TLFeBOOKELECTRONIC STRUCTURE AND MAGNETO-OPTICAL PROPERTIES OF SOLIDS TLFeBOOKThis page intentionally left blank TLFeBOOKElectronic Structure and Magneto-Optical Properties of Solids by Victor Antonov Institute of Metal Physics, Kiev, Ukraine Bruce Harmon Ames Laboratory,Iowa State University, Iowa, U.S.A.and Alexander Yaresko Max-Planck Institute for the Chemical Physics of Solids, Dresden, Germany KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW TLFeBOOKeBook ISBN:1-4020-1906-8 Print ISBN:1-4020-1905-X 2004 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print2004 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at:http://kluweronline.com and Kluwer's eBookstore at:http://ebooks.kluweronline.com TLFeBOOKContentsPrefaceixAcknowledgmentsxv1.THEORETICAL FRAMEWORK 11.1Density Functional Theory (DFT) 41.1.1Formalism 41.1.2Local Density Approximation 61.2Modications of local density approximation 81.2.1Approximations based on an exact equation for Exc 91.2.2Gradient correction 101.2.3Self-interaction correction 121.2.4LDA+Umethod 151.2.5Orbital polarization correction 231.3Excitations in crystals 261.3.1Landau Theory of the Fermi Liquid 261.3.2Greens functions of electrons in metals 301.3.3The GWapproximation 331.3.4Dynamical Mean-Field Theory (DMFT) 351.4Magneto-optical effects 391.4.1Classical optics 401.4.2MO effects 481.4.3Linear-response theory 611.4.4Optical matrix elements 672. MAGNETO-OPTICAL PROPERTIES OFd FERROMAGNETIC MATERIALS712.1Transition metals and compounds 712.1.1Ferromagnetic metals Fe, Co, Ni 71TLFeBOOKviELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS 2.1.2Paramagnetic metals Pd and Pt 752.1.3CoPt alloys 992.1.4XPt3 compounds (X=V, Cr, Mn, Fe and Co)1212.1.5Heusler Alloys 1272.1.6MnBi 1352.1.7Chromium spinel chalcogenides 1382.1.8Fe3O4 and Mg2+-, or Al3+ -substituted magnetite1412.2Magneto-optical properties of magnetic multilayers.1582.2.1Magneto-optical properties of Co/Pd systems1592.2.2Magneto-optical properties of Co/Pt multilayers1782.2.3Magneto-optical properties of Co/Cu multilayers1932.2.4Magneto-optical anisotropy in Fen/Aun superlattices2033. MAGNETO-OPTICAL PROPERTIES OF fFERROMAGNETIC MATERIALS2293.1Lantanide compounds 2293.1.1Ce monochalcogenides and monopnictides2303.1.2NdX (X=S, Se, and Te) and Nd3S42393.1.3Tm monochalcogenides 2483.1.4Sm monochalcogenides 2653.1.5SmB6 and YbB122733.1.6Yb compounds 2863.1.7La monochalcogenides 3083.2Uranium compounds. 3203.2.1UFe23223.2.2U3X4 (X=P, As, Sb, Bi, Se, and Te) 3273.2.3UCu2P2, UCuP2, and UCuAs23323.2.4UAsSe and URhAl 3383.2.5UGa23413.2.6UPd33464. XMCD PROPERTIES OF d ANDfFERROMAGNETIC MATERIALS3574.13d metals and compounds 3584.1.1XPt3 Compounds (X=V, Cr, Mn, Fe, Co and Ni).3594.1.2Fe3O4 and Mn-, Co-, or Ni-substituted magnetite.3794.2Rare earth compounds. 3954.2.1Gd5(Si2Ge2 compound) 3974.3Uranium compounds. 4014.3.1UFe2404TLFeBOOKContents vii4.3.2 US, USe, and UTe4124.3.3 UXAl (X=Co, Rh, and Pt)4204.3.4 UPt3 4284.3.5 URu2Si2 4344.3.6 UPd2Al3 and UNi2Al3 4384.3.7 UBe13 4444.3.8 Conclusions451Appendices453ALinear Method of MT Orbitals453A.1Atomic Sphere Approximation 453A.2MT orbitals 455A.3Relativistic KKRASA 456A.4Linear Method of MT Orbitals (LMTO) 460A.4.1 Basis functions460A.4.2 Hamiltonian and overlap matrices462A.4.3 Valence electron wave function in a crystal463A.4.4 Density matrix463A.5Relativistic LMTO Method 464A.6Relativistic Spin-Polarized LMTO Method 466A.6.1 Perturbational approach to the relativistic spin-polarizedLMTO method467BOptical matrix elements469B.1ASA approximation 469B.2Combined-correction term. 475References477Index527TLFeBOOKThis page intentionally left blank TLFeBOOKPrefaceIn 1845Faraday discovered[1] thatthe polarizationvectorof linearlypo-larized light is rotated upon transmission through a sample that is exposed to a magnetic eld parallel to the propagation direction of the light.About 30 years later,Kerr [2]observedthatwhenlinearlypolarizedlightisreectedfroma magnetic solid, its polarizationplane also becomes rotated over a small angle withrespecttothatoftheincidentlight.Thisdiscoveryhasbecomeknown asthemagneto-optical(MO)Kerreffect.Sincethen,manyothermagneto-opticaleffects,as for example the Zeeman,Voigt and Cotton-Moutoneffects [3], have been discovered.These effects all have in common