Full orbital calculation scheme for materials with strongly correlated electrons

V. I. Anisimov,1 D. E. Kondakov,1 A. V. Kozhevnikov,1 I. A. Nekrasov,1 Z. V. Pchelkina,1 J. W. Allen,2 S.-K. Mo,2

H.-D. Kim,3 P. Metcalf,4 S. Suga,5 A. Sekiyama,6 G. Keller,7 I. Leonov,7 X. Ren,7 and D. Vollhardt71Institute of Metal Physics, Russian Academy of Sciences-Ural Division, 620219 Yekaterinburg GSP-170, Russia

2Randall Laboratory of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA3Pohang Accelerator Laboratory, Pohang 790-784, Korea

4Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA5Division of Materials Physics, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan

6Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan7Theoretical Physics III, Center for Electronic Correlations and Magnetism, University of Augsburg, D-86135 Augsburg, Germany

sReceived 24 July 2004; revised manuscript received 11 November 2004; published 21 March 2005d

We propose a computational scheme for theab initio calculation of Wannier functionssWFsd for correlated

electronic materials. The full-orbital HamiltonianH is projected into the WF subspace defined by the physi-

cally most relevant partially filled bands. The HamiltonianHWF obtained in this way, with interaction param-eters calculated by constrained local-density approximationsLDA d for the Wannier orbitals, is used as anabinitio setup of the correlation problem, which can then be solved by many-body techniques, e.g., dynamical

mean-field theorysDMFTd. In such calculations the matrix self-energyos«d is defined in WF basis which thencan be converted back into the full-orbital Hilbert space to compute the full-orbital interacting Green functionGsr ,r 8 ,«d. UsingGsr ,r 8 ,«d one can evaluate the charge density, modified by correlations, together with a newset of WFs, thus defining a fully self-consistent scheme. The Green function can also be used for the calcula-tion of spectral, magnetic, and electronic properties of the system. Here we report the results obtained with thismethod for SrVO3 and V2O3. Comparisons are made with previous results obtained by the LDA1DMFTapproach where the LDA density of states was used as input, and with new bulk-sensitive experimental spectra.

DOI: 10.1103/PhysRevB.71.125119 PACS numberssd: 71.27.1a, 71.30.1h

I. INTRODUCTION

Model Hamiltonians used in the study of correlation ef-fects in solids have a Coulomb interaction term in a site-centered atomiclike orbital basis set which is not explicitlydefined. When the correlated electrons are well localized, as,for example, 4f states of rare-earth ions, atomic orbitalssoratomic sphere solutions like muffin-tin orbitalsd are a goodchoice. However, the most interesting problems occur in theregime of metal-insulator transitions, where the states of in-terest become partially itinerant and rather extended. Theerror of using atomic orbitals is most severe in the case ofmaterials with strong covalency effects, like late transition-metal oxides, where partially filled bands are formed by themixture of metallic d orbitals and oxygenp orbitals. Forexample, in high-Tc cuprates correlated states have the sym-metry of Cu−3d x2−y2 orbitals, but are actually Zhang-Ricesinglets formed by the combination of oxygenp states cen-tered around the Cu ion and havingx2−y2 symmetry.

In model calculations the problem of defining the corre-lated orbitals is not very important, because it only affectsmodel parameter values, which in any case are consideredfitting parameters. However, any attempt to construct an “abinitio” calculation scheme requires an explicit definition ofthe basis set for the Coulomb interaction term. An importantrequirement for such a choice is that the orbitals must pro-duce the partially filled bands where Coulomb correlationsoccur while preserving the localized, site-centered atomiclikeform. These requirements are fulfilled for Wannier functionssWFsd uWn

Tl defined as a Fourier transformation of the Bloch

functions ucnkl.1 Here and below functions are labeled withband indexn, lattice translation vectorT, and wave vectork.

When there is more than one band crossing the Fermilevel, WFs are not uniquely defined. Anyk-dependent uni-tary transformationUskd of the set of Bloch functionsucnklfor these bands produces a new set which can be used for thecalculation of WFs via Fourier transformationfEq. s5d, Sec.II A g. If one imposes the requirement that the WFs shouldhave the symmetry of atomic orbitals,2,3 this unitary transfor-mation is well defined. The explicit form of the WFs allowsone to compute Coulomb interaction parameters in con-strained local-density approximationsLDA d calculations.

In this way the parameters for theab initio many-bodyHamiltoniansnoninteracting HamiltonianHWF and Coulombinteractiond in the WF basis can be computed by any first-principle electronic structure calculation schemefbelow weuse the linear muffin-tin orbitalsLMTOd methodg. ThisHamiltonian can then be further investigated by one of themethods developed in the many-body community. In thepresent work we use the dynamical mean-field theorysDMFTd.4–7 Within DMFT, the effective impurity problemcorresponding to the many-body Hamiltonian is solved byquantum Monte Carlo simulationssQMCd.8 The DMFT partof the proposed calculation scheme is essentially the same asthe one used in the recently developed LDA1DMFTapproach9 for the ab initio investigations of correlated elec-tron materials.10 However, here we propose a more generalprocedure to compute the Green function using the Hamil-tonian matrix and an integral over the Brillouin zone insteadof the Hilbert transform of the LDA density of statessDOSd.

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This particular method allows one to avoid the uncontrol-lable errors occurring in the computation of the Green func-tion using the Hilbert transform of the LDA DOS. Thus toobtain an insulating solution we need to cut off the longsmetal-oxygend hybridization tails of the DOS, renormalizeit, and shift the Fermi energy to get an integer filling. In thepresent method we overcome the above-mentioned difficul-ties owing to the integer filling of Wannier orbitals. The re-sult of the DMFT calculations is a local matrix self-energy

Ss«d in the WF basis sethWnj which acts in the subspace ofpartially filled bands used for the construction of the WFs.

The paper is structured as follows. In Sec. II the details ofour scheme are presented. In Sec. II A we describe the con-struction of WFs, as well as theab initio Hamiltonian matrixwithin this basis set in terms of Bloch functions. In Sec. II Bwe propose a general method for the construction of WFsusing the Green functionGsr ,r 8 ,«d, which reduces to theresults of Sec. II A in the noninteracting case. The reason fordoing so is that the correlation effects can significantly renor-malize the electronic states of the partially filled bands.Hence the WFs computed from noninteracting Bloch statesare not an optimal choice for the basis set any more. In Sec.II C we discuss how to calculate within DMFT the localGreen function with the input of the Hamiltonian matrix inthe WF basis set instead of the LDA DOSswhich is validonly in the case of degenerate bandsd. In Sec. II D we showthat the matrix self-energy within the WF subspace, which isthe solution of the correlation problem, can be transformedback into the full-orbital Hilbert space, thus enabling thecomputation of the full interacting Green functionGsr ,r 8 ,«d. It can then be used to calculate the spectral, mag-netic, and electronic properties of the system under investi-gation. In addition, to make the calculation scheme fully self-consistent, one can employ theGsr ,r 8 ,«d to calculate thecharge density affected by correlations and thus the newLDA potential. Thereby the feedback from DMFT to LDAcan be incorporated in a well-defined way. This is actuallyone of the great advantages of using the WF basis since inthe LMTO basis the feedback from DMFT to LDA is essen-tially uncontrolled. In Sec. III the results for the electronicstructure of the two vanadium oxides SrVO3 and V2O3 ob-tained by the method developed in this work are presentedand compared with the previous calculations by the simplermethods and new bulk-sensitive spectra. Finally in Sec. IVwe close this work with a conclusion.

II. METHOD

Let us consider the general case of the electronic structure

problem. For the LDA HamiltonianH we have a Hilbertspace of eigenfunctionssBloch statesucikld with the basis setufml defined by particular methodsfe.g., LMTO,11 or linear-ized augmented plane wavessLAPWd,12 etc.g. In this basisset the Hamiltonian operator is defined as

H = omn

ufmlHmnkfnu. s1d

Here and later greek indices are used for full-orbital matri-ces.

