Electronic structure and excitations in oxygen deficient CeO from DFT calculations
The electronic structures of supercells of CeO have been calculated within the Density Functional Theory (DFT). The equilibrium properties such as lattice constants, bulk moduli and magnetic moments are well reproduced by the generalized gradient approximation (GGA). Electronic excitations are simulated by robust total energy calculations for constrained states with atomic core holes or valence holes. Pristine ceria CeO is found to be a non-magnetic insulator with magnetism setting in as soon as oxygens are removed from the structure. In the ground state of defective ceria, the Ce- majority band resides near the Fermi level, but appears at about 2 eV below the Fermi level in photoemission spectroscopy experiments due to final state effects. We also tested our computational method by calculating threshold energies in Ce-M and O-K x-ray absorption spectroscopy and comparing theoretical predictions with the corresponding measurements. Our result that electrons reside near the Fermi level in the ground state of oxygen deficient ceria is crucial for understanding the catalytic properties of CeO and related materials.
pacs:71.28.+d, 71.15.Mb 71.15.Qe 79.60.-i
Mixed-valency cerium oxides (ceria) are technologically important materials trovarelli (); esch (); pirovano2007 () with remarkable properties that are useful for applications in heterogeneous chemical r2 () and electrochemical catalysis r3 (); mogensen (); r5 (); r6 (). In chemical catalysis, ceria is used as an active support. Ceria at an interface catalyzes surface reactions r7 (); r8 (), while the bulk material is used as an oxygen reservoir. In electro-catalysis, on the other hand, mixed ionic and electronic conductivity (with electrons localized around Ce) r9 (); r10 () is essential for making ceria a potentially good electrode in solid oxide fuel cells r11 (); r12 (); r13 (); r14 () with an outstanding electrocatalytic activity even without any metal co-catalyst r15 ().
Ce and oxygen vacancies are thought to be the active sites on ceria surfaces r16 (); r17 (); r18 () in reactions such as hydrolysis, with the surface undergoing a Ce/ Ce redox cycle during the complete catalytic reaction. In the pristine CeO compound, Ce atoms assume a +4 oxidation state, but the phase diagram of ceria contains a continuous range of partially reduced phases CeO in which oxygen vacancies can be easily formed or eliminated. The formation of oxygen vacancies in CeO results in changes in cerium oxidation state similar to those implicated in the cuprates jbmb (). Notably, the Ce valence and defect structure in ceria can change in response to physical parameters such as temperature, voltage, and oxygen partial pressure mogensen ().
The nature of the Ce active site remains not well-understood because studies with ceria are complicated by the fact that the electrons appear to be far from the Fermi level. The degree of participation of -electrons in catalytic reactions watkins (); galea (); hutter (); hellman () is therefore not clear, since the standard theory of catalysis heavily relies on localized orbitals at the Fermi energy, norskov (). Fortunately, the computational description of perfect CeO structure is not so complicated due to the absence of Ce 4 electrons in the insulating material prb_ceo2 (); abri (). However, in CeO, when partially filled orbitals are involved, the ground state predicted by the density-functional theory (DFT) clearly places the electrons in narrow bands piled at , interacting only weakly with other electrons. Since the elusive -transition in pure Ce can be described quite accurately by temperature dependent DFT calculations in which vibrational, electronic and magnetic free energies are taken into account, ce (), DFT should be expected to provide a reasonable description of ceria. In fact, corrections to the value of the chemical expansion coefficient marrocchelli () have been explained via disorder and entropy effects enhancing lattice expansion ce () rather than via correlations beyond the Generalized Gradient Approximation (GGA) gga (). In sharp contrast, however, signatures of bands are often found in spectroscopic measurements not at the Fermi energy , as GGA predicts, but several eV’s above or below depending on the nature of the spectroscopy. These shortcomings may be cured in an advanced DFT approach fexit (), which includes relaxation energies relevant for excitation of occupied and empty states in various spectroscopic probes. These relaxation energies are of the order of the corrections obtained within quasi-particle schemes chantis (); pcs66 () or modeled by adding a Hubbard term cococcioni (). Overall, the published studies that consider CeO cover the range of parameter between 2 and 8 eV, depending on the property of interest andersson (); ldau0 (); ldau1 (); ldau2 (); preda2011 (); spiel2011 (); peles2012 (). Many-body perturbation theory with jiang2009 (), self-interaction corrections gerward2005 (), and hybrid DFT functionals have also been considered graciani (); gillen (). In the present study, we explore the relaxation energy approach fexit () to find a reasonable description of x-ray photoemission (XPS) as well as x-ray absorption spectroscopy (XAS) results in CeO.
