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4

A linear response approach to the calculation of the effective interaction parameters

in the LDA+U method

Matteo Cococcioni∗ and Stefano de GironcoliSISSA – Scuola Internazionale Superiore di Studi Avanzati and

INFM-DEMOCRITOS National Simulation Center,via Beirut 2-4, I-34014 Trieste, Italy

(Dated: February 2, 2008)

In this work we reexamine the LDA+U method of Anisimov and coworkers in the framework ofa plane-wave pseudopotential approach. A simplified rotational-invariant formulation is adopted.The calculation of the Hubbard U entering the expression of the functional is discussed and alinear response approach is proposed that is internally consistent with the chosen definition forthe occupation matrix of the relevant localized orbitals. In this way we obtain a scheme whosefunctionality should not depend strongly on the particular implementation of the model in ab-initiocalculations. We demonstrate the accuracy of the method, computing structural and electronicproperties of a few systems including transition and rare-earth correlated metals, transition metalmonoxides and iron-silicate.

INTRODUCTION

The description and understanding of electronic prop-erties of strongly correlated materials is a very impor-tant and long standing problem for ab-initio calculations.Widely used approximations for the exchange and corre-lation energy in density functional theory (DFT), mainlybased on parametrization of (nearly) homogeneous elec-tron gas, miss important features of their physical behav-ior. For instance both local spin-density approximation(LSDA) and spin-polarized generalized gradient approx-imations (σ−GGA), in their several flavors, fail in pre-dicting the insulating behavior of many simple transitionmetal oxides (TMO), not only by severely underestimat-ing their electronic band gap but, in most cases, produc-ing a qualitatively wrong metallic ground state.

TMOs have represented for long time the most notablefailure of DFT. When the high-Tc superconductors en-tered the scene (their parent materials are also stronglycorrelated systems) the quest for new approaches thatcould describe accurately these systems by first princi-ples received new impulse, and in the last fifteen yearsmany methods were proposed in this direction. Amongthese, LDA+U approach, first introduced by Anisimovand coworkers [1, 2, 3], has allowed to study a large vari-ety of strongly correlated compounds with considerableimprovement with respect to LSDA or σ−GGA results.The successes of the method have led to further develop-ments during the last decade which have produced verysophisticated theoretical approaches[4] and efficient nu-merical techniques.

The formal expression of LDA+U energy functionalis adapted from model hamiltonians (Hubbard model

∗present address: Massachusetts Institute of Technology, 77 Mas-

sachusetts avenue, Cambridge MA, 02139 USA.

in particular) that represent the ”natural” theoreticalframework to deal with strongly correlated materials. Asin these models, a small number of localized orbitals isselected and the electronic correlation associated to themis treated in a special way. The obtained results stronglydepend on the definition of the localized orbitals and onthe choice of the interaction parameters used in the cal-culation, that should be determined in an internally con-sistent with. This is not always done and a widespreadbut, in our opinion, unsatisfactory attitude is to deter-mine the value of the electronic couplings by seeking agood agreement of the calculated properties with the ex-perimental results in a semiempirical way.

In this work a critical reexamination of the LDA+Uapproach is proposed, which starting from the formula-tion of Anisimov and coworkers [1, 2, 3], and its furtherimprovements [5, 6, 7], develops a simpler approxima-tion. This is, in our opinion, the ”minimal” extensionof the usual approximate DFT (LDA or GGA) schemesneeded when atomic-like features are persistent in thesolid environment.

In the central part of this work we describe a method,based on a linear response approach, to calculate inan internally consistent way—without aprioristic as-sumption about screening and/or basis set employed inthe calculation—the interaction parameters entering theLDA+U functional used. In this context our plane-wavepseudopotential (PWPP) implementation of the LDA+Uapproach is presented and discussed in some details. Westress however that the proposed method is basis-set in-dependent.

Our methodology is then applied to the study of theelectronic properties of some real materials, chosen asrepresentative of ”normal” (bulk iron) and correlated(bulk cerium) metals, as well as a few examples ofstrongly correlated systems (iron oxide, nickel oxide andfayalite).

2

STANDARD LDA+U IMPLEMENTATION:

In order to account explicitly for the on-site Coulombinteraction responsible for the correlation gap in Mott in-sulators and not treated faithfully within LDA, Anisimovand coworkers [1, 2, 3] correct the standard functionaladding an on-site Hubbard-like interaction, EHub:

ELDA+U [n(r)] = ELDA[n(r)] +

EHub[{nIσm }] − Edc[{n

Iσ}] (1)

where n(r) is the electronic density, and nIσm are the

atomic-orbital occupations for the atom I experiencingthe ”Hubbard” term. The last term in the above equa-tion is then subtracted in order to avoid double count-ing of the interactions contained both in EHub and, insome average way, in ELDA. In this term the total, spin-projected, occupation of the localized manifold is used:nIσ =

∑

m nIσm .

In its original definition the functional defined in Eq.1 was not invariant under rotation of the atomic-orbitalbasis set used to define the occupancies nIσ

m . A rotation-ally invariant formulation has then been introduced [5, 6]where the orbital dependence of EHub is borrowed fromatomic Hartree-Fock with renormalized slater integrals:

EHub[{nImm′}] =

1

2

∑

{m},σ,I

{〈m,m′′|Vee|m′,m′′′〉nIσ

mm′nI−σm′′m′′′

+(〈m,m′′|Vee|m′,m′′′〉

−〈m,m′′|Vee|m′′′,m′〉)nIσ

mm′nIσm′′m′′′} (2)

with

〈m,m′′|Vee|m′,m′′′〉 =

2l∑

k=0

ak(m,m′,m′′,m′′′)F k

where l is the angular moment of the localized (d or f)electrons and

ak(m,m′,m′′,m′′′) =4π

2k + 1

k∑

q=−k

〈lm|Ykq|lm′〉〈lm′′|Y ∗

kq |lm′′′〉.

The double-counting term Edc is given by:

Edc[{nI}] =

∑

I

U

2nI(nI − 1)

−∑

I

J

2[nI↑(nI↑ − 1) + nI↓(nI↓ − 1)]. (3)

The radial Slater integrals F k are the parameters of themodel (F 0,F 2 and F 4 for d electrons, while also F 6 mustbe specified for f states) and are usually re-expressedin terms of only two parameters, U and J , describingscreened on-site Coulomb and exchange interaction,

U =1

(2l + 1)2

∑

m,m′

〈m,m′|V ee|m,m′〉 = F 0 (4)

J =1

2l(2l+ 1)

∑

m 6=m′,m′

〈m,m′|V ee|m′,m〉 =F 2 + F 4

14,

by assuming atomic values for F 4/F 2 and F 6/F 4 ratios.To obtain U and J , Anisimov and coworkers [3, 8] pro-

pose to perform LMTO calculations in supercells in whichthe occupation of the localized orbitals of one atom isconstrained. The localized orbitals of all atoms in thesupercell are decoupled from the remainder of the basisset. This makes the treatment of the local orbitals anatomic-like problem—making it easy to fix their occupa-tion numbers—and allows to use Janak theorem [9] toidentify the shift in the corresponding eigenvalue withthe second-order derivative of the LDA total energy withrespect to orbital occupation. It has however the effect ofleaving a rather artificial system to perform the screen-ing, in particular when it is not completely intra-atomic.In elemental metallic Iron, for instance, Anisimov andGunnarsson [8] showed that only half of the screeningcharge is contained in the Wigner-Seitz cell. This fact,in addition to a sizable error due to the Atomic SphereApproximation used [8], could be at the origin of the se-vere overestimation of the computed on-site coulomb in-teraction with respect to estimates based on comparisonof spectroscopic data and model calculations[10, 11].

