimed Cr3+. We re-evaluated previously published x-ray absorption near edge structure spectroscopy data and confirmed the computed redox state of Cr. Thermodynamic consideration shows that the favorable structural arrangement of Cr in UO2 is given by a pair of associated Cr2+ and oxygen vacancy. The realism of this doping mechanism is further demonstrated by a match to the measured maximum Cr solubility and small lattice contraction. Uranium dioxide is commonly doped with chromium to improve its performance as a nuclear fuel. Here, with the aid of ab initio simulations and re-evaluation of experimental data, the oxidation state of chromium in the uranium dioxide lattice is identified as +2, not the widely believed +3.

N uclear reactors use mainly UO 2 -based fuel elements. Additions of other trace elements, such as chromium, are considered to improve the microstructural and mechanical properties of fuel matrices 1,2 . Among other desired effects, chromium is known to significantly increase the grain sizes, which due to the elongation of bulk diffusion path limits the release of fission gas products such as xenon or krypton 2 , and increases the fission products retention capability by the fuel elements 3 . However, how Cr incorporates into the UO 2 matrix on the atomic level is not conclusively understood. The impact of Cr doping on the performance of UO 2 matrix depends also on the oxidation state of Cr in the nuclear fuel, which is claimed to be Cr 3+ 4,5 . Other oxidation states, however, are not excluded 6,7 . In particular, by analyzing X-ray absorption spectrum of Cr-doped UO 2 Mieszczynski et al. 7 suggest existence of Cr 2+ , but express a need for the electronic structure calculations to confirm such a possibility.
The most common oxidation states of chromium are Cr 3+ and Cr 6+ . However, the Cr 0 , Cr 1+ , Cr 2+ , Cr 4+ , or Cr 5+ could be also realized 8 . The previous experimental studies of Cr-doped UO 2 show evidence of bulk Cr incorporation as Cr 3+ 4 . At the same time, only small amounts of Cr could be incorporated into UO 2 , with maximum solubility at high temperature (T~2300 K) of up to 0.1 wt%∕UO 2 [3][4][5]9 . The formed diluted solid solution shows a linear decrease in the matrix lattice parameter with content of Cr 1,9 , which however, is significantly smaller than the predictions by most theoretical models 1 .
Besides the experimental effort, the atomistic modeling studies have been used to understand the incorporation of Cr into the UO 2 matrix at atomic level. Most of these studies used force-fields approach, in which interatomic interactions are represented by analytical functions and charge/oxidation states of the involved species are fixed 10 , for chromium atom to Cr 3+ 1,11,12 . Middleburgh et al. 12 and more recently Guo et al. 11 found that Cr incorporates as a pair of Cr atoms with the formation of an oxygen vacancy (on the oxygen site that bridges the two Cr atoms) as a charge balance mechanism. The density functional theory (DFT)-based first principles studies of Middleburgh et al. 12 consider formation of Cr 3+ as dominant Cr species in the same structural arrangement as resulted from the force-field simulations, and discuss a less energetically favorable scenario of Cr 4+ . The more recent ab initio studies of Cooper et al. 6 show the possibility of Cr in "+1" oxidation state. Interestingly, the "+2" state has never been considered as dominant at ambient or hightemperature synthesis conditions (T> 1500 K). The oxidation state chemistry of uranium oxides is very interesting with different oxides realizing uranium in different oxidation states (U 4+ , U 5+ , or U 6+ ), for instance U 3 O 8 and U 4 O 7 as mixtures of (U 6+ , U 5+ ) and (U 5+ , U 4+ ) pairs, respectively [13][14][15] . Potential change in the cations redox states would have a significant impact on the evolution of nuclear fuel, including, for instance, its dissolution rates 16,17 . The real oxidation states are not always well constrained, but could be conclusively deducted by a combination of X-ray absorption near edge structure (XANES) measurements and first-principle atomistic modeling 15 . It is also not conclusively clear, how different impurities or defects affect oxidation states of the UO 2 matrix atoms.
