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Correlated electrons in Fe-As compounds: a quantum chemical perspective L. Hozoi and P. Fulde Max-Planck-Institut fu¨r Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany (Dated: January 12, 2009) State-of-the-artquantumchemicalmethodsareappliedtothestudyofthemultiorbitalcorrelated electronicstructureofaFe-Ascompound,therecentlydiscovered LiFeAs. Ourcalculations predict ahigh-spin,S=2,ground-stateconfigurationfortheFeions,whichshowsthattheon-siteCoulomb interactions are substantial. Also, orbital degeneracy in the (xz,yz) sector and a three-quarter filling of these levels suggest the presence of strong fluctuations and are compatible with a low 9 metallicconductivityinthenormalstate. Thelowest electron-removalstateshaveAs4pcharacter, 0 in analogy to theligand hole states in p-typecupratesuperconductors. 0 2 Electron correlations in transition-metal (TM) solid- In LiFeAs, the Fe2As2 layers are separated from each n state compounds give rise to a variety of less conven- other by double layers of Li ions. a J tional phenomena. Superconductivity in layered copper Thefirststepinourstudyisaground-staterestricted- 2 oxidesattemperaturesashighas100K,forexample,and Hartree-Fock (RHF) calculation for the periodic crystal. 1 the pairing mechanism in these systems are believed to Thiscalculationwascarriedoutwiththecrystal pack- involve strong correlation effects among the valence Cu age [7]. We employed the lattice parameters reported in l] 3d electrons. The discoveryof the Fe2+–As3− supercon- Ref. [4] and Gaussian-type, all-electron basis sets. Basis e ductingcompounds[1]isthelatestsurpriseinthefieldof sets of double-zeta quality from Towler’s crystal data - r d-electron systems. Also in this case, the experimental basis were applied for the As and Li ions [8, 9]. For t s data indicate unconventional superconductivity. “Bad- Fe,weusedabasissetoftriple-zetaquality,withsandp . t metal” conductivity in the normal state, a small carrier functionsfromTowler’sdatabasis[8]andthedfunctions a density, arelativelysmallin-plane coherencelength, and developed by Seijo et al. [10]. m Uemura scaling in the muon spin relaxation spectra [2] TheperiodicRHFcalculationyieldsafinitegapatthe - d are all reminiscent of cuprate superconductors. Fermi level. For the RHF wave function, the xz and yz n How strong correlationsare in ironpnictides is an im- components are the highest among the Fe 3d levels and o portantissue. Inafirstapproximation,thechargedistri- unoccupied;theotherFe3dorbitalsaredoublyoccupied. c bution within the Fe 3d levels and the spin state depend We employ a reference system having the x and y axes [ on the intra-orbital Coulomb interaction, the so-called rotated by 45◦ with respect to the a and b coordinates 1 HubbardU. AU valuemuchlargerthanthecrystal-field of the P4/nmm space group, such that the As NN’s of a v splittingswillfavorahigh-spinarrangementofthesixFe given Fe site are situated either in the xz or yz plane. 4 9 3d electrons, while a low-spin, closed-shell configuration On-site and inter-site correlation effects are in- 5 isexpectedforsmallvaluesofU. Thepicturecomplicates vestigated in direct space by means of multi- 1 when inter-orbital Coulomb and exchange interactions configuration complete-active-space self-consistent-field 1. are considered. Additionally, correlation effects related (CASSCF) and multireference configuration-interaction 0 to ligand p to TM d charge transfer excitations may be (MRCI)calculations. TheCASSCFwavefunctioniscon- 9 important too, as discussed for the case of another 3d6 structed as a full configuration-interaction (CI) expan- 