that they are due to a different interaction of left- and right-hand circularly polarized light with a magnetic solid.The Kerr effect has now been known for more than a century, but it was only in recent times that it became the subject of intensive investiga-tion.The reason for this recent development is twofold:rst, the Kerr effect is relevant for modern data storagetechnology,becauseit can be used to read suitablystoredmagneticinformationin an opticalmanner[4,5] and second, theKerreffecthasrapidlydevelopedintoanappealingspectroscopictoolin materials research.The technological research on the Kerr effect was initially motivated by the search for good magneto-optical materials that could be used as informationstorage media.In the course of this research,the Kerr spectra of many ferromagnetic materials were investigated.An overview of the exper-imentalandtheoreticaldatacollectedontheKerreffectcanbefoundinthe review articlesby Buschow[6],Reim and Schoenes[7],Schoenes[8],Ebert [9], Antonov et al.[10, 11], and Oppeneer [12]. ThequantummechanicalunderstandingoftheKerreffectbeganasearly as 1932 when Hulme [13] proposed that the Kerr effect could be attributed to spin-orbit(SO)coupling(see,alsoKittel[14]).Thesymmetrybetweenleft-and right-hand circularly polarized light is broken due to the SO coupling in a magneticsolid.This leadsto differentrefractiveindicesfor the two kinds of circularlypolarizedlight,sothatincidentlinearlypolarizedlightisreected TLFeBOOKxELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS withellipticalpolarization,andthemajorellipticalaxisisrotatedbytheso called Kerr angle from the original axis of linear polarization.The rst system-atic study of the frequency dependent Kerr and Faraday effects was developed by Argyres [15] and later Cooper presented a more general theory using some simplifying assumptions [16]. The very powerful linear response techniques of Kubo [17] gave generalformulasfor the conductivitytensor which are being widely used now. A general theory for the frequency dependent conductivity of ferromagnetic (FM) metals over a wide range of frequencies and temperatures was developed in 1968 by Kondorsky and Vediaev [18]. The rst ab initio calculation of MO properties was made by Callaway with co-workers in the middle of the 1970s [19, 20].They calculated the absorption parts of the conductivitytensor elements xx and xy for pure Fe and Ni and obtained rather good agreement with experiment.The main problem afterward was the evaluation of the complicated formulas involving MO matrix elements using electronicstates of the real FM system.With the tremendousincreases incomputationalpowerandtheconcomitantprogressinelectronicstructure methods the calculationof such matrix elements became possible,if not rou-tine.Subsequentlymanyearlier,simpliedcalculationshavebeenshownto be inadequate,and only calculationsfrom rst-principleshave provided,on the whole, a satisfactorydescriptionof the experimental results.The existing difculties stem either from problems using the local spin density approxima-tion(LSDA)todescribetheelectronicstructureofFMmaterialscontaining highly correlated electrons,or simply from the difculty of dealing with very complex crystal structures. In recent years, it has been shown that polarized x rays can be used to deter-mine themagneticstructureof magneticallyorderedmaterialsby x-rayscat-tering and magnetic x-ray dichroism.Now-days the investigation of magneto-optical effects in the soft x-ray range has gained great importance as a tool for theinvestigationofmagneticmaterials[21].Magneticx-rayscatteringwas rstobservedinantiferromagneticNiO,wherethemagneticsuperlatticere-ectionsare decoupledfromthestructuralBraggpeaks[22].The advantage over neutron diffraction is that the contributions from orbital and spin momen-tumareseparablebecausetheyhaveadifferentdependenceupontheBragg angle [23].Also in ferromagnets and ferrimagnets, where the charge and mag-netic Bragg peaks coincide, the magnetic structure can be determined because the interference term between the imaginary part of the charge structure factor and the magneticstructurefactorgives a largeenhancementof the scattering cross section at the absorption edge [24]. In1975thetheoreticalworkofErskineandSternshowedthatthex-ray absorptioncouldbe usedtodeterminethex-raymagneticcirculardichroism (XMCD)intransitionmetalswhenleft- andrightcircularlypolarizedx-ray beamsareused[25].In1985Tholeetal.