Ab initio evaluation of Coulomb and exchange parameters for DFT+U calculations

Nicholas J. Mosey and Emily A. Carter*Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics, Princeton University,

Princeton, New Jersey 08544-5263, USA�Received 15 March 2007; revised manuscript received 18 June 2007; published 25 October 2007�

Conventional density functional theory �DFT� fails for materials with strongly correlated electrons, such aslate transition metal oxides. Large errors in the intra-atomic Coulomb and exchange interactions are the sourceof this failure. The DFT+U method has provided a means, through empirical parameters, to correct theseerrors. Here, we present a systematic ab initio approach in evaluating the intra-atomic Coulomb and exchangeterms, U and J, respectively, in order to make the DFT+U method a fully first-principles technique. Themethod is based on a relationship between these terms and the Coulomb and exchange integrals evaluated inthe basis of unrestricted Hartree-Fock molecular orbitals that represent localized states of the extended system.We used this ab initio scheme to evaluate U and J for chromia �Cr2O3�. The resulting values are somewhathigher than those determined earlier either empirically or in constrained DFT calculations but have the advan-tage of originating from an ab initio theory containing exact exchange. Subsequent DFT+U calculations onchromia using the ab initio derived U and J yield properties consistent with experiment, unlike conventionalDFT. Overall, the technique developed and tested in this work holds promise in enabling accurate and fullypredictive DFT+U calculations of strongly correlated electron materials.

DOI: 10.1103/PhysRevB.76.155123 PACS number�s�: 71.27.�a, 71.15.Mb

I. INTRODUCTION

The advent of Kohn-Sham density functional theory1–5

�DFT� in conjunction with rapid advances in computer tech-nology has enabled accurate first-principles simulations ofsystems containing hundreds of atoms. These calculationshave contributed to the development of a better understand-ing of the fundamental properties of molecules and materials,leading to the emergence of simulation as a powerful tool forobtaining information that is difficult, or even impossible, toacquire experimentally. Despite the success of DFT, therestill exist classes of systems and phenomena for which cur-rent forms of the theory fail not just quantitatively but evenqualitatively �e.g., predicting a known insulator to be metal-lic�. As usual, this is due primarily to the use of approximateexchange-correlation �XC� functionals. Accurate first-principles calculations of such systems must await the devel-opment of computationally efficient methods that capture theessential physics missing in approximate XC functionals.The development of such techniques is an active area ofresearch.

An important class of materials poorly described by con-ventional DFT �i.e., DFT within the local density approxima-tion �LDA� or the generalized gradient approximation�GGA� to electron exchange and correlation� is that ofstrongly correlated electron materials �SCEMs�. SCEMs in-clude actinides and middle-to-late transition metal oxides,which contain many electrons in partially filled d or f shells.The d and f electrons are inherently localized on each metalatom, and the associated intra-atomic exchange and Cou-lomb energy terms are both numerous and large, due themany electrons and the localized, tight nature of the wavefunctions involved. Since conventional DFT with approxi-mate exchange and correlation does not properly cancel outthe spurious electron self-interaction, the electron-electronrepulsion is predicted to be far too large. As a result, theelectrons in conventional DFT calculations incorrectly delo-

calize to reduce the repulsion energy, which converts whatshould be an insulator �e.g., in the case of transition metaloxides� to a metal.

A suitable theoretical treatment of SCEMs thus rests onthe ability to accurately account for self-interaction energies,especially those arising from electrons in localized states.Hartree-Fock �HF� theory provides a natural framework forthis because HF calculations contain exact exchange and,hence, are free of self-interaction by design. Unfortunately,the absence of static and dynamic correlation effects in HFcalculations renders this method unsuitable for predictivesimulations. Well-known extensions of HF theory exist thataccount for electron correlation; however, these methodolo-gies are very expensive computationally and, for the mostpart, have not been implemented for extended crystals.6,7

Recently, the search for an electronic structure methodthat is computationally efficient, largely free of self-interaction error, and accounts for a degree of dynamic cor-relation has focused on combining various methods that meetat least one of these criteria. In these approaches, the elec-tronic structure is divided into localized and delocalizedstates, and each electronic subsystem is treated with an ap-propriate electronic structure method. Such hybrid ap-proaches are motivated from model Hamiltonian techniques,such as the Hubbard U8,9 and Anderson impurity10 models,which have been used historically to treat SCEMs in an em-pirical fashion.

Currently, the hybrid technique used most frequently tostudy SCEMs is known as DFT+U.11,12 In this theory, theinteractions between electrons in states localized on the sameatomic center �called on-site interactions� are treated in anHF-like manner, while the remaining interactions are treatedwith DFT. This makes great physical sense because the er-rors incurred by DFT are largely due to the intra-atomic self-interaction error, which should be well corrected by an HFdescription. In practice, the on-site interaction energy isevaluated with a parametrized Hamiltonian instead of an ex-

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plicit HF calculation. The parameters that enter into thisHamiltonian correspond to the average Coulomb and ex-change interactions between electrons of the same angularmomentum that are localized on the same atom. These pa-rameters are designated as UI� and JI�, respectively, where Idesignates the atomic center on which the electrons are lo-calized and � denotes their angular momentum. The key toperforming an accurate DFT+U calculation lies in selectingsuitable values for these parameters. Two methods are com-monly used in this respect. In one case, these values aretreated as tunable parameters, which are chosen to reproduceknown properties of the system of interest.13,14 This empiri-cal approach often may be reasonable but lacks a theoreticalbasis, is subject to bias from using a limited amount of datato select the parameters, and is not applicable in cases whereexperimental data are limited or nonexistent. In the othercase, constrained DFT calculations15,16 are performed inwhich the number of electrons in states localized on a par-ticular atomic site is fixed while the remainder of the elec-tronic structure is relaxed. By monitoring the change in thetotal energy as a function of the number of localized elec-trons, it is possible to extract values for UI� and JI� fromthese calculations while accounting for screening effectsfrom the remainder of the system. Although this approach ismotivated from first principles, the calculated values are ofquestionable accuracy due to the artificial nature of the con-strained systems, the inherent inaccuracy of the DFTexchange-correlation functionals for localized, open-shellelectronic states �as discussed above�, and the observed sig-nificant dependence of the calculated values of UI� and JI� onthe basis set used in such calculations.