0 system, the cobalt oxide perovskite LaCoO3 [3]. sion within a limited set of “active” orbitals [11], i.e., all : v We investigatehere the electronicstructure ofaFe-As possibleoccupationsareallowedfortheseactiveorbitals. Xi compound, the recently discovered LiFeAs [4]. Ab ini- Inthepresentstudy,theactiveorbitalsetcontainsall3d tio,wave-function-basedmethodsfrommodernquantum orbitals at a given number of Fe sites. Not only the CI r a chemistry are used in our study. We characterize the coefficients but also the orbitals are variationally opti- ground-stateelectronconfigurationforthe undopedcase mized in CASSCF, which makes this method quite flex- and provide new insight into the nature of doped car- ible. MRCI wave functions are further constructed by riers. Our results lend credence to the view [5, 6] that adding single and double excitations from the Fe 3s, 3p, correlations are moderate to strong in Fe pnictides. 3dandAs4s,4porbitalsontopofthereferenceCASSCF LiFeAs has a tetragonal crystal sructure, with the wave function, which is referred to as SD-MRCI [11]. P4/nmm space group [4]. Different from other Fe- Thequantumchemicalcomputationsareperformedon As compounds, it exhibits superconductivity at ambi- a finite cluster C including nine FeAs4 tetrahedra – a ent pressure without chemical doping, with Tc ≈18 K. “central”FeAs4unitplusfourNNandfourNNNtetrahe- The common feature of the Fe pnictide superconduc- dra – and 16Li neighbors ofthe As ions of the “central” tors is the Fe2As2 network of FeAs4 tetrahedra. Near- unit. The orbital basis entering the correlation treat- est neighbor (NN) FeAs4 units share edges, while next- ment is a set of projected RHF Wannier functions: lo- nearest-neighbor (NNN) tetrahedra share their corners. calized Wannier orbitals (WO’s) are first obtained with 2 the Wannier-Boys localization module [12] of the crys- are treated by CASSCF, on the same footing with the tal package and subsequently projected onto the set of reference Fe site. Gaussianbasisfunctionsassociatedwiththeatomicsites TheCASSCFcalculationsshowthatateachFesitethe of C [13]. Projected As 4p and Fe 3d WO’s, for ex- 3d electrons are coupled into quintet states. In contrast x xz ample, are plotted in Fig. 1. Moreover, the RHF data totheRHFresults[16],thexzandyzcomponentsarethe is used to generate an embedding potential for the nine- lowest in energy, such that the sixth electron is accomo- tetrahedra fragment C. This potential is obtained from dated into these levels. The ground-state wave function theFockoperatorintheRHFcalculation[13]andmodels is thus doubly degenerate, d3 d1 d1 d1 . The xz,yz xy 3z2−r2 x2−y2 the surroundings of the finite cluster, i.e., the remaining first excited state is also a quintet, at 0.25 eV higher en- of the crystalline lattice. It is added to the one-electron ergy, and corresponds to a (xz,yz)→xy transition [17]. Hamiltonian in the subsequent CASSCF/MRCI calcula- The fact that the relative energies of the Fe 3d lev- tions via an interface program developed in our labora- els are not consistent with simple considerations based tory [14]. The CASSCF and MRCI investigations are onligand-fieldtheoryfortetrahedralcoordination,which carried out with the molpro program [15]. predicts that the eg levels are lower than the t2g states, In a first set of CASSCF calculations, a number of waspointedoutbefore,see,e.g.,Refs.