An outline of this paper is as follows. In Sec. II, we present details of our electronic structure and total energy computations for various CeO supercells where . Results of the calculations are presented and compared with relevant experimental data in Sec. III, and the conclusions are given in Sec. IV.
Ii Experimental setup and Method of calculation
The photoelectron spectroscopy experiments were performed at beamline 9.3.2 at the Advanced Light Source (Berkeley). Detailed description of the AP-XPS endstation used in this study and the ceria thin films sample preparation can be found in Refs. Ref1, ; Ref2, . The Fermi level was determined by using a gold foil. The binding energy was also calibrated by using the Pt 4 core level. The Ce 4 spectrum was collected at a photon energy of 270 eV Ref3 (). In the interest of brevity, we refer to previous publications for further details of measurements. Ref3 (); acscatal ()
In order to determine various equilibrium properties, we used the pseudo-potential projected augmented wave method vasp1 () implemented in the VASP package vasp2 () with an energy cutoff of 520 eV for the plane-wave basis set. CeO has a cubic fluorite lattice (Fmm) with four cerium and eight oxygen atoms per unit cell. The exchange-correlation energy was computed using the GGA functional gga_pbe (), which gives a reasonable agreement with experimental low temperature equilibrium volumes for CeO and CeO. Andersson et al. andersson () have pointed out that this agreement is not maintained if a non-zero Coulomb parameter is deployed in the GGA scheme.
To estimate the XPS and XAS relaxation effects, we have performed self-consistent first-principles calculations using the the Linear Muffin-Tin Orbital (LMTO) method lmto () within the Local Spin Density Approximation (LSDA) lda () as in Ref. fexit () for supercells containing 4 or 8 formula units of CeO. The same LMTO method has been successfully applied previously to study the effect of doping copper oxide high temperature superconductors jbmb (); bbtj (). Here, empty spheres were inserted in the interstitial region opposite to the oxygen atoms, a total of 16 or 32 spheres per supercell. Defective ceria CeO configurations were modeled with the supercell method by considering two concentrations: and , in the 16 and 32 atom supercells, respectively.AB1 () The converged self-consistent results were obtained using a mesh of 286 or 89 -points within the irreducible Brillouin zone for the small and large supercells, respectively. These calculations were made for a lattice constant of 5.45 Å for stoichiometric CeO and 5.54 Å when vacancies are present. The atomic sphere radii in the LMTO calculations are 0.303 for Ce, 0.230 for O and 0.196 for the empty spheres. A precise tetrahedron method was used to determine the density-of-states (DOS) rath ().
In order to calculate the XAS threshold energy, we start with the electronic structure obtained within the LMTO method. Our approach for modeling XAS lerch () assumes that the absorption is essentially a screened single-particle process. A step to account for many-body relaxation effects is to extract an electron from the core shell and add it into the valence states. The electronic structure computations were carried out self-consistently under these constrained conditions. After the system has relaxed, we consider the total energy difference between the unperturbed state and the relaxed state to determine the XAS threshold energy.
The calculation of the excitation energy in x-ray photoemission spectroscopy (XPS) from the occupied Ce- state is made in the same way as in our earlier study of NdCuO fexit (). Excitation energies for localized -electrons are different from those for itinerant electrons, since relaxation effects are smaller for itinerant bands. An electron is removed at an energy lying just below the Fermi level from the occupied majority state on one of the Ce atoms, and it is then spread out uniformly over the cell to account for a final state at high energy. The difference in total energy per electron between this state and the ground state gives the relaxation energy, defined in Appendix A. In particular, the final state will appear shifted by an amount with respect the Fermi level. The procedure for simulating inverse photoemission (or bremsstrahlung isocromat spectroscopy, BIS) is reversed. The final state then has one -electron in an empty Ce- state and the same amount of opposite neutralizing charge is spread uniformly over the cell. These procedures assume large excitation energies because the compensating uniform charges are valid approximations for free electron states ignoring the lattice potential jn ().
