[[abstract]]We show that plane wave ultrasoft pseudopotential methods readily extend to the calculation of the structural properties of lanthanide and actinide containing compounds. This is demonstrated through a series of calculations performed on UO, UO2, UO3, U3O8, UC2, α-CeC2, CeB6, CeSe, CeO2, NdB6, TmOI, LaBi, LaTiO3, YbO, and elemental Lu.[[incitationindex]]SCI[[booktype]]紙本[[booktype]]電子

Pickard, Chris J.

We show that plane wave ultrasoft pseudopotential methods readily extend to the calculation of the structural properties of lanthanide and actinide containing compounds. This is demonstrated through a series of calculations performed on UO, UO2, UO3, U3O8, UC2, α-CeC2, CeB6, CeSe, CeO2, NdB6, TmOI, LaBi, LaTiO3, YbO, and elemental Lu.SCI紙本電子

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VOLUME 85, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 11 DECEMBER 20005122Structural Properties of Lanthanide and Actinide Compounds withinthe Plane Wave Pseudopotential ApproachChris J. Pickard and Björn WinklerInstitut für Geowissenschaften, Universität Kiel, Olshausenstrasse 40 D-24098 Kiel, GermanyRoger K. Chen and M. C. PayneTCM, Cavendish Laboratory, Madingley Road, Cambridge, CB3 0HE, United KingdomM. H. Lee and J. S. LinDepartments of Physics and Chemistry, Tamkang University, Tamsui, Taipei 251, TaiwanJ. A. White and V. MilmanMolecular Simulations Inc., 240/250 The Quorum, Cambridge, CB5 8RE, United KingdomDavid VanderbiltDepartment of Physics and Astronomy, Rutgers University, New Jersey 08855-0849(Received 28 April 2000)We show that plane wave ultrasoft pseudopotential methods readily extend to the calculation of thestructural properties of lanthanide and actinide containing compounds. This is demonstrated through aseries of calculations performed on UO, UO2, UO3, U3O8, UC2, a-CeC2, CeB6, CeSe, CeO2, NdB6,TmOI, LaBi, LaTiO3, YbO, and elemental Lu.PACS numbers: 61.50.Ah, 61.66.FnThe plane wave pseudopotential approach has developedsteadily in applicability. Initially, empirical pseudopoten-tials were constructed for the so-called “easy” elementssuch as silicon and aluminum. These potentials were usedmainly in the prediction and understanding of electronicband structures and were found to be particularly accuratefor semiconductors. With the subsequent development ofab initio pseudopotentials and expressions for the accuratecalculation of forces and stresses, the plane wave pseu-dopotential approach has come into its own as the methodof choice for the first principles prediction of structuralparameters.A combination of improved computing resources,algorithms, and pseudopotential formalisms has allowedthe plane wave pseudopotential technique to be appliedto ever “harder” elements, such as, initially, the first rowelements (including carbon and oxygen), and later thetransition metal elements. In this paper we will showthat the pseudopotential formalism due to Vanderbilt[1], in which the norm-conserving constraint is removedin the pseudisation process, allows the steady marchthrough the periodic table to be taken to completion, inthat we can reliably calculate structural parameters ofcompounds containing f electrons. In support of ourapproach, recent benchmark calculations on atoms by Liuand Dolg [2] find that current density functionals workwell for compact 4f shells, correcting opposite statementsby other authors. A similar conclusion has been reachedby Söderlind et al. who performed calculations on aseries of f-electron metals [3,4] using gradient correctedfunctionals.0031-90070085(24)5122(4)$15.00There have been numerous other all-electron, first prin-ciples studies of f-electron systems [5–11], but, whileaccurate and relevant results have been obtained, the vastmajority of the structures investigated have been of a smallsize and high symmetry. Previous attempts to apply thepseudopotential approach [12,13], which is more suitablefor large, low-symmetry structures, to the calculation