[5,18]. Thisseems nine sites are explicitly correlated,those of the reference toberelatedtothedistortionoftheAstetrahedra,which FeAs4 tetrahedron and the four NN Fe ions. This group aresqueezedinthezdirection,andthepresenceofdirect, of nine sites form the active region of the cluster, which nearest-neigbor Fe d-d orbital overlap[5, 18]. wedenoteasC . TheotherionsinC,i.e.,fourFeNNN’s, Our finding of a high-spin (HS), S = 2 ground-state A 12 As, and 16 Li sites, form a buffer region C , whose configurationagreeswiththe resultsofdynamicalmean- B roleistoensureanaccuraterepresentationofthetailsof field theory (DMFT [19]) investigations in the moderate the WO’s centered in the active part C . For our choice tostrongcouplingregimebyHauleetal. [5]andCracoet A of CB, the norms of the projected WO’s at the central al. [6]. Also,a three-quarterfilling ofthe degeneratedxz tetrahedronandNNplusNNNFesitesarenotlowerthan anddyz bands as found in our calculations is compatible 99.5% of the original crystal WO’s. While the occupied withthe lowmetallic conductivity in the normalstate of orbitals in the buffer zone are frozen, orbitals centered these systems. Systems where a pair of degenerate or- at sites in the C region (and their tails in C ) are al- bitalsaccomodatesoneelectronoroneholeoftendisplay A B lowed to relax and polarize in the CASSCF study. For very rich physics, involving couplings among the charge, this first set of CASSCF calculations, the active orbital lattice,andspindegreesoffreedom. Thestructuraltran- space consists of 25 Fe 3d orbitals, five at each active Fe sition at about 150 K in some of the Fe-As compounds site. Due to the large size of our CAS, i.e., 30 electrons [20, 21] might occur such that the degeneracy of the xz in 25 orbitals, we restrict our calculations to the case and yz orbitals is lifted. This issue remains to be inves- of high-spin(i.e., ferromagnetic)couplingsamong neigh- tigated in future work. boringFeions,althoughtheexperimentsindicateantifer- It would be instructive to determine the relative ener- romagnetic inter-site interactions. It is known, however, gies of states involving low-spin couplings at a given Fe thatthelocalchargedistributionatagivenTMsitedoes site. However,foraCASwithfiveFeionsand25orbitals, not depend on the nature of the inter-site d-d magnetic such investigations are quite difficult. Since the on-site couplings. We observed this in the case of the Cu ox- interactions are much larger than the inter-site d-d spin ide superconducting compounds, for example. Also, a couplings, the states related to low-spin configurations closed-shell representation of the Fe ions in the rest of at a given site are among the highest in a multitude of the crystalis acceptable,giventhe factthat the Fe NN’s low-spinexcitedstates. Identifying andoptimizing those states is a very tedious task. In order to access those states, we reduce then the orbital space in the CASSCF calculations to the set of five 3d functions at the central Fe site. For each of the Fe NN’s, the 3d electrons are “forced” into a t6 closed-shell configuration. 2g With this choice of the CAS, the lowest intermediate- spin (IS), S=1 and low-spin (LS), S=0 states require excitationenergiesof1.91and2.34eV,respectively,with respect to the HS ground-state (see Table I). The lowest a) b) S=0 state corresponds to a t6 orbital occupation and 2g the lowest S=1 state corresponds to a t5 e1 configura- 2g g tion, more precisely d2 d3 d1 . FIG.1: a)As4px andb)Fe3dxz RHFWO’safterprojection xy xz,yz 3z2−r2 onto the finite cluster, see text. The tails at nearby sites are OntopoftheCASSCFwavefunctions,wefurtherper- verysmall. FortheFe3dxzand3dyzWO’s,thelobespointing formed MRCI calculations with single and double exci- towards the NN Assites are less extended. tations from the Fe 3s, 3p, 3d and As 4s, 4p orbitals at 3 a value of0.25eV forthe t2g−eg energyseparation,sub- TABLEI: RelativeenergiesfortheHS,IS,andLSconfigura- stantially