In the present work, a means of evaluating UI� and JI�from unrestricted HF �UHF� calculations is developed andassessed. This ab initio approach is not subject to the diffi-culties and potential bias associated with using limitedamounts of experimental data, as in empirical parameter fit-ting, and although UHF calculations do not capture static anddynamic correlation effects, they do not suffer from the self-interaction errors that are present in constrained DFT calcu-lations. As such, this approach may prove to be an effectivemeans of unambiguously determining these parameters fromfirst principles for use in subsequent DFT+U calculations,offering a fully predictive DFT+U theory. The method de-veloped in this work is based on an approximate relationshipbetween the parameters UI� and JI� and the Coulomb andexchange integrals evaluated in the basis of the UHF mo-lecular orbitals corresponding to the localized states of thesystem. These integrals are evaluated by performing UHFcalculations on embedded cluster models that represent thebulk material of interest and yield approximate values for thetrue interactions between the localized electrons in the realmaterial. The values of UI� and JI� obtained in this approachare treated as constants for the material and do not depend onthe exchange-correlation functional to be used in the DFT+U calculation. Due to the lack of static and dynamic corre-lation effects in UHF calculations, the values of UI� and JI�derived from this method are upper bounds on the truestrengths of these interactions. The results of test calculationson chromia �Cr2O3� demonstrate that converged ab initiovalues of UI� and JI� can be obtained with clusters and basis

sets that are modest in size, with the primary requisite forconvergence being a model that provides an adequate de-scription of the slight delocalization of the “localized” statesover neighboring atoms. The converged values of UI� and JI�evaluated in these cluster calculations are similar to thosedetermined previously with empirical and constrained DFTtechniques, but we believe that the ab initio values rest on afirmer theoretical footing. Subsequent DFT+U calculationsof bulk chromia performed with the ab initio values of UI�and JI� yield results in very good agreement with experi-ments. This indicates that the developed methodology maybe a powerful means of obtaining reliable estimates of UI�and JI� for use in DFT+U calculations of other SCEMs.

The remainder of this paper is as follows. A brief over-view of the DFT+U method is provided in Sec. II. The abinitio evaluation of UI� and JI� is described in Sec. III andapplied in Sec. IV to determine parameters for chromia.DFT+U calculations of bulk chromia are presented in Sec. Vto assess the quality of the ab initio parameters, and theconclusions are given in Sec. VI.

II. DFT+U METHODOLOGY

The DFT+U method has been described in detailelsewhere,11,12 and only a brief introduction will be providedhere. The technique is based on the use of HF theory toevaluate the on-site interactions between localized electronsand the use of DFT to calculate all other electronic terms.This is achieved through the following total energy func-tional:

EDFT+U��,�nI�m��� = EDFT��� + Eon-site��nI�m��� − Edc��nI���� ,

�1�

where EDFT+U is the total energy of the system, EDFT is theDFT energy of the system based on the total electron density�, Eon-site is the HF energy arising from on-site interactionsbetween localized electrons, and Edc is a double-countingterm which corrects for contributions to the total energy thatare included in both EDFT and Eon-site. The on-site interactionenergy Eon-site is dependent on the number of electrons thatoccupy localized orbitals �I�m�, which are centered on atomI and characterized by angular momentum �, magnetic quan-tum number m, and spin �. These atom-centered, localizedorbitals consist of spherical harmonics multiplied by a radialfunction selected to accurately represent the electronic statesof the spheridized non-spin-polarized atom. The occupationnumbers nI�m� are obtained by projecting the Kohn-ShamDFT orbitals for the total system onto a set of these localizedorbitals. The value NI�� that enters into Edc corresponds tothe total number of electrons of a given spin and angularmomentum that are localized on I, i.e., NI��=�mnI�m�.

Evaluating the energy with Eq. �1� requires expressionsfor Eon-site and Edc. Many different forms of these expressionshave been proposed in the literature. Here, we will focus onthe rotationally invariant approach proposed by Dudarev etal.,17 which was employed in the DFT+U calculations dis-cussed below. In this approach, these quantities are expressedas

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Eon-site��nI�m��� = �I,�� �

m,m�

nI�m�nI�m�� + �m,m��m

nI�m�nI�m�� + �m,m��m

nI�m�nI�m��UI��− �

I,�� �

m,m��m

nI�m�nI�m�� + �m,m��m

nI�m�nI�m��JI�� �2�

and

Edc��NI���� = �I,��NI��NI�� +

NI���NI�� − 1�2

+NI���NI�� − 1�

2UI�� − �

I,��NI���NI�� − 1�

2+

NI���NI�� − 1�2

JI�� , �3�

where the limits on the summations are such that each on-site interaction is counted only once. It is important to notethat these expressions only account for interactions betweenelectrons that are localized on the same atomic site and havethe same angular momentum. Justifications for these expres-sions of Eon-site and Edc have been provided elsewhere,17–20

but essentially, they enumerate the allowed pairwise interac-tions. For Coulomb terms UI�, this corresponds to all pair-wise interactions between all electrons of the same angularmomentum on each site, while for exchange terms JI�, onlypairwise interactions between electrons of the same spin oneach site are nonzero.

Inserting Eqs. �2� and �3� into Eq. �1� yields the followingtotal energy functional17

EDFT+U��,�nI�m���

= EDFT��� + �I,�,m,�

�UI� − JI��2

�nI�m� − nI�m�2 � . �4�

The first term on the right hand side of this equation corre-sponds to the DFT energy obtained using the total electrondensity � and includes the on-site interactions, albeit calcu-lated with conventional DFT, which is known to be inaccu-rate in this respect. The second term on the right hand side ofEq. �4� corrects for this behavior in the following sense. Notethat UI� and JI� are positive and nI�m� lies between 0 and 1.Effectively, the second term then serves as a penalty functionthat drives the system toward electron densities in which thelocalized states �I�m� have occupation numbers of either 0 or1. This counters the tendency of DFT calculations to overde-localize these electronic states and leads to an improved de-scription of the electronic structure. In particular, the ener-gies of the localized states will be shifted from their DFTvalues such that the band gap is increased, which increasesthe insulating character of the system. This can be seen if oneevaluates the orbital energies by differentiating Eq. �4� withrespect to the orbital occupation numbers, which demon-strates that the occupied states are shifted to lower energieswhile the unoccupied states are shifted to higher energies.

In order to employ Eq. �4�, one must provide values forUI� and JI�. These values are not known a priori and aresystem dependent. Thus, the challenge in performing mean-ingful DFT+U calculations lies in selecting appropriate val-ues for these parameters. As noted above, this is typicallyachieved through either empirical fitting or constrained DFT

calculations, both of which suffer from significant shortcom-ings. In the next section, an approach will be outlined forevaluating these parameters through ab initio UHF calcula-tions.

