DFT + U Simulation of the X-ray Absorption Near-Edge Structure of Bulk UO2 and PuO2

Hubbard U-corrected density functional theory within the periodic boundary condition model in the WIEN2k code is used to simulate the actinide LIII and O K edge X-ray absorption near-edge structure (XANES) for UO2 and PuO2. Spin-orbit coupling effects are included, as are possible excitonic effects using supercells with a core hole on one of the atoms. Our calculations yield spectra in excellent agreement with previous experiments and superior to previous simulations. Density of states analysis reveals the mechanism behind the XANES peaks: the main contribution to the U/Pu LIII edges comes from the U/Pu d states hybridized with O p states, while as expected, the O p states primarily determine the O K edges of both UO2 and PuO2. The O K edges also feature O p hybridizing with U/Pu d and f states in the low-energy region and with U/Pu s and p states for the higher-energy peaks.


The WIEN2k programme
4] The unit cell is divided into two regions, non-overlapping atomic spheres that are centred at the nuclear sites, and the interstitial region. 5The atomic sphere radii (R MT ) are set automatically, depending on the geometry and the atomic species.The electronic states are decomposed into core and valence states.Core states are defined as having wave functions (densities) which are completely confined inside the atomic spheres; these are constrained to be localized, do not hybridize with neighbouring atoms' states but are recalculated in each self-consistent field (SCF) cycle.The valence electrons have basis functions consisting of APWs, which are plane waves in the interstitial region augmented with an angular momentum expansion with numerical radial wavefunctions u ℓ (r,E) defined at a fixed energy, and LOs which contain the energy derivative of u ℓ to allow the necessary variation of the radial functions.Additional LOs, which contain radial functions expanded at an appropriate energy, are added to treat states having lower energies than the valence states, which are called semi-core states.All energy parameters are chosen automatically and dynamically updated during the SCF cycle.Spin-orbit coupling (SOC) can be included in a second variational step using the scalar relativistic orbitals as a basis. 4,6  Figure S1

Searching for optimum U and J
A wide range of U values (0.2 -0.6 Ry) are tested for AFM UO 2 unit cell; band gap and magnetic moment of U atoms are compared with experiments to search for a suitable U value.Spin-orbit (SO) coupling has negligible influence on band gap (Figure S2 (a)), while has obvious influence on magnetic moment on U atoms (Figure S2 (b)).Without SO coupling, calculations predict slightly higher magnetic moment for U atoms than the experimental value 1.74  B , 10 and the magnetic moment (including spin moment and orbital moment) of U decreases by about 0.4  B by taking SO coupling into consideration.We then did the same calculations with J (U/7 -U/4) (Figure S2  We also adopt an AFM state for PuO 2 , as it is predicted as the lowest energy state with the DFT + U method, although previous experiments and our previous DFT + U + OMC simulation support the nonmagnetic ground state for PuO 2 .As we will consider the core-hole effects, the ground state is different for PuO 2 with and without core-hole.Besides, the density of states (DOS) of unoccupied states is the most important thing we need to take into consideration for XANES simulation.Therefore, we use AFM PuO     The choice of U has little influence on the beyond edge peaks in the O K edge XANES for both UO 2 and PuO 2 ; the peaks' positions (Table S1) and relative intensities (Figure S9) are very close for XANES simulated with different U values.In the lower energy region, the overall structures of the simulated spectra also do not show much dependence upon U, e.g.peak d of the UO 2 O K edge is always sharp in our simulated spectra, and the left shoulder peak around 9 eV does not disappear with larger or smaller U values.Similarly, peak a' in the PuO 2 O K edge XANES is found in all simulated spectra (Figure S9 (b)).However, there is some effect of U on the positions and relative intensities of the lower energy peaks.The relative intensities of peaks a and b of UO 2 and peaks a and peak a' of PuO 2 increase with U value.All up to the edge peaks move to higher energies with increasing U value.U = 0.3 Ry, which was chosen based on the ground state properties, also seems most appropriate for the O K edge, as simulations with this U value give the smallest mean absolute deviations with the experimental peak positions.Overall, we conclude that the choice of U slightly influences the intensities and positions of the low energy peaks at the O K edge of UO 2 and PuO 2 , but this variation is not sufficient to eliminate the small differences between our simulations and previous experiments.
Table S1: energies of the O K edge peaks for UO 2 (S = 0.2 eV, G = 0.8 eV) and PuO 2 (S = 0.5 eV, G = 0.6 eV), Our simulations are performed with different U values (J = U/7) for a supercell with 1CH, generated by moving one spin up 1s electron of O to the valence band or to background for UO 2 and PuO 2 , respectively.All XANES are shifted such that all peaks d are at 10.0 eV.The mean absolute deviation (MAD) of our simulations from the experimental data (Table 1) are given in the last column.