Here, we present the atomistic modeling studies combined with an extensive analysis of the relevant experimental data (XANES, the maximum solubility and the lattice parameter), aimed at understanding the atomic structure and redox chemistry of Cr-doped UO 2 material. We have been especially interested in the resulting oxidation state of Cr and U cations, derivation of the conclusive structural model of Cr incorporation and testing the ability of such a model structure to match the measured solubility limit of Cr in UO 2 and the lattice parameter of Cr-doped UO 2 matrix.

Results
The computed Cr-doped UO 2 . In order to perform the DFT+U calculations, we first derived the Hubbard U parameter for the d and f elements in the considered compounds (Cr and U oxides, see Fig. 1 for the computed structure of UO 2 ). The computed value of Hubbard U parameter for Cr in Cr 2 O 3 (i.e., Cr 3+ ) is 4.0 eV, which is consistent with the one used in previous studies of Cr-bearing systems (~3.5 eV, e.g., refs. 6,[18][19][20][21]. For Cr 2+ we used the value of 6.5 eV, which we derived for CrO. The same value we were getting for Cr 2+ in UO 2 and we observed that the derived Hubbard U parameter depends strongly on the Cr oxidation state, but is weakly dependent on the local environment of Cr. Therefore, we decided to use the oxidation state-dependent values derived for Cr-oxides in all the calculations of Cr-doped UO 2 . For UO 2 we used 1.7 eV derived for UO 2 by Beridze and Kowalski 22 , which is also representative for U 4+ species 15,22,23 . The detailed information and discussion on the lattice parameter, the band gap and the X-ray related spectral signatures obtained for UO 2 , as well as other uranium oxides with the applied computational methodology is provided in the previous studies 15,22 . Initially, we computed the two most plausible cases of Crdoped UO 2 , namely: (1) the cation exchange with the formation of U 5+ species and (2) the two neighboring site cation exchange with the formation of O-vacancy on the bridging oxygen site attached to the Cr atoms as proposed by Guo et al. 11 . Both configurations are visualized in Fig. 2 and are marked as 1Cr and 2Cr + 1Ov, respectively. Following the interpretation of XANES measurements by Riglet-Martial et al. 4 , in both cases we initially assumed formation of Cr 3+ species. In the case (1) we assumed the charge balance through formation of one U 5+ per one Cr atom. In addition, for the case (1) we tried to compute systems with Cr 2+ and Cr 4+ species, by enforcing the initial occupation of d-orbitals of Cr (4, 3, and 2 d electrons for Cr 2+ , Cr 3+ , and Cr 4+ , respectively). Surprisingly, in all the calculations, the resulting electronic configuration was always that of Cr 2+ (i.e., four electrons), with charge balanced by the formation of U 5+ species. Fig. 1 The structure of computed UO 2 supercell. Arrows indicate the computed transverse 1k anti-ferromagnetic spin arrangement, which was found by Pegg et al. 52 as the preferred magnetic structure in UO 2 and adopted in our studies.
For instance, in case (2) with the formation of two Cr 2+ species we got a pair of U 5+ .
Such an unexpected theoretical result requires sounding justification and support from the experimental data. The simplest thing to do, when considering oxidation state of a species substituting another host matrix element, i.e., chromium substituting uranium atom in our case, is to check the sizes of involved cations. This is because it is easier to exchange cations of similar sizes due to minimization of the associated strain resulting from sizes mismatch. In Table 1, we list the ionic radii and volumes of the considered species 24 . Cr 3+ is much smaller than U 4+ (by 66% in volume) and even U 5+ (by 51% in volume). The size of Cr 2+ is much similar to that of U 4+ (smaller by only 32 % in volume) and nearly the same as that of U 5+ . As excess strain effects arise as quadratic function of volume mismatch (e.g., Kowalski and Li 25 ), considering the sizes of cations, the scenario with Cr 2+ is more probable. However, even with appealing theoretical arguments in hand, one can not easily neglect the results of XANES studies of Riglet-Martial et al. 4 that seem rather conclusively indicate formation of Cr 3+ in Cr-doped UO 2 . Therefore, we decided to perform re-analysis of these data using the reference XANES for Cr 2+ , the case that is missing in the analysis of Riglet-Martial et al. 4 .