BASIS SET INDEPENDENT FORMULATION OF

LDA+U METHOD

Some aspects of currently used LDA+U formulation,and in particular of the determination of the parametersentering the model, have been so far tied to the LMTOapproach. This is not a very pleasant situation and someefforts have been done recently [7, 12] to reformulate themethod for different basis sets. Here we want to elabo-rate further on these attempts and provide an internallyconsistent, basis-set independent, method for the calcu-lation of the needed parameters.

Localized orbital occupations

In order to fully define how the approach works the firstthing to do is to select the degrees of freedom on which“Hubbard U” will operate and define the correspondingoccupation matrix, nIσ

mm′ . Although it is usually straight-forward to identify in a given system the atomic levels tobe treated in a special way (the d electrons in transitionmetals and the f ones in the rare earths and actinidesseries) there is no unique or rigorous way to define occu-pation of localized atomic levels in a multi-atom system.Equally legitimate choices for nIσ

mm′ are i) projections onnormalized atomic orbitals, or ii) projections on Wannierfunctions whenever the relevant orbitals give raise to iso-lated band manifolds, or iii) Mulliken population or iv)

3

integrated values in (spherical) regions around the atomsof the angular-momentum-decomposed charge densities.Taking into account the arbitrariness in the definition ofnIσ

mm′ no particular significance should be attached to anyof them (or other that could be introduced) and the use-fulness and reliability of an approximate DFT+U method(aDFT+U), and of its more recent and involved evolu-tions like the aDFT+DMFT method, should be judgedfrom its ability to provide a correct physical picture ofthe systems under study irrespective of the details of theformulation, once all ingredients entering the calculationare determined consistently.

All above mentioned definitions for the occupation ma-trices can be put in the generic form

nIσmm′ =

∑

k,v

fσkv〈ψ

σkv|P

Imm′ |ψσ

kv〉 (5)

where ψσkv is the valence electronic wavefunction corre-

sponding to the state (kv) with spin σ of the system andfσkv is the corresponding occupation number. The P I

mm′ ’sare generalized projection operators on the localized-electron manifold that satisfy the following properties:

∑

m′ P Imm′P I

m′m′′ = P Imm′′ ; P I

mm′ = (P Im′m)†;

P Imm′P I

m′′m′′′ = 0 when m′ 6= m′′. (6)

In particular P I =∑

m P Imm is the projector on the com-

plete manifold of localized states associated with atom atsite I and therefore

nI =∑

σ

∑

k,v

fσkv〈ψ

σkv|P

I |ψσkv〉 =

∑

σ,m

nIσmm (7)

is the total localized-states occupation for site I. Orthog-onality of projectors on different sites is not assumed.

In the applications discussed in this work we will definelocalized-level occupation matrices projecting on atomicpseudo-wavefunctions. The needed projector operatorsare therefore simply

P Imm′ = |ϕI

m〉〈ϕIm′ | (8)

where |ϕIm〉 is the valence atomic orbital with angular

momentum component |lm〉 of the atom sitting at site I(the same wavefunctions are used for both spins). Sincewe will be using ultrasoft pseudopotentials to describevalence-core interaction, all scalar products between crys-tal and atomic pseudo-wavefunctions are intended to in-clude the usual S matrix describing orthogonality in pres-ence of charge augmentation [13].

As already mentioned, other choices could be used aswell and different definitions for the occupation matriceswill require, in general, different values of the parameterentering the aDFT+U functional, as it has been pointedout recently also by Pickett et al. [7] where, for instance,the value of Hubbard U in FeO shifts from 4.6 to 7.8 eVwhen atomic d-orbitals for Fe2+ ionic configuration are

used instead of those of the neutral atom. In an earlystudy [19] the U parameter in La2CuO4 varies from 6.8to 7.7 eV upon variation of the atomic sphere radius em-ployed in the LMTO calculation. As pointed out in theseworks it is not fruitful to compare numerical values ofU obtained by different methods but rather comparisonshould be made between results of complete calculations.

A simplified rotationally invariant scheme and the

meaning of U

In order to simplify our analysis and gaining a moretransparent physical interpretation of the ”+U” correc-tion to standard aDFT functionals we concentrate on themain effect associated to on-site Coulomb repulsion. Wethus neglect the important but somehow secondary ef-fects associated to non sphericity of the electronic inter-action and the proper treatment of magnetic interaction,that in the currently used rotational invariant method isdealt with assuming a screened Hartree-Fock form. [5].

We are therefore going to assume in the following thatparameter J describing these effects can be set to zero, oralternatively that its effects can be mimicked redefiningthe U parameter as Ueff = U − J , a practice that havebeen sometime used in the literature [14].

The Hubbard correction to the energy functional, Eqs.2 and 3, greatly simplifies and reads:

EU [{nIσmm′}] = EHub[{n

Imm′}] − Edc[{n

I}]

=U

2

∑

I

∑

m,σ

{nIσmm −

∑

m′

nIσmm′nIσ

m′m}

=U

2

∑

I,σ

Tr[nIσ(1 − nIσ)]. (9)

Choosing for the localized orbitals the representationthat diagonalizes the occupation matrices

nIσ

vIσi = λIσ

i vIσi (10)

with 0 ≤ λIσi ≤ 1, the energy correction becomes

EU [{nIσmm′}] =

U

2

∑

I,σ

∑

i

λIσi (1 − λIσ

i ). (11)

from where it appears clearly that the energy correctionintroduces a penalty, tuned by the value of the U param-eter, for partial occupation of the localized orbitals andthus favors disproportionation in fully occupied (λ ≈ 1)or completely empty (λ ≈ 0) orbitals. This is the ba-sic physical effect built in the aDFT+U functional andits meaning can be traced back to known deficiencies ofLDA or GGA for atomic systems.

An atom in contact with a reservoir of electrons canexchange integer numbers of particles with its environ-ment. The intermediate situation with fractional number

4

LDAexactLDA+U correction

Tot

al e

nerg

y

Number of electronsN-1

E(N+2)

E(N+1)

E(N)

E(N-1)

N N+1 N+2

FIG. 1: Sketch of the total energy profile as a function ofnumber of electrons in a generic atomic system in contactwith a reservoir. The bottom curve is simply the differencebetween the other two (the LDA energy and the ”exact” resultfor an open system).

of electrons in this open atomic system is described notby a pure state wave function, but rather by a statisti-cal mixture so that, for instance, the total energy of asystem with N + ω electrons (where N is an integer and0 ≤ ω ≤ 1) is given by:

En = (1 − ω)EN + ωEN+1 (12)

where EN and EN+1 are the energies of the system cor-responding to states with N and N + 1 particles respec-tively, while ω represents the statistical weight of thestate with N +1 electrons. The total energy of this openatomic system is thus represented by a series of straight-line segments joining states corresponding to integer oc-cupations of the atomic orbitals as depicted in fig. 1. Theslope of the energy vs electron-number curve is insteadpiece-wise constant, with discontinuity for integer num-ber of electrons, and corresponds to the electron affinity(ionization potential) of the N (N+1) electron system.

Exact DFT correctly reproduce this behavior [15, 16],which is instead not well described by the LDA or GGAapproach, which produces total energy with unphysicalcurvatures for non integer occupation and spurious min-ima in correspondence of fractional occupation of the or-bital of the atomic system. This leads to serious problemswhen one consider the dissociation limit of hetero-polarmolecules or an open-shell atom in front of a metallic sur-face [15, 16], and is at the heart of the LDA/GGA failurein the description of strongly correlated systems[1]. Theunphysical curvature is associated basically to the incor-rect treatment by LDA or GGA of the self-interactionof the partially occupied Kohn-Sham orbital that gives a

non-linear contribution to the total energy with respectto orbital occupation (with mainly a quadratic term com-ing from the Hartree energy not canceled properly in theexchange-correlation term).