iii.1 Ground-state properties
The VASP calculation on a CeO cell gives an equilibrium lattice constant Å and a bulk modulus GPa, which compare well with the corresponding low-temperature experimental values Å eyring () GPa nakajima (). By removing an oxygen atom and by letting the volume and the atomic positions relax, the lattice constant was found to expand to Å , the bulk modulus reduced to GPa, and the total spin magnetic moment was 2 . This predicted ferromagnetic structure is consistent with experiments fernandes1 (); fernandes2 (); chen () and other first principles studies han (). We note that by removing all the oxygen atoms, one recovers the fcc-phase of Ce, and the calculated bulk modulus is only 51.4 GPa for the non-magnetic phase at an equilibrium lattice constant of 4.714Å. Our results for CeO as a function of the number of oxygen vacancies are summarized in Fig. 1. The calculated chemical expansion coefficient of CeO is given by . However, our results for O-vacancies in the small unit cell CeO show that is not at all linear. In fact, between and , is , while between and , , which is slightly below the experimental value marrocchelli (). Part of the anomalous behavior of can be explained in term of oxygen vacancy ordering leading to lattice contractions opposite to the chemical expansionkuru (). The GGA calculation for one vacancy in a large supercell with atoms gives a larger dilute vacancy limit () as shown by Marrocchelli et al.. Therefore, calculated values of are not easy to compare with experiment within large ranges of and . The discrepancy between DFT and the experiment could be explained by disorder and entropy effects leading to larger high-temperature equilibrium volumes ce (). The electronic entropy will increase the lattice constant of oxygen-deficient ceria, since the DOS near the Fermi level is higher at large volume and large , although this effect is rather weak. Lattice disorder is caused both by zero-point motion gga () and thermal vibrations fesi (), which produce a pressure given by , where the phonon frequency is proportional to . Also the effect of spin and orbital magnetic fluctuations usually produce lattice expansions as shown in Ref. ce (). The relevant temperature range for catalytic applications is rather high (i.e. C) Ref3 () and the effects from lattice vibrations on magnetic fluctuations in this range make the total entropy balance complicated.
The LMTO electronic structure of CeO is found to be non-magnetic and insulating as shown in Fig. 2. The distance between the valence band and the Ce 4 edge is about 1.6 eV, to be compared with 3 eV in experiments prb_ceo2 (); delley (). Ferromagnetism (FM) is not expected because of the absence of occupied Ce 4 electrons. An oxygen atom has 4 valence electrons. But all 3 O-2 bands are below and can harbor 6 electrons (2 spins in each band). Therefore, each removal of an O-atom removes 3 occupied -bands, but since the system has only 4 fewer electrons, this means that will rise, and one more band will be occupied to account for the two additional electrons. Thus, an oxygen vacancy introduces partially filled Ce states and the FM ordering sets in because the high DOS of Ce- states produces a Stoner exchange splitting. The calculated moment is 0.52 per Ce atom in CeO and the FM state has a lower total energy () than the non-magnetic state by 0.14 eV per formula unit. The induced moment per oxygen is negative, about 0.01 , and the total moment per CeO cell is exactly 2.0 , or 0.50 per CeO unit. The FM state is half-metallic with no minority bands at , and as expected from the qualitative discussion above, the charge transfer to the majority states is exactly 2 spin states per cell, see Figs. 3 and 4. Consequently, there also are exactly 2 more majority states than minority states in the calculation for the 32-site cell CeO, which corresponds to CeO. Here there are two types of Ce sites, the 4 closest to the O-vacancy, Ce, has each a moment of 0.33 and the 4 towards the interior, Ce, have 0.19 each charge (). Together with the small negative moments on the oxygen this gives exactly 2.0 per CeO cell, or 0.25 per CeO unit. In other words, the removal of one oxygen atom gives rise to a spin magnetic moment of in the dilute vacancy limit ().
iii.2 Excited-state properties
Table 1 compares measured positions of several Kohn-Sham core energy levels with the corresponding calculated XAS threshold energies. Clearly, the total energy calculations give XAS threshold energies in much better agreement with experiments yagci (); melcher (); chen () compared to just taking the Kohn-Sham energy of the core level relative to . Note that we are not looking for absolute values, but rather for relative differences between FM and non-magnetic configurations with or without vacancies. In particular, the value of the Ce M-edge threshold is reduced by about 1 eV for an oxygen vacancy in the small supercell CeO and this value is consistent with the displacements of the CeO melcher () and of the metallic Ce yagci () threshold energies toward lower values. In the large super-cell, CeO, the Ce-3 Kohn-Sham core levels differ by 0.44 eV between the two types of Ce sites. The threshold of oxygen K-edge position seems less affected by the formation of O vacancies in ceria in agreement with measurements performed in oxygen-deficient CeO nanoparticles chen ().