The re-evaluation of the experimental XANES. The oxidation state of an atom in a solid matrix is usually measured with the help of XANES technique 26 . For that purpose, the measured data are compared to a set of reference data of a species in different oxidation state. In the previous studies of Riglet-Martial et al. 4 , the references for Cr 0 (Cr metal), Cr 3+ (Cr 2 O 3 , FeCrO 4 ), and Cr 6+ (CaCrO 4 ) were used for analysis of XANES of Cr-doped UO 2 . The authors concluded that their XANES of Cr-doped UO 2 resembles best the reference for Cr 3+ , which they use as a definitive evidence for the existence of Cr 3+ in UO 2 . The conclusion is mainly supported by the best match of the positions of two preedges, which follows from the in-depth characterization of XANES spectra of various Cr-bearing compounds by Farges 27 . However, the match to the reference XANES is not ideal, which suggest possible other interpretation. In similar studies, Mieszczynski et al. 7 , although assumed Cr 3+ , also discuss possibility of existence of Cr 2+ state, which could be indicated by a shoulder feature at 5996 Å, marked by the arrow in Fig. 3. In order to confirm or reject the possible Cr 2+ state scenario they expressed a need for ab initio calculations. As such simulations performed here show definitely the Cr 2+ state, we made an extensive literature search for a relevant reference for Cr 2+ state. We found such data on Cr 2+ in synthetic enstatite (Mg 0.9 Cr 0.1 SO 3 ) 28 . With these, in Fig. 3 we plot the XANES of Cr-doped UO 2 of Riglet-Martial et al. 4 , which are consistent with the data of Mieszczynski et al. 7 , together with the reference data for Cr 3+ (Cr 2 O 3 ) and for the aforementioned Cr 2+ . The comparison of the entire spectra shows that the XANES measured by Riglet-Martial et al. 4 resembles as closely the Cr 3+ as Cr 2+ references. This refers to the spectral shape and the distribution of peaks and dips, that are, anyway, not perfectly reproduced by the references. This "imperfectness" is rather expected and, as explained by Farges 27 , comes from the different local environments of Cr in all the considered phases. Following analysis of Farges 27 , applied also by Riglet-Martial et al. 4 , it is the positions of the two pre-edges, or weighted centroids (of the peaks located at~5991 eV and~5994 eV in the case of Cr 2 O 3 , Fig. 3) that discriminate between the different oxidation states of Cr. There is a significant and progressive shift in the second pre-edge peak (the higher energy peak) and centroid positions as a function of Cr oxidation state (~1 eV per increase in the oxidation state). For different Cr species, the centroid should be located as follows:~5993 eV (Cr 2+ ),~5994 eV (Cr 3+ ),~5995 eV (Cr 4+ ),~5995.7 eV (Cr 5+ ) and~5996.7 eV (Cr 6+ ) (see Fig. 4 of Farges 27 ). We thus have checked the position of that centroid in XANES of Cr-doped UO 2 and compared it to the references for Cr 2+ and Cr 3+ . The positions of the first, lower energy pre-edge peak are very similar for both oxidation states of Cr 7,27 (See Fig. 3 and Supplementary  Fig. 1). We note that the comparison to the Cr 2+ reference was not performed by Riglet-Martial et al. 4 . Fig. 2 The computed supercells representing different considered Cr incorporation configurations. In cases of 1Cr+1Ov (a) and 2Cr+2Ov (a) Cr is shifted towards empty cation interstitial site and the two structures resulted from classical molecular simulation runs. Table 1 The ionic radius (r), and volume (V) of Cr and U cations in different oxidation state. The values are given in Å and Å 3 , respectively. Cases of 6-and 8-fold coordinated cations are reported with coordination number given in parentheses.