Hybrid-DFT simulations for O K edge:
5] The screening parameter  in YS-PBE0 is set to 0.165  (the screening length in HSE), but uses an exponential instead of the error function for screening in WIEN2k.In hybrid functionals a fraction  of semi-local (SL) exchange is replaced by Hartree-Fock (HF) exchange:   ℎ =    + (   −    ) We retained PBE for the SL exchange and a value of 0.25 for .We first performed the hybrid simulations with a reduced k-mesh (2 X 2 X 3 for the supercell), then redid the calculations from the converged calculation with a finer 4 X 4 X 6 k-mesh.Hybrid DFT simulations are compared with DFT + U simulations in Figure S10.To save computational time, only the low energy regions were studied, as the differences are mainly found in this region.The hybrid and DFT + U simulations predict almost the same structure for the up to edge peaks, with and without CH.
Figure S1: (a) unit cell of AFM UO 2 or PuO 2 (b) 2 X 2 X 1 supercell of AFM UO 2 or PuO 2 .Grey and red balls are U/Pu and O, respectively, light grey U/Pu denotes dominance of spin up, dark grey U/Pu of spin down electrons.
Figure S2: (a) band gap and (b) magnetic moment on U atoms for UO 2 calculated with PBE + U without and with spin-orbit (SO) coupling; (c) band gap and (d) magnetic moment on U atoms for UO 2 calculated with PBE + U + J with spin-orbit (SO) coupling.
2 in this work and search a U value by DOS.The U = 0.3 Ry give the best DOS, 11 so U = 0.3 Ry (4.1 eV) and J = U/7 = 0.04 Ry (0.5 eV) is used in this work for all PuO 2 simulations.

Figure
Figure S3: density of state (DOS) of AFM PuO 2 , simulated with U value range from 0.1 Ry to 0.6 Ry and J = U/7.

Figure
Figure S5: (a) simulated Pu L III edge of ground state PuO 2 (unit cell) and with different corehole treatment with a spectrometer broadening factor S = 0.2 eV and G = 1.0 eV.(b) ground state calculation with life-time broadening factor S = 2.4 eV and G as given in the legend.The highest peaks of XANES in figures are aligned and shifted to 10 eV, the spectra are shifted vertically for better visibility.

Figure
Figure S6: (a) dipole matrix elements of Pu s and d states; (b) density of state (DOS, broadening factor is 0.003 mRy) of Pu s (10 fold amplified) and d states and Pu L III edge XANES of PuO 2 with S = 2.4 eV and G = 9.0 eV.(c) DOS of Pu d states and amplified O s (4-fold), p (2-fold) and d (4-fold) states with a broadening factor of 0.003 mRy, and Pu L III edge XANES of PuO 2 with S = 2.4 eV and G = 9.0 eV.All calculations are done without CH.The highest XANES peak in figure (b) and (c) is shifted to 10 eV, the energy of DOS in figure (b) and (c) and dipole matrix elements in figure (a) are also shifted to match the XANES.Spectra and DOS in figure (b) and (c) are shifted vertically for better visibility.

Figure
Figure S8: (a) simulated O K edge of PuO 2 with different core-hole treatment and broadening factor S = 0.2 eV and G = 0.5 eV.(b) simulated O K edge of PuO 2 for supercell with 1 CH: up-BG; with broadening factor S = 0.5 eV and G is given in the legend.The highest peaks of XANES in figures are aligned and shifted to 10 eV, the spectra are shifted vertically for better visibility.

Figure
Figure S9: (a) simulated O K edge XANES with different U value for UO 2 supercell with 1 CH: up-VB (S = 0.2 eV, G = 0.8 eV); (b) simulated O K edge XANES with different U value for PuO 2 supercell with 1 CH: up-BG (S = 0.5 eV, G = 0.6 eV).The highest peaks of the XANES in all figures are aligned and shifted to 10 eV, intensities are scaled to 1, and spectra are shifted vertically for better visibility.

Figure
Figure S10: simulated O K edge of PuO 2 supercell (S = 0.5 eV, G = 0.6 eV), and previous experimental O K edge of PuO 2 (reference).For CH simulation, the CH is generated by removing a O 1s electron and adding to the background charge.