IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 21 (2009) 045604 (14pp) doi:10.1088/0953-8984/21/4/045604

Self-interaction correction in multiplescattering theory: application to transitionmetal oxidesM Däne1,2, M Lüders3, A Ernst4, D Ködderitzsch5,W M Temmerman3, Z Szotek3 and W Hergert1

1 Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, Von-Seckendorff-Platz 1,D-06120 Halle, Germany2 Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge,TN 37831, USA3 Daresbury Laboratory, Daresbury, Warrington WA4 4AD, UK4 Max Planck Institut für Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany5 Ludwig-Maximilians-Universität München, Physikalische Chemie, Butenandtstraße 11,D-81377 München, Germany

E-mail: markus.daene@physik.uni-halle.de

Received 15 July 2008, in final form 20 October 2008Published 8 January 2009Online at stacks.iop.org/JPhysCM/21/045604

AbstractWe apply to transition metal monoxides the self-interaction corrected (SIC) local spin densityapproximation, implemented locally in the multiple scattering theory within theKorringa–Kohn–Rostoker (KKR) band structure method. The calculated electronic structureand in particular magnetic moments and energy gaps are discussed in reference to the earlierSIC results obtained within the linear muffin-tin orbital atomic sphere approximation bandstructure method, involving transformations between Bloch and Wannier representations, inorder to solve the eigenvalue problem and calculate the SIC charge and potential. Since theKKR method can be easily extended to treat disordered alloys, by invoking the coherentpotential approximation (CPA), in this paper we compare the CPA approach and supercellcalculations to study the electronic structure of NiO with cation vacancies.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Density functional theory (DFT) forms the basis of mostfirst-principles electronic structure calculations of solids [1].In particular, the commonly used local (spin) densityapproximation (L(S)DA) to DFT has been extremelysuccessful in describing bonding properties of solids,especially the conventional metals and their alloys. However,due to the local approximation, describing correlations at thelevel of the homogeneous electron gas, LSDA (LSD: localspin density) or even the semi-local, generalized gradientapproximation (GGA) often fails for systems like transitionmetal monoxides (TMOs) containing partially filled TM d-shells. The same is true for rare earths and heavier actinidesand their compounds, where the f-shells are partially filled. Itis the strong Coulomb repulsion among the electrons in those

partially filled d- and f-shells that is not well represented withinLSDA and GGA. For TMOs, LSDA either fails to describetheir insulating ground state or predicts much too small bandgaps and magnetic moments [2–4]. This is associated with anunphysical self-interaction of an electron with itself, occurringin the Hartree term of the LSDA energy functional on accountof the local approximation applied to the exchange–correlationenergy functional. This self-interaction becomes importantfor localized electrons like d electrons of TM elements intheir monoxides. In the latter, the self-interactions pushthe localized electron orbitals into the valence band, usuallyresulting in too strong a hybridization with the other valenceelectrons. This problem was recognized many years agoand a remedy was proposed by Perdew and Zunger [5] tosimply subtract the spurious self-interactions from the LSDAfunctional, orbital by orbital, for all the localized states.

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J. Phys.: Condens. Matter 21 (2009) 045604 M Däne et al

The resulting SIC–LSDA approach treats both localized anditinerant electrons on equal footing, leading to split d- and f-manifolds and describing the dual character of an electron.

The SIC–LSDA method has been successfully appliedto numerous systems containing localized d and/or felectrons [6–16]. In most cases the so-called full SIC (FSIC)implementation was used [17, 18], involving repeatedtransformations between Bloch and Wannier representationsto, respectively, solve the generalized eigenvalue problem,and calculate orbital dependent charges and potentials. Sincethe self-interaction correction is local in nature with theelectronic states affected by SIC localized to a high degreeon a site, the so-called local SIC (LSIC) has recently beenimplemented and successfully applied to some f electronmaterials [15, 16]. This LSIC approach, taking advantageof multiple scattering theory, has been implemented withinthe Korringa–Kohn–Rostoker (KKR) band structure method,which offers straightforward generalizations to treating varioustypes of chemical, charge and spin disorder through thecoherent potential approximation (CPA) and disordered localmoments (DLM) [19] theory. Due to an easy access tothe Green’s function, the KKR method is also well suitedfor studying surfaces, interfaces and other low dimensionalsystems. Finally, the LSIC implementation in KKR allows forfinite temperature calculations and therefore is very useful forstudying finite temperature phase diagrams [15].

The motivation for the present paper is to explore theconsequences of applying LSIC to study the ground stateproperties of TMOs. The latter are the prototypical d electronmaterials for studying strong correlation effects and have beenextensively studied both experimentally and theoretically by anumber of methods [7, 20–25]. One of the goals of this paperis to demonstrate that the LSIC scheme can be as successfullyapplied to systems containing localized 3d electrons, as itwas for the systems with localized 4f electrons. In the caseof TMOs, these are the 3d electrons of the transition metalelements that are affected by SIC.

As already mentioned, a number of theoretical methodshave been used to study TMOs the most popular being theLDA + U method [20, 26–28]. It treats Coulomb interactionsbetween electrons better than LSDA, by explicitly adding to theLSDA Hamiltonian a Hubbard term with the on-site Coulombinteraction U . The latter is usually treated as an adjustableparameter and chosen to optimize agreement with experiment.For TMOs, the approach works well and for NiO, with theappropriate choice of the U parameter, the results have beenshown to be in very good agreement with those of the FSICapproach [21]. The LDA + U method has also been appliedto study surface properties of NiO [29, 30] and MnO [31], aswell as exchange interactions in some TMOs [31, 32]. BesidesLDA + U and SIC schemes, a class of hybrid functionals hasrecently been introduced. These functionals treat Coulombcorrelations beyond LSDA, accomplished by incorporatingsome fraction of the exact exchange. The most well knownamong them is the B3LYP functional [33]. There have been anumber of recent applications of various hybrid functionals toTMOs [22–25].

Whereas all the above mentioned approaches provide astatic treatment of the correlation problem, the dynamical mean

field theory (DMFT) [34, 35] allows to systematically includelocal dynamical correlations into lattice models. However,most applications to date invoke the Hubbard Hamiltonian,via the so-called LDA + DMFT implementation, and thusinherit the uncertainties associated with the LDA + Umethod. Recent LDA + DMFT calculations [36] for NiO,with a suitable choice of U , provide good agreement withangle resolved photoemission spectroscopy. Finally, the GWmethod, in various implementations, has also been used tostudy TMOs [37–40]. However, the degree of its successdepends on the starting band structure and the LDA bandstructure is a poor starting point.