III. AB INITIO EVALUATION OF U AND J

As discussed in the preceding section, DFT+U calcula-tions employ a parametrized Hamiltonian to evaluate the on-site interaction energy Eon-site between electrons in localizedstates. In what follows, a means of systematically calculatingthe parameters used to evaluate Eon-site is developed. Thisapproach involves performing a UHF calculation on the ma-terial and calculating the two-electron energy arising frominteractions between electrons occupying molecular orbitals�MOs� localized on the same atom. The starting point for thisapproach is to write the two-electron UHF energy EUHF for aset of MOs ��i

�i�:

EUHF���i�i�� = �

i,jCij

�� + �i,j�i

Cij�� + �

i,j�i

Cij��

− �i,j�i

Xij�� − �

i,j�i

Xij��,

Cij�i�j =� � �i

�i�r1��i�i�r1�� j

�j�r2�� j�j�r2�

r1 − r2 dr1dr2,

Xij�i�j =� � �i

�i�r1�� j�j�r1�� j

�j�r2��i�i�r2�

r1 − r2 dr1dr2, �5�

where Cij�i�j and Xij

�i�j represent Coulomb and exchange inte-grals involving UHF canonical MOs i and j, which are oc-cupied by electrons with spins �i and � j, respectively. Anunconventional notation has been employed in Eq. �5� toavoid confusion with terms used earlier �conventionally,Coulomb integrals would be designated with U or J andexchange integrals would be labeled as J or K�.

EUHF is the HF energy arising from all two-electron inter-actions in the system. Meanwhile, Eon-site only includes inter-actions between electrons of the same angular momentumthat are localized on the same atom. In order to extract theon-site energy from Eq. �5�, it will prove useful to recognizethat, in principle, it is possible to decompose the occupationnumbers of the UHF MOs into contributions from atomicorbitals of well-defined angular momentum that are centered

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on specific atoms. That is, the occupation number of orbital�i

�i can be expressed as Ni�=�I,�nI�i�, where nI�i� designatesthe contribution to the MO of electrons with spin � residingin atomic orbitals of angular momentum � centered on atomI. The values of nI�i� can be obtained by projecting the UHFcanonical MOs onto a set of atom-centered basis functionsthat have well-defined values of I and �.

Since Ni�=1 for occupied states, it is valid to rewrite Eq.�5� as

EUHF = �i,j�

I,�nI�i� �

I�,��

nI���j�Cij��

+ �i,j�i

�I,�

nI�i� �I�,��

nI���j�Cij��

+ �i,j�i

�I,�

nI�i� �I�,��

nI���j�Cij��

− �i,j�i

�I,�

nI�i� �I�,��

nI���j�Xij��

− �i,j�i

�I,�

nI�i� �I�,��

nI���j�Xij�� . �6�

If we retain in Eq. �6� only the terms corresponding to theon-site interactions between electrons with the same angularmomentum, i.e., those where I= I� and �=��, we obtain

Eon-siteUHF = �

I,��

i,jnI�i�nI�j�Cij

�� + �i,j�i

nI�i�nI�j�Cij��

+ �i,j�i

nI�i�nI�j�Cij�� − �

I,� �

i,j�i

nI�i�nI�j�Xij��

+ �i,j�i

nI�i�nI�j�Xij�� , �7�

which corresponds to the two-electron energy arising fromon-site interactions between electrons in atomic orbitals ofthe same angular momentum � that are centered on the sameatom I. If one restricts I and � to values consistent with thelocalized electrons in the system and restricts i and j to UHFMOs that are composed primarily of atomic basis functionsof angular momentum � that are centered on I, i.e., the UHFMOs that correspond to the localized states, it is possible todefine Eon-site

UHF as the on-site energy of interest in DFT+Ucalculations. Setting Eon-site=Eon-site

UHF , with the appropriate re-strictions on I, �, i, and j, forms the basis for evaluating UI�and JI� from the Coulomb and exchange integrals evaluatedin the basis of localized canonical UHF MOs.21

In order to obtain expressions for UI� and JI� from Eq. �7�,a weighting factor, nI�i�i

nI�j�j, is assigned to each Coulomb

and exchange integral. The Coulomb and exchange terms inEq. �7� are then multiplied and divided by the sum of thesefactors to yield

Eon-siteUHF ���i��,�nI�i��� = �

I,� ��i,j nI�i�nI�j� + �i,j�i

nI�i�nI�j� + �i,j�i

nI�i�nI�j�

��i,j

nI�i�nI�j�Cij�� + �

i,j�i

nI�i�nI�j�Cij�� + �

i,j�i

nI�i�nI�j�Cij��

�i,j

nI�i�nI�j� + �i,j�i

nI�i�nI�j� + �i,j�i

nI�i�nI�j� ��− �

I,� � �i,j�i

nI�i�nI�j� + �i,j�i

nI�i�nI�j�� �i,j�i

nI�i�nI�j�Xij�� + �

i,j�i

nI�i�nI�j�Xij��

�i,j�i

nI�i�nI�j� + �i,j�i

nI�i�nI�j� �� . �8�

This expression is similar to that in Eq. �2�. Specifically, within each of the brackets, there exists a summation over productsof occupation numbers multiplied by an expression representing the average on-site Coulomb or exchange energies. EquatingEq. �2� with Eq. �8� and setting nI�i�=nI�m� lead to the following expressions for UI� and JI�:

UI� =

�i,j

nI�i�nI�j�Cij�� + �

i,j�i

nI�i�nI�j�Cij�� + �

i,j�i

nI�i�nI�j�Cij��

�i,j

nI�i�nI�j� + �i,j�i

nI�i�nI�j� + �i,j�i

nI�i�nI�j�

,

JI� =

�i,j�i

nI�i�nI�j�Xij�� + �

i,j�i

nI�i�nI�j�Xij��

�i,j�i

nI�i�nI�j� + �i,j�i

nI�i�nI�j�

. �9�

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These expressions relate UI� and JI� to the localized canoni-cal MOs obtained in a UHF calculation, thereby providing anab initio means of evaluating the parameters required forDFT+U calculations.

It can be shown that Eq. �2� is equal to Eq. �8� only if it ispossible to transform the subset of canonical UHF MOs usedto evaluate the Coulomb and exchange integrals into a formwhere each orbital can be mapped onto an atomiclike orbitalwith distinct values of I, �, m, and �. Values of UI� and JI�are invariant to such rotations provided that the transforma-tion is applied to orbitals of the same symmetry. If, for somereason, the UHF orbitals used to calculate UI� and JI� couldnot be represented in a form where only one MO correspondsto a given value of m, the calculated values of UI� and JI�would contain contributions from electrons defined by thesame values of I, �, m, and �, which are not included in thedefinition of Eon-site in Eq. �2�. This requirement cannot bestrictly met—even for localized states of interest in DFT+U calculations—due to the delocalized nature of UHFMOs. However, in practice, the fact that the canonical UHFMOs of interest are highly concentrated on specific atomsand the requirement that the MOs remain orthogonal is suf-ficient to ensure that MOs of the same spin that are charac-teristic of the same angular momentum and concentrated onthe same atom will be projected primarily onto sphericalharmonic basis functions with different values of m. Thus,while this is an approximation that is inherent to the methodoutlined above, it is relatively minor for systems with local-ized states appropriate for studying with DFT+U methods.