[26]predictedastrongmagnetic TLFeBOOKxiPREFACE dichroismintheM4,5 x-rayabsorptionspectraofmagneticrare-earthmate-rials,for which they calculatedthe temperatureand polarizationdependence. A yearlaterthis MXD effectwas conrmedexperimentallyby van der Laan etal.[27]attheTbM4,5-absorptionedgeofterbiumirongarnet.Thenext year Schtz et al.[28] performed measurements using x-ray transitions at the Kedge of ironwith circularlypolarizedx-rays,wherethe asymmetryin ab-sorptionwas found to be of the order of 104 .This was shortlyfollowed by theobservationofmagneticEXAFS[29].Atheoreticaldescriptionforthe XMCDattheFeK-absorptionedgewasgivenbyEbertetal.[30]usinga spin-polarizedversion of relativistic multiple scattering theory.In 1990 Chen etal.[31]observedalargemagneticdichroismattheL2,3 edgeofnickel metal.Also cobaltand ironshowedhugeeffects,which rapidlybroughtfor-wardthestudyofmagnetic3dtransitionmetals,whichareoftechnological interest.Full multiplet calculations for 3d transition metal L2,3 edges by Thole and van der Laan [32] were conrmed by several measurements on transition metal oxides.First considered as a rather exotic technique, MXD has now de-veloped as an important measurement technique for local magnetic moments. WhereasopticalandMO spectraareoftenswampedby too many transitions between occupied and empty valence states, x-ray excitations have the advan-tagethatthecorestatehasapurelylocalwavefunction,whichofferssite, symmetry,and elementspecicity.XMCD enablesa quantitativedetermina-tionofspinandorbitalmagneticmoments[33],element-specicimagingof magneticdomains[34]orpolarizationanalysis[35].Recentprogressinde-vices for circularly polarized synchrotron radiation have now made it possible to explore the polarization dependenceof magnetic materials on a routine ba-sis.Results of correspondingtheoretical investigations published before 1996 can be found in Ebert review paper [9]. The aim of this book is to review of recent achievementsin the theoretical investigationsof the electronicstructure,optical,MO, and XMCD properties of compounds and multilayered structures. Chapter 1 of this book is of an introductory character and presents the the-oretical foundations of the band theory of solids such as the density functional theory (DFT) for ground state properties of solids. We also present the most frequently used in band structure calculationslo-cal density approximation (LDA) and some modications to the LDA (section 1.2),such as gradientcorrection,self-interectioncorrection,LDA+U method and orbital polarization correction.Section 1.3 devoted to the methods of cal-culating the elementary excitations in crystals.Section 1.4 describes different magneto-optical effects and linear response theory. Chapter 2 describes the MO properties for a number of 3d materials. Section 2.1 is devoted to the MO properties of elemental ferromagnetic metals (Fe, Co, and Ni) and paramagnetic metals in external magnetic elds (Pd and Pt). Also TLFeBOOKxiiELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS presentedareimportant3dcompoundssuchasXPt3 (X=V,Cr,Mn,Fe,and Co), Heusler alloys, chromium spinel chalcogenides, MnB and strongly corre-lated magnetite Fe3O4.Section 2.2 describes the recent achievements in both the experimental and theoretical investigations of the electronic structure, opti-cal and MO properties of transition metal multilayered structures (MLS). The most important from the scientic and the technological point of view materi-als are Co/Pt, Co/Pd, Co/Cu, and Fe/Au MLS. In these MLS, the nonmagnetic sites(Pt,Pd,CuandAu)exhibitinducedmagneticmomentsduetothehy-bridization with the transition metal spin-polarized 3d states.The polarization isstrongatPtandPdsitesandweakatnoblemetalsitesduetocompletely occupied d bands in the later case.Also of interest is how the spin-orbit inter-action of the nonmagnetic metal (increasing along the series of Cu, Pd, Pt, and Au) inuences the MO response of the MLS. For applications a very important question is how the imperfection at the interface affects the physical properties of layered structures including the MO