The paper is organized as follows. In section 2, weelaborate on the local implementation of the SIC formalismwithin the KKR band structure method. In section 3, numericaldetails of the calculations are discussed. Section 4 is devotedto an extensive discussion of the LSIC results for NiO todemonstrate LSIC at work. In section 5, we review the groundstate properties of all the 3d TMOs studied with LSIC, incomparison with experiment and the early FSIC results. Insection 6, we consider cation vacancies in NiO using CPA,in addition to the earlier supercell study within the full SICimplementation [10]. The paper is summarized in section 7.

2. Formalism

2.1. Basic SIC–LSDA equations

To present the basics of the SIC–LSDA method, we startfrom the LSDA total energy functional, in the Kohn–Shamrepresentation [1], which is (in Rydberg units) given by

ELSDA[n↑, n↓] =occ∑

α

〈φα| − ∇2|φα〉 + Eext[n]

+ EH[n] + ELSDAxc [n̄], (1)where φαs are the Kohn–Sham orbitals, giving rise to theorbital densities nα = |φα|2, with α being a combined indexlabelling the orbital and spin σ (↑ or ↓), n̄ stands for the spindensities n↑, n↓, and the total density is defined as n = n↑+n↓.Here Eext is the external potential energy functional due to ions,EH is the Hartree energy functional

EH[n] =∫

d3r∫

d3r ′n(r)n(r′)|r − r′| , (2)

and ELSDAxc is the LSD approximation to the exchange–correlation energy functional, which is the source of thespurious self-interaction [5]. The exact exchange–correlationenergy has the property that it cancels exactly the Hartreeenergy for any single-electron density nα , namely

EH[nα] + Eexactxc [n̄α] = 0, (3)where n̄α represents the pair of spin densities (nα, 0) or (0, nα),depending on the spin σ of the state α. Correcting the LSDAenergy functional for the unphysical self-interaction error, bysubtracting explicitly the self-Coulomb and self-exchange andself-correlation energy of all the occupied orbitals, restores

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J. Phys.: Condens. Matter 21 (2009) 045604 M Däne et al

the above property, but leads to an orbital dependent energyfunctional

ESIC−LSDA[{nα}] = ẼLSDA[n̄]−

occ∑

α

(EH[nα] + ELSDAxc [n̄α]), (4)

with ẼLSDA[n↑, n↓] having the same form as ELSDA[n↑, n↓],however, the orbitals setting up the orbital densities, nα , inequation (4) are no longer the usual Kohn–Sham orbitals.The latter are replaced by φ̃α orbitals that minimize the self-interaction corrected energy functional. Varying the aboveSIC–LSDA energy functional with respect to the orbital spindensities, with the constraint that the φ̃αs form a set oforthonormal functions, one gets the SIC–LSDA generalizedeigenvalue equations

Hα|φ̃α〉 =(−∇2 + V SIC−LSDAeff,ασ (r)

) |φ̃α〉= (H0σ + V SICα (r)

) |φ̃α〉 =∑

α′λαα′ |φ̃α′ 〉, (5)

with H0σ being the orbital independent LSDA Hamiltonian.The Lagrangian multipliers λαα′ are used to secure thefulfilment of the orthonormality constraint. Due to the orbitaldependent potential, the SIC energy functional is not stationarywith respect to infinitesimal unitary transformations among theorbitals. The so-called localization criterion

〈φ̃β |V SICα − V SICβ |φ̃α〉 = 0 ∀(α, β) (6)has to be fulfilled to ensure that the solutions of the SIC–LSDAequations (5) are most optimally localized to reach the absoluteminimum of the SIC–LSDA functional (4).

The orbital dependent potential V SIC−LSDAeff,ασ is given by

V SIC−LSDAeff,ασ (r) = Vext(r) + VH[n](r) + V LSDAxcσ [n̄](r)︸ ︷︷ ︸V LSDAeff,σ

− VH[nα](r) − V LSDAxc,σ [n̄α](r)︸ ︷︷ ︸V SICα (r)

, (7)

the external lattice potential Vext(r), and

VH[n](r) = 2∫

d3r ′n(r′)

|r − r′| , (8)

V LSDAxc,σ [n↑, n↓](r) =δELSDAxc [n↑, n↓]

δnσ. (9)

In order to implement the above SIC–LSDA methodology,and solve the underlying generalized eigenvalue equations,one can either choose the direct minimization of the energyfunctional (4) with respect to orbitals, using the steepestdescent method [6, 18], or the so-called unified Hamiltonianformulation [7, 18]. Both approaches have been implementedwithin the linear muffin-tin orbital (LMTO) band structuremethod in the atomic sphere approximation (ASA) [41, 42],i.e., both approaches start from the band representation of theelectronic structure. This, as mentioned earlier, means thatrepeated transformations between the Bloch representation, tosolve the generalized eigenvalue problem, and the Wannierrepresentation, to calculate nα and construct V SICα (r), have

to be invoked in every step of the self-consistency cycle. Itis this change of representations which complicates the self-consistency problem of the self-interaction corrected LSDA.The technical details of these SIC–LSDA implementations,referred to as FSIC, can be found elsewhere [18].

2.2. Multiple scattering theory and LSIC implementation

The so-called ‘local’ implementation of the SIC–LSDAmethodology, in the framework of multiple scattering theory,has already been discussed by Lüders et al [15]. Here werepeat the main concepts and formulae for completeness.The important difference of this LSIC implementation, withrespect to the FSIC approach, is working with the scatteringphase shifts, describing scattering properties of the individualatoms in a solid, treated within multiple scattering theoryimplemented in the Korringa–Kohn–Rostoker (KKR) method.Core electron states, represented as bound states at negativeenergies, are characterized by abrupt jumps by π of theirgeneralized complex phase shifts at those energies. Thelocalized valence states, like e.g. 3d electron states in transitionmetal elements, also show sharp jumps by π but at positiveenergies, referred to as resonances. The itinerant states, on theother hand, are characterized by slowly varying phase shifts.The idea behind the local implementation of SIC–LSDA in theKKR band structure method is to associate SIC with the on-site scattering potential through modifying the correspondingresonant scattering phase shifts. It is the realization of this ideathat lies at the heart of LSIC scheme.