Another approximation inherent to the method involvesthe evaluation of the occupation numbers nI�i�, as there doesnot exist an exact means of dividing the occupation numberof an MO into contributions from orbitals of a specific an-gular momentum located on a specific atomic center. In prac-tice, this decomposition is achieved by projecting each MOonto a set of atom-centered basis functions, which are com-posed of a set of spherical harmonics multiplied by appro-priate radial functions. Thus, the occupation numbers willultimately be dependent on the nature of these radial func-tions. Fortunately, the localized UHF MOs used to evaluateany particular values of UI� and JI� will be projected ontobasis functions with the same value of I and � and, hence,will be subjected consistently to any variations in the radialcomponents. That is, if the radial function used in conjunc-tion with spherical harmonics of angular momentum � local-ized on atom I is changed such that the occupation numbernI�i� associated with one of the localized UHF MOs is de-creased by one-half, the occupation numbers of the otherlocalized MOs associated with the same angular momentumand atomic center will also be reduced by approximatelyone-half. Furthermore, the similarities between the localizedUHF MOs associated with specific values of I and � ensurethat the occupation numbers associated with these MOs willalso be similar to one another. Thus, the occupation numbersassociated with the MOs will be similar to one another andwill scale uniformly with the basis functions onto which theyare projected. It can be demonstrated that if these conditionsare met, the variations in the occupations numbers with thebasis set will cancel when UI� and JI� are calculated with theexpressions in Eq. �9�, and, hence, the calculated values of

UI� and JI� will be relatively insensitive to changes in thebasis set used to evaluate the occupation numbers. Numericaltests given below will demonstrate this assertion.

IV. COULOMB AND EXCHANGE PARAMETERSFOR CHROMIA

In the preceding section, an approach was developed forevaluating UI� and JI� from the set of canonical UHF MOsrepresenting the localized states in a strongly correlated sys-tem. In this section, this scheme will be used to determinevalues of UI� and JI� for chromia. Since the only localizedelectrons in chromia reside in the d states of the chromiumatoms ��=2 and I=Cr�, the subscripts on UI� and JI� will bedropped in what follows. Furthermore, efforts will focus ondetermining �U−J�, which is the relevant quantity in theDFT+U energy functional in Eq. �4�. The details of the UHFcalculations are given in Sec. IV A, and the results are dis-cussed in Sec. IV B.

A. Details of the unrestricted Hartree-Fock calculations

1. Model systems

The repulsion between electrons in orbitals localized on aspecific atomic center is highly influenced by the environ-ment around that atom. This is primarily due to screeningeffects which reduce the repulsive interactions between theseelectrons. Indeed, preliminary UHF calculations on indi-vidual Cr ions, i.e., neglecting all screening from the envi-ronment, yielded values of �U−J� that were between 17 and23 eV, depending on the ionic charge. Meanwhile, typicalvalues of �U−J� used in calculations of transition metal ox-ides are in the range of 2–10 eV,22 with a value of 4 eV forchromia being determined empirically in a previous study.14

The large differences between the values obtained in the iso-lated ion calculations and those determined �either empiri-cally or with constrained DFT calculations� for extended sys-tems highlight the need to incorporate environmental aspectsinto the calculation of �U−J�.

In order to include screening effects, calculations wereperformed on finite-sized CrxOy clusters embedded in a fieldof point charges representing bulk chromia. The clusterswere constructed by starting with a single CrO6

9− unit �con-sistent with the formal oxidation states of Cr and O in chro-mia of Cr3+ and O2−� and adding successive shells of Cr3+

ions along with enough O2− ions to ensure that each chro-mium was surrounded by a total of six oxygens. Altogether,clusters containing between seven atoms �CrO6

9−� and 63atoms �Cr15O48

51−� were considered. The positions of the at-oms in these clusters corresponded to those in the experi-mental geometry of bulk chromia. The influence of the sizeof the embedded cluster on the calculated values of �U−J� isdiscussed below.

The clusters were embedded in a set of point charges withpositions corresponding to the locations of Cr and O atomsin an 882 representation of the experimental hexagonalunit cell of Cr2O3, where 882 refers to the number ofrepeated hexagonal unit cells along each lattice vector. Thesystem was only repeated twice along the last direction due

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to the increased length of the associated lattice vector. Alto-gether, this model contains a total of approximately 4000point charges, depending on the specific cluster under con-sideration, and provides an approximately isotropic represen-tation of the point charges about the central position of thepoint charge field where the embedded cluster was placed.Test calculations demonstrated that the distance between theoutermost atoms of the embedded cluster and the outer edgeof the point charge field �at least 8 Å� was sufficient to yieldconverged values of �U−J�. The points at the locations of Cratoms were assigned charges of +3e, while those at the po-sitions of O atoms were assigned charges of −2e. The pointcharges at the locations of Cr ions nearest to the embeddedcluster were represented by Al pseudopotentials �PPs�,23

which have a net charge of +3e. The purpose of the PPs is toact as a barrier between the cluster and the set of pointcharges, which prevents the unphysical drift of electron den-sity from the outer oxygen atoms of the cluster into the pointcharge field. This is achieved with PPs because they providea degree of exchange repulsion which effectively containsthe electron density within the region associated with theembedded cluster. This approach has been used extensivelyin embedded cluster calculations.24–27 An example of one ofthe clusters considered in these calculations surrounded bythe set of PPs is shown in Fig. 1.

2. Electronic structure calculations

Chromia is an antiferromagnetic material in which the netspin on a given chromium atom is the opposite of that on itsneighboring four Cr atoms.28–30 Obtaining an antiferromag-netic electronic structure is a challenging task in quantumchemical calculations, particularly for large systems in whichone must control the net spin at many atomic sites. However,an accurate representation of the electronic structure is criti-cal for evaluating reliable values of �U−J�. In this work, the

following approach was employed to generate electronicstructures consistent with the experimental antiferromagneticspin ordering of chromia. First, a UHF calculation was per-formed with the system in a ferromagnetic state, and theresulting canonical UHF MOs were localized according tothe procedure of Pipek and Mezey.31 Analysis of the local-ized MOs permitted the identification of orbitals that werecomposed primarily of d-type atomic basis functions cen-tered on an individual Cr atom. The localized MOs were thenused to construct the initial guess wave function for a subse-quent UHF calculation, with the localized MOs characterizedas d states associated with specific Cr atoms being assignedeither � or � spin as required to achieve an antiferromagneticelectronic structure. The electronic structure obtainedthrough this second UHF calculation exhibited the correctantiferromagnetic spin ordering. Moreover, a subset of thecanonical UHF MOs obtained in this calculation was com-posed of orbitals that were highly concentrated on specific Cratoms and exhibited a high degree of d character. An ex-ample of a set of these orbitals is shown in Fig. 2.