Ce-3 in CeO
|856.2||871.6||877 Ref. yagci ()|
|O-1 in CeO||501.0||523.7||528 Ref. chen ()|
|Ce-3-maj. in CeO||856.0||870.8|
|Ce-3-min. in CeO||856.0||870.3|
|O-1 in CeO||502.0||523.5|
The results for the XPS and BIS excitation energy per electron given in Table 2 show trends similar to those given in Ref. fexit () for NdCeCuO. In particular, the relaxation corrections for defective ceria split the single peak in LSDA to an occupied and an unoccupied band. The former falls below and the latter lies above . Our XPS calculations predict that the occupied peak appears at about 1 eV below the Fermi level while experimentally the position of this peak seems to be even lower. Typical in situ or in operando Ce 4 and Ce 3 x-ray photoelectron spectra of a ceria electrode are shown in Fig. 3a of Ref. Ref3 (). Figure 7 of Ref. acscatal () provides the spatially resolved spectral image of the Ce 4 valence band, allowing visualization of regions of electrochemical activity in a ceria electrode. In fact, the presence of Ce species is revealed by the intensity of the Ce 4 occupied peak at about 2 eV binding energy as already demonstrated by ex situ results exp_xps (). The Ce 3 XPS core-level spectra display different final-state populations of 4, which lead to the peak splitting shown in Fig. 3b of Ref. Ref3 (). The final state effects lead to an upward shift of the lowest binding Ce-3 peak attributed to the Ce state [see also Ce 3 XPS spectra in Fig.1 of Ref. exp_xps ()], which is consistent with the shift of the calculated Ce 3 XAS threshold AB2 () given in Table 1.
For the smaller unit cell, CeO, where all the Ce atoms are equivalent, we find that the occupied band shifts only by eV, indicating that the degree of localization of the orbital plays an important role in the value of the energy shift. If we take the atomic positions relaxed by VASP, the energy shift becomes eV. Therefore, the correction due to atomic relaxation is of the order of eV. Calculations were also made for the large cell, CeO, where both Ce and Ce give the same position for the occupied peak at about 1 eV below footnote1 (). This is consistent with the observation that the Ce 4 binding energy does not change much with the vacancy concentration, but that different contributions from different sites lead to some broadening. Therefore, the excess electrons left behind by the removal of neutral oxygen atoms produce occupied states with practically the same 2 eV binding energy. The reason for the underestimation of the theoretical XPS relaxation is not known, but several mechanisms may be involved. As shown above, part of the correction is due to lattice relaxation near the vacancies. Other possible modifications of the Ce potential would result at surface sites or due to atomic vibrations. Interestingly, the calculated shift for the states in NdCeCuO fexit () has also been found to be about 2 eV. It is also possible that the approximation of a completely delocalized excited state is less appropriate for Ce at these energies.
|excitation||Ce in CeO||Ce in CeO||Ce in CeO|
We have obtained electronic structures of supercells of CeO within the framework of the DFT. The experimental equilibrium lattice constants, bulk moduli and magnetic moments are well reproduced by the generalized gradient approximation (GGA) without the need to introduce a large Coulomb parameter . The computed value of lattice chemical expansion as a function of O-vacancy concentration is not linear for ranging from 0 to 1. Pristine CeO is found to be a non-magnetic insulator with magnetism setting in as soon as oxygens are removed from the structure. Excitation properties are simulated via constrained total energy calculations, and Ce-M and O-K edge x-ray absorption threshold energies are discussed. Our study shows that the way the ground state is probed by different spectroscopies can be modified significantly through final state effects. fexit () In particular, these relaxation effects yield a renormalization of -levels away from the Fermi Level for electron excitation spectroscopies. Our result that electrons reside near the Fermi level in the ground state of oxygen deficient ceria is crucial for understanding catalytic properties of CeO and related materials.norskov ().