lower than our SD-MRCI result of 0.71 eV for tions by CASSCF and SD-MRCI calculations, see text. The the (xz,yz)→3z2−r2 excitation energy. In order to re- HS state is always the lowest. The energy of the HS state in theSD-MRCI calculation was taken as reference. produce the experimental spectra, a larger value for the crystal-fieldsplitting wouldimply a largervalue for U in Relativeenergy (eV) HS IS LS the model-Hamiltonian calculations. CASSCF 13.16 15.07 15.50 ThattheCoulombinteractionsaresubstantialinthese SD-MRCI;Fe 3s,3p,3d, As4s,4p 0 1.30 1.74 systems is best illustrated by the nature of the lowest electron-removal states. Our calculations show that for these states the additional holes populate the ligand p levels,which resembles the situationin layeredcuprates. the central tetrahedron. The SD-MRCI treatment de- In a simple picture, the formation of oxygen 2p hole creases the HS-IS and HS-LS energy splittings to 1.30 states in p-type cuprates is due to the largeCoulombre- and 1.74 eV, respectively (lowest line in Table I). Now, pulsion at the Cu 3d9 sites [27]: in order to minimize the IS-LS energy difference, for example, can be used the interaction with the Cu 3d holes, extra holes in the to extract information on the magnitude of parameters doped system enter the O 2p levels. Ab intio quantum such as the intra-orbital and inter-orbital Coulomb in- ′ chemical calculations show indeed that the first (N−1) teractions U and U and the inter-orbital exchange coupling J . The t5 e1−t6 energy difference can be ex- states have O 2p character in copper oxides, see, for ex- H 2g g 2g ′ ample, the analysis in Ref. [28]. The situation is quite pressed as ∆E = U +JH −U −∆CF. For the sake of similar in LiFeAs, where we find that for the lowest ion- simplicity, we assume that the Coulomb repulsion terms ized state the extra hole is accommodated into the As between different t2g and eg orbitals have all the same 4p orbitals. Some details are, however,different. Due to value. Auniquevalueisalsoassumedforthet2g−eg inter- the xz and yz degeneracy, the ligand hole is distributed orbitalexchangecouplings[22]. Todeterminethecrystal- field splitting ∆CF between the (xz,yz) and 3z2−r2 withequalprobabilityovertwolinearcombinationsofAs 4p orbitals, i.e., a “+” combination of slightly tilted p components, we perform SD-MRCI calculations for the x (xz,yz)→3z2−r2 excitationenergyinthe HS,S=2con- functions at As sites in the xz plane, see Fig. 2, and a combination of tilted p functions at As sites in the yz figuration. By SD-MRCI, this splitting is 0.71 eV. Since y plane. Not only the on-site Coulomb repulsion, but also ∆E=0.44 eV, see the lowest line in Table I, it follows ′ ′ the inter-site correlations are effective: in the CASSCF thatU+J −U =1.15. UsingtherelationU =U−2J , H H ′ wavefunction|Ψiforthelowestionizedstate,thelargest we find U−U ≈0.8 eV and J ≈0.4 eV. H weightisaquiredbyconfigurationswherethetwoholesin ′ As expected, U−U does not depend on the electron the(xz,yz)sectorresideinpairsof“orthogonal”orbitals, configuration at the NN Fe sites: if in the CASSCF and SD-MRCI calculations for the central tetrahedron either dyz andpx12=px1+px2 ordxz andpy34=py3+py4. Those are the first two terms in the CASSCF expan- we adopt a e4d2 configuration at the NN Fe sites, as g xy sion |Ψi=0.63|p1 p2 d2 d1 i+0.63|p2 p1 d1 d2 i+ foundintheperiodicRHFcalculation[16],thedifference x12 y34 xz yz x12 y34 xz yz 0.33|p2 p2 d1 d1 i−0.31|p1 p1 d2 d2 i+...,wherethe between U and U′ remains 0.8 eV. Our results provide a x12 y34 xz yz x12 y34 xz yz subscripts 1,2 and 3,4 refer to As sites in the xz and yz