off-electron systems have lacked one or more of the ingre-dients described below, in particular this is the first timethat ultrasoft pseudopotentials have been used.The extension of the plane wave pseudopotential methodto f-electron systems requires nothing more than a timelycombination of already widely used techniques. The planewave pseudopotential technique is reviewed in Ref. [14].It is one of the density functional [15] based methods. Inthis paper we use both the local-density approximation(LDA) parametrization due to Perdew and Zunger [16]and the generalized gradient approximation (GGA) dueto Perdew and Wang [17]. Many of the systems treatedhere are spin polarized, so we use the spin-polarized ver-sions of these functionals unless otherwise stated. Someof the systems are also metallic, and so we use the den-sity mixing scheme described by Kresse and Furthmüller[18] to efficiently reach self-consistency. Within the con-straints of the imposed space group symmetry we auto-matically determine the occupancy and spin state of theground state. To describe the electron-ion interaction weuse the ultrasoft pseudopotentials developed by Vanderbiltand co-workers [2,19], extended so that nonlocal projectorswere available up to l  3 allowing felectrons to be accu-rately described. This requires charge augmentation terms© 2000 The American Physical SocietyVOLUME 85, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 11 DECEMBER 2000up to and including l  6. To obtain good pseudopoten-tials, so-called “shallow-core” electrons are included as va-lence electrons (for example, the 5s and 5p electrons forthe lanthanides). In order to optimize the structures it isessential that we calculate the first derivatives of the totalenergy — the forces and stresses. The expressions requiredfor the case of ultrasoft pseudopotentials have been sum-marized by Focher and Chiarotti [20]. The first Brillouinzone is sampled using the Monkhorst-Pack scheme [21]so that distances between sampling points are always lessthan 0.04 Å21 and the largest plane wave cutoff required isabout 400 eV. The particular implementation used in thiswork is CASTEP [22], and the pseudopotentials were takenfrom the standard database distributed with this code. Itsuse in the first principles study of inorganic crystals hasbeen recently reviewed by Milman et al. [23]. No attemptis made to model the spin-orbit terms that are thought tobe large in these systems.In order to build confidence in this approach we havechosen a spectrum of test compounds, ranging fromuranium and cerium oxides to elemental lutetium metal.This includes examples of both metals and insulators,lanthanides and actinides, and open- and closed-shellsystems.By far the most economically important uranium-bear-ing mineral is uraninite [24], UO2, which crystallizes inthe fluorite-type structure with space group Fm3¯m. It hasa lattice constant of aexp  5.458 Å, depending on the de-gree of partial oxidation of U41 to U61 [25]. The calcu-lated value for the ideal structure is atheo  5.474 Å, andis thus in good agreement with experiment. While this Let-ter focuses on structural properties it is interesting to notethat density-functional theory (DFT) predicts UO2 to bemetallic, while it is in fact an insulator. This confirms thecommon view that the electronic properties of lanthanide-and actinide-containing compounds require a treatment be-yond DFT.A useful uranium pseudopotential must also be able todescribe other oxidation states, such as U21 and U61. Forcubic UO, the experimental lattice parameter is 4.92(2) Å[25], while the calculated value is 4.961 Å, again in goodagreement. Several polymorphs of UO3 are known, andthey are interesting from a crystallographic point of view.The experimentally determined and computed structuralparameters of three structurally different polymorphs aregiven in Table I. With one exception, namely, the freeoxygen coordinate in the hexagonal polymorph, all struc-tural