If we consider a certain subset of the Hamiltonian eigen-functions, for example Bloch states of partially filled bandsucnkl, we can define a corresponding subspace in the totalHilbert space. The Hamiltonian matrix is diagonal in theBloch states basis. However, physically more appealing is abasis set which has the form of site-centered atomic orbitals.That is, a set of WFsuWn

Tl defined as the Fourier transforma-tion of a certain linear combination of Bloch functions be-longing to this subspacefsee below Eq.s6dg. The Hamil-

tonian operatorHWF defined in this basis set is

HWF = onn8T

uWn0lHnn8sTdkWn8

T u. s2d

The total Hilbert space can be divided into a direct sum ofthe above introduced subspacesof partially filled Blochstatesd and the subspace formed by all other states orthogonalto it. Those two subspaces are decoupled since they are theeigenfunctions corresponding to different eigenvalues. TheHamiltonian matrix in the WF basissi.e., a collection of thebases of the specific subspacesd is block diagonal so that thematrix elements between different subspaces are zero. Theblock matrix Hnn8 in Eq. s2d corresponding to the partiallyfilled bands can be considered as a projection of the full-orbital Hamiltonian operators1d onto the subspace definedby its WFs.

All this concerns the noninteractingsor LDAd Hamil-tonian. To treat Coulomb correlations one also needs a defi-nition of the localized orbitals where the electrons interact.WFs are a natural choice for such a definition. This choiceleads to an important flexibility in the size of the basis set inthe sense that the number of WFs can be changed by chang-ing the set of Bloch bands considered. The simplest case is aset of partially filled bands, for example thet2g bands ofvanadium oxides. This is a physically justified approximationbecause the Coulomb interaction happens mainly betweenelectronssor holesd in the partially filled bands. If the prob-lem to be solved concentrates on the excitation spectrum in asmall energy window around the Fermi level, this basis set issufficient. However, if the excitations to higher lying statessreal or virtuald are also important, the set of Bloch bandsused to construct the WFs need to be extended so that theCoulomb interaction will be treated in a larger Hilbert sub-space.

Practically this means that the correlation problem is

solved using a noninteracting few-orbital HamiltonianHWF

fEq. s2dg instead of the full Hilbert space HamiltonianH fEq.s1dg. The interaction matrix elements of the model Hamil-tonian can be determined from constrained LDA calculationsfor the specific WF basis sets25d.

Projecting the full orbital Hilbert space HamiltonianHfEq. s1dg onto the subspace of the partially filled bands gives

us a few-orbital HamiltonianHWF fEq. s2dg. This signifi-cantly decreases the complexity of the correlation problem,thus permitting its explicit solution. The many-body problemwith a local intraorbital Coulomb interactionsHubbard inter-

actiond then leads to a local matrix self-energySWFs«d which

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is naturally defined in the basis of WFs centered on the samesite:13

SWFs«d = onn8

uWn0lSnn8s«dkWn8

0 u. s3d

We note that, in contrast to other “basis-reducing” methods,the information about the states corresponding to the bandsbelow and above the projected ones is not lost. In fact, theinformation is stored in thek-dependent projection matrixbetween the full orbital basis set and the orthonormalizedWFss16d. The definitions3d allows one to convert the matrixself-energySnn8s«d back to the full Hilbert space basis setsSec. II Dd. With this the interacting Green function can alsobe calculated in the full-orbital Hilbert spacesSec. II Bd

A. Definition and construction of Wannier functions

The concept of WFs has a very important place in theelectron theory in solids since its first introduction in 1937by Wannier.1 WFs are the Fourier transformation of Blochstatesucikl,

uWiTl =

1ÎN

ok

e−ikT ucikl, s4d

whereN is the number of discretek points in the first Bril-louin zone sor, the number of cells in the crystald. Theseextremely convenient orthogonal functions were widely in-vestigated in the seventies.14 The strongly localized nature ofthe WFs together with all advantages of the atomic functionsmakes them a very useful tool where the atomic character ofthe electrons is highlighted. Thus, using the WF method,significant progress was achieved in the fields of narrow-band superconductors, disordered systems, solid surfaces,etc. Several methods for calculating WFs for single and mul-tiple bands in periodic crystals and their generalization tononperiodic systems were proposed. The problem of nonu-nique definition of WFs in these methods was resolved by aniterative optimization of trial functions which have the samereal and point-group symmetry properties as WFs. Amongthese methods, there are the variational Koster-Parzenprinciple15,16 which was generalized by Kohn,20–25 the gen-eral pseudopotential formalism proposed by Anderson,26 andthe projection operator formalism by Cloizeaux.27–29 How-ever, all these computational schemes are restricted to simpleband structures.

Wannier functions are not uniquely defined because for acertain set of bands any orthogonal linear combination ofBloch functionsucikl can be used in Eq.s4d. In general itmeans that the freedom of choice of Wannier functions cor-responds to the freedom of choice of a unitary transformationmatrix Uji

skd for corresponding Bloch functions:2

ucikl = oj

Ujiskduc jkl. s5d

The resulting Bloch functionucikl will generally not be aneigenfunction of the Hamiltonian but has the meaning of a

Bloch sum of Wannier functionsfsee belowuWnkl in Eq.

s6dg. There is no rigorous way to defineUjiskd. This calls for an

additional restriction on the properties of WFs. Among oth-ers, Marzari and Vanderbilt2 proposed the condition of maxi-mum localization for WFs, resulting in a variational proce-dure to calculateUji

skd. To get a good initial guess the authorsof Ref. 2 proposed choosing a set of localized trial orbitalsufnl and projecting them onto the Bloch functionsucikl. Itwas found that this starting guess is usually quite good. Thisfact later led to the simplified calculating scheme proposed inRef. 3 where the variational procedure was abandoned andthe result of the aforementioned projection was considered asthe final step. The approach of Ref. 2 has recently been usedfor the investigation of the row of 3d transition metalssFe,Co, Ni, and Cud within the simplest many-body approxima-tion, namely the unscreened Hartree-Fock approximation.30

Another possibility to construct WFs was recently devel-oped by Andersenet al.31 They proposed theNth-ordermuffin-tin orbital sNMTOd scheme in which Wannier-likelow-energy MTOs can be designeda priori. Using a differ-ent implementation of the LDA1DMFT approach they per-formed an investigation of the Mott transition in orthorhom-bic 3d1 perovskites.34 In this approach a realisticHamiltonian constructed with Wannier orbitalsson sym-metrically orthonormalized NMTOsd was solved by DMFT,including the nondiagonal part of the on-site self-energy.

Our projection procedure works as follows. First of allone needs to identify the physically relevant bands whichwill then be projected onto a WF basis. For example, inperovskites one usually takes the partially filledd shell orsome particulard bands of transition metals, since they aremainly responsible for the physical properties of thesystem.10 These orbitals are well separated and are, in ourapproach, easily extracted from the full orbital space as willbe shown later. Moreover, the projection method is appli-cable even in the case where the bands of interest differ andare strongly hybridizedsfor example, Cu-3d and O−2pstates in high-Tc superconductors32d.

To project bands of particular symmetry onto the WFsbasis one can select either the band indices of the corre-sponding Bloch functionssN1,… ,N2d, or choose the energyintervalsE1,E2d in which the bands are located. Nonorthogo-

nalized WFs in reciprocal spaceuWnkl are then the projectionof the set of site-centered atomiclike trial orbitalsufnl on theBloch functionsucikl of the chosen bandsfband indicesN1 toN2, energy intervalsE1,E2dg:

uWnkl ; oi=N1

N2

uciklkcikufnl = oisE1,«iskd,E2d

uciklkcikufnl.

s6d

Then the real space WFsuWnTl are given by

uWnTl =

1ÎN

ok

e−ikT uWnkl. s7d

In the present work the trial orbitalsufnl are LMTOs. Note

that in the multiband case a WF in reciprocal spaceuWnkldoes not coincide with the Bloch functionucnkl due to the

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summation over band indexi in Eq. s6d. One can considerthem as Bloch sums of WFs analogous to the basis functionBloch sumsf j

ksr d fEq. s9dg.The coefficientskcik ufnl in Eq. s6d definesafter orthonor-

malizationd the unitary transformation matrixUjiskd in Eq. s5d.