The central quantity of the multiple scattering theory is theGreen’s function which for the scalar-relativistic case is givenby

Gσ (r, r′; ) =∑

L L ′Z̄ iLσ (ri ; ) τ i jσ L L ′() Z jL ′σ (r′j; )

−∑

L

Z̄ iLσ (r; )δi j . (10)

Here r is given by r = Ri + ri , where ri is a vector insidethe cell at Ri , L = (l, m) denotes the combined index forthe angular momentum l and magnetic m quantum numbers,while r) is the vector smaller (larger) in magnitude fromthe pair (r, r′). The functions Z̄ iLσ , Z

iLσ , J

iLσ are expressed by

the regular (Z iLσ ) and irregular (JiLσ ) solutions of the radial

Schrödinger equation at a given (complex) energy and thecomplex spherical harmonics YL (r̂) as

Z iLσ (ri ; ) = Z ilσ (ri ; )YL(r̂i ) (11)Z̄ iLσ (ri ; ) = Z ilσ (ri ; )Y ∗L (r̂i) (12)J iLσ (ri ; ) = J ilσ (ri ; )YL(r̂i ). (13)

The scattering-path matrix τ is given by

τ () = [t−1() − g()]−1, (14)where the underlined symbols stand for matrices in angularmomentum, L, and atomic site indices. The structural Green’sfunction, g(), describes the free propagation between thescattering centres, and the t matrix denotes the single-site

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J. Phys.: Condens. Matter 21 (2009) 045604 M Däne et al

scattering and is related to the phase shifts, δil , discussed above,via

t il () = −1

κsin δil ()e

iδil (), (15)

with κ equal to√

.The total valence charge density, per spin σ , can be

calculated from the Green’s function as

nσ (r) = − 1π

∫ EF

EB

d Im Gσ (r, r; ), (16)

where EB and EF denote the bottom of the valence bandand the Fermi energy, respectively. In the standard LSDAcalculations, this charge density is used to construct the neweffective potential V LSDAeff,σ (cf equation (7)) for the next iterationof the self-consistency cycle.

In the SIC–LSDA method, one aims to extract fromequation (16) those contributions which originate fromlocalized electron states, characterized by sharp resonances inthe scattering phase shifts and therefore long Wigner delaytimes, as they will be associated with large self-interactionerrors. These are the scattering states to which SIC will beapplied. Since the angular momentum decomposition usedin (10) is adapted to spherical symmetry, we need to workin the representation which reflects the local symmetry of thescattering centre under consideration. To accomplish this, weapply a unitary transformation to the spherical harmonics inthe angular momentum decomposition, such that the angularmomentum diagonal block of the on-site scattering-path matrixbecomes diagonal. This unitary transformation is defined by

∑

m1,m2

U †(lm̃,lm1)

τ i i(lm1,lm2)() U(lm2,lm̃′) = δm̃m̃′ τ̃ iiL̃,L̃ ′(), (17)

while the regular solution in this symmetry adaptedrepresentation assumes the form

Z iL̃σ

(ri ; ) =∑

m1

U †(lm̃,lm1)

{Z ilm1σ (ri ; )Ylm1 (r̂i)}, (18)

and similarly for the irregular solution. This corresponds toan expansion into lattice harmonics, i.e., linear combinationsof spherical harmonics of the same quantum number l,forming the basis functions of the respective irreduciblerepresentations.

Due to the employed muffin-tin or atomic sphereapproximation in KKR, only the spherically symmetric parts ofthe SIC charges and potentials are considered. Because of this,only the spherical part of the Green’s function matters, whichin the symmetrized representation becomes diagonal withrespect to L̃ = (l, m̃), which defines the so-called scatteringchannel. Hence, we can decompose the spin resolved chargedensity (equation (16)) into its L̃ components and define thecharge of a given channel, characterized by its site index i ,symmetry index L̃ and spin σ , as

nSICi L̃σ

(r) = − 1π

∫ E2

E1

d Im GL̃,σ (r, r; ), (19)

where the energies E1 and E2 lie, respectively, slightly belowand above the energy of the resonance in the channel L̃σ

at site i . In principle, the integration range should enclosethe localized state only, but for simplicity we use the energycontour encompassing the whole valence band, namely rangingfrom the bottom of the valence band to the Fermi energy. Fromthese channel densities, we can construct the correspondingSIC potentials, which together with the LSDA potential, definethe effective SIC–LSDA potential, namely

V SIC−LSDAeff,i L̃σ

(r) = V LSDAeff,σ (r)−VH[nSICi L̃σ ](r)−V LSDAxc [nSICi L̃σ , 0](r).(20)

This gives rise to the single scattering matrix of the form

t i,corrL̃σ

= t iL̃σ

(1 − δL̃,L̃cδσ,σ c) + t i,SIC−LSDAL̃cσ δL̃,L̃cδσ,σ c, (21)

with some of the channels, marked by L̃cσ c, being self-interaction corrected, and t i

L̃σbeing the t-matrix calculated

from the effective LSDA potential, V LSDAeff,σ (r). Here ti,SIC−LSDAL̃σ

is calculated from the effective SI-corrected LSDA potential,V SIC−LSDA

eff,i L̃σ(r). The corrected tcorr-matrix is then used in

equation (14) to calculate the new, SI-corrected, scattering-path matrix τ̃ . From the latter the new SIC–LSDA chargedensity is calculated, and the process is iterated until self-consistency is reached. Since we are dealing with resonances,the localization criterion used for optimizing Wannier orbitalsin FSIC, giving rise to the additional self-consistency cycle, isnot in operation in LSIC.

One of the important advantages of the SIC–LSDAformalism is that one can realize and study different valenceconfigurations of the elements under consideration, and this isvalid for both FSIC and LSIC schemes. The nominal valence,Nval, is defined as

Nval = Z − Ncore − NSIC, (22)where Z is the atomic number, Ncore is the number of core(and semicore) states and NSIC is the number of self-interactioncorrected states. The ground state valence is the one defined bythe ground state energy. One has to mention that it has to betested which states need to be corrected (e.g. equation (21))in order to find the ground state and the ground state valency.How it is done in practice will be demonstrated for NiO insection 4.

Finally, as already mentioned, the multiple scatteringformulation of SIC, naturally lends itself to various extensionslike the coherent potential approximation (CPA), [43–46]allowing study of random alloys, within the resulting KKR–CPA method. In section 6, we will discuss an applicationof LSIC, in combination with CPA, to study the influenceof vacancies on the Ni-sublattice in NiO on its electronicstructure.