parameters are in good agreement with experimentalvalues. We believe that our theoretical value for Oz inthe hexagonal polymorph is likely to be more reliable thanthe experimentally determined value, as the latter has beenderived from estimated film intensities [26].A more subtle test is the modeling of a mixed-valencecompound, such as U3O8. For a hexagonal polymorphwith space group symmetry P6¯m2 we obtain atheo 6.829 Å and ctheo  4.145 Å, which agrees well withthe corresponding experimental parameters of aexp 6.817 Å and cexp  4.145 Å [27]. This structure has threeinternal degrees of freedom, with experimental valuesUx  0.3526, O1x  0.7453, and O2x  0.3609.The corresponding values of the relaxed structure are0.35057, 0.74799, and 0.35893, respectively. In summary,the structures of the uranium oxides are very well repro-duced, irrespective of the oxidation state of uranium. Thisimplies that we might expect to be able to reliably modelother uranium compounds. Indeed, this is found to be truefor UC2 (see Table III below).To show that it is not just equilibrium properties whichare reliably reproduced, the elastic constants of one ofthe uranium oxides, UO2, were also calculated usingthe method of imposed strains. The results are given inTable II, and are consistent with those expected for thedensity-functional theory approach in general [28] andwith previous all-electron studies [29,30].TABLE I. Polymorphs of UO3. A (–) indicates that the parameter is fixed by symmetry.Space group Pm3¯ma P3¯m1b P212121ca Å expt. 4.165(8) 3.971 7.511(9)a Å theory 4.152 3.804 7.585b Å expt. – – 5.466(8)b Å theory – – 5.516c Å expt. – 4.168 5.224(8)c Å theory – 4.136 5.274Ux, y, z expt. 0 0 0 0 0 0 0.0699 0.1329 0.0350Ux, y, z theory – – – – – – 0.0672 0.1167 0.0410O1x, y, z expt. 12 0 0 0 012 0.4947 0.1263 0.3284O1x, y, z theory – – – – – – 0.4945 0.1351 0.3203O2x, y, z expt. 1323 0.17 0.1604 0.3872 0.2124O2x, y, z theory – – 0.1087 0.1530 0.3810 0.2220O3x, y, z expt. 0.6735 0.6176 0.6654O3x, y, z theory 0.6777 0.6267 0.6634aExpt. data from [37].bExpt. data from [26].cExpt. data from [38].5123VOLUME 85, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 11 DECEMBER 2000TABLE II. The elastic constants (in GPa) of UO2, where C0 12 C11 2 C12, B 23 C11 1 2C12.C11 C12 C44 C0 Bexpt. [39] 389.3 118.7 59.7 135.3 208.9calc. 318.2 96.0 43.1 111.1 170.1We calculated the ground state structures of a-CeC2,CeB6, CeSe, and CeO2. a-CeC2 crystallizes in spacegroup I4mmm. The structure was determined by Atoji[31], and the experimental and theoretical data are com-pared in Table III. The structure of CeB6 has space groupPm3¯m and one free internal structural parameter, deter-mining the position of the boron atoms. A comparison ofexperimental and theoretical data is presented in Table IV.Svane et al. [11] found that within the SIC-LSD approachit is necessary to “localize” a single f electron per Ceatom to obtain good structural parameters for CeSe in therock-salt structure. In contrast, our calculations using aspin-polarized GGA require no special treatment of indi-vidual electronic states. The experimental lattice parame-ter is 5.990 Å [32] while we calculated a value of 6.043 Å.In CeO2 all atomic positions are fully constrained by thespace group symmetry. The experimentally determinedlattice parameter of this cubic structure with space groupFm3¯m is 5.411 Å [33], while the geometry-optimizedstructure had a lattice parameter of 5.421 Å. For all ofthese cerium compounds, the pseudopotential approachagrees closely with experiment.The accuracy of this approach is sufficient to resolve thedifference in the unit cell volumes for the Fm3¯m structuresof CeO2 and UO2. Experimentally the lattice parameterof CeO2 is about 1.1% smaller than that of UO2, while acomparison of the theoretical values gives 1.0%. However,smaller differences are not resolved. For example, NdB6(Table IV) experimentally has a lattice parameter about0.2-0.3% smaller than that of