However, the projection procedure defined in Eq.s6d is moregeneral than the unitary transformations5d. Namely, thenumber of bandssN2−N1+1d can be larger than the numberof trial functions. In this case the projections6d will produce

N new functionsuWnkl which define a certain subspace of theoriginal sN2−N1+1d-dimensional space. This subspace willhave the symmetry of the set of trial functions. In the nextsubsection we propose a way to determine WFs from theGreen function of the systems28d rather than from a set ofBloch states as in Eq.s6d. In this alternative projection pro-cedure, trial functions are projected onto the subspace de-fined by the Green function in a certain energy interval.

The Bloch functions in the LMTO basissor any otheratomic orbital-like basis setd are defined as

ucikl = om

cmik ufm

k l, s8d

where m is the combined index representingqlm sq is theatomic number in the unit cell,lm are orbital and magneticquantum numbersd, fm

k sr d are the Bloch sums of the basisorbitalsfmsr −Td,

fmk sr d =

1ÎN

oT

eikTfmsr − Td, s9d

and the coefficients have the property

cmik = kfmucikl. s10d

If n in ufnl corresponds to the particularqlm combinationsin other wordsufnl is an orthogonal LMTO basis set or-bitald, thenkcik ufnl=cni

k* , and hence

uWnkl = oi=N1

N2

uciklcnik* = o

i=N1

N2

om

cmik cni

k* ufmk l = o

m

bmnk ufm

k l,

s11d

with

bmnk ; o

i=N1

N2

cmik cni

k* . s12d

For a nonorthogonal basis set, see Appendix C.In order to orthonormalize the WFss11d one needs to

calculate the overlap matrixOnn8skd,

Onn8skd ; kWnkuWn8kl = oi=N1

N2

cnik cn8i

k* , s13d

and its inverse square rootSnn8skd which is defined as

Snn8skd ; Onn8−1/2skd. s14d

In the derivation of Eq.s13d the orthogonality of Bloch stateskcnk ucn8kl=dnn8 was used.

From Eqs.s11d and s14d, the orthonormalized WFs inkspaceuWnkl can be obtained as

uWnkl = on8

Snn8skduWn8kl = oi=N1

N2

uciklcnik* = o

m

bmnk ufm

k l,

s15d

with

cnik* ; kcikuWnkl = o

n8

Snn8skdcn8ik* , s16d

bmnk ; kfm

k uWnkl = oi=N1

N2

cmik cni

k* . s17d

The real space site-centered WFs at the originuWn0l are

given by the Fourier transform ofuWnkl with T =0. FromEqs.s15d and s9d one finds

Wnsr d =1

ÎNok

kr uWnkl = oT,m

S 1

Nok

eikTbmnk Dfmsr − Td

= oT,m

w8sn,m,Tdfmsr − Td = os

wsn,sdfassdsr − Tsd,

s18d

where w8 and w are the expansion coefficients of WF interms of the corresponding LMTO orbitals, in particular,

wsn,sd =1

Nok

eikTsbassdnk . s19d

Here s is an index counting the orbitals of the neighboringcluster for the atom where orbitaln is centeredfTs is thecorresponding translation vector,assd is a combinedqlm in-dexg. The explicit form of the real-space WFs18d can beused to produce, e.g., shapes of chemical bonds.

For other applications only the matrix elements of thevarious operators in the basis of WFs15d are needed. FromEqs. s15d, s16d, and s18d the matrix elements of the Hamil-

tonian HWF in the basis of WF in real space where bothorbitals are in the same unit cell are

Hnn8WFs0d = kWn

0uS 1

Nok

oi=N1

N2

ucikleiskdkcikuDuWn80 l

=1

Nok

oi=N1

N2

cnik cn8i

k* eiskd. s20d

eiskd is the eigenvalue for a particular band.If, on the other hand, one of the orbitals corresponds to

the WF for the atomn8 shifted from its position in the pri-mary unit cell by a translation vectorT, then the correspond-ing Hamiltonian matrix element is

Hnn8WFsTd = kWn

0uHuWn8T l =

1

Nok

oi=N1

N2

cnik cn8i

k* eiskde−ikT .

s21d

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Matrix elements of the density-matrix operatorsoccupa-tion matrix Qnm

WFd in the basis of WFs can be calculated as

Qnn8WFsTd = kWn

0uS 1

Nok

oi=N1

N2

uciklusEf − eiskddkcikuDuWn8T l

=1

Nok

oi=N1

N2

cnik cn8i

k* ufEf − eiskdge−ikT ; s22d

usxd is the step function,Ef is the Fermi energy.

Finally, the matrix elements of the HamiltonianHWF inreciprocal space are

Hnn8WFskd = kWnkuS 1

Nok8

oi=N1

N2

ucik8leisk8dkcik8uDuWn8kl

= oi=N1

N2

cnik cn8i

k* eiskd. s23d

Equations23d is valid only if the WFs are computed by Eqs.s15d–s17d. If the WFs were obtained in one calculation andthen used to compute the Hamiltonian matrix in anothersasis the case for the WFss28d in the Green-function formalismssee Sec. II Bd then Eq.s23d is not valid any more and thegeneral expression must be used:

Hnn8WFskd = o

i=N1

N2

eiskdom

bmnk* cmi

k on

bnn8k cni

k* . s24d

Thus the transformation from LMTO to the WF basis setis defined by the explicit form of WFss15d ands17d, and bythe expressions of the matrix elements of the Hamiltonianand density matrix operators in WF basiss20d ands22d. Theback transformation from the WF to the LMTO basis canalso be defined using Eq.s15d ssee Sec. II Dd.

Finally, the Coulomb matrix elementU needs to be cal-culated in the same WF basis. This requires a method similarto the constrained LDA,33 but now for WFs. To this end theWF energys20d is computed as a function of its occupancys22d for a given WFn. Then the corresponding Coulombinteraction parameterUn in the WF basis is given by

Un ;dHnn

WFs0ddQnn

WFs0d. s25d

As one can see,Un depends on the WFs via Eqs.s20d ands22d. Once the WFs have been recalculatedsfor example, insome self-consistent loopd the interaction has to be recalcu-lated as well.

WFs not only containd orbitals but also states which areusually considered delocalized, such as O-2p orbitals. Thequestion is then: which ones are the localized functions thatdescribe the interacting electrons? In our definition these areWFs describing the partially occupied bands andnot puredorbitals. The corresponding Coulomb interaction strengthcalculated in constrained LDA will give values ofU forWannier functions smaller than the corresponding value forpured orbitals due to the admixture of O-2p states.

B. Wannier functions in the Green-function formalism

In many-body theory the system is usually not describedby Bloch functionsucikl fEq. s8dg and their energieseiskd butby the Green function

Gsr ,r 8,«d =1

Nok

Gksr ,r 8,«d =1

Nok

omn

fmk sr dGmn

k s«dfn*ksr 8d.

s26d

The matrix Green functionGmnk s«d is defined via the nonin-

teracting Hamiltonian matrixHmnskd and the matrix self-energySmn

k s«d fEq. s39dg as

Gmnk s«d = f« − Hskd − Ss«,kd + ihgmn

−1. s27d

We define the nonorthonormalized WF obtained by project-ing the trial orbitalfnsr d on the Hilbert subspace defined bythe Green functions26d in the energy intervalsE1,E2d,namely,

Wnksr d = −1

pImE

E1

E2

d«E dr 8Gksr ,r 8,«dfnksr 8d

= om

bmnk fm

k sr d s28d

and

bmnk ; −

1

pImE

E1

E2

d«Gmnk s«d. s29d

In the noninteracting case, the self-energySs« ,kd is ab-sent, and hence we have

Gmnk s«d = o

i

cmik cni

k*

« − eiskd + ih, s30d

wherecmik are the eigenvectorss10d, andeiskd are the eigen-

values ofHskd. Thusbmnk in Eq. s29d becomes

bmnk = o

i=N1

N2

cmik cni

k* , s31d

whereN1,N2 are the band numbers which correspond to theenergy intervalsE1,E2d. Since this recovers the result of Eq.s12d, we demonstrated that our general definition of WFss28d via Green functions reduces to that in terms of Blochfunctionss11d in Sec. II A.