As shown in Fig. 3, the second pre-edge peak that discriminates the different Cr species is located at~5993 eV for Cr 2+ and 5994 eV for Cr 3+ species, and the offset of~0.7 eV is clearly visible. On the other hand, that peak is barely visible, but clearly detectable (Fig. 3) in the spectra of Cr-doped UO 2 of Riglet-Martial et al. 4 . In order to check the right position of that preedge peak, we isolated the relevant peak from the spectra by subtracting out a linear baseline that spans the base of the feature. The obtained profiles of the peaks for the Cr 2+ and Cr 3+ references and the Cr-doped UO 2 XANES spectra are compared in Fig. 4. The three peaks extracted from XANES are symmetric, which is demonstrated with good fits by the Gaussian (see Supplementary Table 1 for the fitted parameters). This also validates the applied data processing procedure. It is clearly visible that the position and height of the peaks of Cr 2+ and the Cr-doped UO 2 are similar and the position of the peak in the case of Cr 3+ is significantly shifted rightwards by~0.7 eV. The so derived second pre-edge positions obtained by fitting with Gaussian function are given in Table 2. Here, we also report the positions of centroid (i.e., a weighted average of the all preedge peaks) obtained using the multi-peaks fitting procedure proposed by Farges 27 and Wilke et al. 29 (see Supplementary  Note 1 and Supplementary Tables 2 and 3 for details).
The obtained positions of the pre-edge centroid were then compared with the results obtained by Farges 27 for the set of Crmaterials containing Cr in different oxidation states ranging from "+2" to "+6". The results are given in Fig. 5, which resembles Fig.  4 of Farges 27 . The positions of centroids we got for the Cr 2+ and Cr 3+ references ( Table 2) fell within the given ranges of Farges 27 , indicating agreement with that data. The centroid position in the case of Cr-doped UO 2 falls in the range corresponding to Cr 2+ . The careful analysis of the pre-edge XANES spectrum of Crdoped UO 2 shows thus rather clearly the "2+" oxidation state of chromium in Cr-doped UO 2 .
Both, the above-discussed atomistic simulations and the experimental XANES data thus clearly indicate incorporation of chromium into UO 2 matrix as Cr 2+ species. This must be associated with the formation of U 5+ species or O vacancies. This may affect the overall performance of such material, including its dissolution rate, which strongly depends on the U oxidation states 30,31 . In order to predict the most probable structural arrangement of Cr in UO 2 matrix we computed a set of simple, different Cr incorporation arrangements involving single, as well as pair of dopants with charge balancing through formation of oxygen vacancies (Ov) or U 5+ species. The eight computed configurations are visualized in Fig. 2 and some of them (e.g., 1Cr+1Ov (a) and 2Cr+2Ov (a)) were obtained with the aid of classical molecular dynamics simulations. All the constructed supercells were then carefully analyzed regarding the enthalpy and the free energy of relevant cation exchange reaction and the change in the matrix lattice parameter. The obtained results were analyzed against the available experimental data on the maximum solubility (requiring the reaction enthalpy and the free energy as parameters) 1,4,9,32,33 and change in the lattice parameter of Cr-doped UO 2 1,9 .
The Cr solubility limit in UO 2 . Riglet-Martial et al. 4 derived a simple thermodynamic model of maximum Cr solubility in UO 2 , which when fitted to the measured temperature dependent maximum Cr solubility in UO 2 4,9,32,33 resulted in the estimation of Cr solution enthalpy and entropy (Table 5 of Riglet-Martial et al. 4 ). The result depends on the Cr-oxide reference phase and for Cr 2 O 3 they obtained the reaction enthalpy of 92.4 kJ mol −1 , assuming Cr 3+ incorporation into UO 2 . In Table 3 we provide the enthalpy of single Cr atom solution computed ab initio, assuming the following cation exchange reactions (having the size of computed supercell as 32 UO 2 units), realizing eight considered structural arrangements of Cr in UO 2 (Fig. 2): In addition, in Table 3 we report an estimate for the Cr solution free energy (ΔG), which we derived by adding/ subtracting the high-temperature entropy of molecular oxygen gas (S(O 2 ) = 270 J mol −1 K −1 , Chase 34 ) and the oxygen partial pressure contribution to the free energy. Depending on the involvement of molecular oxygen as product or reactant, the free energy is estimated from: where ΔS ≃ qS (O 2 ) and ΔH is the enthalpy of single cation exchange reaction (Eq. (1)). The temperature dependent oxygen partial pressure was estimated using the model of Toker et al. 35 and the values used in the calculations are reported in caption of Table 3.