Acknowledgements.We acknowledge useful discussions with Dario Marrocchelli. The work at Northeastern University is supported by the US Department of Energy (USDOE) Contract No. DE-FG02-07ER46352. The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the USDOE under Contract No. DE-AC02-05CH11231. We benefited from computer time from Northeastern University’s Advanced Scientific Computation Center (ASCC) and USDOEÕs NERSC supercomputing center.
Appendix A Details of Constrained DFT Computations
In the XPS final state, one whole electron is transferred from the ground-state to a homogeneous plane-wave single-particle state. We can simulate this process by creating a hole obtained by removing states from the local DOS (LDOS) over a narrow energy window , where is the Fermi energy in the excited state and is a cut-off energy. The electronic density associated with the hole at the site is
Here, is the self-consistent LDOS and is the radial wave function component at the site with angular momentum . The total electron density for the excited state thus is
where the last term in Eq. 2 imposes charge neutrality within the simulation cell of volume .
The charge density in the final state allows us to determine the total energy of the excited state. The electrostatic and exchange-correlation contributions are evaluated straightforwardly from . The kinetic energy corresponding to the excited charge density does not involve a single energy level of the solid but rather a group of states and it can be calculated using the standard expression given by Janak janak (), which involves the Kohn-Sham energy average
In this way, the total energy can be obtained in terms of the Kohn-Sham energy average and the hole density exactly as in the XAS threshold energy calculations within the self-consistent-field (SCF) method lerch (); hedin (). Finally, the energy of the XPS photoelectron is given by fujikawa ()
where is the photon energy, is the ground state energy, and is the Fermi level for the excited state. If KoopmansÕ theorem applies
where the Kohn-Sham energy average
is calculated in the ground state. The difference between Eq. 4 and Eq. 5 defines the relaxation energy . DFT is expected to give a reasonable description of the energy difference involved in the photoemission processdabo ().
In order to focus on the excitation corresponding to a given DOS peak, in actual computations, we considered a narrow energy interval containing a fraction of an electron per site , and renormalized the result to obtain to account for a whole electron involved in the XPS process footnote1 (). We have performed test computations using a range of values and found that the relaxation energy is very insensitive to the value of used, which is also anticipated from the analysis of Ref. ley (). Note that placing the hole on only one of the sites in the unit cell is an approximation for removing a band electron in that it neglects the overlap of the wave function with neighboring sites. This however is expected to be a reasonable approximation for localized -electrons of interest here. To check this point, we computed the excitation energy self-consistently in CeO where we removed 1/4th of an -electron from each of the 4 Ce atoms in the unit cell, i.e. a total of one electron from the unit cell. The value of the relaxation energy so obtained was 0.75 eV compared to 0.7 eV shown in Table II, which is within the intrinsic error of 0.05 eV in our total energy computations.
- (1) A. Trovarelli, Catalysis by Ceria and Related Materials, Catalytic Science Series, vol. 2, (Imperial College Press, London, 2002).
- (2) Friedrich Esch, Stefano Fabris, Ling Zhou, Tiziano Montini, Cristina Africh, Paolo Fornasiero, Giovanni Comelli, Renzo Rosei, Science 309, 752 (2005).
- (3) M. Veronica Ganduglia-Pirovano, Alexander Hofmann, Joachim Sauer, Surface Science Reports 62, 219 (2007).
- (4) A. Trovarelli, Cat. Rev. - Sci. Eng. 38, 439-520 (1996).
- (5) A. Atkinson, S Barnett, R.J. Gorte, J.T.S. Irvine, A.J. McEvoy, M.B. Mogensen, S Singhal, J. Vohs, Nature Mat. 3, 17 (2004).
- (6) Mogens Mogensen, Nigel M. Sammes, Geoff A. Tompsett, Solid State Ionics 129, 63 (2000).
- (7) E.P. Murray, T. Tsai, and S.A. Barnett, Nature 400, 649 (1999).
- (8) S. Park, J. M. Vohs and R. J. Gorte, Nature 404, 265 (2000).
- (9) J.B. Park, J. Graciani, J. Evans, D. Stacchiola, S. Ma, P. Liu, A. Nambu, J. F. Sanz , J. Hrbek, and J. A. Rodriguez, Proc. Natl. Acad. Sci. 163, 4975 (2009).