properties. Chapter3ofthebookpresentstheMOpropertiesoffbandferromag-netic materials.Sections 3.1 devoted to the MO properties of 4fcompounds: Tm, Nd, Sm, Ce, and La monochalcogenides, some important Yb compounds, SmB6 and Nd3S4.In Section 3.2 we consider the electronic structure and MO propertiesofthefollowinguraniumcompounds:UFe2, U3X4 (X=P, As,Sb, Bi, Se, and Te), UCu2P2, UCuP2, UCuAs2, UAsSe, URhAl, UGa2, and UPd3. Withinthetotalgroupofalloysandcompounds,wediscusstheirMOspec-trainrelationshipto:thespin-orbitcouplingstrength,themagnitudeofthe localmagneticmoment,the degreeof hybridizationin the bonding,the half-metallic character, or, equivalently, the Fermi level lling of the bandstructure, the intraband plasma frequency, and the inuence of the crystal structure. In chapter 4 results of recent theoretical investigations on the MXCD in var-ious representative transition metal 4fand 5fsystems are presented.All these investigations deal exclusively with the circular dichroism in x-ray absorption assumingapolargeometry.Section4.1presentstheXMCD spectraofpure transitionmetals,someferromagnetictransition-metalalloysconsistingofa ferromagnetic3d elementandPt atomas wellas Fe3O4 compoundandMn-,Co-,orNi-substitutedmagnetite.Section4.1brieyconsiderstheXMCD spectra in Gd5(Si2Ge2), a promising candidate material for near room temper-ature magnetic refrigeration.Section 4.3 contains the theoreticallycalculated electronic structure and XMCD spectra at M4,5 edges for some prominent ura-nium compounds, such as, UPt3, URu2Si2, UPd2Al3, UNi2Al3, UBe13, UFe2, UPd3,UXAl (X=Co,Rh,andPt),andUX (X=S, Se,andTe).The rst ve compoundsbelongtothefamilyofheavy-fermionsuperconductors,UFe2 is widelybelievedto be an exampleof compoundwith completelyitinerant5f electrons, while UPd3 is the only known compound with completely localized 5felectrons. TLFeBOOKPREFACExiii Appendix A provides a description of the linear mufn-tin method (LMTO) of band theory including its relativistic and spin-polarized relativistic versions based on the Dirac equation for a spin-dependentpotential.Appendix B pro-videsadescriptionoftheopticalmatrixelementsintherelativisticLMTO formalism. TLFeBOOKThis page intentionally left blank TLFeBOOKAcknowledgmentsThe authorsare greatly indebtedto Dr.A. Perlov from Mnich University and Prof.Dr.S. Uba and Prof.Dr.L. Uba fromthe Instituteof Experimen-tal Physics of Bialystok University,a long-standingcollaborationwith whom stronglycontributedto creatingthe pointof viewon thecontemporaryprob-lems in themagneto-opticsthatis presentedinthisbook.V.N. Antonovand A.N.YareskowouldliketothankProf.Dr.P.Fuldeforhisinterestinthis work and for hospitality received during their stay at the Max-Planck-Institute in Dresden.We are also gratefull to Prof. Dr. P. Fulde and Prof. Dr. H. Eschrig for helpful discussions on novel problems of strongly correlated systems. This work was partly carried out at the Ames Laboratory, which is operated for the U.S. Department of Energy by Iowa State University under Contract No. W-7405-82.This work was supported by the Ofce of Basic Energy Sciences of the U.S. Department of Energy.V.N. Antonov greatfully acknowledges the hospitality during his stay at Ames Laboratory. TLFeBOOKThis page intentionally left blank TLFeBOOKChapter 1 THEORETICAL FRAMEWORK Determination of the electronic structure of solids is a many-body problem thatrequirestheSchrdingerequationtobesolvedforanenormousnumber of nuclei and electrons.Even if we managed to solve the equation and nd the complete wave function of a crystal, we face, the not less complicated problem of determining how this function should be applied to the calculation of phys-ically observablevalues.While the exact solution of the many-body problem is impossible,it is also quite unnecessary.To theoreticallydescribe the quan-tities of physical interest, it is required to know only the energy spectrum and severalcorrelationfunctions(electrondensity,paircorrelationfunction,etc.) which depend on a few variables. Since only lower excitation branches of the crystal energy spectrum are im-portantforourdiscussion,wecanintroducetheconceptofquasiparticlesas theelementaryexcitationsofthesystem.Therefore,ourproblemreducesto deningthe dispersioncurvesofthequasiparticlesandanalyzingtheirinter-actions.Two typesof