To orthonormalizeWnksr d defined in Eq.s28d, one canjust follow the orthonormalizing procedure made in Sec. II AfEqs. s13d–s17dg, which will not be repeated here. But itshould be pointed out that in the Green-function formalismthe overlap matrixOnn8skd is defined as

Onn8skd = kWnkuWn8kl = om

bmnk* bmn8

k .

The occupancy matrix in the orthogonalized WF basiss28d isdefined as

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Qnn8sTd = −1

pImE

−`

EF

d«E E drdr 81

Nok

Wnk* sr d

3Gksr ,r 8,«dWn8ksr 8de−ikT . s32d

By using Eq.s26d and orthogonalized Eq.s28d, one finds

Qnn8sTd =1

Nok

omn

bmnk* bnn8

k Qmnk e−ikT , s33d

with

Qmnk = −

1

pImE

−`

EF

d«Gmnk s«d. s34d

The energy matrix can be defined similarlysexcept thatthe integral over energy is calculated in thesE1,E2d intervalwhere the corresponding WFs are definedd as

Enn8sTd = −1

pImE

E1

E2

«d«E E drdr 81

Nok

Wnk* sr d

3Gksr ,r 8,«dWn8ksr 8de−ikT

=1

Nok

omn

bmnk* bnn8

k Emnk e−ikT , s35d

with

Emnk = −

1

pImE

E1

E2

«d«Gmnk s«d. s36d

While Eq. s35d looks similar to the noninteracting Hamil-

tonian in WFs basiss21d, it includes correlations viaSs«d inEq. s27d and hence isinteracting.

C. DMFT in the Wannier function formalism

In the previous subsection we showedfEqs. s26d–s28dgthat the matrix self-energy is needed to construct the WFs interms of the full interacting Green function. The DMFTsRefs. 4–6d was recently found to be a powerful tool to nu-merically solve multiband Hubbard models. To define pa-rameters of the correlated model Hamiltonianshoppings,screened Coulomb integralsd, density-functional theorywithin the LDA was used.9 The combined LDA1DMFTcomputational scheme was successfully applied to a widerange of compounds with degeneratesor almost degeneratedorbitals sfor more details, see Ref. 10d. In these cases thenoninteracting LDA DOS was used to obtain the Green func-tion of the system through a Hilbert transformation. Further-more, the screened Coulomb interaction parametersU andJwere calculated by constrained LDA.33

Quite generally, this scheme needs to be improved in tworespects:sid instead of the LDA DOS an LDA Hamiltonianwith a few, relevant orbitals should be used to calculate theGreen function, andsii d a feedback from DMFT to LDAshould be incorporated. Both of these problems are solved bythe approach proposed in this work. In this method theHamiltonian matrix in the WF basis setHnn8

WFskd is calculatedfrom the LDA Hamiltonian via the projection procedures6dand s23d.

In DMFT the lattice problem becomes an effective single-site problem which has to be solved self-consistently for the

matrix self-energyS and the local matrix Green function

Gnn8s«d =1

VBZE dksfs« + Ef

sNdd1 − HWFskd − SWFs«dg−1dnn8.

s37d

The integration can actually be restricted to the irreduciblepart of the Brillouin ZonesBZd via the analytical tetrahedronmethod38 with a subsequent symmetrization of the matrixGreen function. The chemical potentialEf

sNd is determined bythe number of electrons on theN interacting orbitals ofinterest.35

The DMFT is based on the fact that in thed=` limit theself-energy is local.36,37 Its matrix Snn8s«d sn, n8 are WFindicesd is defined in WF basiss3d. If the trial functions inEqs. s6d and s28d are chosen as the basis functions of theirreducible representation of the point symmetry group ofsome particular real system,39 the matrix Green functions27dand hence the matrix self-energys3d can be made diagonal40

in the n index for on-site matrix elements. However, thesymmetry may be so low that the matrix self-energy isstrongly off-diagonal. In this case one needs to employ ageneral formalism which is dealing with off-diagonal matrixGreen functions and self-energies.34 In the present work weinvestigate systemssSrVO3 and V2O3d where the symmetryis high enough to result in a diagonal matrix Green functionand self-energy. For this reason we use a DMFT computa-tional scheme10 assuming diagonal matrices.

The DMFT single-site problem may be formulated as aself-consistent single-impurity Anderson model.6 The corre-

sponding local one-particle matrix Green functionG can bewritten as a functional integral6 involving an action wherethe Hamiltonian of the correlation problem under investiga-tion, including the interaction term with the Hubbard inter-action and Hund’s rule couplings, enters.10 The action de-

pends on the bath matrix Green functionG through

sGd−1 = sGd−1 + S. s38d

To solve the functional integral of the effective single impu-rity Anderson problem, various methods can be used: quan-tum Monte CarlosQMCd, numerical renormalization groupsNRGd, exact diagonalizationsEDd, noncrossing approxima-tion sNCAd, etc. sfor a brief overview of the methods seeRef. 10d.

D. Converting back to the full-orbital Hilbert space

The matrix self-energySWFs«d obtained as a solution ofDMFT in Sec. II C is defined in the WF basis sets3d. Inorder to compute the interacting Green function in the full-orbital Hilbert spaces26d ands27d one has to convert it backto the full-orbitalsLMTOd basis set. This can be easily doneby using the linear-expansion form of the WFs in terms ofthe full-orbital basis sets15d and s17d,

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Smnk s«d = kfm

k uSs«dufnkl

= on

kfmk uWnklSnn8s«dkWn8kufn

kl

= on

bmnk Snn8s«dbnn8

k* . s39d

Here we use the local form of the matrix self-energy as ob-tained in DMFT, but the formalism can be easily generalized.In the following we refer to this matrix self-energy as the“full-orbital” self-energy.

The matrix elements of the self-energySmnk s«d fEq. s39dg

together with the noninteracting Hamiltonian matrixHmnk al-

low one to calculate the matrix Green functionGmnk s«d fEq.

s27dg and thus the full-orbital interacting Green functionGsr ,r 8 ,«d fEq. s26dg. Gsr ,r 8 ,«d contains the full informa-tion about the system, and various electronic, magnetic, andspectral properties can be obtained from it. In Sec. III B weuse the full-orbital interacting Green function computedwithin DMFT sQMCd to calculate the photoemission andx-ray-absorption spectra for the strongly correlated vana-dium oxides SrVO3 and V2O3, and to compare them withnew bulk-sensitive experimental spectra.

One can also calculate the charge-density distributionmodified by correlation effects via

rsr d = −1

pImE

−`

EF

d«Gsr ,r ,«d. s40d

With this rsr d one can recalculate the LDA potentialswhichis a functional of electron densityd. From the full-orbitalGreen functions26d one can recalculate new WFss28d ands29d which together with the new LDA Hamiltonian allowsone to obtain new parameters for the noninteracting Hamil-tonians24d. With Eq. s25d one can then compute a new Cou-lomb interaction parameterU. The set of new LDA potential,WFs, and Coulomb interaction parameters calculated from

the interacting Green functions26d defines the input for thenext iteration step and hence closes the self-consistency loopin the proposed computation scheme. For the feedback fromDMFT to LDA in the particular case of the LMTO method11

one needs a set of moments for the partial densities of statesMql

smd for every atomic sphereq and the orbital momentl,41 inorder to calculate the new charge density and hence the newLDA potential:

Mqlsmd =E

−`

EF

d««mNqls«d,

Nqls«d = −1

pNImo

kom

Gqlm,qlmk s«d. s41d

E. Summary of the WF scheme

For clarity, in Fig. 1 the essential steps of the WF schemepresented here are summarized. There are four intercon-nected parts in this scheme:sid the basis of WF,sii d thematrix elements of the Hamiltonian and the self-energy inthe WF basis,siii d the Coulomb interaction between elec-trons on the WFs, andsivd the projection into the few-orbitalbasis and back transformation to the full-orbital basis whichretains the information about all orbitals. First the matrixelements of the noninteracting Hamiltonian in reciprocalspaceHnn8

WFskd fEq. s23dg and the interaction termSnn8sivdfEq. s3dg are written in the basis of explicitly defined WFsuWnkl fEq. s15dg. The actual correlation problem, defined bythe sum of these two termss37d, is then solved within theLDA1DMFT sQMCd approach.10 The local self-energySnn8sivd obtained thereby is then transformed back from theWannier basis to the full-orbital spacessee Sec. II Dd. Fur-thermore, with the full-orbital self-energySmnsivd fEq. s39dgthe full-orbital Green functionGmnsivd fEqs.s26d and s27dgfor the correlated electrons is calculated by ak integration

FIG. 1. Scheme of theab initio fully self-consistent LDA1DMFT scheme based on theWF formalismssee textd.