The derived reaction enthalpies suggest that the 2Cr+1Ov incorporation mechanism is the most preferable, which is in line with the prediction of Guo et al. 11 . However, the reaction free energy-a more relevant parameter-points towards 1Cr+1Ov and 2Cr+2Ov scenarios, with the 1Cr+1Ov being the most preferable. We thus assume that Cr incorporates as an associated pair of Cr and Ov, and focus on checking the performance of the two structures (1Cr+1Ov (a) and 1Cr+1Ov (b), Fig. 2) in matching the measured solubility data. For that case we consider the following cation exchange reaction: The activity, a, of Cr in UO 2 is related to the oxygen gas partial pressure and the cation exchange reaction enthalpy and entropy as (Riglet-Martial et al. 4 , Eq. (6)): log 10 ðaðCr : UO 2 ÞÞ ¼ q log 10 P O 2 þ 1 Rln10 ðS À H=TÞ: The 1Cr+1Ov structure represents a pair of Cr atom and associated Ov. In that case, for cubic fluorite structure, the activity of Cr in UO 2 is a(Cr:UO 2 ) = (y∕8), where y is the concentration of Cr in UO 2 . This is because an associated Ov has eight possibilities to arrange around the Cr atom (the case of a cube with Cr atom in the center and eight oxygen atoms in the corners), This gives the following equation for the maximum solubility of Cr in the case of 1Cr+1Ov structural arrangements: ðS À H=TÞ þ log 10 8: ð5Þ The 2Cr+2Ov structure represents a pair of associated Cr atoms with removed two oxygen atoms that bridge the Cr atoms. In that case, for cubic fluorite structure, the activity of Cr in UO 2 is a(Cr:UO 2 ) = (y∕12) 1∕2 . This is because for a pair of associated Cr atoms, second Cr atom has 12 possibilities to be arranged around the first Cr atom (the two Cr atoms are connected by a cube edge and there are 12 such arrangements (edges)), and the activity of a single atom is just the square root of the activity of a pair. This gives the following equation for the maximum solubility of Cr in the case of 2Cr+2Ov structural arrangement: ðS À H=TÞ þ log 10 12: The resulted Cr solubility, y, estimated for the conditions of various measurements reported in the literature, by applying the two discussed structural arrangement models and the computed cation exchange reaction enthalpies (Table 3) along with the experimental oxygen partial pressures, are given in Table 4. We note that such an estimate in the case of the preferred 1Cr+1Ov model, with the computed here ΔH, only slightly overestimates the The values are given in kJ mol −1 and computed per single Cr atom. The bold face font indicates the thermodynamically favorable structure. The oxygen partial pressures used in the estimation of the free energies are log 10 PðO 2 Þ ¼ À12:85 (T = 1800 K) and log 10 PðO 2 Þ ¼ À9:01 (T = 2200 K) 35 Fig. 5 The pre-edge centroid positions for the Cr in different oxidation states varying from "+2" to "+6". The data represented by black triangles come from ref. 27 . The blue diamond is the result for Cr-doped UO 2 and the green circle and red square are the result for Cr 3+ and Cr 2+ references. Table 2 The position of the second pre-edge peak feature obtained by fits by Gaussian function (A expðBðx À CÞ 2 Þ) as illustrated in Fig. 4 and the position of the centroid. measured solubilities and that the perfect match could be obtained with the reaction enthalpy increased by only~21 kJ mol −1 . This is an excellent agreement, keeping in mind computational uncertainty of~50 kJ mol −1 22 . On the other hand, in the case of 2Cr +2Ov model, the computed ΔH is by~90 kJ mol −1 larger than the value that matches the maximum solubility data. The good description of the Cr solubility data with the 1Cr+1Ov model is yet another argument supporting the Cr 2+ scenario found in our atomistic modeling studies and by the re-evaluation of the XANES data, pointing towards 1Cr+1Ov structural arrangements as a realistic case. In the next step, we will check how that structure matches the measured lattice parameters of Cr-doped UO 2 .