- (10) J. A. Rodriguez, S. Ma, P. Liu, J. Hrbek, J. Evans, M. Perez, Science 318, 1757 (2007).
- (11) M. V. Ganduglia-Pirovano, J. L. F. Da Silva and J. Sauer, Phys. Rev. Lett. 102, 026101 (2009).
- (12) Hui-Ying Li, Hai-Feng Wang, Xue-Qing Gong, Yang-Long Guo, Yun Guo, Guanzhong Lu, and P. Hu, Phys. Rev. B 79, 193401 (2009).
- (13) W. Lai and S. M. Haile, J. Am. Ceram. Soc. 88, 2979-2997 (2005).
- (14) C. Lu, W. L. Worrell, J. M. Vohs, and R. J. Gorte, J. Electrochem. Soc. 150, A1357-A1359 (2003).
- (15) Chunjuan Zhang, Michael E. Grass, Anthony H. McDaniel, Steven C. DeCaluwe, Farid El Gabaly, Zhi Liu, Kevin F. McCarty, Roger L. Farrow, Mark A. Linne, Zahid Hussain, Gregory S. Jackson, Hendrik Bluhm and Bryan W. Eichhorn, Nature Mat. 9, 944 (2010).
- (16) S. Adler, Chem. Rev. 104, 4791 (2004).
- (17) W.C. Chueh, H. Yong, W. Jung, and S. M. Haile, Nature Mater. 11, 155 (2012).
- (18) B. Murugan and A. V. Ramaswamy, J. Am. Chem. Soc. 129, 3062 (2007).
- (19) M. Nolan, S. Parker, and G. W. Watson, Phys. Chem. Chem. Phys. 8, 216(2005).
- (20) T. X. T. Sayle, S. C. Parker, and C. R. A. Catlow, Surf. Sci. 316, 329 (1994).
- (21) T. Jarlborg, B. Barbiellini, R.S. Markiewicz and A. Bansil, Phys. Rev. B 86, 235111, (2012); T. Jarlborg, A. Bianconi, B. Barbiellini, R.S. Markiewicz, A. Bansil, J. Supercond. Nov. Magn. 26, 2597 (2013).
- (22) Matthew B. Watkins, Adam S. Foster, and Alexander L. Shluger, J. Phys. Chem. C 111, 15337 (2007).
- (23) Natasha M. Galea, David O. Scanlon, Benjamin J. Morgan, Graeme W. Watson, Molecular Simulation 35, 577 (2009). DOI 10.1080/08927020802707001
- (24) Konstanze R. Hahn, Marcella Iannuzzi, Ari P. Seitsonen, and Jürg Hutter, J. Phys. Chem. C 117, 1701 (2013).
- (25) O.Hellman, N.V.Skorodumova, and S.I.Simak, Phys. Rev. Lett. 108, 135504 (2012).
- (26) B. Hammer and J.K. Norskov, Advances in Catalysis 45, 71 (2000).
- (27) N. V. Skorodumova, R. Ahuja, S. I. Simak, I. A. Abrikosov, B. Johansson, and B. I. Lundqvist, Phys. Rev. B 64, 115108 (2001).
- (28) N.V. Skorodumova, S. I. Simak, B. I. Lundqvist, I. A. Abrikosov, and B. Johansson, Phys. Rev. Lett. 89, 166601 (2002).
- (29) T. Jarlborg, E.G. Moroni and G. Grimvall, Phys. Rev. B 55, 1288, (1997).
- (30) Dario Marrocchelli, Sean R. Bishop, Harry L. Tuller, Graeme W. Watson and Bilge Yildiz, Phys. Chem. Chem. Phys. 14, 12070 (2012).
- (31) B. Barbiellini, E. G Moroni, T. Jarlborg Journal of Physics: Condensed Matter 2, 7597 (1990);
- (32) T. Jarlborg, B. Barbiellini, H. Lin, R. S. Markiewicz, and A. Bansil, Phys. Rev. B 84, 045109 (2011).
- (33) A. N. Chantis, M. van Schilfgaarde, and T. Kotani, Phys. Rev. B 76, 165126 (2007).
- (34) B. Barbiellini and A. Bansil, J. Phys. Chem. Solids 66, 2192 (2005).
- (35) M. Cococcioni and S. de Gironcoli, Phys. Rev. B 71, 035105 (2005).