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over the Brillouin zone.38 The Green function allows one todetermine a new charge-density distributions40d fin theLMTO case see Eq.s41dg and a new set of WFsfvia Eqs.s28d ands29dg. This is used to construct a new LDA potentialand new noninteracting Hamiltonian. Together with the newCoulomb interaction parametersUn fEq. s25dg they serve asthe input for the next iteration of calculations, thus complet-ing the self-consistency loop. It should be stressed that in thisscheme self-consistency involves not only the self-energybut also the basis of WFs in which it is defined, the chargedensity and LDA potential used for constructing the nonin-teracting Hamiltonian, and the interaction strength betweenelectrons in the WFs.

After convergency is reached the maximum entropymethodsMEMd sRef. 46d can be used to obtain imaginarypart of Green functionGnn8sivd. Then, using the Kramers-Kronig transformation, the Green function on the real axisGnn8svd is computed. With that one can construct the WFsbasis self-energySnn8svd ssee Appendix Bd. By the backtransformation ofSnn8svd with a subsequentk integrationthe full-orbital Green functionGmnsvd is obtained, whichnow contains information not only about the states for whichcorrelations are considered, but also for all other orbitals ofthe system.Gmnsvd is used to obtain orbitally resolved spec-tral functions. This allows one, for example, to investigatethe influence of the correlated orbitalsse.g., partially filledt2g orbitals of V in SrVO3 and V2O3d on other orbitalssoxy-gen 2p and vanadiumeg orbitalsd. It also makes possible thecomputation of spectral functions in a wide energy regionand not only in the vicinity of the Fermi level.

III. RESULTS AND DISCUSSION

In our earlier LDA1DMFT calculation scheme the spe-cific properties of a material entered only via the LDA partialdensities of statessDOSsd for the orbitals of interest. Thisprocedure is valid for systems where the bands of interest aredegeneratesas in cubic crystals; see Appendix Ad. For morecomplicated systems with lower symmetry one needs to em-ploy the scheme proposed in this work, where the noninter-

acting HamiltonianHWFskd fEq. s23dg sprojected on WFsddescribing theN orbitals under consideration is used for thecalculation of Gnn8s«d fEq. s37dg within DMFT ssee Sec.II Cd.

In this section we present results of LDA1DMFT calcu-

lations using the projected HamiltonianHWFskd. The schemewas applied to SrVO3 which has a cubic perovskite crystalstructure, and to the trigonally distorted V2O3 sboth in theinsulating and metallic phased. The results are compared withprevious LDA1DMFT calculations where the Hilbert trans-formation of the LDA DOSs was used for SrVO3 sRef. 42dssee Sec. III Ad. A similar scheme was used by Liebsch43 tocompute surface properties of SrVO3. One should note thatthe DOSs used for DMFT calculations of V2O3 were ob-tained by the TB-LMTO-ASA code version 47,11 in contrastto Ref. 44 where the DOSs were calculated by the ASWmethod.45 However, the DOSs obtained in both methods arevery similar and do not produce much different LDA1DMFT results.

The full-orbital calculation scheme proposed in this workallows one to answer an important question: how do Cou-lomb correlations between some orbitals affect the other or-bitals, and in particular, how does the interaction between thepartially filled t2g orbitals of the V3d shell influence the filledO-2p and the unoccupied V−3dseg

sd bands? To answer thesequestions the matrix of the self-energySnn8s«d s3d was con-verted back to the full-orbital basis set from the WF basis,and the full-orbital interacting Green functions27d was cal-culated. From that, total and partial DOSs were computed toproduce theoretical spectra for comparison with the experi-mental photoemission and x-ray-absorption data.

In this work, QMC simulations8 were used to solve theeffective single impurity Anderson problem in the DMFTloop. The result of the DMFTsQMCd calculation is the self-energy on the imaginary energy axisSnn8sivd. To find thefull-orbital self-energySmn

k s«d on the real energy axis onehas to perform an analytical continuation. The procedure isdescribed in Appendix B. A self-consistent computation ofthe charge density as described in section II D was not yetperformed. Investigations of correlation effects on the unoc-cupied cubiceg

s states for both SrVO3 and V2O3 are also amatter of future calculations.

A. Comparison of DMFT results obtained by Hilberttransformation of the LDA DOS and the

projected Hamiltonian HWF„k…

The cubic perovskite SrVO3 can serve as a test case forour method. For the cubicOh point symmetry group, thethree 3d orbitalsxy, xz, yz transform according to the triplydegeneratet2g irreducible representation of this group. Hencethe corresponding Green function and self-energy matriceshave diagonal form with equal diagonal elements. As shownin Appendix A, the results of the Hilbert transformation us-ing LDA DOS must coincide with the results of the proce-dure of integration over the Brillouin zone with the projected333 t2g Hamiltonian.

We use the same LDA DOS and interaction parametersfor SrVO3as in our previous papers.42 The V−3dst2gd statesform a partially filled band in SrVO3. All t2g orbitalssxy,xz,zyd are equivalent due to the cubic symmetry of thelattice, so only the results for one of thet2g orbitals are pre-sented in Fig. 2. In this figure V−3dst2gd spectral functions,calculated using the LDA DOS and the projected Hamil-tonian are shown. It is easy to see that both results are almostidentical. The small differences between these two curves aredue to the MEMsRef. 46d used for the calculation of thespectral function on the real energy axis from the DMFTGreen function.

We also applied our Hamiltonain scheme to a more com-plicated system with lower symmetry where one can expectdeviations from the results obtained using the LDA DOS. Weperformed LDA1DMFT sQMCd calculations for the insulat-ing and metallic phases of V2O3 with the projected 333 t2gHamiltonian and severalU valuessunless stated otherwisethe values ofU are the same as in Ref. 44d. Here and in thefollowing we use an exchange Coulomb parameterJ=0.93 eV for V2O3.

44 In contrast to SrVO3, V2O3 has a non-

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cubic trigonal symmetrysspace groupR3cd. Therefore weuse the basis functions of the trigonalD3d point group in-stead of the cubicOh group. In this basis set the V-3d shell issplit into two groups of bands. The threet2g bands are lo-cated around the Fermi energy, the two degenerateeg

s bandsare at higher energies. Interesting to us are the partially filledt2g bands which are formed by onea1g and two degenerateeg

p

orbitals.In Fig. 3, we present V−3dst2gd spectral functions result-

ing from DOSsLMTO and ASWd and Hamiltonian calcula-tions atU=4.5 eV, averaged over the threet2g bands. Thespectrum computed with ASW input is taken from Ref. 44.

As mentioned before, the differences between the curvescalculated with ASW and LMTO DOS as input are small;there are only minor deviations in the peak height. In thecomparison between the DOS calculations and the Hamil-tonian calculation, it is interesting to note that for these av-eraged spectra, the differences are relatively small. We findthe typical four peak structureswith lower Hubbard, quasi-particle peak, and double-peaked upper Hubbard band splitby Hund’s rule couplingd for all three calculations. Only the

position and the height of the quasiparticle peak are quitedifferent for the Hamiltonian calculation, indicating a moreinsulating solution than for the DOS calculations at the sameU value. The similarity between the results for the DOS andHamiltonian calculations is not surprising, because the trigo-nal distortion in V2O3 is relatively small. Therefore the cen-ter of gravity and the bandwidth of the threet2g bands do notchange much and the DOS calculations can still produceaccurate results.

The differences between DOS and Hamilton calculationsare more pronounced when one compares the band-resolvedt2g spectra. In Figs. 4 and 5, thea1g and eg

p spectra for theinsulating and metallic crystal structure are presented forU=4.5, 5.0, and 5.5 eV.