The values of U and J were calculated with the expres-sions in Eq. �9� using the canonical UHF MOs correspondingto the d states that were localized on the central Cr atom ofeach CrxOy cluster. In general, there were three of these or-bitals, which corresponded to the t2g states of this atom. Theorbitals used in the calculations of U and J were restricted tothose localized on the central Cr atoms because the environ-ment around this atom will become increasingly bulklike asthe cluster size is increased, while the orbitals on the outer Cratoms experience edge effects and may not be screened prop-erly. The number of d electrons associated with each of theMOs used to evaluate U and J, i.e., the values of nI�i� thatenter into Eq. �9�, were calculated by using the MO coeffi-cients to construct Mulliken populations of the spherical har-monic Gaussian basis functions used in the UHF calculationsand summing over populations of all basis functions with thecorrect values of I and �. This was repeated for each of thethree MOs used to evaluate U and J.

The UHF calculations were performed using a version ofthe GAMESS-US code,32,33 which we modified to calculate andprint out the Coulomb and exchange integrals involving anyspecific pair of MOs. These individual integrals, evaluated inthe MO basis, must be known to calculate U and J with Eq.�9�. A series of atom-centered Gaussian basis sets was con-sidered in these calculations to explore the dependence of�U−J� on the basis set size and ensure that a convergedvalue of this quantity was obtained. Three different basis sets

FIG. 1. �Color online� One of the clusters considered in thisstudy. The Cr2O9

12− cluster is surrounded by a set of Al PP cores,and the whole cluster is embedded in a series of point chargescorresponding to an 882 representation of the experimentalhexagonal unit cell of chromia. This set of point charges is notshown. Chromium atoms are purple, oxygen atoms are red, andpoint charges represented by Al PPs are pale blue. Solid lines indi-cate bonds within the cluster, and dotted lines designate interactionsbetween the Al PPs and the outer oxygen atoms of the cluster.

FIG. 2. �Color online� Canonical UHF MOs that are highly lo-calized on one of the Cr atoms of the Cr2O9

12− cluster. The orbitalsare plotted at a contour value of 0.075 a.u. The different colors ofthe orbitals indicate opposite signs of the wave function. The chro-mium atoms are colored purple and the oxygens are red. Note thatthe structures were rotated to clearly illustrate each orbital.

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were used to treat the electrons on the Cr atoms in the em-bedded cluster. Two of these, the LANL2DZ basis set of Hayand Wadt23 and the Stuttgart basis set developed by Dolg etal.,34 treated the 3s, 3p, 3d, and 4s valence electrons explic-itly and represented the remaining ten electrons with a PP.The third was the all-electron basis set of Wachters.35,36 Fivedifferent basis sets were considered for oxygen. These werethe well-known 6-31G�d�, 6-311G�d�, 6-31+G�d�, cc-pVDZ, and DZP basis sets.37–39 The first three basis sets inthis series permit an investigation of the effect of systemati-cally altering the basis set on O by increasing the number ofvalence basis functions or adding diffuse functions, while thelatter two allow for a consideration of the effect of employ-ing non-Pople basis sets in the calculations. In all cases, theangular dependence of the basis functions was representedwith spherical harmonics, which facilitated the evaluation ofthe occupation numbers nI�i� that enter into Eq. �9� throughMulliken population analysis as described above.

B. Results

The procedure and models described above were used toevaluate �U−J� for chromia. As noted above, the values of Uand J were calculated with the expressions in Eq. �9� usingcanonical UHF MOs that were of high d character and local-ized on the central Cr atom of the cluster. The dependence of�U−J� on the basis set and cluster size is explored in whatfollows, along with the requisites for obtaining convergedvalues of this quantity.

Values of �U−J� were calculated using the series of basissets described above for a small range of cluster sizes. Theresults of these calculations are summarized in Fig. 3 andshow that the data form two groups, which are distinguishedby the basis set used on Cr. Specifically, changing the basisset on Cr from LANL2DZ to either of the Stuttgart or Wacht-ers basis sets can alter the calculated value of �U−J� by asmuch as �2.5 eV, which is sufficiently large to have a sig-nificant effect on the results of a DFT+U calculation. Mean-while, changing the basis set on O alters the results by0.5 eV at most, which is sufficiently converged for DFT

+U calculations. These observations are true across the rangeof clusters considered in the calculations.

The results thus indicate that the calculated value of �U−J� is relatively insensitive to the basis set used on O yet ishighly dependent on that applied to Cr, which is not particu-larly surprising, since �U−J� is intrinsically a Cr-based prop-erty. In light of this, the following calculations will employ a6-31G�d� basis on O and the LANL2DZ and Stuttgart basissets on Cr. Although the Stuttgart and LANL2DZ basis setsprovide quite different results, it is likely that a more reliableestimate of �U−J� is obtained with the Stuttgart basis set dueto its greater number of basis functions and the fact thatsome of its basis functions are quite diffuse. This is sup-ported by the agreement between the results obtained withthe Stuttgart basis and those calculated with the larger, all-electron Wachters basis set, which is evident from Fig. 3where the data evaluated with the Wachters basis set essen-tially overlap those obtained with the Stuttgart basis. This isparticularly true when these basis sets are used in conjunc-tion with the 6-31G�d� basis set on oxygen, where the largestdiscrepancy between the values obtained with the Stuttgartand Wachters basis sets was 0.1 eV for the CrO6

9− system.The importance of employing a flexible, sufficiently diffusebasis set on Cr is discussed further below.

The values in Fig. 3 are not converged with respect tosystem size. To obtain converged values of �U−J�, additionalcalculations were performed on large clusters using the6-31G�d� basis on O and either the LANL2DZ or Stuttgartbasis sets on Cr. The results of these calculations are sum-marized in Fig. 4 and demonstrate that �U−J� is convergedto within approximately ±0.3 eV or better for clusters largerthan Cr11O39

45−. This is true for both sets of data presented inthat figure. Assuming that the Stuttgart basis set provides amore accurate representation of the electronic structure, aconverged value of �U−J� is taken as 7.7 eV. This is in linewith values previously determined empirically for chromia�4 eV� �Ref. 14� and through constrained DFT+U calcula-tions for Cr in perovskites �7–9 eV�.22

Cr: LANL2DZ, O: 6-31G(d)

Cr: LANL2DZ, O: 6-31+G(d)

Cr: LANL2DZ, O: 6-311G(d)

Cr: LANL2DZ, O: cc-pVDZ

Cr: LANL2DZ, O: DZP

Cr: Stuttgart, O: 6-31G(d)

Cr: Stuttgart, O: 6-311G(d)

Cr: Stuttgart, O: cc-pVDZ

Cr: Stuttgart, O: DZP

Cr: Wachters, O: 6-31G(d)

CrO 6

9-

Cr2O912-

Cr5O2127-

Cr8O3036-

7.5

10.0

12.5

15.0(U–J)[eV]

FIG. 3. Dependence of �U−J� on the basis set for a series ofCrxOy clusters. The results demonstrate that the calculated value of�U−J� is relatively insensitive to the basis set used on the oxygenatoms yet can be altered significantly by changing the basis set usedon chromium.