quasiparticlesare of interest:fermions(electrons)and bosons (phonons and magnons). Theproblemthusformulatedisstillrathercomplicatedandneedsfurther simplication.The rst is to note that the masses of ions M , forming the lat-tice,considerablyexceedtheelectronmass.Thisgreatdifferenceinmasses givesrisetoalargedifferenceintheirvelocitiesandallowsthefollowing assumption:anyconcentrationofnuclei(evenanon-equilibriumone)may reasonablybeassociatedwithaquasi-equilibriumcongurationofelectrons which adiabaticallyfollow the motion of the nuclei.Hence, we may consider theelectronstobeinaeldofessentiallyfrozennuclei.ThisistheBorn Oppenheimer approximation,and it justies the separationof the equation of motionfor theelectronsfromthatof thenuclei.Althoughexperienceshows thattheinteractionsbetweenelectronsandphononshavelittleeffectonthe TLFeBOOK 2ELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS electronenergyandtheshapeoftheFermisurface,thereexistmanyother properties which require that the electronphonon interaction be accounted for evenintherstapproximation.Thesepropertiesinclude,transportandthe phenomenon of superconductivity. In this book, we shall use the adiabatic approximation and consider only the electronsubsystem.The readerinterestedin the electronphononinteraction in crystals may refer to Ref. [36]. Wealsousetheapproximationofanideallattice,meaningthattheions constitutingthe lattice are arranged in a rigorously periodic order.Hence, the problemsrelatedtoelectronstatesinrealcrystalswithimpurities,disorder, and surfaces are not considered. In these approximations, the non-relativistic Hamiltonian of a many-electron system in a crystal is 2 H= i 2 +V (ri) +,(1.1) |ri rj |iii,j wheretherst termisthesumofthekineticenergiesoftheindividualelec-trons,theseconddenestheinteractionofeachoftheseelectronswiththe potentialgeneratedbythenuclei,andthenaltermcontainstherepulsive Coulomb interaction energy between pairs of electrons. Two important properties of our electron subsystemshould be pointed out. First, the range of electron density of all metals is such that the mean volume for one electron, proportional to re = [3/4]1/3 , is in the range 1 6.It can be shown thatthis valueis approximatelythe ratioof the potentialenergyof particles to their mean kinetic energy.Thus, the conduction electrons in metals are not an electron gas but rather a quantum Fermi liquid. Second,theelectronsinametalarescreenedataradiussmallerthanthe lattice constant.After the papers by Bohm and Pines [37, 38], Hubbard [39], GellMann,Brueckner[40]werepublished,itbecameclearthatthelong rangeportionof the Coulombinteractionis responsiblemainlyfor collective motionsuchasplasmaoscillations.Theirexcitationenergiesarewellabove the ground state of the system.As a result, the correlated motion of electrons due to their Coulomb interactions is important at small distances (in some cases as small as 1 ), but at larger distances an average or mean eld interaction is a good approximation. Therstofthesepropertiesdoesnotallowustointroducesmallparame-ters.Hence,wecannotusestandardperturbationtheory.Thismakestheo-retical analysis of an electron subsystem in metals difcult and renders certain approximationspoorly controllable.Thus, the comparisonof theoreticalesti-mations with experimental data is of prime importance and a long tradition. The secondpropertyof the subsystempermitsus to introducetheconcept of weakly interacting quasiparticles, thus, to use Landaus idea [41] that weak TLFeBOOK3THEORETICAL FRAMEWORK excitationsofanymacroscopicmany-fermionsystemexhibitsingleparticle likebehavior.Obviously,for varioussystems,theenergyrangewherelong lived weaklyinteractingparticlesexist will be different.In many metalsthis range is rather signicant, reaching 5 10 eV. This has enabled an analysis of the electronic properties of metals based on single-particle concepts. Incalculatingbandstructures,thecrucialproblemischoosingthecrystal potential.WithintheHartree-Fockapproximationthepotentialmustbede-termined self-consistently.However, the exchange interactionleads to a non-local potential,which makes the calculationsdifcult.To avoid the difculty Slater [42](1934)proposedto use a simpleexpression,whichis validin the case of a free-electron gas when the electron density is uniform.Slater sug-gested