Nevertheless, it is well known [17] that total energy dif-ferences between different states can be reproduced quiteaccurately by the LDA (or GGA) approach, if the oc-cupation of the orbitals is constrained to assume integervalues. As an alternative, we can recover the physical sit-uation (an approximately piece-wise linear total energycurve) by adding a correction to the LDA total energywhich vanishes for integer number of electrons and elim-inates the curvature of the LDA energy profile in everyinterval with fractional occupation (bottom curve of fig.1). But this is exactly the kind of correction that is pro-vided by eq. 9 if the numerical value of the parameter Uis set equal to the curvature of the LDA (GGA) energyprofile.

This clarifies the meaning of the interaction parame-ter U as the (unphysical) curvature of the LDA energyas a function of N which is associated with the spuri-ous self-interaction of the fractional electron injected intothe system. From this analysis it is clear that the nu-merical value of U will depend in general not only, asnoted in the preceding section, on the definition adoptedfor the occupation matrices but also on the particularapproximate exchange-correlation functional to be cor-rected, and should vanish if the exact DFT functionalwere used.

The situation is of course more complicated in solidswhere fractional occupations of the atomic orbitals canoccur due to hybridization of the localized atomic-likeorbitals with the crystal environment and the unphysi-cal part of the curvature has to be extracted from thetotal LDA/GGA energy, which contains also hybridiza-tion effects. In the next section this problem is discussedand a linear response approach to evaluate Hubbard Uis proposed.

Internally consistent calculation of U

Following previous seminal works [8, 18, 19] we com-pute U by means of constrained-density-functional cal-culations [20]. What we need is the total energy as afunction of the localized-level occupations of the ”Hub-bard” sites:

E[{qI}] = minn(r),αI

{

E[n(r)] +∑

I

αI(nI − qI)

}

, (13)

where the constraints on the site occupations, nI ’s fromEq. 7, are applied employing the Lagrange multipliers,αI ’s. From this dependence we can compute numeri-cally the curvature of the total energy with respect to

5

the variation, around the unconstrained values {n(0)I } ,

of the occupation of one isolated site. A supercell ap-proach is adopted in which occupation of one representa-tive site in a sufficiently large supercell is changed leav-ing unchanged all other site occupations. This curvaturecontains the energy cost associated to the localization ofan electron on the chosen site including all screening ef-fects from the crystal environment, but it is not yet theHubbard U we want to compute. In fact, had we com-puted the same quantity from the total energy of the non-interacting Kohn-Sham problem associated to the samesystem,

EKS [{qI}] = minn(r),αI

{

EKS [n(r)] +∑

I

αKSI (nI − qI)

}

,

(14)we would have obtained a non vanishing results as wellbecause by varying the site occupation a rehybridizationof the localized orbitals with the other degrees of freedomis induced that gives rise to a non-linear change in theenergy of the system. This curvature coming from re-hybridization, originating from the non-interacting bandstructure but present also in the interacting case, hasclearly nothing to do with the Hubbard U of the inter-acting system and should be subtracted from the totalcurvature:

U =∂2E[{qI}]

∂q2I−∂2EKS [{qI}]

∂q2I. (15)

In Ref. [8] Anisimov and Gunnarsson, in order to avoiddealing with the above mentioned non-interacting curva-ture, exploited the peculiarities of the LMTO method,used in their calculation, and decoupled the chosen lo-calized orbitals from the remainder of the crystal by sup-pressing in the LMTO hamiltonian the correspondinghopping terms. This reduced the problem to the one ofan isolated atom embedded in an artificially disconnectedcharge background. Thanks to Janak theorem [9] thesecond order derivative of the total energy in Eq. 13 canthen be recast as a first order derivative of the localized-level eigenvalue. In our approach the role played in Refs.[3, 8] by the eigenvalue of the artificially isolated atomis taken by the Lagrange multiplier, used to enforce leveloccupation[20]:

∂E[{qJ}]

∂qI= −αI ,

∂2E[{qJ}]

∂q2I= −

∂αI

∂qI, (16)

∂EKS[{qJ}]

∂qI= −αKS

I ,∂2EKS [{qJ}]

∂q2I= −

∂αKSI

∂qI.

At variance with the original method of Refs. [3, 8], inour approach we need to compute and subtract the band-structure contribution, −∂αKS

I /∂qI , from the total cur-vature, but, in return, Hubbard U is computed in exactlythe same system to which it is going to be applied and

the screening from the environment is more realisticallyincluded. The present method was inspired by the linearresponse scheme proposed by Pickett and coworkers [7]where however the role of the non-interacting curvaturewas not appreciated.

In actual calculations constraining the localized orbitaloccupations is not very practical and it is easier to pass,via a Legendre transform, to a representation where theindependent variables are the αI ’s

E[{αI}] = minn(r)

{

E[n(r)] +∑

I

αI nI

}

, (17)

EKS [{αKSI }] = min

n(r)

{

EKS [n(r)] +∑

I

αKSI nI

}

.

Variation of these functionals with respect to wavefunc-tions shows that the effect of the αI ’s is to add to thesingle particle potential a term, ∆V =

∑

I αIPI (or

∆V =∑

I αKSI P I for the non-interacting case), where

localized potential shifts of strength αI (αKSI ) are ap-

plied to the localized levels associated to site I.It is useful to introduce the (interacting and non-

interacting) density response functions of the system withrespect to these localized perturbations:

χIJ =∂2E

∂αI∂αJ

=∂nI

∂αJ

, (18)

χ0IJ =

∂2EKS

∂αKSI ∂αKS

J

=∂nI

∂αKSJ

.

Using this response-function language, the effective in-teraction parameter U associated to site I can be recastas:

U = +∂αKS

I

∂qI−∂αI

∂qI=

(

χ−10 − χ−1

)

II(19)

that is reminiscent of the well known random-phase ap-proximation [21] in linear response theory giving the in-teracting density response in terms of the non-interactingone and the Coulomb kernel. A similar result is obtainedwithin DFT linear response [22] where the interactionkernel also contains an exchange-correlation part.

The response functions, Eq. 18, needed in Eq. 19are computed taking numerical derivatives. We per-form a well converged LDA calculation for the uncon-strained system (αI = 0 for all sites in the supercell)and—starting from its self-consistent potential—we addsmall (positive and negative) potential shifts on eachnon equivalent ”Hubbard” site J and compute the vari-ation of the occupations, nI ’s, for all sites in the super-cell in two ways: i) letting the Kohn-Sham potential ofthe system readjust self-consistently to optimally screenthe localized perturbation, ∆V = αJPJ , and ii) with-out allowing this screening. This latter result is nothingbut the variation computed from the first iteration in

6

the self-consistent cycle leading eventually to the former(screened) results. The site-occupation derivatives calcu-lated according to i) and ii) give the matrices χIJ andχ0

IJ respectively.

Further considerations

Before moving to examine some specific examples inthe next section, let’s end the present one discussing afew additional technical points.

As mentioned earlier, Hubbard U is computed, ideally,from variation of the site occupation of a single site in aninfinite crystal and in practice adopting a supercell ap-proach where periodically repeated sites are perturbedcoherently. In order to speed up the convergence of thecomputed U with supercell size it may result useful toenforce explicitly charge neutrality for the perturbation,that is to be introduced in the response functions, thusenhancing its local character and reduce the interactionwith its periodic images. In this procedure we introducein the response functions, χ and χ0, —in addition tothe degrees of freedom associated to the localized sites—also a ”delocalized background” representing all otherdegrees of freedom in the system. This translates in onemore column and row in the response matrices, whose el-ements are determined imposing overall charge neutralityof the perturbed system for all localized perturbations,(∑

I χIJ = 0,∑

I χ0IJ = 0, ∀J) and absence of any

charge density variation upon perturbing the system witha constant potential (

∑

J χIJ = 0,∑

J χ0IJ = 0, ∀I).

From a mathematical point of view both χ and χ0 acquirea null eigenvalue, corresponding to a constant potentialshift, and the needed inversions in Eq. 19 must be takenwith care. It can be shown that their singularities cancelout when computing the difference χ−1

0 − χ−1 and thefinal result is well defined. We stress that in the limitof infinitely large supercell the coupling with the back-ground gives no contribution to the computed U , but wefound that this limit is approached more rapidly whenthis additional degrees of freedom is included.