Lattice parameter of Cr-doped UO 2 . It is established experimentally that the UO 2 matrix doped with Cr undergoes contraction. The two available experimental studies of Leenaers et al. 9 and Cardinaels et al. 1 show linear decrease of the lattice parameter with content of Cr, with the amount of incorporated chromium of up to 1000 μg g −1 . The measured relative decrease of lattice parameter, reported in Table 5, varies by a factor of 2 between the two experimental studies, but in both cases is smaller than the one expected from consideration of sizes of the involved species 9 . In Table 5 we also report the simulated change in the lattice parameter obtained for all the considered structures (Fig. 2). Interestingly, the 1Cr+1Ov (a) and 2Cr+2Ov (a) structures result in very small shift and the result obtained for 1Cr +1Ov (a) structure matches the measured data. The other structures, including the 2Cr+1Ov configuration, result in significantly, at least order of magnitude larger change. We note however, that most of the configurations, except 2Cr+1Uv and 2Cr+2Ov (a) cases, predict decrease of the lattice parameter. The small, and well consistent with the measurements, lattice contraction with 1Cr+1Ov (a) structure is thus yet another argument that this is the correct Cr incorporation model.

Discussion
Using the accurate first principles method we performed intensive ab initio calculations of chromium-doped UO 2 matrix that represents an enhanced nuclear fuel case. In spite of trying to obtain the Cr 3+ oxidation state, either by co-formation of U 5+ or creation of oxygen vacancies, we always ended up with the Cr 2+ state. The consideration of U 4+ , Cr 3+ , and Cr 2+ cations sizes shows that Cr 2+ has much similar size to U 4+ and Cr 3+ is much smaller, indicating plausibility of Cr 2+ scenario. The goodness of such, nevertheless, rather-unexpected prediction has been further demonstrated by the re-evaluation of the measured XANES with the previously omitted reference for Cr 2+ species. The calculations show that the most thermodynamically favorable structure is with an associated pair of Cr atom and oxygen vacancy. In such a configuration, Cr atom is shifted towards cations sublattice vacant site, which results in reduced, pseudo sixfold coordination, as opposite to the eightfold coordinated uranium site. The thermodynamic consideration of such a structure results in a good match to the measured maximum solubility of Cr in UO 2 and the small lattice contraction of Cr-doped UO 2 . This strongly indicates that we not only obtained a correct description of Cr oxidation state as "+2" valence state, but also good understanding of the structural incorporation of Cr into the UO 2 matrix. In these aspects the obtained results should be very helpful for correct interpretation of the measurements of Cr-doped UO 2 nuclear fuel. We hope that our studies will trigger supplementary investigations of not always intuitive redox solid state chemistry and structural/thermodynamic parameters of doped nuclear fuel matrices.
The reported results come from the synergy of atomistic modeling and data collected by different experimental techniques. A careful choice of the accurate computational method allowed for making an unexpected prediction regarding the oxidation state of Cr, which initiated the re-evaluation of the experimental data. This shows the need of atomistic modeling support for correct interpretation of the measurements and thus in-depth, correct understanding of challenging electronic systems such as, for instance, the doped UO 2 matrices.