FIG. 4. Dependence of �U−J� on the size of the CrxOy clustersfor two different basis sets on Cr and a 6-31G�d� basis set onoxygen. The results demonstrate that converged values of �U−J�are obtained for clusters larger than Cr11O39

45−. The results obtainedwith the Stuttgart basis set on Cr demonstrate that a convergedvalue of �U−J� is 7.7 eV.

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The values of �U−J� evaluated through the UHF calcula-tions are sensitive to both the basis set and the size of theembedded cluster. Analysis of the electronic structure indi-cates that this dependence is related to the ability of themodel to account for the slight degree to which the localizedcanonical UHF MOs are delocalized over neighboring at-oms. To investigate the �de�localized nature of the MOs, thefollowing localization metric is defined:

LIi =��

=dAO

on I

ci2

�=all AOs

ci2

, �10�

where ci is coefficient of atomic orbital �AO� in MO i.This metric measures the contribution of d-type AOs on aspecific atom, I, to a particular MO. In the event that LIi=1,it is possible to characterize MO i as being a d orbital that iscompletely localized on atom I. At the other extreme, i.e.,when LIi=0, no d-type AOs on atom I contribute to MO i. Ingeneral, the UHF MOs used in the evaluation of �U−J� willlie between these extremes, with values of LIi that are closerto 1 than to 0.

LIi values were determined for the UHF MOs used tocalculate the data presented in Fig. 3, with a focus on thecontribution from AOs centered on the central Cr atom of thecluster, i.e., I=central Cr. The results are plotted in Fig. 5,where each data point represents an average over the LIivalues for each of the three UHF MOs that were localized onthe central Cr atom of the cluster and used to evaluate �U−J�. The data exhibit trends that are quite similar to the �U−J� values plotted in Fig. 3. Specifically, the values are con-verged beyond the Cr11O39

45− cluster, and the values ob-tained with the Stuttgart basis set on Cr are lower than thoseobtained with the LANL2DZ basis set. The convergence ofthe LIi values beyond a certain cluster size irrespective of thebasis set is consistent with the fact that the orbitals used to

calculate �U−J� are only delocalized over a few of the atomsnearest to the central Cr atom, and, hence, adding atoms tothe outside of the cluster will have a minimal effect on theresults. The differences between the results obtained with theStuttgart and LANL2DZ basis sets arise from the inclusionof diffuse basis functions in the former, which allow the dstates on chromia to more easily interact with orbitals onneighboring atoms, leading to delocalization. The increaseddelocalization observed with the Stuttgart basis set is consis-tent with the lower values of �U−J� obtained with this basis.Specifically, delocalization decreases on-site repulsion,yielding lower values for this quantity. Overall, these resultsindicate that the requisite for obtaining converged values of�U−J� is to employ a structural and theoretical model thatadequately accounts for the short-range delocalization of thelocalized states in the system.

V. DFT+U CALCULATIONS OF BULK CHROMIA

The calculations performed in the previous section ledto a converged ab initio value of �U−J�=7.7 eV. In thissection, DFT+U calculations will be performed on bulk-chromia to determine whether this value of �U−J� yieldsresults consistent with experiments. The details of the calcu-lations are given in Sec. V A, and the results are discussed inSec. V B.

A. Calculation details

DFT+U calculations of bulk chromia were performedwith the VASP simulation package.40,41 The particular DFT+U formalism used in these calculations was that developedby Dudarev et al.17 in which the electron density is opti-mized to minimize the total energy functional in Eq. �4�. Forpurposes of comparison, calculations were performed forvalues of �U−J� equal to 0, 2, 4, 6, 7.7, and 8 eV. When�U−J�=0.0 eV, the calculation is equivalent to a standardKohn-Sham DFT calculation. The range of values for �U−J� considered in the calculations is sufficient to examinethe dependence of various calculated quantities on the valueof �U−J�, empirically identify the optimum value of �U−J�,and determine if the ab initio value of �U−J�=7.7 eV yieldsresults consistent with experiments. All calculations wereperformed on a single hexagonal unit cell of chromia, whichis shown in Fig. 6 with labels corresponding to quantitiescalculated below.

The calculations were performed using two different XCfunctionals: the LDA functional of Perdew and Zunger42 andthe GGA functional of Perdew-Burke-Ernzerhof �PBE�.43

The nuclei and core electrons were described with the pro-jector augmented wave44 �PAW� potentials provided withVASP for each of these functionals. The 1s electrons on oxy-gen were included in the core �six valence electron� and the1s, 2s, 2p, and 3s electrons on chromium were describedby the PAW potential �12 valence electron�. The valencestates were described by a set of plane waves expanded up toa kinetic energy cutoff of 550 eV, and a �-point-centered331 k-point grid was used in all calculations. The fastFourier transform mesh used to evaluate the augmentation

FIG. 5. Dependence of LIi on the size of the CrxOy clusters fortwo different basis sets on Cr. The values of LIi were obtained withEq. �10�, setting I equivalent to the central Cr atom in the cluster.The plotted data were obtained by averaging over the values of LIi

for the three UHF MOs used to evaluate �U−J� in each cluster.

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charges contained twice the number of grid points along eachdirection as that used to represent the wave function. Thistheoretical model converged the total energies to better than1 meV/atom and the elements of the stress tensor to betterthan 1 kbar with respect to higher plane wave cutoffs andlarger k-point grids. Gaussian smearing was used in all opti-mizations with a smearing width of 0.1 eV. The only devia-tions from this procedure were in the calculation of the den-sity of states �DOS�, band gap, and magnetic moment, wherethe tetrahedron method45,46 with corrections by Blöchl etal.47 was used with a �-point-centered 993 k-point grid,which increased sampling of the Brillouin zone for the accu-rate evaluation of the DOS.

The lattice constants and atomic structure of the hexago-nal unit cell were optimized as follows. First, a series ofsystems with different volumes was constructed by alteringuniformly the lattice vectors of the experimental hexagonalunit cell of chromia. At each volume, the lattice constantsand atomic positions were relaxed, subject to the constraintthat the volume remained constant. The lattice constants and

atomic positions of the lowest energy structure were opti-mized further without any constraints placed on the volumeof the system, yielding the optimized unit cell and structure.This procedure was performed for both the LDA and PBEfunctionals over the range of �U−J� values listed above. Theresults of the optimization provided the optimal lattice con-stants and atomic structure for each functional and value of�U−J�. Furthermore, a set of energy vs volume data wasobtained in each case, which provides access to the bulkmodulus of the system.