- (36) D. A. Andersson, S. I. Simak, B. Johansson, I. A. Abrikosov, and N. V. Skorodumova, Phys. Rev. B 75, 035109 (2007).
- (37) C. W. M. Castleton, J. Kullgren, and K. Hermansson, J. Chem. Phys. 127, 244704 (2007).
- (38) Christoph Loschen, Javier Carrasco, Konstantin M. Neyman, and Francesc Illas, Phys. Rev. B 75, 035115 (2007).
- (39) Talgat M. Inerbaev, Sudipta Seal, Artëm E. Masunov, J. Mol. Model. 16, 1617 (2010).
- (40) Gloria Preda and Gianfranco Pacchioni, Catalysis Today 177, 31 (2011).
- (41) Christian Spiel, Peter Blaha, Yuri Suchorski, Karlheinz Schwarz, and Günther Rupprechter, Phys. Rev. B 84, 045412 (2011).
- (42) Amra Peles, J. Mater. Sci. 47, 7542 (2012). DOI 10.1007/s10853-012-6423-1
- (43) Hong Jiang, Ricardo I. Gomez-Abal, Patrick Rinke, and Matthias Scheffler, Phys. Rev. Lett. 102, 126403 (2009).
- (44) L. Gerward, J. Staun Olsen, L. Petit, G. Vaitheeswaran, V. Kanchana and A. Svane, Journal of Alloys and Compounds 400, 56 (2005).
- (45) Jesus Graciani, Antonio M. Marquez, Jose J. Plata, Yanaris Ortega, Norge C. Hernandez, Alessio Meyer, Claudio M. Zicovich-Wilson, and Javier Fdez. Sanz, J. Chem. Theory Comput. 7, 56 (2011).
- (46) Roland Gillen, Stewart J. Clark, and John Robertson, Phys. Rev. B 87, 125116 (2013).
- (47) Michael E. Grass, Patrik G. Karlsson, Funda Aksoy, Mans Lundqvist, Bjorn Wannberg, Bongjin S. Mun, Zahid Hussain, and Zhi Liu, Review of Scientific Instruments 81, 053106 (2010).
- (48) Funda Aksoy, Michael E. Grass, Sang Hoon Joo, Naila Jabeen, Young Pyo Hong, Zahid Hussain, Bongjin S. Mun and Zhi Liu, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 645, 260 (2011).
- (49) William C. Chueh, Anthony H. McDaniel, Michael E. Grass, Yong Hao, Naila Jabeen, Zhi Liu, Sossina M. Haile, Kevin F. McCarty, Hendrik Bluhm, and Farid El Gabaly, Chemistry of Materials 24, 1876 (2012).
- (50) Chunjuan Zhang, Michael E. Grass, Yi Yu, Karen J. Gaskell, Steven C. DeCaluwe, Rui Chang, Gregory S. Jackson, Zahid Hussain, Hendrik Bluhm, Bryan, W. Eichhorn and Zhi Liu, ACS Catal. 2, 2297 (2012).
- (51) G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
- (52) G. Kresse, G and J. Furthmuller, Phys. ReV. B 54, 11169 (1996); G. Kresse and J. Hafner, Phys. Rev. B 48, 13115 (1993).
- (53) J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
- (54) O.K. Andersen, Phys. Rev. B12, 3060 (1975); B. Barbiellini, S.B. Dugdale and T. Jarlborg, Comput. Mater. Sci. 28, 287 (2003).
- (55) W. Kohn and L.J. Sham, Phys. Rev. 140, A1133 (1965); O. Gunnarsson and B.I Lundquist, Phys. Rev. B 13, 4274 (1976).
- (56) B. Barbiellini and T. Jarlborg, Phys. Rev. Lett. 101, 157002 (2008).
- (57) Non-stoichometric compositions could be investigated via first-principles KKR-CPA methodology: A. Bansil, S. Kaprzyk, P.E. Mijnarends and J. Tobola, Phys. Rev. B 60, 13396 (1999); A. Bansil, Phys. Rev. B20, 4025(1979); A. Bansil, Phys. Rev. B20, 4035(1979); L. Schwartz and A. Bansil, Phys. Rev. B 10, 3261 (1974).
- (58) J. Rath and A.J. Freeman, Phys. Rev. B 11, 2109 (1975).