Methods
Density functional theory (DFT) simulations. The DFT-based ab initio calculations were performed with the DFT-based plane-wave Quantum-ESPRESSO package 36 . In DFT calculations, we specifically applied the gradient-corrected PBEsol exchange correlation functional 37 because it results in a good description of structures of f-element-bearing materials 22,38,39 and a good prediction of their vibrational 40,41 and thermodynamic parameters 25,42 . We applied the scalar Table 4 The computed and measured maximum solubility (y) of Cr in UO 2 reported for the experimental conditions. The experimental oxygen chemical potential and oxygen partial pressure are also reported. The computed values refer to the 1Cr+1Ov (M1) and 2Cr+2Ov (M2) models and we used the computed reaction enthalpy, ΔH = 327 kJ mol −1 and 384 kJ mol −1 , respectively. In addition we report the ΔH values fitted to the experimental data, and the resulting maximum solubility (y e ) obtained with these values. The uncertainties in the measured solubilities are estimated at 7% 4 . Table 5 The computed and measured relative change of lattice parameter a (Δa∕a) in Cr-doped UO 2 with Cr content of 1000 μg g −1 . Configuration +0.71 Exp. 9 −0.12 Exp. 1 −0.22 The bold face font indicates the result for thermodynamically favorable structure (see Table 3).
relativistic ultrasoft pseudopotentials to mimic the presence of core electrons of the considered atoms 43 . We note that in the case of uranium, the spin-orbit effects are of secondary importance and can be neglected 22,44 . As a result of our previous extensive tests studies, we applied the plane-wave cut-off energy of 50 Ryd 22 . The computed 2 x 2 x 2 supercell of UO 2 contained 96 atoms and the Brillouin zone was sampled with the 2 x 2 x 2 Methfessel-Paxton k-points grid 45 . The size of the computed system and the k-points sampling are similar to the simulation setup used in the previous studies 6, [46][47][48] . The lattice parameters and the atomic positions in all the computed structures were relaxed to the equilibrium positions, assuming constant pressure of P = 0 GPa with a tolerance of 0.1 GPa. The computed atomic structure is visualized in Fig. 1. It is of cubic fluorite-type with eightfold coordinated U atoms. As U and Cr atoms contain strongly correlated electrons, 5f and 3d, respectively, we applied the DFT+U method and derived the Hubbard U parameter from first principles using the linear response method 49 . This method is known to give accurate values for 3d elements 50,51 and has been extensively tested by us for f elements and subsequently applied to describe the electronic structure of uranium oxides, including UO 2 (e.g., refs. 15,22,38 ). Following the recent studies of Pegg et al. 52 , all the calculations were performed assuming anti-ferromagnetic transverse 1k spins arrangement in UO 2 . This resulted in a small distortion towards tetrahedral symmetry (~0.3% of the lattice parameter value). Computation of the correct ground state of UO 2 is a challenge by itself and to assure the convergence to the correct ground state we applied the occupation matrix control technique 53 . After the tests performed by Beridze and Kowalski 22 for uranium-based compounds, we estimate the uncertainty of the so computed solid state reaction enthalpies at the level of~50 kJ mol −1 .
Molecular dynamic simulations. In order to probe the possible structural arrangements of Cr atoms within the UO 2 matrix, focusing on the potential distortions of the computed atomic arrangements, we performed a set of test, qualitative level classical molecular dynamics simulations runs using GULP code 54 56 . We note that simple Buckingham-type potentials can not be expected to provide accurate (e.g., at the ab initio calculations level) structural parameters or energies and that mixing of the interaction potentials of different types is not always well grounded. However, we used the molecular dynamics simulations only to explore potential structural distortions of the computed structural arrangements of Cr in UO 2 and these simulations were not used to derive any numbers or structural/thermodynamic parameters discussed in the paper. The molecular dynamics NPT (constant pressure-temperature) runs were performed using standard approach (e.g., ref. 12 ). These were 20 ps long, with time step of 0.001 ps, and were performed at T = 1000 K and P = 0 GPa, both controlled by the thermostat.