B. Results

Bulk properties of chromia were calculated using the pro-cedure described in the preceding section. The results ofthese calculations are discussed in what follows and focus ondetails related to the unit cell �volume and lattice constants�,atomic structure �atomic positions and interatomic dis-tances�, bulk modulus, and electronic structure �band gapand magnetic moment�. Emphasis will be placed on assess-ing the ability of the ab initio value of �U−J� to reproduceexperimental results, identifying the optimal empirical valueof �U−J� and determining which type of XC functional isbetter able to provide an accurate description of chromia inDFT+U calculations. This will aid in the development of amodel for future calculations of chromia.

1. Unit cell

The unit cell volume and lengths of lattice vectors a and care plotted in Fig. 7 as a function of �U−J� for the LDA andPBE functionals along with experimental values.48 The re-sults demonstrate that the PBE functional provides a reason-able estimate of the volume when �U−J�=0.0 eV; however,the volume is severely overestimated when PBE is employedwith higher values of �U−J�. Meanwhile, LDA underesti-mates the volume by 7% when �U−J�=0.0 eV yet yieldsresults in good agreement with experiment �±2.5% � for val-ues of �U−J� between 2 and 8 eV. In terms of the latticeconstants, the PBE functional provides a good estimate of a yet overestimates c by 0.22 Å when �U−J�=0.0 eV. Theagreement with experiment worsens with this functional as�U−J� is increased. With the LDA functional, both latticeconstants are in poor agreement when �U−J�=0.0 eV; how-ever, the results are brought in line with experiment as �U−J� is increased. For instance, the calculated value of a agrees to within 1% of the experimentally observed value for�U−J� between 2 and 8 eV, while c agrees with experimentto within 0.5% over the same range of �U−J�.

Overall, the results indicate that the LDA functional pro-vides a better description than the PBE functional of the unitcell for finite values of �U−J�. Specifically, values of V, a ,and c that are in good agreement with experiment can beobtained in LDA+U calculations when �U−J��4.0 eV. Onthe other hand, the PBE functional does not yield resultswhere all of these quantities are in good agreement withexperiment for a single value of �U−J�. The results alsoindicate that the ab initio value of �U−J�=7.7 eV yields rea-sonably accurate values for the unit cell volume and lattice

a

c

dCr-Cr-A

dCr-Cr-B

FIG. 6. �Color online� Hexagonal unit cell of chromia. The cellcontains six Cr2O3 units. The lattice vectors a and c are labeled,along with the interatomic distances Cr-Cr-A and Cr-Cr-B, whichcorrespond to quantities evaluated below. Chromium atoms are grayand oxygens are red.

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vectors when used in conjunction with the LDA functional.

2. Atomic structure

Bulk chromia has a corundum structure and belongs to the

R3̄c space group, with 6 formula units in the hexagonal cell.The Cr atoms lie along the threefold axis at positions of±�0,0 ,ZCr� and ±�0,0 , 1

2 +ZCr�, and the O atoms reside at±�XO,0 , 1

4�, ±�0,XO, 1

4�, and ±�−XO,−XO, 1

4�, where ZCr and

XO are fractions of the lattice vectors.48 Since these param-eters define the positions of the atoms within a given hex-agonal unit cell, monitoring their values will shed light onwhether changing �U−J� causes significant changes in thefundamental structure of the material. However, since ZCrand XO are expressed relative to the lattice vectors, they donot provide insight into how the interatomic distanceschange with �U−J�. To explore these changes, the inter-atomic distances labeled Cr-Cr-A and Cr-Cr-B in Fig. 6 willalso be monitored as a function of �U−J�. The results of theanalysis of the atomic structure are summarized in Fig. 8.

The data in Fig. 8 show that XO decreases with increasingvalues of �U−J�, while ZCr increases with this parameter.These observations are true for both functionals. The errorsin these quantities are relatively large when �U−J�=0.0 eV,but the agreement is improved significantly ��1% relativeerrors at most� for values of �U−J� between 2 and 8 eV.Overall, these results indicate that the positions of the ionsrelative to the lattice vectors are not very sensitive to eitherthe XC functional or value of �U−J� used in the calculations.However, the results presented above regarding the unit celldemonstrate that the lattice vectors and volume are sensitiveto changes in �U−J�. For instance, error associated with thevolume ranges from −7% to +2% with the LDA functionalas �U−J� is varied from 0.0 to 8.0 eV. The PBE functionaloverestimates the volume by 2%–11% over the same rangeof �U−J�. Thus, although the positions of the ions relative tothe lattice vectors remain relatively constant, the interatomicdistances may vary significantly with �U−J�. This is re-flected in the values of dCr-Cr-A and dCr-Cr-B provided in Fig.8, which indicate that these distances increase with �U−J�for both functionals. When �U−J�=0.0 eV, the PBE func-tional provides an accurate estimate of the interatomic dis-tances, while the agreement with experiment decreases forthis functional with higher values of �U−J�. Meanwhile, theLDA functional underestimates dCr-Cr-A and dCr-Cr-B by 0.13and 0.14 Å, respectively, when �U−J�=0.0 eV, although theresults are brought into much better agreement with experi-ment for larger values of �U−J�. Overall, the best simulta-neous agreement with experiment for all of the parametersXO, ZCr, dCr-Cr-A, and dCr-Cr-B is achieved with the LDA func-tional when �U−J�=4.0 eV, while the ab initio value of �U−J� also yields results that are in quite good agreement withthe experimental data when used with the LDA functional.The PBE functional again does not yield values of XO, ZCr,dCr-Cr-A, and dCr-Cr-B that are all in good agreement with ex-periment for a specific value of �U−J�.

3. Bulk modulus

The bulk modulus B was evaluated by fitting the E vs Vdata obtained above to a third-order Birch-Murnaghan equa-tion of state.49,50 The calculated values of B are plotted inFig. 9 as function of �U−J� for both the LDA and PBEfunctionals along with the experimental value of thisproperty.51 The data exhibit a clear distinction between theLDA and PBE values, with the LDA results being in betteragreement with experiment over the entire range of �U−J�.The large increase in the value of B when moving from �U

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FIG. 7. �a� Volume and lengths of lattice vectors �b� a and �c� cfor the hexagonal unit cell of chromia as a function of �U−J�. Dataare provided for the LDA and PBE functionals, along with theexperimental values of these quantities �Ref. 48�.

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−J�=0.0 eV to �U−J�=2.0 eV to with LDA is most likelyrelated to the dramatic change in c mentioned above for thesame change in �U−J�. Once again, a value of �U−J�=4.0 eV used in conjunction with the LDA functional yieldsresults that are most consistent with experimental results;however, the ab initio value of �U−J� also provides a goodestimate of B with this functional.

4. Electronic structure

Properties directly related to the electronic structure of thesystem are most sensitive to the value of �U−J�. This is notsurprising given that the motivation for the DFT+U tech-nique was to correct for the inability of Kohn-Sham DFTcalculations to accurately describe the electronic structure ofstrongly correlated systems. In the following, the influenceof �U−J� on the band gap and magnetic moment will beexplored.