Here the curves calculated by different schemes are dis-tinctively different. This is especially clear from the upperpart of Figs. 4 and 5 corresponding to thea1g orbital of theV−3dst2gd subband. This is due to the fact that the hybrid-ization effects are better accounted for by the projectedHamiltonian, which includes not only intraband hoppings butalso interband on-site and intersite hoppings. The latter ef-

FIG. 2. Comparison of V-3dst2gd spectralfunctions for SrVO3 calculated via LDA1DMFTsQMCd using: LDA DOS—light line; projectedHamiltonian—black line. The Fermi level corre-sponds to zero.

FIG. 3. Comparison of V-3dst2gd spectralfunctions for V2O3 in the metallic phase calcu-lated via LDA1DMFT sQMCd using: LDA DOSsLMTOd—light line sRef. 44d; LDA DOSsASWd—dashed line; projected Hamiltonian—black line. The Fermi level corresponds to zero.

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fectively increases the bandwidth of thea1g orbital. The cor-responding spectral functions have a different peak structureand different intensities.

Another conclusion from this series of figures is that theresults obtained with the Hamiltonian procedure show amore insulating behavior for the same values ofU than theresults with the scheme using the LDA DOS.

B. Results of full-orbital calculations: Comparison ofcalculated spectra with experimental photoemission

and x-ray-absorption data

The full-orbital calculations scheme proposed in this workproduces an interacting Green functionGsr ,r8 ,«d fEq. s26dg.The knowledge of the full Green function allows us to cal-culate the spectral function not only for the states with cor-relations fV−3dst2gd orbitals in the case of SrVO3 andV2O3g, but also their effect on the lower-lying occupied oxy-gen 2p states and the higher-lying unoccupied V-3dseg

sdstates, thus facilitating a comparison of the calculated andexperimental spectra in a wide energy region.

Valence-band photoemission spectroscopysPESd usinghigh photon energies and O-1s→2p x-ray-absorption spec-troscopysXASd were performed on the beamline BL25SU atthe SPring-8 synchrotron radiation facility. The PES spectrawere taken using a GAMMADATA-SCIENTA SES-200electron energy analyzer and XAS spectra were measured bythe total electron yield. The overall energy resolution was setto 0.2 eV. The pressure in the analyzer chamber was about4310−8Pa. Single crystals of SrVO3 were cooled to 20 Kand clean surfaces were obtained by fracturingin situ for thePES spectra, and by scrapingin situ for the much more bulksensitive XAS spectra. Well-oriented single-crystalline V2O3samples were cleavedin situ at a temperature near the metal-insulator transition, yielding clean specular surfaces. The sur-face cleanliness was confirmed before and after the spectralrun.

We start with the results for the SrVO3 system. In Fig. 6,the self-energy on the real energy axis for SrVO3 calculatedvia Eq. sB2d is shown. Them* /m ratio is calculated viam* /m=1/h1−f] ReSsvd /]vgj. This self-energy was used forthe calculation of total and partial DOSs in the full-orbitalHilbert spacessee Figs. 7 and 8d. In Fig. 7, the total spectral

FIG. 4. Comparison of V-3dst2gd spectralfunctions for V2O3 in the insulating phase calcu-lated via LDA1DMFT sQMCd using: LDADOS—light line; projected Hamiltonian—blackline. Upper figures—a1g; lower figures—eg

p orbit-als. The Fermi level corresponds to zero.

FIG. 5. Comparison of V-3dst2gd spectralfunctions for V2O3 in the metal phase calculatedvia LDA1DMFT sQMCd using: LDA DOS—light line; projected Hamiltonian—black line.Upper figures—a1g; lower figures—eg

p orbitals.The Fermi level corresponds to zero.

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functions of SrVO3 are presented. Differences between LDAand full-orbital LDA1DMFT sQMCd spectral functionsmainly occur near the Fermi level. In particular, the LDAspectral function has a more pronounced quasiparticle peaknear 1 eV. The DOS calculated using the full-orbital self-energyssee Fig. 8d has a three-peak structure: lower Hubbardband from −4 to −2ssuppressed by oxygen statesd, quasipar-ticle peak and upper Hubbard band located at about 3 eV.The origin of the upper Hubbard band becomes clear fromFig. 8. One can see that this broad peak is the sum of V-3dseg

sd and V-3dsegsd states. It should be noted that correla-

tions on the V-3dsegsd orbitals were not explicitly included

here. We find, however, that the V-3dsegsd states are slightly

modified by the full-orbital self-energy due to the hybridiza-tion with the correlated V-3dseg

sd orbitals. The question ofhow correlations affect the position and width of V-3dseg

sdstates directly can only be answered by employing the full3d-shell 535 projected Hamiltonian.

Introducing correlations betweent2g states changes sig-nificantly the total and partialsis not shown here directly butcan be conjectured from Fig. 9d LDA DOSs of SrVO3. Themain modification is a transfer of spectral weight from theenergy region near the Fermi level to the lower and upperHubbard bands, and the reduction of the weight of the qua-

siparticle peak. Our calculations suggest a strongly corre-lated but still metallic ground state for SrVO3.

To compare our results with the experimental PES wecalculated a weighted sum of V-3d and O-2p spectral func-tions according to the photoemission cross section ratio 3:1,corresponding to an experimental photon energy 900 eV. Thetheoretical spectra were multiplied with the Fermi functioncorresponding to 20 K and broadened with a 0.2-eV Gauss-ian to take into account the instrumental resolution. In Fig. 9one can see that the full-orbital spectra obtained in this waydescribe not only the quasiparticle peak, but also the peak at26 eV and the shoulder at23.5 eV of the PES spectra.Since previous LDA1DMFT sQMCd results were takinginto account only thet2g states they were not able to describePES spectra below22 eV. It is interesting to note that pre-vious experimental studies did not find any states at23.5eV; only the new spectra reported here show this feature. Theagreement between the experimental and theoretical spectrais expected to improve wheneg states are included in thecorrelation problem and when also the charge and LDA po-tential self-consistency is fully implemented.

The influence of correlation effects on the electronicstructure will be more pronounced for systems close to theMott insulator transition. For this purpose we compare thetotal and partial DOS’s for V2O3 in the insulating and me-

FIG. 6. Self-energy on real energy axis forSrVO3. Real part—full black line; imaginarypart—full light line. The Fermi level correspondsto zero.

FIG. 7. Comparison of total spectral functionsof SrVO3 calculated via: LDA—full light line;using the full-orbital self-energy from LDA1DMFT sQMCd—dashed black line. The Fermilevel corresponds to zero.

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tallic phases calculated via LDA and full-orbitalLDA1DMFT.48,49 Since the V-3dst2gd orbitals in trigonalV2O3 split into nonequivalenta1g and eg

p states, the self-energy will be different for these orbitalsssee Fig. 10d.

In Fig. 11 one can see that the introduction of correlationeffects changes the total DOS drastically. Whereas the LDADOS is metallic for both metallic and insulating phases, theLDA1DMFT spectra clearly show the metal-insulator tran-sition. Although the change near the Fermi level is the mostprominent, modifications between23 to 27 eV are alsoseen. Moreover, there are also changes in theeg

s states whichare caused by the hybridization with the correlatedt2g bands,and are not due to a direct calculation of correlations in theeg

s states. In Fig. 12 the totald-state DOS above the Fermienergy, consisting oft2g andeg

s states, is shown.In Fig. 13 a comparison of experimental PES and calcu-

lated LDA1DMFT sQMCd st2g only and full-orbitald spectrais presented. The full-orbital spectrum is a weighted sum of

the calculated V-3d and O-2p spectral functions, accordingto the photoemission cross-section ratio 2:1, correspondingto the experimental photon energy of 500 eV. The theoreticalspectra were multiplied with the Fermi function and broad-ened with a Gaussian of 0.2 eV.47 Below the Fermi energythe LDA1DMFT sQMCd spectral functionssfor t2g only andthe full-orbital schemed agree quite well with the PES spec-tra. However, the theory curves do not yet describe fine de-tails. For example, the PES spectrum of the insulator phaseshows two definite slope changes, at roughly20.6 and21.5eV, producing a rather flat topped spectrum centered on21eV, whereas the theory curves show a single peak centeredon 21 eV.