lowerlimitforthevalueoftheHubbardU. Wenotethat planes, respectively, and the HS-coupled electrons in the estimates based on density-functional (DF) calculations Fe xy, x2−y2, and z2 orbitals are omitted. Also dif- stronglydepend onthe type andsize ofthe Wannier-like ferent from cuprates, the ligand p and TM d holes are orbitalbasis. ConstrainedDFcomputationsbyAnisimov HS-coupledforthe lowest(N−1)state,with atotalspin et al. [23] using a WO basis restricted to the Fe 3d or- S=5/2. Configurationswhereboth(xz,yz)-likeholesre- bitals yield U=0.55 and J =0.5 eV. With an extended H orbitalbasisincluding As4pfunctions,itwasfoundthat U=3÷4 and J =0.8 [23]. Constrained random-phase- H approximation(RPA) calculations [24] on top of the DF data lead to U=2.2÷3.3 and J =0.3÷0.6. Values of 4 H eV were used for U in DMFT investigations by Haule et al. [5] and Craco et al. [6]. A value of U =0.3 eV was employed by Korshunov and Eremin [25] in RPA calculations for the spin response in the normal state of Fe a) b) pnictide compounds. InRef.[26],Krolletal. foundthatmodel-Hamiltonian multiplet calculations with U=1.5 and J =0.8÷0.9re- FIG.2: a)Linearcombination of4pholeorbitals atAssites H Rpreogdaurdceinwgetlhlethceryxs-traal-yfiealbdsosprplittitoinngFs,etLh2e,s3e-eadugtehosrpsecutsread. isntatthe,esxeeztpelxatn.ebf)orFeth3edxlozwoersbtiteallecfotrrotnh-erelmowoevsatl(qNua−s1ip)asrttaictlee. 4 sideattheTMsitecontributeaswelltothe(N−1)wave analysis of the (N−1) states and for a careful reading of function, see the third term in the above expansion, as themanuscript. Wealsoacknowledgefruitfuldiscussions also found in cuprates for the two holes of x2−y2 sym- with M. S. Laad and M. Gulacsi. metry [28, 29]. In the Fe-As system, however, the two holes donot occupythe samed orbital. The fourthterm in the expression of |Ψi refers to a configuration where both holes have As 4p character. [1] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, The CASSCF results for the lowest (N−1) state |Ψi J. Am.Chem. Soc. 130, 3296 (2008). wereobtainedwithaCASthatcontainsfiveFedandtwo [2] See,e.g.,H.Luetkensetal.,Phys.Rev.Lett.101,097009 As p orbitals. The 3d orbitals of the Fe NN’s were kept (2008). again in a t62g configuration. Also of interest is the na- [3] L.Hozoi, U.Birkenheuer,H.Stoll,andP.Fulde,NewJ. tureofthe nexthigher-lyingionizedstate. Thatinvolves Phys., accepted (also available at arXiv:0804.2626). a hole in a composite As 4p orbital of x2−y2 symme- [4] J. H.Tapp et al.,Phys. Rev.B 78, 060505 (2008). try extending over all four ligands around the TM site, [5] K. Haule, J. H. Shim, and G. Kotliar, Phys. Rev. Lett. 100, 226402 (2008). like the Zhang-Rice hole in cuprates [27], LS-coupled to [6] L. Craco, M. S. Laad, S. Leoni, and H. Rosner, Phys. the hole in the Fe dx2−y2 orbital. The effective occupa- Rev. B 78, 134511 (2008). tion numbers at the Fe site are d3xz,yzd1xyd13z2−r2d1x2−y2, [7] crystal 2000, University of Torino, Italy. with a HS on-site coupling, such that the total spin for [8] Availableathttp://www.tcm.phy.cam.ac.uk/∼mdt26/crystal.html. the FeAs4 tetrahedron is S=3/2. By CASSCF, the en- [9] Thecoefficientsofthemostdiffusespanddfunctionsfor ergy separation between the two hole states is 0.59 eV, the As basis set were increased from 0.1259 to 0.22 and where the active space for the higher state |Ψ′i contains from 0.407 to 0.45, respectively. [10] L. Seijo, Z. Barandiáran, and S. Huzinaga, J. Chem. six orbitals. MRCI calculations with single and double Phys. 91, 7011 (1989). excitationsfromallFe3dandAs4porbitalsattheFeAs4 [11] For a monograph, see T. Helgaker, P. Jørgensen, and tetrahedron yield a splitting of 0.75 eV. We also