3. Computational details

The 3d transition metal monoxides crystallize in the rocksaltstructure (B1, Fm3̄m, space group 225), consisting oftwo fcc sublattices, shifted with respect to each other by[a/2, 0, 0]. One of the sublattices is occupied by theoxygen atoms and the other one by the transition metal

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J. Phys.: Condens. Matter 21 (2009) 045604 M Däne et al

elements. At low temperatures, TMOs show a small(

J. Phys.: Condens. Matter 21 (2009) 045604 M Däne et al

Table 1. The summary of all studied 2+ and 3+ SIC configurations,specifying the spin and symmetry of the chosen localized orbitals foreach. The two digits separated by a hyphen in the first columnidentify the configurations. The first digit denotes the valency and theother simply numbers the consecutive configurations within a givenvalence group, namely separately for the divalent and trivalent sets.

Config. Valency Choice of orbitals

2-1 2+ 3t2g↑ + 2eg↑ + 3t2g↓2-2 2+ 3t2g↑ + 2eg↑ + 2t2g↓ + 1eg↓2-3 2+ 3t2g↑ + 1eg↑ + 3t2g↓ + 1eg↓2-4 2+ 3t2g↑ + 2eg↑ + 1t2g↓ + 2eg↓2-5 2+ 3t2g↑ + 1eg↑ + 2t2g↓ + 2eg↓2-6 2+ 2t2g↑ + 2eg↑ + 2t2g↓ + 2eg↓3-1 3+ 3t2g↑ + 2eg↑ + 2t2g↓3-2 3+ 2t2g↑ + 2eg↑ + 3t2g↓3-3 3+ 3t2g↑ + 1eg↑ + 3t2g↓3-4 3+ 3t2g↑ + 2eg↑ + 1t2g↓ + 1eg↓3-5 3+ 2t2g↑ + 2eg↑ + 2t2g↓ + 1eg↓3-6 3+ 3t2g↑ + 1eg↑ + 2t2g↓ + 1eg↓3-7 3+ 3t2g↑ + 2eg↑ + 2t2g↓3-8 3+ 3t2g↑ + 1eg↑ + 1t2g↓ + 2eg↓3-9 3+ 2t2g↑ + 2eg↑ + 1t2g↓ + 2eg↓

dzx , while the two eg states are referred to as dx2−y2 andd3z2−r2 , for both spin-up and spin-down channels. Of course,as we have spherical potentials and work in the symmetryadapted representation, all the states within a given subbandare energetically equivalent. Thus in what follows we referonly to t2gs and egs for both spin channels. Based on this,one can consider localized orbitals of different symmetry andspin and for these orbitals the self-interaction correction isimplemented when describing their electronic structure. In theLSIC formulation there are the resonant phase shifts associatedwith these localized d states that define the scattering channelsto be self-interaction corrected. With respect to differentvalency states, one can realize them by varying the number oflocalized d states, e.g. for a divalent Ni-ion one would considereight localized d electrons, but only seven for a trivalent ion.For the monovalent Ni-ion one would need to assume ninelocalized d electron states, which however is hard to stabilizeas far as self-consistency is concerned.

As mentioned earlier, the SIC–LSDA functional is notstationary with respect to mixing among the orbitals anddifferent sets of orbitals will lead to different SIC. Thus onehas to study different configurations of localized orbitals andminimize the SIC–LSDA total energy with respect to theseconfigurations to find the ground state energy and valence.Naturally, one needs to consider not only different symmetriesof the localized orbitals, but also configurations with differentnumbers of localized orbitals, giving rise to different valencestates. The reason is that some valence states may lie closein energy and valence transitions may readily occur e.g. as afunction of pressure. For NiO all the studied configurations oflocalized states, namely six giving rise to the 2+ valence andnine resulting in 3+ valence, are given in table 1.

In order to find the ground state total energy, equilibriumvolume and valence state, we have performed LSICcalculations for all the configurations listed in table 1. Theresults are presented in figure 1, where the total energy

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

4.1 4.15 4.2 4.25 4.3 4.35 4.4

2+ (2-1)2+ (2-2)2+ (2-3)2+ (2-4)2+ (2-5)2+ (2-6)

3+ (3-1)3+ (3-2)3+ (3-3)3+ (3-4)3+ (3-5)3+ (3-6)3+ (3-7)3+ (3-8)3+ (3-9)

Figure 1. Total energies of different SIC configurations in NiOcalculated as a function of the lattice constant. The energy zero is setto the ‘global’ minimum among the different configurations. Thelowest energy is found for the high symmetry state(3 t2g↑ + 2eg↑ + 3t2g↓), resulting in the valency of 2+.

curves are plotted for all the configurations as a function oflattice constants. The equilibrium lattice constants of all theconfigurations are defined by the minima of the correspondingenergy curves. Specifically, the energies are plotted relativeto the global energy minimum, the 2+ state, obtained when thefive majority Ni ds and three minority Ni t2g states are occupiedand treated as localized [(2-1)]. This result is in agreement withthe FSIC calculations [6, 7, 21] and reflects the first Hund’srule in maximizing the spin magnetic moment. This Hund’srule ground state is very robust and well separated in energywith respect to the other 14 curves seen in figure 1, showingthat in the lowest energy configuration, the Ni d states are verywell localized, nearly atomic-like, in the sense that they evenobey Hund’s rules. Furthermore, the robustness of this Hund’srule driven ground state comes about due to the crystal fieldsplitting and the resulting preference to fill the lower lying t2gstates first.

Looking at the remaining 14 curves in figure 1, wecan observe many valence transitions between divalent andtrivalent states as a function of pressure. The generalobservation, for all the studied configurations, is thatthe configurations which follow the first Hund’s rule inmaximizing the spin moment, independently within the t2gand eg subbands, have usually lower energies than theconfigurations violating this rule. This can be easily seenin table 2, where all the configurations are listed in theorder starting from the most energetically favourable to mostenergetically unfavourable, separately for the divalent andtrivalent scenarios. Among the configurations with equalnumber of t2g states occupied, the configurations giving riseto higher magnetic moments have in general lower energies.

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J. Phys.: Condens. Matter 21 (2009) 045604 M Däne et al

Table 2. The summary of the main characteristics of all the studied2+ and 3+ SIC configurations in NiO (cf figure 1). The six 2+configurations appear as the top six rows and are marked by twodigits separated by a hyphen. The first digit is always ‘2’, markingthe valence, while the second one changes from 1 to 6, marking theorder in which the 2+ configurations appear in the legend of figure 1.The larger the digit, the more energetically unfavourable theconfiguration is. For the 3+ configurations, the first digit is naturally‘3’, while the second one changes from 1 to 9, again correspondingto the order in which these configurations appear in the legend of thefigure, reflecting the gradual decrease in energy preference. Here �Edenotes the total energy differences per formula unit relative to thetotal energy of the ground state configuration marked as 2-1. �agives the relative shift (in per cent) of the equilibrium latticeconstants of all the configurations with respect to the equilibriumlattice parameter of the ground state configuration. The valencycolumn is followed by the columns giving the total numbers oflocalized states, separately for the (t2g↑, t2g↓) and (eg↑, eg↓)subbands. Finally, the corresponding spin moments for the twosubbands, as well as the total one, are also given.