- (59) P. Lerch, T. Jarlborg, V. Codazzi, G. Loupias and A.M. Flank, Phys. Rev. B45, 11481 (1992).
- (60) T. Jarlborg and P.O. Nilsson, J. Phys. C 12, 265 (1979).
- (61) L. Eyring, in Handbook on the Physics and Chemistry of Rare Earths, edited by K. A. Gschneider and L. Eyring (North- Holland, Amsterdam, 1979), Vol. 3, Chap. 27.
- (62) A. Nakajima, A. Yoshihara, and M. Ishigame, Phys. Rev. B 50, 13 297 (1994).
- (63) V. Fernandes, R. J. O. Mossanek, P. Schio, J. J. Klein, A. J. A. de Oliveira, W. A. Ortiz, N. Mattoso, J. Varalda, W. H. Schreiner, M. Abbate, and D. H. Mosca, Phys. Rev. B 80, 035202 (2009).
- (64) V. Fernandes, P. Schio, A. J. A. de Oliveira, W. A. Ortiz, P. Fichtner, L. Amaral, I. L. Graff, J. Varalda, N. Mattoso, W. H. Schreiner and D. H. Mosca, J. Phys.: Condens. Matter bf 22, 216004 (2010).
- (65) Shih-Yun Chen, Yi-Hsing Lu, Tzu-Wen Huang, Der-Chung Yan and Chung-Li Dong, J. Phys. Chem. C 114, 19576 (2010).
- (66) Xiaoping Han, Jaichan Lee, and Han-Ill Yoo, Phys. Rev. B 79, 100403 R (2009).
- (67) Y. Kuru D. Marrocchelli, S. R. Bishop, D. Chen, B. Yildiz, and H. L. Tuller, Journal of The Electrochemical Society 159 F799 (2012).
- (68) T. Jarlborg, Phys. Rev. B59, 15002, (1999).
- (69) E. Wuilloud, B. Delley, W.-D. Schneider and Y. Baer, Phys. Rev. Lett. 53, 202 (1984).
- (70) The valence charge is 0.05 electrons larger on the Ce with small moment compared to the other Ce site.
- (71) O. Yagci, J. Phys. C: Solid State Phys. 19 3487 (1986).
- (72) C. L. Melcher, S. Friedrich, M. A. Spurrier, P. Szupryczynski, R. Nutt, Nuclear Science, IEEE Transactions on 52, 1809 (2005). DOI 10.1109/TNS.2005.856594
- (73) D.R. Mullins, P.V. Radulovic, S.H. Overbury, Surface Science 409, 307 (1998).
- (74) We should keep matrix element effects in mind in comparing theoretical and experimental spectral intensities in various highly resolved spectroscopies. See, e.g., R. S. Markiewicz and A. Bansil, Phys. Rev. Lett. 96, 107005 (2006); J. Nieminen, H. Lin, R. S. Markiewicz, and A. Bansil, Phys. Rev. Lett. 102, 037001 (2009); J. Mader, S. Berko, H. Krakauer and A. Bansil, Phys. Rev. Letters 37, 1232(1976); A. Bansil, M. Lindroos, S. Sahrakorpi, and R. S. Markiewicz, Phys. Rev. B 71, 012503 (2005).
- (75) J.F. Janak, Phys. Rev. B 12, 7165 (1978).
- (76) L. Hedin and A. Johansson, J. Phys. B 2, 1336 (1969).
- (77) Takashi Fujikawa, J. Phys. Soc. Japan 51, 2619 (1982).
- (78) I. Dabo, A. Ferretti, N. Poilvert, Y. Li, N. Marzari, and M. Cococcioni, Phys. Rev. B 82, 115121 (2010).
- (79) This does not mean that we are exciting a fraction of an electron since each Bloch electron is spread over a macroscopic number of unit cells in the crystal, and its spectral weight on any atom is negligibly small. All results in Table II are based on using in between 0.125 and 0.45.
- (80) L. Ley, F. R. McFeely, S. P. Kowalczyk, J. G. Jenkin, and D. A. Shirley, Phys. Rev. B 11, 600 (1975); S. P. Kowalczyk, L. Ley, R. L. Martin, F. R. McFeely, and D. A. Shirley, Faraday Discuss. Chem. Soc. 60, 7 (1975).