In the same spirit we found that the spatial localityof the response matrices can be rather different from theone of their inverse and a supercell sufficient to decou-ple the periodically repeated response may be too smallto describe correctly the inverse in eq. 19. As a prac-tical procedure, therefore, after evaluating the responsefunction matrices in a given supercell, we extrapolate theresult to much larger supercells assuming that the mostimportant matrix elements in χ0 and χ involve the atomsin the few nearest coordination-shells accessible in theoriginal supercell. The corresponding matrix elements ofthe larger supercell are filled with the values extractedfrom the smaller one while all other, more distant, inter-actions are neglected. Again, when a sufficiently largesupercell to extract the matrix elements of the response

functions is considered, the effect of this extrapolationvanishes, but, as we will see in the following, this schemecapture a large fraction of the system-size dependence ofthe calculated U and it may allow to reach more rapidlythe converged result.

As a final remark we notice that the electronic struc-ture of a system described within the LDA+U approachmay largely differ from the one obtained within the LDAused to compute U . In a more refined approach onemight seek internal consistency between the band struc-ture used in the calculation of U and the one obtainedusing it. We have not addressed this issue here, but onecan imagine performing the same type of analysis leadingto the U determination for a functional already contain-ing an LDA+U correction. The computed U would inthat case be a correction to be added to the original Uand internal consistency would be reached when the cor-rection vanishes.

EXAMPLES

Metals: Iron and Cerium

In their seminal paper Anisimov and Gunnarsson [8]computed the effective on site Coulomb interaction be-tween the localized electrons in metallic Fe and Ce. ForCe the calculated Coulomb interaction was about 6 eVin good agreement with empirical and experimental esti-mates ranging from 5 to 7 eV [20, 23, 24], while the resultfor Fe (also about 6 eV) was surprisingly high since U wasexpected to be in the range of 1-2 eV for elemental tran-sition metals, with the exception of Ni [10, 11]. Let usapply the present approach to these two system, startingwith Iron.

In its ground state elemental Iron has a ferromag-netic (FM) spin arrangement and a body-centered cu-bic (BCC) structure. Gradient corrected exchange-correlation functional are needed in order to stabilize theexperimental structure as compared with non-magneticface-centered cubic (FCC) structure preferred by LDA.The Perdew-Burke-Ernzherof (PBE) [25] GGA func-tional was employed here. Iron ions were representedby ultrasoft pseudopotential and kinetic energy cutoffsof 35 Ry and 420 Ry were adopted for wavefunction andcharge density Fourier expansion. Brillouin Zone inte-grations where performed using 8×8×8 Monkhorst andPack special point grids [26] using Methfessel and Paxtonsmearing technique [27] with a smearing width of 0.005Ry in order to smooth the Fermi distribution.

The calculation of the effective Hubbard U followed theprocedure outlined in preceding section: a supercell wasselected containing a number of inequivalent Iron atoms;then, after a well converged self-consistent calculation,we applied to one of these atoms small, positive and neg-ative, potential shifts, ∆V = αPd (with α = ±0.2-0.5

7

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0number of atoms per unit cell

0.0

1.0

2.0

Hub

bard

U (

eV)

first n.n.second n.n.third n.n.

FIG. 2: Calculated Hubbard U in metallic Iron for differentsupercells. Lines connect results from the cell-extrapolationprocedure described in the text and different symbols corre-spond to inclusion of screening contributions up to the indi-cated shell of neighbors of the perturbed atom.

eV), where Pd is the projector on the localized d electronof the selected atom. From the variation of the d-leveloccupations of all Iron atoms in the cell one column ofχ and χ0 response functions was extracted and all othermatrix elements were reconstructed by symmetry, includ-ing the background as explained previously. Hubbard Uwas then calculated from Eq. 19.

In order to describe response for an isolated pertur-bation four supercells were considered: i) a simple cu-bic (SC) cell containing two inequivalent iron atoms, theperturbed atom and one of its nearest neighbors; ii)a 2×2×2 BCC supercell containing 8 inequivalent Ironatoms, 4 in the nearest-neighbor shell of the perturbedatom and 3 belonging to the second shell of neighbors;iii) a 2×2×2 SC cell containing 16 atoms, including alsosome third nearest-neighbor atom and iv) a 4×4×4 BBCsupercell containing 64 inequivalent Iron atoms; we usedthis largest cell just to extrapolate the results from thesmaller ones.

The convergence properties of the effective U of bulkiron with the size of the used supercell are shown in fig. 2.

The Hubbard U obtained from the SC 2-atom cell,once inserted in the 64-atom supercell, captures most ofthe effective interaction; second nearest neighbors shellbrings some significant corrections to the final extrap-olated result, while third nearest neighbor shell has asmaller effect. We believe that contributions from fur-ther neighbor rapidly vanish and that an accurate valueof U can be extracted from the SC supercell containing16 atoms. The extrapolation from this cell to larger cells

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0Lattice spacing (a.u.)

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

Hub

bard

U (

eV)

FIG. 3: Lattice spacing dependence of the calculated HubbardU parameter for Iron.

brings only minor variations which are within the finitenumerical accuracy that we estimate within a fraction ofan eV. From this analysis our estimate for the HubbardU in elemental Iron at the experimental lattice parameteris therefore 2.2 ± 0.2 eV.

This results is in very good agreement with the exper-imental estimates [10, 11], but disagrees with Anisimovand Gunnarsson result [8]. We can only recall here thatmany technical details differ in the two approaches. Inparticular i) in the original approach the perturbed atomis disconnected from the rest of the crystal by removingall hopping terms, thus leaving a rather unphysical envi-ronment to perform the screening, while in our approachthe actual system is allowed to screen the perturbation,ii) the Atomic Sphere Approximation (ASA) was em-ployed in the original LMTO calculation while no shapeapproximation is made in our case.

In order to further test our approach on this element weinvestigate the dependence of the Hubbard parameter oncrystal structure. The dependence of the calculated in-teraction parameter on the lattice spacing of the unit cellis shown in fig. 3 where a marked increase of the HubbardU can be observed when the lattice parameter is squeezedbelow its experimental value. Despite this may appearcounterintuitive, as correlation effects are expected to be-come less important when atoms gets closer, one shouldactually compare the increasing value of U with the muchsteeper increase of bandwidth when reducing the inter-atomic distance. Upon increase of the lattice parameterthe Hubbard parameter should approach the atomic limitthat can be estimated from all-electron atomic calcula-tions where the local neutrality of the metallic system ismaintained: U = E(d8s0) +E(d6s2)− 2×E(d7s1) = 2.1eV, in reasonable agreement with the results of fig. 3.

8

TABLE I: Comparison between the calculated lattice con-stant (a0), bulk modulus (B0) and magnetic moment (µ0)within several approximate DFT schemes and experimentalresults quoted from [28].

a0 (a.u.) B0 (Mbar) µ0 (µB)

Expt. 5.42 1.68 2.22

LSDA 5.22 2.33 2.10

σ-GGA 5.42 1.45 2.46

LDA+U 5.53 2.12 2.60

LDA+U (AMF) 5.34 1.53 2.00

Using the calculated volume dependent Hubbard U pa-rameter we have studied the effect of the LDA+U ap-proximation on the structural properties of Iron. Resultsare reported in table I where they are compared with re-sults obtained within LSDA and σ-GGA(PBE) approxi-mation and with experimental data. From these data itappears that, although simple σ-GGA(PBE) approxima-tion appears to be superior in this case, LDA+U providesa reasonable description of the data, of the same qual-ity as LSDA. In weakly correlated metals it has beensuggested [29] that a formulation of LDA+U in termsof occupancy fluctuations around the uniform occupancyof the localized level could be more appropriate than thestandard one. This ”around mean field” (AMF) LDA+Uapproach has been revisited recently [30, 31] and an ”op-timally mixed” scheme has also been proposed [31]. Wedon’t want to enter in this discussion here, but we men-tion that by following the AMF recipe the description ofstructural and magnetic properties of metallic Iron im-proves as it is evident from table I.