Localized electrons Moments

Config.�E(eV)

�a(%) Valency t2g eg t2g eg Tot.

2-1 0.00 ±0.00 2+ 6 (3, 3) 2 (2, 0) 0 2 ↑ 2 ↑2-2 0.69 +2.05 2+ 5 (3, 2) 3 (2, 1) 1 ↑ 1 ↑ 2 ↑2-3 0.84 +0.14 2+ 6 (3, 3) 2 (1, 1) 0 0 02-4 1.31 +4.03 2+ 4 (3, 1) 4 (2, 2) 2 ↑ 0 2 ↑2-5 1.47 +2.12 2+ 5 (3, 2) 3 (1, 2) 1 ↑ 1 ↓ 02-6 2.16 +3.93 2+ 4 (2, 2) 4 (2, 2) 0 0 03-1 0.67 −0.90 3+ 5 (3, 2) 2 (2, 0) 1 ↑ 2 ↑ 3 ↑3-2 0.81 −0.78 3+ 5 (2, 3) 2 (2, 0) 1 ↓ 2 ↑ 1 ↑3-3 1.03 −0.46 3+ 6 (3, 3) 1 (1, 0) 0 1 ↑ 1 ↑3-4 1.15 +0.37 3+ 4 (3, 1) 3 (2, 1) 2 ↑ 1 ↑ 3 ↑3-5 1.50 +0.99 3+ 4 (2, 2) 3 (2, 1) 0 1 ↑ 1 ↑3-6 1.55 −0.68 3+ 5 (3, 2) 2 (1, 1) 1 ↑ 0 1 ↑3-7 1.77 +2.10 3+ 3 (3, 0) 4 (2, 2) 3 ↑ 0 3 ↑3-8 2.01 +0.48 3+ 4 (3, 1) 3 (1, 2) 2 ↑ 1 ↓ 1 ↑3-9 2.17 +2.97 3+ 3 (2, 1) 4 (2, 2) 1 ↑ 0 1 ↑

Also, within the configurations with the same number oflocalized d states, the ones that have more localized t2g statesthan eg states have generally lower energies.

Inspecting both figure 1 and table 2, one can see thatthere is quite a strong dependence of the lattice parameter onthe type of the configuration. Among the 2+ configurationsthe tendency for the larger lattice constants is observed.Among the 3+ valence configurations, we observe no clearpreference as some of them have smaller and others largerlattice constants. In general, configurations with a completefilling of eg subbands show a strong dependence of their totalenergies on the lattice parameter with an apparent shift towardslarger lattice parameters.

As can be seen in figure 2, there exists some sensitivityto the equilibrium lattice parameter of the amount of chargeassociated with a selected scattering channel for which the SICis applied. The charge is calculated using equation (19), andfrom this the SIC potential of the given channel is obtained andadded to the effective LSDA potential. One should rememberthat in LSIC these charges are calculated only within theatomic sphere corresponding to the site of the selected channel.Thus, it is likely that, as seen in the figure, charges of less thanone electron per channel will be obtained. This is unlike in the

Figure 2. The SIC charges, per channel, for the Ni t2g and eg statescorresponding to the ground state configuration of NiO as a functionof lattice constant. The vertical lines denote the experimental(dotted) and theoretical (dashed) lattice constants, respectively.

FSIC approach where the charges of the orbitals consideredfor SIC are calculated from exponentially decaying Wannierfunctions optimized on a large cluster of atoms surrounding agiven site. In such a case, charges very close to or even ofexactly one electron are obtained. In the LSIC approach, weget slightly less than one electron for the majority and minorityNi t2g-channels, but more than one electron for the majority egchannel. The latter is caused by hybridization with the oxygen2s and 2p channels as demonstrated in figure 3, showingrespectively the phase shifts of both majority and minority Nit2g and eg channels, and the corresponding DOS and integratedDOS. In particular, the latter (see figure 3(c)) shows that indeedthere is a small contribution to the SIC charge of the majority egstate due to the aforementioned hybridization. This behaviourmight also be elucidated based on the analysis of molecularorbitals (MOs). An MO with the eg symmetry constructed fromthe oxygen environment of a given cation and the eg state at thiscation site will have more hybridized character than the MOswith the t2g symmetry.

The phase shifts displayed in figure 3(a) characterize thescattering properties of the constituent atoms in the system.The stronger the scatterer, the more resonant are the phaseshifts, as seen for the Ni d states. In the LSD approximationdue to the spherical symmetry of the scattering potential, thereis only one phase shift associated with all the Ni d electrons(see figure 3(a)). The situation is different in the SIC approachdue to the orbital dependent potential breaking the symmetry.By applying SIC to the specific scattering channels, here theNi d states, their respective phase shifts get shifted downwardsin energy and the resonances sharpen up considerably ascompared to the LSDA counterparts (figure 3(a)). This agreeswell with the physical picture of localization and a long Wignerdelay time for such SIC states. However, these states donot become quasi-bound states because their resonances, afterapplying the SI correction, still lie well above the muffin-tinzero. The positions of these resonances coincide with therespective d peaks in the DOS (figure 3(b)). As can be seenin the figure, the eg resonance lies lower in energy than the t2g

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Figure 3. The phase shifts, densities of states, and integrateddensities of states at the experimental lattice constant of NiO. Here(a) shows the Ni d phase shifts calculated within LSDA andSIC–LSDA. The LSDA phase shifts (brown/violet) are degeneratefor all the m channels, the SIC–LSDA phase shifts split due to theorbital dependent potential (breaking LSDA symmetry). Themuffin-tin zero VMTZ is marked. In (b) the total DOS (black),together with the symmetry resolved partial DOS of the Ni 3d states,is presented. Panel (c) displays the integrated density of all the Ni dstates. The colour scheme is identical in all the subfigures.

resonance due to a more attractive SIC potential resulting froma larger SIC charge (see figure 2). The shift of the occupiedd states down in energy is associated with an effective upwardshift of the unoccupied states, giving rise to opening up of aconsiderable charge transfer band gap seen in figure 3(b). Thisis in line with experiment and contrary to the LSDA band gapwhich is of the Mott–Hubbard-type (see figure 4). The latter,as already discussed by Terakura et al [2] is the result of asmall exchange splitting which is comparable to a small crystalfield splitting. The width of the eg band is crucial in stabilizingthe AFII structure in LSDA because the band gap opens upbetween the t2g and eg bands.