A comparison of the calculated LDA1DMFT sQMCdfull-orbital spectral functions and the O 1s→2p XAS spec-trum is shown on Fig. 14. The full-orbital spectraswhich arethe partial O-2p spectra for the unoccupied V-3d statesd are

FIG. 8. Partial V-3d spectral functions forSrVO3 calculated using the full-orbital self-energy from LDA1DMFT sQMCd. Total d—fulllight line; t2g—full black line; eg—dashed blackline. The Fermi level corresponds to zero.

FIG. 9. Comparison of photoemission spectra of SrVO3 withspectral functions calculated via LDA1DMFT sQMCd: taking intoaccount onlyt2g—light line; using full-orbital self-energy—blackline. Theoretical spectra are multplied with the Fermi function cor-responding to 20 K and broadened with a 0.2-eV Gaussian to ac-count for the instrumental resolution. Intensities are normalized toyield the same height for the quasiparticle peaks. The Fermi levelcorresponds to zero.

FIG. 10. Self-energy on real energy axis for V2O3. Real part ofthe self-energy from LDA1DMFT sQMCd—full black line; imagi-nary part—full light line. The Fermi level corresponds to zero.

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found to agree with the experimental spectrum as to the rela-tive weights of the part nearest the Fermi energy and theparts at higher energy. This is due to the inclusion of O-2pand V-3dseg

sd states in the calculations. The strong hybrid-ization of the O-2p with the d states is described more cor-rectly in the full-orbital calculations and the inclusion ofeg

s

statesseven without correlationsd significantly changes theV-3d spectrum in the energy region above the Fermi edge.The inclusion of correlations in theeg

s states within a 535projected V-3d Hamiltonian is expected to add spectralweight in the energy interval between 3 and 5 eVssee Fig.14d.

IV. CONCLUSION

We formulated a fullyab initio and self-consistent com-putational scheme based on Wannier functionssWFsd for thecalculation of the electronic structure of strongly correlatedcompounds. The WF formalism provides an explicit strategyfor the construction of the matrix elements of the requiredoperatorssHamiltonian, self-energy, etc.d, both in full-orbitaland few-orbital bases, in real and reciprocal representations.The WF formalism allows one to project these operatorsfrom the full-orbital space to the few-orbital space and back,keeping the complete information about all orbitals. Theseprojections are the essential ingredients of the computationalscheme presented here. The self-consistency involves notonly the self-energy but also the WF basis itself, the chargedensity with the LDA potential and the interaction strengthparameters between the electrons on the WFs. The spectraobtained thereby found to be in good agreement with newbulk-sensitive experimental data.

In the present work we did not yet employ the full schemessee Fig. 1d to investigate spectral functions of strongly cor-

FIG. 11. Comparison of total spectral functions of V2O3 calcu-lated via: LDA—full light line; using full-orbital self-energy fromLDA1DMFT sQMCd—dashed black line. The Fermi level corre-sponds to zero.

FIG. 12. Partial V-3d spectral functions for V2O3 calculatedusing full-orbital self-energy from LDA1DMFT sQMCd. Totald—full light line; t2g—full black line; eg—dashed black line. TheFermi level corresponds to zero.

FIG. 13. Comparison of photoemission spectra of V2O3 withspectral functions calculated via LDA1DMFT sQMCd: taking intoaccount onlyt2g—light line; using full-orbital self-energy—blackline. Theoretical spectra are multiplied with the Fermi function andbroadened with a 0.2-eV Gaussian to simulate instrumental resolu-tion. Intensities are normalized for peaks situated around21 eV.The Fermi level corresponds to zero.

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related systems but used only few-orbital Hamiltonians witht2g symmetry. Clearly, fulld-shell DMFT sQMCd results in-cluding eg states will provide additional information aboutcorrelation effects in the system. Such studies, as well asconstrained LDA calculations in the WF basis and investiga-tions of the feedback from the DMFT to the LDA part, are inprogress now.

ACKNOWLEDGMENT

This work was supported by Russian Basic ResearchFoundation Grant Nos. RFFI-GFEN-03-02-39024-a, RFFI-04-02-16096, by the joint UrO-SO Project No. N22, by theU.S. NSF Grant No. DMR-03-02825, and at POSTECH byKOSEF through eSSC and by the Deutsche Forschungsge-meinschaft through Sonderforschungsbereich 484. I.N. ac-knowledges support from the Grant of President of RussianFederation for young scientistssGrant No. MK-95.2003.02d,Dynasty Foundation, and International Center for Fundamen-tal Physics in Moscow program for young scientists 2004d,Russian Science Support Foundation program for youngPh.D. of Russian Academy of Science 2004. We would liketo acknowledge John von Neumann Institut für Computing,Jülich for providing and support of computing time andJapan Synchrotron Radiation Research Institute for supportof soft x-ray-absorption and photoemission studies onSPring-8.

APPENDIX A: HILBERT TRANSFORMATION

For cubic systems the matrix of the self-energysfor ex-ample, fort2g orbitalsd is diagonal and all diagonal elementsare equal. Therefore the calculation of the Green functionwithin DMFT by integration of the Hamiltonian over the BZis equivalent to the Hilbert transformation of the noninteract-ing sLDA d DOS N0sed:

Ssvd =1ssvd 0 … 0

0 ssvd … 0

A A � A0 0 … ssvd

2⇒ Gsvd

=EIBZ

fv − Ssvd − HLDA0 skdg−1dk

= G0fv − ssvdg

=E N0s«dv − ssvd − e

de. sA1d

APPENDIX B: SELF-ENERGY ON THE REALENERGY AXIS

DMFT produces Green functions and self-energies on theimaginary energy axis. With the maximum entropy method,46

the spectral function on the real energy axis is calculated,which yields the imaginary part of the Green function. Inorder to obtain the self-energy on the real energy axis, thefull complex Green functionGs«d is calculated using itsimaginary part obtained by MEM:

Gs«d = −1

pE

−`

` Im Gs«8dd«8

« − «8 + ih. sB1d

The self-energy for real energies is then calculated by solv-ing the following two equations with the two variablesReSs«d and ImSs«d:

Re,ImhGs«dj = Re,ImHEBZ

f« − Hskd − Ss«dg−1dkJ ,

sB2d

where

Ss«d = ReSs«d + i Im Ss«d. sB3d

APPENDIX C: NONORTHOGONAL BASIS SET

Equations11d is valid only for orthogonal LMTO orbitals.In the case of general nonorthogonal LMTOssor any otheratomic-type orbital basis setd, an orthogonalization procedure

can be used by defining an orthogonal HamiltonianH and

the corresponding eigenvectorsC for the nonorthogonalHamiltonianH and overlapping matrixO:

FIG. 14. Comparison of O-1s→2p x-ray-absorption spectros-copy sXASd spectra of V2O3 with spectral functions calculated viaLDA1DMFT sQMCd using full-orbital self-energy. Theoreticalspectra are multiplied with the Fermi function and broadened with a0.2-eV Gaussian to simulate instrumental resolution. Intensities ofthe theory curves are normalized to the correct number of V-3delectrons in the unit cellseight electronsd. The experimental curve isnormalized to the same peak height as the full orbital curve. TheFermi level corresponds to zero.

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H = O−1/2HO−1/2, sC1d

C = O1/2C.

This orthogonalization is equivalent to the basis set transfor-mation:

ufnl = on8

Onn8−1/2ufn8l. sC2d

Then for the nonorthogonal LMTO the trial functionufnl hasto be replaced byufnl and the eigenvectors with coefficientscji

k by cjik .

1G. H. Wannier, Phys. Rev.52, 191 s1937d.2N. Marzari and D. Vanderbilt, Phys. Rev. B56, 12 847s1997d; N.

Marzari, I. Souza, and D. Vanderbilt, Psi-k Newsletter57, 112s2003d fpsi-k.dl.ac.uk/newsletters/News_57/Highlight_57.pdfg.

3Wei Ku, H. Rosner, W. E. Pickett, and R. T. Scalettar, Phys. Rev.Lett. 89, 167204s2002d.

4D. Vollhardt, inCorrelated Electron Systems, edited by V. J. Em-ery sWorld Scientific, Singapore, 1993d, p. 57.