Using the calculated value of U we have obtained theelectronic structure of Iron at the experimental latticespacing. The theoretical band structure obtained usingthe AMF version of LDA+U is reported in fig. 4 togetherwith some experimental results [32]. The overall agree-ment is rather good for this scheme. However, when usingthe standard LDA+U scheme a somehow worse agree-ment with experimental data was obtained, mainly dueto a rigid downward shift of the majority spin bands ofabout 1 eV. This is an indication that LDA+U approx-imation may still require some fine tuning in order todescribe accurately both strongly and weakly correlatedsystems [31].

Let us proceed to examine the Cerium case. Elemen-tal cerium presents a very interesting phase diagram witha peculiar isostructural α − γ phase transition betweena low volume (α) and a high volume (γ) phase, bothFCC. This phase transition has attracted much experi-mental and theoretical interest and in the last 20 years[33], many interpretations have been put forward to ex-plain its occurrence. It is clear now that standard LDA

Ene

rgy

[eV

]

-10

-5

0

5

H P Γ H N P N Γ

FIG. 4: Band structure of bulk iron obtained within the AMFLDA+U approach. Green lines are for minority spin states,black ones for majority spin levels. Photoemission resultsfrom [32] are also reported for comparison.

or GGA approximations do not describe the transitionand it appears that a treatment of the correlation at theDMFT level might be required [34], however a full un-derstanding of the nature of the transition is still underdebate [35]. Here, we do not want to address this deli-cate topics but we simply want to follow Anisimov andGunnarsson [8] by computing the Hubbard U parameterfor elemental cerium in the high volume γ phase.

The interaction of valence-electrons with Ce nuclei andits core electrons was described by a non-local ultrasoftpseudopotential [13] generated in the 5s25p65d14f1 elec-tronic configuration. Kinetic cutoffs of 30 Ry and 240 Rywere adopted for wavefunction and charge density Fourierexpansion. The LSDA approximation was adopted forthe exchange and correlation functional. Brillouin Zoneintegrations where performed using 8×8×8 Monkhorstand Pack special point grids [26] using Methfessel andPaxton smearing technique [27] with a smearing widthof 0.05 Ry.

To obtain the response to an isolated perturbation wehave perturbed a Cerium atom in three different cells: i)the fundamental face-centered cubic (FCC) cell contain-ing just one inequivalent atom, ii) a simple-cubic (SC)cell containing 4 atoms (giving access to the first nearest-neighbor response) and iii) a 2×2×2 FCC cell (8 in-equivalent atoms) including also the response of second-nearest neighbor atoms. The result of these calculationsand their extrapolation to very large SC cells is reportedin Fig. 5 where it can be seen that the converged value

9

0 50 100 150 200 250

number of atoms per unit cell

1

2

3

4

5H

ubba

rd U

(eV

)

on-sitefirst n.n.second n.n.

FIG. 5: Calculated Hubbard U in metallic Cerium fordifferent supercells. Lines connect results from the cell-extrapolation procedure and different symbols correspond toinclusion of screening contributions up to the indicated shellof neighbors of the perturbed atom.

for U approaches 4.5 eV.

The screening in metallic cerium is extremely localized,as can be seen from the fact that inclusion of the first-nearest neighbor response is all is needed to reach con-verged results. This is at variance with what we found inmetallic Iron where third nearest-neighbor response wasstill significant (see Fig. 2). The calculated value is notfar from the value (5-7 eV) expected from empirical andexperimental estimates [20, 23, 24], especially if we con-sider that the parameter U we compute plays the roleof U − J in the simplified rotational invariant LDA+Uscheme adopted [14].

As a check, we performed all-electron atomic calcula-tions for Ce+ ions where localized 4f electrons were pro-moted to more delocalized 6s or 5d states and obtainedU = E(f3s0) + E(f1s2) − 2 × E(f2s1) = 4.4 eV, orU = E(f2s0d1) + E(f0s2d1) − 2 × E(f1s1d1) = 6.4 eV,depending on the selected atomic configurations. Thisconfirms the correct order of magnitude of our calculatedvalue in the metal.

The present formulation is therefore able to providereasonable values for the on-site Coulomb parameterboth in Iron and Cerium, at variance with the originalscheme of ref. [8] where only the latter was satisfactorilydescribed. We believe that a proper description of theinteratomic screening, rather unphysical in the originalscheme where atoms were artificially disconnected fromthe environment, is important to obtain a correct valuefor Hubbard U parameter, especially in Iron where thisresponse is more long-ranged.

FIG. 6: The unit cell of FeO: blue spheres represent Oxygenions, red ones are Fe ions, with arrows showing the orienta-tion of their magnetic moments. Ferromagnetic (111) planesof iron ions alternate with opposite spins producing type IIantiferromagnetic order and rhombohedral symmetry.

Transition metal monoxides: FeO and NiO

The use of the LDA+U method for studying FeO ismainly motivated by the attempt to reproduce the ob-served insulating behavior. In fact, as for other tran-sition metal oxides (TMO), standard DFT methods, asLDA or GGA, produce an unphysical metallic characterdue to the fact that crystal field and electronic struc-ture effects are not sufficient in this case to open a gapin the three-fold minority-spin t2g levels that host oneelectron per Fe2+ atom. As already addressed in quiteabundant literature on TMO (and FeO in particular), abetter description of the electronic correlations is neces-sary to obtain the observed insulating behavior and thestructural properties of this compound at low pressure[36, 37, 38, 39]. The application of our approach to thismaterial will thus allow us to check its validity by com-parison of our results with the ones from experiments andother theoretical works.

The unit cell of this compound is of rock-salt type,with a rhombohedral symmetry introduced by a type IIantiferromagnetic (AF) order (see fig. 6) which sets inalong the [111] direction below a Neel temperature of 198K, at ambient pressure.

The calculations on this materials were all performedin the antiferromagnetic phase starting from the cubic(undistorted) unit cell of fig. 6 with the experimentallattice spacing. We used a 40 Ry energy cut off for the

10

0.0 50.0 100.0 150.0 200.0 250.0Number of Fe ions per unit cell

2.0

3.0

4.0

5.0H

ubba

rd U

(eV

)

C1C4C16

FIG. 7: Convergence of Hubbard U parameter of FeO withthe number of iron included in the supercell used in the ex-trapolation. Lines connect results including the screening con-tributions extracted from the indicated cell.

electronic wavefunctions (400 Ry for the charge densitydue to the use of ultrasoft pseudopotentials [13] both forFe and O) and a small smearing width of 0.005 Ry whichrequired a 4×4×4 k-points mesh.

To compute the Hubbard effective interactions, we per-formed GGA calculations with potential shifts on oneHubbard site in larger and larger unit cells, that wenamed C1, C4, and C16, containing 2, 8, and 32 ironions respectively, and extrapolated their results up toa supercell containing 256 magnetic ions (called C128).The result for the undistorted cubic cell at the exper-imental lattice spacing is reported in fig. 7. We canobserve that the effective interaction obtained from C4is already very well converged, when extrapolated to thelargest cell, with respect to inclusion of screening fromadditional shells of neighborers,

The final result for the Hubbard U is 4.3 eV which issmaller than most of the values obtained (or simply as-sumed) in other works [37, 38, 39]. If we use this valuein a LDA+U calculation we can obtain the observed in-sulating behavior as shown in the band structure plot offig. 8 where a comparison is made with GGA (metallic)results.