In figure 4 (panels (a), (b), and (c)), we present thetotal and Ni d densities of states for NiO, calculated in theLSD approximation as well as from the FSIC and LSICapproaches for the ground state configuration (figure 1 andtable 2). Contrary to LSDA where all the Ni d bands arefound around a tiny band gap, in both the LSIC and FSICapproaches the occupied t2g↑, t2g↓, and eg↑ bands have movedto higher binding energies and have become narrower as aresult of the applied self-interaction corrections. Anothereffect of SIC is the observed considerable change in thecrystal field splitting as compared to LSD. From figures 4(b)and (c) one can see that the band gaps, relative positions ofdifferent bands, and their hybridizations agree between both

Figure 4. The symmetry resolved partial DOS of the self-interactioncorrected 3d electron states of Ni: (a) the LSDA calculation,(b) FSIC (LMTO), (c) LSIC (KKR), and (d) LSIC (KKR) where theSlater transition state concept has been implemented.

the FSIC and LSIC implementations. The slight shifts inthe positions of the SIC d bands may arise from such detailslike, e.g., possible small differences in the calculated SICcharges. In any case, a fully quantitative agreement cannot beexpected due to the completely different implementations ofthe two SIC approaches, and this is in addition to the differentnumerical accuracies of the two band structure methods usedfor these two implementations. It is reassuring, however, thatdespite these apparent differences in both the FSIC and LSICimplementations, the most relevant trends in DOS and bandgaps are correctly reproduced in both approaches.

Regarding comparison of the LSIC and FSIC resultswith experiment, one has to keep in mind that they areprimarily total energy ground state approaches. Whilst theband gaps calculated using SIC–LSDA are of the correctcharge transfer character [53, 54], the latter is substantiallyoverestimated. The reason is that the calculated removalenergies of the localized Ni d states do not agree with thephotoemission spectroscopy, lying a few eV lower in energythan observed experimentally. Consequently, there is notenough Ni d character hybridized into the predominantlyO p valence band. Now, a question arises whether onecan expect correct removal energies from such an effectiveone electron ground state theory like SIC–LSDA. AlthoughDFT is, strictly speaking, a theory of the ground state fromwhich spectroscopic information is not easily extracted, theLDA based band structure is often compared to photoemissionexperiments. This is because the effective Kohn–Shampotentials can be viewed as an energy independent self-energyand hence the Kohn–Sham energy bands correspond to the

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mean field approximation for the spectral function. In theSIC–LSDA, this argument only applies to the itinerant statesthat are not self-interaction corrected. The localized statesthat have been self-interaction corrected respond to a differentpotential [55] and the solution to the generalized SIC–LSDAeigenvalue problem, which is different from the solution tothe Kohn–Sham equations in the LDA, no longer correspondsto a mean field approximation of the spectral function [56].To extract spectroscopic information for the localized states,one needs to take into account screening/relaxation effects.One of the well known approaches is to perform the so-called �SCF calculations [57], where one has to compute totalenergies of the systems with N , N − 1, and N + 1 electronswhich, however, is computationally non-trivial. To obtainelectron removal energies for the localized Ni d states just fromthe self-consistent ground state calculation, a transition stateconcept [58] can be used. According to this approximation, theremoval energy of a Ni d state is defined as the average of thecalculated SIC–LSDA and LSDA d state expectation values,formally

TS = 12 (〈d|HLSDA + VSIC|d〉 + 〈d|HLSDA|d〉). (23)In effect, the SIC potential is only counted with half of itsstrength in the transition state approximation to the removalenergies. The transition state concept was implemented byFilippetti and Spaldin [59], albeit in a different manner, byinvoking the averaging factor of 12 already in the total energyfunctional. To compare with photoemission spectra, one canimplement equation (23) after the self-consistency has beenreached in the step of calculating the densities of states, wehave done this for NiO.

In figure 4(d) we show the LSIC DOS obtained for theNiO ground state configuration with the transition state conceptimplemented. The corresponding Bloch spectral functionis presented in figure 5. Although the calculations for theBloch spectral function have been performed for the complexenergies with a small constant imaginary part, the figure showsa well resolved band structure, albeit with a small smearing,along a number of symmetry directions in the Brillouin zoneof the AFII structure. The density of states calculated in thetransition state approximation shows considerably hybridizedvalence band with a substantial amount of Ni d characterat the top of the valence band, thus correcting for the SICoverestimate of the charge transfer character of the band gap(compare figures 4(b)–(d)). This strong hybridization effect isalso seen in the band structure of figure 5. The unoccupiedminority Ni eg states are very slightly hybridized with theO p states. Still, it is rather amazing to see how localizedthe unoccupied Ni d states are, considering that no SIC isimplemented for unoccupied band states. Of course, thereexists an indirect effect of SIC on the unoccupied bands dueto the fact that the potential they respond to is the LSDAeffective potential calculated from the charge density of allthe occupied states, of which some have been self-interactioncorrected, notably all the occupied Ni 3d states. However, theflatness of these two minority Ni eg bands is a consequence ofthe considerably reduced phase space for hybridization.

Figure 5. The calculated Bloch spectral function, corresponding tothe Slater transition state concept along a number of symmetry linesin the AFII Brillouin zone, giving rise to a well resolved bandstructure of NiO. The visible small smearing of bands is due to asmall imaginary part of energies in the calculations.

Table 3. SIC configurations minimizing the total energy for theseries of 3d transition metal oxides.

Compound SIC configuration

MnO 3t2g↑ + 2eg↑FeO 3t2g↑ + 2eg↑ + 1t2g↓CoO 3t2g↑ + 2eg↑ + 2t2g↓NiO 3t2g↑ + 2eg↑ + 3t2g↓CuO 3t2g↑ + 2eg↑ + 3t2g↓ + 1eg↓

5. TMO series

In this section we discuss the LSIC results obtained for thewhole TMO series. We restrict the discussion to the groundstate configurations and properties of these compounds. Theseground state configurations lead to the ground state energiesgiven in table 3. As in the case of NiO, these energyminimizing configurations satisfy the first Hund’s rule. Theway they come about may be understood based on simplephysical considerations. For example, in MnO the exchangesplitting on the Mn site is much larger than the ligand fieldsplitting, and therefore all five Mn d electrons occupy fully themajority d band, namely the three t2g↑ and two eg↑ band states,leaving the minority bands empty. For the other compounds inthe series, the additional d electrons start filling the minoritysubband with one minority t2g state in FeO, two in CoO, andall three in NiO; while in CuO, in addition the first minority egstate is also filled, as seen in table 3.