inves- J. Olsen, Molecular Electronic-Structure Theory (Wiley, tigated the nature of the lowest electron-addition states. Chichester, 2000). Our calculations show they have Fe (xz,yz) character. [12] C. M. Zicovich-Wilson, R. Dovesi, and V. R. Saunders, Ab initio quantum chemical calculations for determin- J. Chem. Phys. 115, 9708 (2001). [13] See, e.g., U. Birkenheuer, P. Fulde, and H. Stoll, Theor. ing the dispersion of the (N∓1)quasiparticle bands and Chem. Acc. 116, 398 (2006) and L. Hozoi et al., Phys. the Fermi-surface topology of Fe pnictide systems are Rev. B 76, 085109 (2007). left for future work. A delicate issue is in this context [14] crystal-molpro interface, Max-Planck-Institut fu¨r the treatment of renormalizationeffects due to inter-site Physik komplexerSystemeDresden, Germany. spin interactions and spin-polaron physics. In cuprates, [15] molpro 2006, Cardiff University,United Kingdom. we found [28] that such effects lead to a renormalization [16] Wedo not excludethepossibility that theRHFcalcula- of the NN hoppings by a factor of 4. In Fe pnictides, tion converged to a local minimum, i.e., the e4gd2xy state does not correspond to the global RHFminimum. the experiments indicate effective mass renormalization [17] SD-MRCI calculations were also performed for the low- factors of about 2 as compared to the DF data [30, 31]. est HS (xz,yz)→xy excited state. To make the compu- To summarize, we apply multiconfiguration CASSCF tations manageable, the Fe NN’swere treated as closed- and multireference CI methods to the study of the mul- shellions,eitherinat62g ore4gd2xy configuration.Depend- tiorbital correlated electronic structure of a Fe-As com- ingontheconfigurationoftheFeNN’s,the(xz,yz)−xy pound, the recently discovered LiFeAs. On-site and Fe- CASSCF splitting at the central site varies between 0.2 Asinter-sitecorrelationeffects aretreatedonequalfoot- and0.4eV.Ineachcase,theSD-MRCItreatmentreduces the CASSCFresult by about 0.1 eV. ing in our approach. Our calculations predict a HS [18] L. Boeri, O. V. Dolgov, and A. A. Golubov, Phys. Rev. ground-state configuration for the Fe ions, in agreement Lett. 101, 026403 (2008). withDMFTcalculations[5,6]forsystemsfromthesame [19] G. Kotliar et al.,Rev.Mod. Phys. 78, 865 (2006). family and simulations of the x-ray absorption spectra [20] C. dela Cruz et al.,Nature(London) 453, 899 (2008). [26]. The lowest electron-removal quasiparticle states [21] M. Rotter et al., Phys.Rev.B 78, 020503 (2008). haveAs4pcharacter,inanalogytotheligandholestates [22] Obviously, the Coulomb and exchange interactions are in p-type high-T cuprate superconductors. The results orbital dependentin thequantumchemical calculations. [23] V. I. Anisimov et al., arXiv:0810.2629 (unpublished). indicate that the on-site Coulomb interactions are sub- ′ [24] K.Nakamura,R.Arita,andM.Imada,J.Phys.Soc.Jpn. stantial. We find that U−U ≈0.8eV, which provides a 77, 093711 (2008). lower bound for U, and the Hund coupling constant JH [25] M. M. Korshunov and I. Eremin, Europhys. Lett. 83, is about 0.4 eV. Also, orbital degeneracy in the (xz,yz) 67003 (2008). sector and a three-quarter filling of these levels suggest [26] T. Kroll et al., Phys.Rev.B 78, 220502 (2008). the presence of strong fluctuations and are compatible [27] F. C. Zhang and T. M. Rice, Phys. Rev. B 37, 3759 with a “bad-metal” conductivity in the normal state. (1988). [28] L. Hozoi, M. S. Laad, and P. Fulde, Phys. Rev. B 78, WethankL.Cracoforusefulsuggestionsregardingthe 5 165107 (2008). [31] D. H. Lu et al.,Nature (London) 455, 81 (2008). [29] L. Hozoi and P. Fulde(unpublished). [30] A.I.Coldeaet al.,Phys.Rev.Lett.101, 216402 (2008).