A gap opens around the Fermi level whose minimalwidth is about 2 eV. The band gap is direct and locatedat the Γ point. The corresponding transition, of 3d(Fe)-2p(O)→4s(Fe) character, should be quite weak due tothe vanishing weight of Iron s states at the bottom ofthe valence band (fig. 9, bottom picture). We can ex-pect that a stronger absorption line will appear insteadaround 2.6 eV due to the transition, of 3d(Fe)-2p(O) →3d(Fe) character, among two pronounced peaks of thedensity of states around the Fermi level. This picture

Ene

rgy

[eV

]

-10

-5

0

5

Γ L K T Γ X

Ene

rgy

[eV

]

-10

-5

0

5

Γ L K T Γ XLDA+U

GGA

FIG. 8: The band structure of FeO in the undistorted (cubic)AF configuration at the experimental lattice spacing obtainedwithin GGA (top panel) and LDA+U using the computedHubbard U of 4.3 eV (bottom panel). The zero of the energyis set at the top of the valence band.

is in very good agreement with experiments (and othertheoretical results [39, 40]) where a first weak absorp-tion is reported between 0.5 and 2 eV and a stronger lineappears around 2.4 eV [41]. The large mixing betweenmajority-spin Iron 3d states and the Oxygen 2p manifoldover a wide region of energy and the finite contributionof the Oxygen states at the top of the valence band—afeature not present within σ-GGA (see top panel in fig.9)— are also in good agreement with experiments, whichindicate for FeO a moderate charge transfer character ofthe insulating state.

Despite our U is smaller than the ones used in litera-ture, we find a good agreement of our results about theelectronic structure of the system with experiments andother theoretical works. These findings confirm the va-lidity of our internally consistent method to compute U .We now want to extend its application to the study of

11

Fe d states (majority spin)Fe d states (minority spin)Fe s statesO p states

-10.0 -5.0 0.0 5.0Energy (eV)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0D

OS

(st

ates

/eV

/cel

l)

-10.0 -5.0 0.0 5.0Energy (eV)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

DO

S (

stat

es/e

V/c

ell)

GGA

LDA+U

FIG. 9: Projected density of states of FeO in the undis-torted (cubic) AF configuration at the experimental latticespacing obtained within GGA (top panel) and LDA+U usingthe computed Hubbard U of 4.3 eV (bottom panel).

structural properties. This is indeed a very importanttest because a good ab-initio method should be able todescribe the true ground state of a system and providea complete description of both electronic and structuralproperties. Furthermore the plane-wave implementationwe use allows a straightforward calculation of Hellmann-Feynman forces and stresses, thus giving easily access toequilibrium crystal structure.

As observed in experiments [42], the cubic rock saltstructure of FeO shown in fig. 6 becomes unstable un-der a pressure of 16 GPa (at room temperature) towarda rhombohedral distortion. In the distorted phase theunit cell is elongated along the [111] direction with aconsequent shrinking of the interionic distances on the(111) planes. This transition is driven by the onset of theAFII magnetic order [42] (the Neel temperature reachesroom value at about 16 GPa) which imposes a rhombohe-dral symmetry even in the cubic phase. Upon increasingpressure above the threshold value the distortion of theunit cell is observed to increase producing more elongated

0 50 100 150 200 250Pressure (kbar)

52

54

56

58

60

62

64

Rho

mbo

hedr

al a

ngle

(de

gree

s)

GGAExpLDA+ULDA+U(BSP)

FIG. 10: The pressure dependence of the rhombohedral anglein FeO for the various approximations described in the text iscompared with experimental results. These latter results wereextracted extrapolating the data for the non stoichiometriccompound Fe1−xO up to the stoichiometric composition [42,43].

structures [42].We have computed the Hubbard U on a grid of possible

values for the rhombohedral distortion and cell parame-ter and then from the corresponding total energy calcu-lations we determined the rhombohedral distortion andthe enthalpy of the system as a function of the pressureup to 250 Kbar.

As evident from fig. 10, while GGA overestimates therhombohedral distortion and his pressure dependence,LDA+U method—in the standard electronic configura-tion examined so far–overcorrects the GGA results andintroduces even larger errors with respect to experimen-tal results. In fact not only we obtain a distortion withthe wrong sign (of compressive character along the [111]direction), but also the wrong pressure dependence. Thereason for this failure can be traced back to the differentoccupation of the orbitals around the gap/Fermi level inthe two cases. Even in the undistorted cell, the rhombo-hedral symmetry, induced by the antiferromagnetic or-der, lifts the degeneracy of the minority spin t2g statesof iron and split them in one state of A1g character—which is essentially the m=0 (z2) state along the [111]quantization axis—and two states of eg symmetry local-ized on the iron (111) planes. Within GGA, the Ironminority-spin 3d electrons partially occupy the two equiv-alent eg orbitals giving rise to two half filled bands and a(wrong) metallic state which is delocalized on the (111)plane. The system gains energy by filling the lowest halfof the eg states and tends to elongate in the [111] direc-tion, shrinking in the plane, because this increases theoverlap of the eg states and their bandwidth. WithinLDA+U, fractional occupation of orbitals is energeticallydisfavored and the system would like to have completelyfilled or empty 3d states. In the standard unit-cell con-sidered so far in the literature—and used by us in the

12

iron [111] plane

FIG. 11: Lattice distortion in the (111) iron planes used toinduce symmetry breaking in the electronic configuration ofFeO.

calculation above—this can be accomplished only by fill-ing the non-degenerate A1g level, corresponding to wave-functions elongated along [111], and pushing upward inenergy the in-plane eg states, leaving them empty. Asa consequence, the system tends to pull apart the ionson the same (111) plane, so that the bandwidth of thestate in the plane is reduced, and increases instead theinter-plane overlap of the A1g states. This simple picturegives an explanation of the fact that GGA overestimatesthe elongation of the unit cell in the [111] direction, aswell as the (wrong) compressive behavior of the standardLDA+U solution. We are thus left with the paradoxicalsituation that a correct pressure dependence of the struc-tural properties can be obtained from the wrong bandstructure and viceversa.

We have found that it is possible to solve this para-dox by allowing the possibility that the system partiallyoccupies, as within GGA, the eg levels, thus maintainingthe driving force for the right rhombohedral deformation,and still opens a gap, as in standard LDA+U, by someorbital ordering that breaks the equivalence of the ironions in the (111) plane. This possibility has been some-times proposed in literature [39, 44] but has never beenclearly addressed.

From a simple tight-binding picture one finds that theoptimal broken symmetry phase would be the one whereoccupied eg orbitals have the highest possible hoppingterm with unoccupied eg orbitals in nearest-neighboratoms in the plane, in order to maximize the kineticenergy gain coming from delocalization, and the low-est possible hopping term with neighboring occupied eg

orbitals, in order to minimize bandwidth that tends todestroy the insulating state. In bipartite lattice this issimply achieved by making occupied orbitals in nearest-neighbor sites orthogonal but, in the triangular lattice,formed by iron atoms in (111) planes, this is not exactly

FIG. 12: The projected density of states of FeO as obtainedin the ”standard” LDA+U ground state (top panel) and inthe proposed broken symmetry phase (bottom panel). On theright of each DOS is a picture of the corresponding occupiedFe-3d minority states.

possible, the system is topologically frustrated and somecompromise is necessary.

It is generally believed [45] that Heisenberg model inthe triangular lattice, to which our system resemble insome sense, displays a three-sublattice 120◦ Neel long-range order. We thus imposed a symmetry breaking tothe system where three nearest-neighbor atoms in the(111) plane were made inequivalent by slightly displac-ing them from the ideal positions in the way shown infig. 11. This induced the desired symmetry breaking ofthe electronic structure and opened a gap that was ro-bust and persisted when the atoms were brought backinto the ideal positions. We found, quite satisfactorily,that the new broken symmetry phase (BSP) correspondsto a lower energy minimum than the ”standard” LDA+Usolution and that therefore it is, to say the least, a moreconsistent description of the ground state of FeO. The onedepicted in fig. 11 is, of course, only one of three equiva-lent distortions we could have imposed to the electronicstructure of the system and three symmetry related BSPscould be defined. In the actual system an effective equiv-alence of the ions in the (111) planes is probably restoredby a (dynamical) switching among equivalent states butconsidering the atoms as strictly equivalent, as in thestandard solution, leads to incorrect results.