5Th. Pruschke, M. Jarrell, and J. K. Freericks, Adv. Phys.44, 187s1995d.

6A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev.Mod. Phys.68, 13 s1996d.

7G. Kotliar and D. Vollhardt, Phys. Today57 s3d, 53 s2004d.8J. E. Hirsch and R. M. Fye, Phys. Rev. Lett.56, 2521s1986d; M.

Jarrell,ibid. 69, 168s1992d; M. Rozenberg, X. Y. Zhang, and G.Kotliar, ibid. 69, 1236s1992d; A. Georges and W. Krauth,ibid.69, 1240 s1992d; M. Jarrell, in Numerical Methods for LatticeQuantum Many-Body Problems, edited by D. Scalapino,sAddison-Wesley, Reading, MA, 1997d.

9V. I. Anisimov, A. I. Poteryaev, M. A. Korotin, A. O. Anokhin,and G. Kotliar, J. Phys.: Condens. Matter9, 7359s1997d; A. I.Lichtenstein and M. I. Katsnelson, Phys. Rev. B57, 6884s1998d.

10K. Held, I. A. Nekrasov, N. Blümer, V. I. Anisimov, and D.Vollhardt, Int. J. Mod. Phys. B15, 2611s2001d; K. Held, I. A.Nekrasov, G. Keller, V. Eyert, N. Blümer, A. K. McMahan, R.T. Scalettar, T. Pruschke, V. I. Anisimov, and D. Vollhardt, inQuantum Simulations of Complex Many-Body Systems: FromTheory to Algorithms, edited by J. Grotendorst, D. Marks, andA. Muramatsu, NIC Series No.10sNIC Directors, Forschun-szentrum Jülich, 2002d, p. 175–209; K. Held, I. A. Nekrasov, G.Keller, V. Eyert, N. Blumer, A. K. McMahan, R. T. Scalettar,Th. Pruschke, V. I. Anisimov, and D. Vollhardt, Psi-k Newsletter56, 65 s2003d fpsi-k.dl.ac.uk/newsletters/News_56/Highlight_56.pdfg; A. I. Lichtenstein, M. I. Katsnelson, G. Kot-liar, in Electron Correlations and Materials Properties, 2nd ed.,edited by A. Gonis, Nicholis Kioussis, and Mikael CiftansKlu-wer Academic/Plenum, New York, 2002d, p. 428.

11O. K. Andersen, Phys. Rev. B12, 3060s1975d; O. Gunnarsson,O. Jepsen, and O. K. Andersen,ibid. 27, 7144s1983d.

12L. F. Mattheiss and D. R. Hamann, Phys. Rev. B33, 823s1986d.13The self-energy is, in general, a frequency-dependent, nonlocal

quantity which describes the effect of interactions on the nonin-teracting Hamiltonian. Here it enters into the theory as the dif-ference between the inverse matrix Green functions of the bathand the impurity, Eq.s38d.

14General properties of the Wannier functions have been investi-gated in Refs. 15–20 and 27–29.

15G. F. Koster, Phys. Rev.89, 67 s1953d.16G. Parzen, Phys. Rev.89, 237 s1953d.17P. O. Löwdin, J. Chem. Phys.18, 365 s1950d.18G. G. Hall, Philos. Mag.43, 338 s1952d.19R. I. Blount, in Solid State Physics, edited by F. Seitz and D.

Turnbull sAcademic, New York, 1962d, Vol. 13, p. 305.20W. Kohn, Phys. Rev.115, 809 s1959d.21W. Kohn and J. R. Onffroy, Phys. Rev. B8, 2485s1973d.22W. Kohn, Phys. Rev. B7, 4388s1973d.23W. Kohn, Phys. Rev. B10, 382 s1974d.24J. J. Gehr and W. Kohn, Phys. Rev. B10, 448 s1974d.25M. R. Geller and W. Kohn, Phys. Rev. B48, 14 085s1993d.26P. W. Anderson, Phys. Rev. Lett.21, 13 s1968d.27J. Des Cloizeaux, Phys. Rev.129, 554 s1963d.28J. Des Cloizeaux, Phys. Rev.135, A685 s1964d.29J. Des Cloizeaux, Phys. Rev.135, A698 s1964d.30I. Schnell, G. Czycholl, and R. C. Albers, Phys. Rev. B68,

245102s2003d.31O. K. Andersen and T. Saha-Dasgupta, Phys. Rev. B62, R16 219

s2000d.32Z. V. Pchelkina, D. E. Kondakov, I. V. Leonov, V. I. Anisimov, V.

A. Gavrichkov, and S. G. Ovchinnikov, Phys. Met. Metallogr.sto be publishedd.

33O. Gunnarsson, O. K. Andersen, O. Jepsen, and J. Zaanen, Phys.Rev. B 39, 1708 s1989d; V. I. Anisimov and O. Gunnarsson,ibid. 43, 7570s1991d.

34E. Pavarini, S. Biermann, A. Poteryaev, A. I. Lichtenstein, A.Georges, and O. K. Andersen, Phys. Rev. Lett.92, 176403s2004d.

35In the calculations performed in the present paper we assumed theposition of the Fermi level to be unchanged when going fromLDA to DMFT sQMCd. It was found in previous investigationssRef. 10d that this is a good approximation for vanadates.

36W. Metzner and D. Vollhardt, Phys. Rev. Lett.62, 324 s1989d.37E. Müller-Hartmann, Z. Phys. B: Condens. Matter74, 507

s1989d; , Z. Phys. B: Condens. Matter76, 211 s1989d.38Ph. Lambin and J. P. Vigneron, Phys. Rev. B29, 3430s1984d; O.

Jepsen and O. K. Andersen, Solid State Commun.9, 1763s1971d; P. E. Blöchl, O. Jepsen, and O. K. Andersen, Phys. Rev.B 49, 16 223s1994d.

39For example, thet2g and eg orbitals of thed shell sN=5d in thecase of theOh point group of symmetry for a cubic lattice.

40Or nearly diagonal, if the symmetry is lowered by a relativelysmall distortion of the crystal structure.

41H. L. Skriver,The LMTO Method, Springer Series on Solid StateScience. No. 41sSpringer, NewYork, 1984d.

42I. A. Nekrasov, G. Keller, D. E. Kondakov, A. V. Kozhevnikov,Th. Pruschke, K. Held, D. Vollhardt, and V. I. Anisimov, cond-

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mat/0211508; A. Sekiyama, H. Fujiwara, S. Imada, H. Eisaki, S.I. Uchida, K. Takegahara, H. Harima, Y. Saitoh, and S. Sugacond-mat/0206471; A. Sekiyama, H. Fujiwara, S. Imada, S.Suga, H. Eisaki, S. I. Uchida, K. Takegahara, H. Harima, Y.Saitoh, I. A. Nekrasov, G. Keller, D. E. Kondakov, A. V.Kozhevnikov, Th. Pruschke, K. Held, D. Vollhardt, and V. I.Anisimov, Phys. Rev. Lett.93, 156402s2004d.

43A. Liebsch, Phys. Rev. Lett.90, 096401s2003d.44K. Held, G. Keller, V. Eyert, V. I. Anisimov, and D. Vollhardt,

Phys. Rev. Lett.86, 5345s2001d.45V. Eyert, Int. J. Quantum Chem.77, 1007s2000d sSpecial Issue:

Electronic Structure of Materials, edited by M. Defranceschid.

46For a review on the maximum entropy method, see M. Jarrell andJ. E. Gubernatis, Phys. Rep.269, 133 s1996d.

47S.-K. Mo, J. D. Denlinger, H.-D. Kim, J.-H. Park, J. W. Allen, A.Sekiyama, A. Yamasaki, K. Kadono, S. Suga, Y. Saitoh, T.Muro, P. Metcalf, G. Keller, K. Held, V. Eyert, V. I. Anisimov,and D. Vollhardt, Phys. Rev. Lett.90, 186403s2004d.

48G. Keller, K. Held, V. Eyert, D. Vollhardt, and V. I. Anisimov,cond-mat/0402133.

49For illustration purposes we chose the most insulatingsU=5.5 eVd for insulating V2O3 and the most metallic solutionssU=4.5d for metallic V2O3.

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