CeB6, while our calculationsindicate that it should be about 0.9% larger. This is thelimit of reliability of the GGA-DFT structures, whether ornot f electrons are present.That equally accurate results can also be obtained forsystems with many f electrons is demonstrated by our cal-culations on cubic YbO, which crystallizes in space groupTABLE III. Comparison of the experimentally determinedstructural parameters of a-CeC2 and UC2 to the correspondingtheoretical values.CeC2a UC2bExpt. Theory Expt. Theorya Å 3.875 3.906 3.509–3.522 3.524c Å 6.477 6.486 5.980–5.988 5.946Cx 0.401 0.401 0.388–0.4 0.3860dC-C Å 1.281 1.280 1.321 1.312dCe,U-C Å 2.598 2.603 2.334 2.317dCe,U-C Å 2.814 2.835 2.577 2.577aExpt. data from [31].bExpt. data from [25,40,41].5124TABLE IV. Comparison of the experimentally determinedstructural parameters of CeB6 and NdB6 to the correspondingtheoretical values. Ce and Nd are chosen to occupy the Wyckoffposition 1b, which implies that the carbon atoms are onposition 6f.CeB6a NdB6bExpt. Theory Expt. Theorya Å 4.140 4.098 4.127 4.136Bz 0.1992 0.200 0.1989 0.2023dB-B Å 1.76 1.737 1.755 1.741dCe,Nd-B Å 3.04 3.012 3.032 3.042aExpt. data from [42,43].bExpt. data from [44].Fm3¯m with aexp  4.86 Å [34], while the pseudopoten-tial calculations gives atheo  4.8405 Å. The structure ofa thulium-containing compound, TmOI, given in Table V,is also well reproduced.We performed calculations on the closed f-shelllutetium metal with similarly good results. The ex-perimental [35] ac ratio for the hexagonally closepacked structure is 0.630 and the unit cell volume, V0,is 59.80 Å3. The corresponding LDA results are ac 0.640, V0  53.373 Å3 and the GGA ones are ac 0.638, V0  59.014 Å3. The gradient corrected functionalis shown to be essential for close agreement to experiment.For the sake of completeness, we have also checkedsome lanthanum-bearing compounds. Cubic LaBi hasaexp  6.57 Å [34], the theoretical value is atheo 6.648 Å. For the orthorhombic low-temperature studyof LaTiO3 the calculated structure is in similarly goodagreement with the experimental values (Table VI).In this Letter we have shown that the structural prop-erties of lanthanide- and actinide-containing systems canbe investigated on the same footing as the rest of theperiodic table using density-functional theory, ultrasoftpseudopotentials, and a modest number of plane waves.As a result of the inherent efficiency of the plane wavemethods, the theoretical study of complex, low-symmetrylanthanide- and actinide-containing compounds becomespossible. This is a significant advance compared to previ-ous all-electron approaches which, while accurate, are lessefficient computationally. The ability to accurately calcu-late forces and stresses allows efficient structural optimiza-tion. We confirm that the LDA is not always sufficient todescribe the exchange and correlation energies, but that theuse of the GGA produces results as good as those found forTABLE V. Comparison of the experimentally determined andcalculated structure of TmOI.Expt.a Theorya Å 3.887 3.917c Å 9.166 9.175Iz 0.68 0.6751Oz 0.125 0.1167aExpt. data from [45].VOLUME 85, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 11 DECEMBER 2000TABLE VI. Comparison of the experimentally determined lowtemperature (10 K) and calculated structure of orthorhombicLaTiO3.Expt.a Theorya Å 5.630 5.602b Å 5.584 5.712c Å 7.901 7.899Lax 0.99269 0.98626La y 0.04705 0.04399O1x 0.07902 0.07262O1 y 0.49339 0.48587O2x 0.70941 0.71763O2 y 0.29237 0.28028O2z 0.04204 0.03882aExpt. data from [46].the rest of the periodic table. 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Physical Review Letters

Structural Properties of Lanthanide and Actinide Compounds within the Plane Wave Pseudopotential Approach

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Structural Properties of Lanthanide and Actinide Compounds within the Plane Wave Pseudopotential Approach

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