The comparison of the projected density of state in the

13

”standard” LDA+U solution and in the novel BSP phaseis shown in fig. 12 where also a pictorial representationof the occupied minority-spin orbitals in the two casesis shown. As we can observed, no remarkable qualita-tive difference in the DOS appears apart from the differ-ent ordering of the d states around the gap. In fact theminority-spin d electron is now accommodated on a statelying on the (111) plane (shown on the right panel) whilethe one with A1g (z2) character has been pushed abovethe energy gap. The gap width and the charge transfercharacter of the system do not change significantly andare still in very good agreement with the experiments.

We repeated the structural calculations (according tothe same procedure described above) in the BSP, andobtained the LDA+U (BSP) curve reported in fig. 10.The agreement with experiments is much improved withrespect to both GGA and LDA+U ”standard” groundstates. The mechanism leading to the pressure behaviorin BSP case is basically the same already producing thecorrect evolution of distortion in the GGA calculations.When the unit cell elongates along the cubic diagonalthe iron ions in the (111) plane get closer and the hop-ping between nearest-neighbor orbitals increased with aconsequent lowering of the electronic kinetic energy.

We therefore conclude that LDA+U, not only improvesthe description of the structural and electronic propertieswith respect to GGA, but that a close examination ofboth electronic and structural properties is in this casenecessary in order to describe the correct ground state ofthe system.

Another classical example of TMO we want to studyin order to test the present implementation of LDA+Uis Nickel Oxide. It is a very well studied material andthere is a good number of theoretical [12] and experi-mental works, including some photoemission experiments[46, 47], our results can be compared with. At variancewith FeO, no compositional instability is observed forNiO so that the stoichiometric compound is easy to studyand is much better characterized than iron oxide. It hascubic structure with the same AF spin arrangements ofrhombohedral symmetry as FeO, but does not show ten-dencies toward geometrical distortions of any kind and istherefore easier to study.

In this case we did not perform any structural relax-ation and calculated the value of U at the experimentallattice spacing for the cubic unit cell imposing the rhom-bohedral AF magnetic order which is the ground statespin arrangement for this compound. The GGA approx-imation (in the PBE prescription) was used in the calcu-lation. US pseudopotentials for Nickel and Oxygen (thesame as in FeO) were used with the same energy cutoffs(of 40 and 400 Ry respectively) for both the electronicwavefunctions and the charge density as for FeO and alsothe same 4×4×4 k-point grid for reciprocal space integra-tions.

In the calculation of the Hubbard U of NiO we did not

studied the convergence properties of U with system sizeas we did in FeO but, assuming a similar convergencealso in this case, we performed a constrained calculationonly in the C4 cell and then extrapolated the obtainedresult to the C128 supercell. The calculated value of theU parameter is 4.6 eV. This value is smaller than litera-ture values for the same parameter that are rather in therange of 7-8 eV [1], however it has been recently pointedout [12, 14] that in obtaining these values self-screeningof d electrons is neglected and that better agreement withexperimental results is obtained using an effective Hub-bard U of the order of 5-6 eV.

The magnetic moment of the Ni ions is correctly de-scribed within the present GGA+U approach which givesa value of 1.7 µB well within the experimental range ofvalues ranging from 1.64 and 1.9 µB [48, 49], better thanthe value of 1.55 µB obtained within GGA.

In fig. 13 and fig. 14 the band structure and atomic-state projected density of states of NiO obtained with thisvalue of U is shown, along with the results of standardGGA, and compared with the photoemission data in theΓX direction extracted from ref. [46, 47].

Despite the agreement with the experimental band-dispersion is not excellent—the valence band width issomehow overestimated by both GGA and GGA+Ucalculations—, GGA+U band structure reproduces wellsome features of the photoemmission spectrum for thiscompound and gives a much larger band gap than theone obtained within GGA approximation. A very im-portant feature to be noticed in the density of states re-ported in fig. 14 is the fact that GGA+U modifies quali-tatively the nature of the states at the top of the valenceband, and hence the nature of the band gap: in GGAapproximation the top of valence band is dominated byNickel d-states while in the GGA+U calculation the Oxy-gen p-states give the most important contribution. Inboth approaches the bottom of the conduction band ismainly Nickel d-like and therefore the predicted bandgap is primarily of charge-transfer type within GGA+U,in agreement with experimental and theoretical evidence[40, 50, 51], while it is wrongly described as of Mott-Hubbard type according GGA approximation.

Our GGA+U value for the optical gap is ≈ 2.7 eVaround the T point, smaller that commonly acceptedexperimental values that range from 3.7 to 4.3 eV[52, 53, 54, 55]. More recently however, a re-examination[56] of the best available optical absorption data [52]pointed out that optical absorption in NiO starts at pho-ton energy as low as 3.1 eV, not far from our theoreticalresult. Indeed, Bengone and coworkers [12] reported re-cently an LDA+U calculation in NiO where different em-pirical values of U were employed. When U = 5 eV wasused—a value close to our present first-principles result—, they obtained an optical gap of 2.8 eV, very close toour results, and an excellent agreement between the cal-culated and experimental [52] optical absorption spectra.

14

E

nerg

y [e

V]

-10

-5

0

5

Γ L K T Γ X

Ene

rgy

[eV

]

-10

-5

0

5

Γ L K T Γ X

LDA+U

GGA

FIG. 13: The band structure of NiO in the undistorted (cu-bic) AF configuration at the experimental lattice spacing ob-tained within GGA (top panel) and with the computed Hub-bard U of 4.6 eV (bottom panel). The zero of the energy isset at the top of the valence band. Experimental data fromref. [46] (empty symbols) and [47] (solid symbols) are alsoreported.

The same calculation with the literature value of U = 8eV, gave a larger value for the optical gap but a very pooragreement with the experimental absorption spectrum.

Minerals: Fayalite

As a final example we want to apply the presentmethodology to Fayalite, the iron-rich end member of(Mg,Fe)2SiO4 olivine (orthorhombic structure), one ofthe most abundant minerals in Earth’s upper mantle.Recently [57] we showed that, although good structuraland magnetic properties could be obtained for this min-eral within LDA or GGA, its electronic properties wereincorrectly described as metallic, confirming the corre-lated origin of the observed insulating behavior.

-10.0 -5.0 0.0 5.0Energy (eV)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

DO

S (

stat

es/e

V/c

ell)

Ni d states (majority spin)Ni d states (minority spin)Ni s statesO p states

-10.0 -5.0 0.0 5.0Energy (eV)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

DO

S (

stat

es/e

V/c

ell)

GGA

LDA+U

FIG. 14: Projected density of states of NiO in the undistortedAF configuration at the experimental lattice spacing obtainedwith U = 4.6 eV.

From x-rays diffraction studies it is known that Fay-alite has an orthorhombic cell, whose experimental latticeparameters are (in atomic units) a = 19.79, b = 11.50,c = 9.11. The unit cell (depicted in fig. 15) containsfour formula units, 28 atoms: 8 iron, 4 silicon, and 16oxygen atoms. Silicon ions are tetrahedrally coordinatedto oxygens, whereas iron ions occupy the centers of dis-torted oxygen octahedra. The point group symmetry ofthe non magnetic crystal is mmm (D2h in the Schoenfliesnotation) and the space group is Pnma. The magnetiza-tion of iron reduces the original symmetry and only halfof the symmetry operations survive. The general expres-sion for the internal structural degrees of freedom is givenin table II in the Wyckoff notation [58].