The calculated band gap and magnetic moment are plottedin Fig. 10 across the full range of �U−J� values for both XCfunctionals along with the experimental values.28–30,52 Egap isunderestimated by 2.5 eV with LDA and 1.9 eV with PBEwhen �U−J�=0.0 eV, illustrating the inability of DFT to ac-curately describe the electronic structure of SCEMs such aschromia. A clear improvement in this quantity is observedfor both functionals as �U−J� is increased, and agreementwith experiment is obtained for values of �U−J� in the rangeof 4–6 eV. The ab initio value of �U−J� leads to a slightoverestimation of the band gap for both functionals. How-ever, the experimental value is based on photoemission/inverse photoemission spectroscopy and thus corresponds toa measure of the difference between the ionization energyand electron affinity. DFT calculations do not directly calcu-late these properties but approximate them by the Kohn-

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a) b)

d)c)

FIG. 8. �a� XO, �b� ZCr, �c� dCr-Cr-A, and �d� dCr-Cr-B values as a function of �U−J� for the hexagonal unit cell of chromia. The valuescalculated with the LDA and PBE functionals are shown along with the experimental values for these quantities �Ref. 48�.

190

200

210

220

230

240

250

LDAPBEExperiment

86420(U – J) [eV]

Bulkmodulus[GPa]

FIG. 9. Bulk modulus of chromia as a function of �U−J�. Thevalues calculated with the LDA and PBE functionals are shownalong with the experimental value �Ref. 5�.

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Sham eigenvalues �Koopman’s theorem�. This approxima-tion neglects final state effects that will shift the band gap. Assuch, it is not surprising that calculated band gap differs fromthe gap derived from photoemission.

The data for the magnetic moment demonstrate that theDFT+U results differ significantly from the experimental re-sults, regardless of the functional or value of �U−J�. Thediscrepancies between the experimental and calculated val-ues may be due to the fact that the experimental value wasderived assuming a purely ionic description of chromia,while the calculations and experiments indicate a significantdegree of covalent bonding involving the 3d electrons of Crand the 2p electrons of oxygen.52 Such covalent interactionswould be expected to reduce the residual magnetic momenton Cr due to the low-spin coupling inherent to chemicalbonding, consistent with the lower values we calculate. Are-evaluation of the experimental data incorporating a par-tially covalent bonding model may result in closer agreementbetween the calculated and measured values.

5. Summary

The results presented above shed light on how the valueof �U−J� and the XC functional in DFT+U calculations af-fect various bulk properties of chromia. Taken as a whole,

the data indicate that superior results are obtained with theLDA functional. That is, using a value of �U−J�=4.0 eV inconjunction with LDA yields values for a variety of proper-ties that are in good agreement with experiments. This valuefor �U−J� had been selected previously through empiricalmeans in a DFT+U study of chromia.14 Meanwhile, whenDFT+U calculations are performed with the PBE functional,there does not exist a single value of �U−J� that simulta-neously provides accurate estimates of chromia bulk proper-ties.

The ab initio value of �U−J�=7.7 eV yields results thatare in good agreement with experiment when used with theLDA functional. This is quite remarkable, considering thatno experimental information was used in the selection of thisparticular value of �U−J�. As such, the ab initio approachoutlined above may prove useful in cases where minimalexperimental information is available for empirically select-ing a value of �U−J�.

VI. CONCLUSIONS

In this work, an ab initio means of evaluating the param-eters UI� and JI� was developed and tested. This approachrests on an approximate relationship between these param-eters and the Coulomb and exchange integrals evaluated inthe basis of the UHF MOs corresponding to the localizedstrongly correlated electronic states. This method does notrely on experimental information and is free of the self-interaction errors inherent in DFT calculations. Thus, thetechnique developed in this work has clear advantages overthe constrained DFT calculations and empirical fitting proce-dures commonly used to select values of UI� and JI�.

The method was used to evaluate ab initio values of UI�and JI� for the d electrons in chromia, which are localized onthe Cr atoms. This was achieved through UHF calculationson embedded CrxOy clusters. The requirements for obtainingconverged values of UI� and JI� were explored, with the keydetail being to employ a theoretical and structural modelwhich adequately captures the degree to which the localizedstates are delocalized over their neighbors. In particular, caremust be taken to ensure that the basis set used to representthe valence electrons of the atom on which the localizedstates reside contains basis functions that are sufficiently dif-fuse to interact with orbitals on neighboring atoms. A second,more obvious requirement is that the cluster employed in thecalculations must be large enough to ensure that the environ-ment in the central portion of the cluster is a good represen-tation of that in the bulk material.

The UHF calculations yielded a converged ab initio valueof �U−J�=7.7 eV for chromia. DFT+U calculations wereperformed to investigate whether this value was capable ofaccurately reproducing experimental results. The calculationswere performed over a range of values of �U−J� and withtwo different exchange-correlation functionals. Superior re-sults were clearly obtained with the LDA functional, with avalue of �U−J��4.0 eV providing the best agreement withexperiment. The results obtained when the ab initio value of�U−J�=7.7 eV was used with the LDA functional were alsoin good agreement with experiment, which is quite remark-

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

LDAPBEExperiment

86420(U – J) [eV]

E gap[eV]

Magneticmoment[μ B]

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

LDAPBEExperiment

86420(U – J) [eV]

E gap[eV]

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

LDAPBEExperiment

86420(U – J) [eV]

E gap[eV]

Magneticmoment[μ B]

2.0

2.5

3.0

3.5

4.0

LDAPBEExperiment

86420(U – J) [eV]

2.0

2.5

3.0

3.5

4.0

LDAPBEExperiment

86420(U – J) [eV]

2.0

2.5

3.0

3.5

4.0

LDAPBEExperiment

86420(U – J) [eV]

Magneticmoment[μ B]

b)

a)

FIG. 10. �a� Band gap and �b� magnetic moment of chromia asa function of �U−J�. The values calculated with the LDA and PBEfunctionals are shown along with the experimental values �Refs.28–30 and 52�.

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able considering that the value of this parameter was deter-mined without any reference to experimental data. The dem-onstrated ability of the ab initio value of �U−J� to yieldaccurate results is promising for the future application of thistechnique as a means of unambiguously selecting values of�U−J� for use in DFT+U calculations. This is particularlytrue for cases in which there is little experimental informa-tion available for empirical fitting. Currently, we are usingthis method to evaluate �U−J� for other transition metal ox-

ides. The results of those calculations will be reported infuture publications.

ACKNOWLEDGMENTS

We are grateful to the Army Research Office �E.A.C.� andthe Natural Sciences and Engineering Research Council�NSERC� of Canada �N.J.M.� for funding.

*Author to whom correspondence should be addressed;eac@princeton.edu

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