The results for the equilibrium lattice constants and thecorresponding magnetic moments and band gaps calculatedwith LSIC for all the TMO series are summarized in table 4and compared with the LSDA values and the experimentaldata. Also, the results of much earlier FSIC (LMTO–ASA) calculations are quoted, mostly for a qualitativecomparison [6, 7]. The reason is that in those calculations,minimal basis sets were used and no so-called empty sphereswere considered for improving space filling. To make thecomparison more quantitative and meaningful, for NiO wealso present results of later, more numerically advanced FSIC

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Table 4. The equilibrium lattice constants, local magnetic moments and the band gaps for the 3d transition metal monoxides calculated withLSIC–KKR and compared to two full SIC–LMTO implementations as well as experimental results. The values for the band gap quoted in theparenthesis are calculated at the experimental lattice constant. The calculated magnetic moments are spin moments only, the experimentalvalues are total moments, including orbital contributions which are known to be substantial for FeO and CoO, and non-negligible even forNiO.

Compound

MnO FeO CoO NiO CuO

Lattice constant (Å)

LSDA (KKR) 4.27 4.13 4.05 4.01 4.08LSIC (KKR) 4.49 4.39 4.31 4.24 4.27FSIC (LMTO) [21] 4.18Expt. 4.446a, 4.44b 4.326c 4.26b,d 4.176e, 4.17b 4.245d

Local magnetic moment on TM (μB)

LSDA (KKR) 4.11 (4.27) 3.26 (3.40) 2.20 (2.33) 0.85 (0.97) 0.00 (0.00)LSIC (KKR) 4.63 (4.61) 3.68 (3.66) 2.69 (2.68) 1.68 (1.67) 0.76 (0.76)Expt. (total) 4.79k, 4.58l 3.32m 3.35f, 3.8m 1.77k, 1.90m, 2.20g 0.65n

FSIC (LMTO) [6] 4.49 3.54 2.53 1.53 0.65FSIC (LMTO) [7] 4.64 3.55 2.59 1.49 0.64FSIC (LMTO) [21] 1.61

Band gap (eV)

LSDA (KKR) 0.83 (0.73) 0.00 (0.00) 0.00 (0.00) 0.23 (0.23) 0.00 (0.00)LSIC (KKR) 3.07 (3.25) 3.38 (3.54) 2.78 (2.81) 3.56 (3.76) 2.52 (2.57)Expt. 3.6–3.8o 2.4h, 2.5i 2.4p 4.3q, 4.0r, 4.3j 1.37s

FSIC (LMTO) [6] 3.98 3.07 2.81 2.54 1.43FSIC (LMTO) [7] 3.57 3.25 2.51 2.66 1.00FSIC (LMTO) [21] 3.15

a Reference [63]. b Reference [64].c Reference [65].d Reference [66].e Reference [67]. f Reference [60].g Reference [61]. h Reference [68].i Reference [69]. j Reference [53].k Reference [70]. l Reference [71].m Reference [72]. n Reference [73].o Reference [74]. p Reference [75].q Reference [76]. r Reference [77].s Reference [78].

(LMTO–ASA) calculations [21]. In addition, and for an easyoverview, the LSIC results are plotted in figure 6, togetherwith the LSDA (KKR) and experimental data for all the TMmonoxides and properties collected in table 4.

Inspecting figure 6 in some detail, one can see that ingeneral LSIC provides much better agreement with experimentthan does LSDA, although, in more cases than not, the LSICvalues are larger than the experimental data. This is definitelythe case for the equilibrium lattice constants, which are alsolarger than the values obtained from the FSIC approach asseen in table 4 for NiO [21]. In fact, for NiO both FSIC andLSDA+U give very comparable lattice constants (respectively4.18/4.19 Å cf [21]). The lattice constants calculated in theLSD approximation are considerably smaller than those fromexperiments, and therefore also from LSIC. This can be easilyexplained due to the fact that LSDA treats all the Ni d electronsas delocalized and responding to the same effective potential,leading to the overestimate of the bonding in NiO.

Concerning the magnetic moments calculated in LSIC,they are just spin moments, and thus do not always comparewell with experiment which measure total magnetic moments,

namely, the sum of spin and orbital moments. In the earlierFSIC (LMTO–ASA) calculations, Svane and Gunnarssonevaluated not only the spin moments, but also the orbitalmagnetic moments for the whole TMO series [6] and they aresubstantial in particular for FeO and CoO. Although the orbitalmoments of TM ions in solids are usually quenched, in the caseof TMOs, it is expected that correlation effects will preservethe orbital moments up to a certain value due to a reduction ofligand crystal field effects at the TM ion sites. For example,in CoO, the experimentally observed total magnetic momentof 3.4 μB [60], is quite well reproduced by FSIC with thecalculated orbital moment, μL , of 1.19 μB and spin moment of2.53 μB [6]. LSIC–KKR gives a slightly higher spin magneticmoment for CoO (see table 4) than obtained from FSIC(LMTO) calculations, but by comparison with experimentwould still imply a substantial orbital contribution of about0.7 μB for CoO. Magnetic x-ray scattering experiments ofFernandez et al [61] indicate that also the orbital momentin NiO is not quenched. They observe a spin moment of1.90 μB and an orbital moment of 0.32 μB, which comparesquite well with the FSIC value of 0.27 μB [6]. Looking across

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Figure 6. The calculated equilibrium lattice constants, band gaps and local magnetic moments on TM sites for the TMO series (AFIIstructure). The band gaps and local magnetic moments shown are those corresponding to the equilibrium lattice constant. In contrast to theexperimental magnetic moments the theoretical magnetic moments are spin moments only.

the series, the spin magnetic moments obtained with LSIC–KKR are slightly higher than the moments from the full SICcalculations, which may not necessarily be due to the local SICimplementation but due to the different band structure methodsused in both implementations.

The small size or lack of the band gap is what bestcharacterizes the breakdown of the LSDA description of thephysics of TMOs. Only for MnO and NiO small gaps areobtained in LSDA calculations. The LSIC approach improvesthe agreement with experiment dramatically and the trendsacross the series are predicted correctly, in line with the FSICapproaches although the LSIC values are in general larger thanthose from FSIC.