Iron sites can be divided into two classes (see fig. 15and tab. II): Fe1 centers which are structured in chainsrunning parallel to the b, [010], side of the orthorhom-bic cell, and Fe2 sites which belong to mirror planes forthe non magnetic crystal structure perpendicular to theb side and cutting it at 1/4 and 3/4 of its length. Themain structural units are the iron centered oxygen octa-

15

a

b

bc

a

Fe2 sites

Fe1 sites

Si O

FIG. 15: The unit cell of Fayalite. Large dark ions are Fe,small dark ions are O, light ions are Si.

TABLE II: Definition of the Wyckoff structural parametersappropriate for Fayalite structure

Ion Class Coordinates

Fe1 4a (0,0,0), (1/2,0,1/2)

(0,1/2,0), (1/2,1/2,1/2)

Fe2, Si, O1, O2 4c ±(u,1/4,v),

±(u+1/2,1/4,1/2-v)

O3 8d ±(x,y,z), ±(x,1/2-y,z),

±(x+1/2,1/2-y,1/2-z),

±(x+1/2,y,1/2-z)

hedra which are distorted from the cubic symmetry andtilted with respect to each other both along the chainsand on nearest Fe2 sites. Fayalite is known to be an anti-ferromagnetic (AF) compound with slightly non collineararrangement of spin on Fe1 iron site (this non collinear-ity will not be addressed here). Magnetic moments alongthe central and the edge Fe1 chains are antiferromagnet-ically oriented and from our previous work [57] the moststable spin configuration is the one in which the magne-tization of Fe2 ion is parallel to the one of the closestFe1 iron. This magnetic structure is consistent with an

0.0 50.0 100.0 150.0 200.0 250.0number of iron ions per unit cell

3.5

4.0

4.5

5.0

Hub

bard

U (

eV)

U1U2

FIG. 16: Convergence of Hubbard parameters of Fayalite withthe number of iron included in the supercell used in the ex-trapolation. U1 is the value obtained for Fe1 ions, U2 the onefor Fe2.

iron-iron magnetic interaction via a superexchange mech-anism through oxygen p orbitals.

The calculation of U was performed for the experimen-tal geometry, in the above mentioned spin configuration.As the primitive unit cell of fayalite is already quite large,we performed the constrained calculation only in this celland used larger supercells only to extrapolate the results.We considered three supercells in addition to the prim-itive one: i) a cell duplicated in the [0, 1, 0] chain direc-tion (a 1×2×1 supercell), containing 16 iron atoms; ii)a cell, containing 64 iron ions, obtained by duplicatingthe primitive structure in all directions (a 2×2×2 super-cell) and iii) a 4×4×2 supercell (256 iron ions). Othercomputational details where similar to those used in ourprevious work [57]. As GGA approximation provided aslightly better description of the system than LDA, weassumed this functional as the starting point to be im-proved; the same pseudopotentials used in ref. [57] forFe, O and Si were adopted here; somehow larger energycutoff for the electronic wave functions and charge den-sity (36 and 288 Ry respectively) and a small smearingwidth of 0.005 Ry were used. A 2×4×4 Monkhorst-Packgrid of k-points in the primitive cell was found sufficientfor the BZ integration.

The results of the U calculation for the two differentfamilies of iron sites (Fe1 and Fe2) are reported in fig.16 where the rapid convergence with respect supercelldimension can be seen. The final results for the on-siteCoulomb parameters are U1 = 4.9 eV for Fe1 ions andU2 = 4.6 eV for Fe2, which are in fairly good agreementwith the approximate (average) value of 4.5 eV obtainedin ref. [57] from a rather crude estimate.

The GGA+U band structure of Fayalite is shown in fig.

16

Ene

rgy

[eV

]

-10

-5

0

5

Γ X S Y Γ Z U R T Z

FIG. 17: The band structure of Fayalite obtained within thepresent LDA+U approach. The zero of the energy is set tothe top of the valence band. Complete degeneracy amongspin up and spin down states is present.

17 while in fig. 18 some atomic-projected density of statesare reported. At variance with the GGA results reportedin ref. [57] a band gap of about 3 eV now separates thevalence manifold from the conduction one, in reasonableagreement with the experimental result of about 2 eV[59] at zero pressure.

The minority spin t2g manifold of iron ions, that withinGGA cross the Fermi energy, is split into two subgroupsby the gap opening. The conduction-band states areshrunk to a narrow energy range and moved above thebottom of the iron s-states band which remains almostunaffected; the lower-energy minority-spin d-states, in-stead, merge in the group of states below the Fermi levelwhere they mix strongly with states originating fromOxygen p orbitals: the two sets of states, well separatedin the GGA results, collapse into a unique block. Themost evident consequence of the gap opening consists ina pronounced shrinking of the d states of iron which be-come flatter than in the GGA case. This is evident onthe top of the valence band, but also for states well belowthis energy level, which thus reveal a more pronouncedatomic-like behavior. Beside the gap opening betweenthe two groups of the minority-spin states, a strong mix-ing occurs among the oxygen p states and the iron dlevels over a rather large region extending down to 8 eV

-10.0 -5.0 0.0 5.0Energy (eV)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

DO

S (

stat

es/e

V/c

ell)

O p statesFe1 d states (majority spin)Fe1 d states (minority spin)

FIG. 18: Some atomic-projected density of states of Fayaliteobtained within the present LDA+U approach. Contributionsfrom majority- and minority-spin 3d states of one of the Fe1iron ions and from the total 2p manifold of one oxygen ionare shown.

TABLE III: Comparison of the experimental and LDA+Ucalculated values for the Wyckoff structural parameters ofFayalite as defined in table II

Ion u v x y z

Exp.

Fe2 0.780 0.515

Si 0.598 0.071

O1 0.593 0.731

O2 0.953 0.292

O3 0.164 0.038 0.289

GGA + U

Fe2 0.779 0.515

Si 0.597 0.072

O1 0.593 0.735

O2 0.951 0.289

O3 0.165 0.036 0.286

below the top of the valence band. A finite contribu-tion of the oxygen states is present close to the top ofthe valence manifold showing that the gap is mainly ofMott-Hubbard type with a partial charge-transfer char-acter.

We have then relaxed the geometric structure of thesystem (both internal and cell degrees of freedom) assum-ing no dependence of U1 and U2 on the atomic configu-ration. The resulting structural parameters (a = 20.18,b = 11.75, c = 9.29 atomic unit) as well as the internal co-ordinates reported in table III are in very good agreementwith the experimental results, even better than the al-ready satisfactory agreement obtained in ref. [57] withinGGA.

17

Although we did not studied other spin-configurations,magnetic properties seam to improve slightly in theGGA+U approximation. The magnetic moment on eachiron (both Fe1 and Fe2) was found to be 3.9 µB, in closeragreement with the spin-only value (4 µB) of the exper-imental result (4.4 µB) than the one obtained by GGAonly (3.8 µB). This improvement is probably due to theenhanced atomic-like character of iron d states, which isconsequence of the gap opening.

In conclusion, the GGA+U provides a quite good de-scription of structural, magnetic and electronic propertiesof fayalite, reproducing the observed insulating behaviorwith a reasonable value for its fundamental band gap.

SUMMARY

In this work we have reexamined the LDA+U approxi-mation to DFT and a simplified rotational-invariant formof the functional was adopted. We then developed amethod, based on a linear response approach, to calculatein an internally consistent way the interaction param-eters entering the LDA+U functional, without makingaprioristic assumption about screening and/or basis setemployed in the calculation. Our methodology was thensuccessfully tested on a few systems representative of nor-mal and correlated metals, simple transition metal oxidesand iron silicates. In all cases we obtained rather accu-rate results indicating that our scheme allows us to studyboth electronic and structural properties of strongly cor-related material on equal footing, without resorting toany empirical parameter adjustment.

This work has been supported by the MIUR under thePRIN program and by the INFM in the framework of theIniziativa Trasversale Calcolo Parallelo.

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