To summarize this section, we have to say that LSICoften overestimates the physical properties calculated hereas compared to the respective experimental data. Onepossible reason for this may be the use of the atomic sphereapproximation in KKR. While in the LMTO method ASAworks quite well, due to some error cancellations, in KKRthis is most likely not the case. The KKR method is basedon the multiple scattering theory and invokes the so-called ‘onenergy-shell’ formulation implying that a scattering processis finished before another one will start. With the ASA inuse, this condition would have most likely not been fulfilled.Of course, more developments are needed, e.g., full potentialimplementation or at least the so-called ASA-muffin-tin (MT)correction [62] to check this possibility further. In this paper,however, our aim has been to demonstrate that LSIC works forcorrelated 3d electron systems, with more than one atom perunit cell.

6. Vacancy-induced half-metallicity in NiO

In this section we discuss an application of LSIC to studydoping of NiO with cation vacancies. This has been motivatedby a recent study of Ködderitzsch et al [10] who, using FSICimplemented in the LMTO–ASA method, showed that for

3.125% of cation vacancies in one of the Ni-sublattices of theAFII NiO realized in a supercell geometry, NiO switches froman insulator to a half-metal.

As mentioned earlier, being a Green’s function method,KKR lends itself naturally to such extensions as CPA andDLM. Thus, with the present application, we aim not onlyto compare the LSIC–KKR and FSIC–LMTO approaches, butalso explore how well CPA can reproduce the supercell results.Using CPA instead of the supercell geometry would not onlyreduce the computing effort but also allow access to a wholerange of concentrations that would be difficult to realize in thesupercell approach due to huge sizes required.

Like Ködderitzsch et al [10], we have used a (2 × 2 ×2) AFII supercell, comprising 32 formula units [10]. Also,the ASA radii of 2.70 au for Ni atoms and 2.16 au foroxygen atoms are the same as in the other study [10]. Theconcentration of 3.125% of cation vacancies has been realizedby replacing a single Ni cation by an empty sphere in one ofthe Ni-sublattices of the supercell. To reduce computationalcosts, the calculations have been performed without spacefilling empty spheres, with the effect of decreasing the size ofthe band gap by about 0.3 eV as compared to the case whereempty spheres were included in the basic AFII unit cell. TheCPA calculations have been performed for the latter cell, butalso without space filling empty spheres and the same ASAradii as for the supercell approach. To improve on the ASAin the KKR implementation, we have included the so-calledASA-MT correction [62] while in LMTO–ASA, the combinedcorrection term has been used for this purpose [41, 42].

In figure 7, the resulting DOS for Ni0.97O, calculated bothfor the supercell [LSIC (KKR)] and with CPA [LSIC (KKR–CPA)], are compared to the FSIC–LMTO–ASA calculation.In all the cases we observe a fairly convincing half-metallicstate. The smearing of the CPA curve reflects both thenature of the effective medium and a small imaginary partof the energy. In addition to a qualitatively similar DOS, thetotal magnetic moments calculated with LSIC (KKR) method

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Figure 7. The spin-decomposed total density of states for NiO,calculated with LSIC–KKR–CPA for 3.125% of cation vacancies inone of the Ni-sublattices of the AFII NiO. Presented are DOS curvescalculated in the supercell geometry using respectively, LSICimplemented in the KKR–ASA and FSIC implemented inLMTO–ASA. Also shown is the DOS obtained from the KKR–CPA(ASA) for the same vacancy concentration. The majority spin DOSis printed on the positive side, while the minority one on the negativeside of the DOS-axis.

in both supercell and CPA approaches are exactly zero, inagreement with the FSIC (LMTO) calculations. The way thezero magnetic moment comes about is that by removing a TMcation one create moment of −2μB, which compensates the Nispin moment of 2 μB.

In summary, we have demonstrated that LSIC (KKR)method provides an adequate description of the vacancy-induced half-metallicity in NiO, both with the supercell andCPA approaches. This gives us confidence that one will be ableto explore the whole range of vacancy concentrations not onlyon one but both Ni-sublattices, concentrating on energetics andthe existence and origin of half-metallicity. However, such astudy lies outside the scope of the present paper and will bepublished elsewhere.

7. Summary and conclusions

We have presented and discussed in some detail a localimplementation of the SIC–LSDA formalism in the multiplescattering theory. By applying it to 3d transition metaloxides we have demonstrated that this approach can describetrends for many physical properties of these compounds andcompares well with earlier implementations of SIC–LSDAin LMTO–ASA, exploiting repeated transformations betweenBloch and Wannier representations to solve the eigenvalueproblem and reproduce the localized nature of TM d electronsin these compounds. With the present study we have shownthat LSIC works equally well for the 3d TMOs as for the 4fsystems [15]. The great potential of LSIC lies in its flexibilityand ability to study various types of disorder in combinationwith CPA and DLM, opening up possibilities to considerthermal fluctuations and, through them, finite temperature

phase diagrams and finite temperature magnetism, includingordering temperatures [15, 16, 52]. Applying the combinationof LSIC with CPA to study changes in the electronic structureof NiO induced by doping cation vacancies into Ni-sublattices,we have shown that LSIC can describe chemical disorderin addition to spin and valence disorder demonstrated inearlier applications, also at finite temperatures [15, 16, 52].It is very encouraging that with LSIC one can study alsosurfaces, interfaces and even clusters on surfaces, based on theGreen’s functions without the need of invoking large supercellgeometries. In the present paper we have explored onlythe ASA implementation in KKR although in the applicationto vacancies in NiO, the so-called ASA-MT correction hasalso been included [62]. However, in future, we need toperform a systematic study to compare ASA against the moreappropriate MT approximation, the ASA-MT correction [62],and eventually the full potential (no shape approximation)implementation. In this paper, however, our prime goal wasto explain the LSIC method and demonstrate that it works forcorrelated 3d electron compounds as well as for the 4f electronsystems.

Acknowledgments

This work was supported by the DFG SFB 762 ‘Funktionalitätoxidischer Grenzflächen’. The calculations were performed atthe John von Neumann Institute in Jülich (Germany). Oneof us (WH) thanks the DAAD (Deutscher AkademischerAustauschdienst) for financial support.

Research at the Oak Ridge National Laboratory wassponsored by the Division of Materials Science andEngineering, Office of Basic Energy Science, US Departmentof Energy, under Contract DE-AC05-00OR22725 with UT-Battelle, LLC.

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1. Introduction2. Formalism2.1. Basic SIC--LSDA equations2.2. Multiple scattering theory and LSIC implementation

3. Computational details4. Application to NiO5. TMO series6. Vacancy-induced half-metallicity in NiO7. Summary and conclusionsAcknowledgmentsReferences