that the same expressionfor the local potential can also be used in the case of the non-uniformdensity (r).Subsequently(1965),Slater [42] intro-ducedadimensionlessparametermultiplyingthelocalpotential,whichis determinedbyrequiringthetotalenergyoftheatomcalculatedwiththelo-cal potential be the same as that obtained within the Hartree-Fock approxima-tion.This method is known as the X-method.It was widely used for several decades.It was aboutthistime thata rigorousaccountof theelectroniccor-relation became possible in the framework of the density functional theory.It was proved by Hohenberg and Kohn [43] (1964) that ground state properties of a many-electron system are determined by a functional depending only on the densitydistribution.Kohn and Sham [44,45] (1965,1966)then showed that the one-particle wave functions that determine the density (r) are solutions of a Schrdingerlikeequation,thepotentialbeingthe sumof theCoulombpo-tential of the electron interacting with the nuclei, the electronic charge density, and an effective local exchange-correlation potential, Vxc. It has turned out that in many casesof practicalimportancetheexchange-correlationpotentialcan bederivedapproximatelyfromtheenergyoftheaccuratelyknownelectron-electroninteractioninthehomogeneousinteractingelectrongas(leadingto the so called local density approximation, LDA). Thedensityfunctionalformalismalongwiththelocaldensityapproxima-tion has been enormously successful in numerous applications, however it must be modied or improved upon when dealing with excited state properties and with strongly correlated electron systems, two of the major themes of this book. Therefore after briey describing the DFTLDA formalism in section 1.1, we describein section1.2 several modicationsto the formalismwhich are con-cerned with improving the treatment of correlated electron systems.In section 1.3 several approaches dealing with excited state properties are presented. The approachespresentedin sections1.2 and 1.3 are not general,nal so-lutions.Indeedthetopicsofexcitationsincrystalsandcorrelatedelectron systems continue to be highly active research areas.Both topics come together in the study of magnetsoptical properties. TLFeBOOK 4ELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS 1.1Density Functional Theory (DFT) Themotionofelectronsincondensedmediaishighlycorrelated.Atrst glance,thisleadstotheconclusionthatitisimpossibletodescribesucha system in an approximation of essentially independent particles.However, we can use a model system (electron gas) of interacting particles,where the total energyEandtheelectrondensity(r) approximatesimilarfunctionsofthe real system, and the effects of interactions among electrons are then described by an effective eld.This is the essence of practical approaches until utilizing density functional theory (DFT). 1.1.1Formalism TheDFTisbasedontheHohenbergandKohntheorem[43]wherebyall properties of the ground state of an interactingelectron gas may be described by introducingcertainfunctionalsof theelectrondensity(r).The standard Hamiltonian of the system is replaced by [44] E[] = dr(r)vext (r) +drdr (r)(r

)+ G[],(1.2) |r r

|wherevext(r)istheexternaleldincorporatingtheeldofthenuclei;the functionalG[]includesthekineticandexchangecorrelationenergyofthe interacting electrons.The total energy of the system is given by the extremum of the functional E[]=0 (r) = 0, where 0 is the distributionof the ground state electron charge.Thus, to determine the total energy Eof the system we need not know the wave function of all the electrons, it sufces to determine a certain functional E[] and to obtain its minimum.Note that G[] is universal and does not depend on any external elds. This concept was further developed by Sham and Kohn [45] who suggested a form for G[] G[] = T [] + Exc[].(1.3) Here T [] is the kinetic energy of the system of noninteractingelectrons with density(r);thefunctionalExc[] containsthemanyelectroneffectsofthe exchange and the correlation. Let us write the electron density as N2(r) = |i(r)| ,(1.4) i=1 whereNisthenumberofelectrons.Inthenewvariablesi (subjecttothe usual normalization conditions), 2ZI (r

) 2 + 2dr + Vxc(r)i = ii .(1.5) |r RI | |r r

|I TLFeBOOK 5THEORETICAL FRAMEWORK Here,RI isthepositionofthenucleusIofchargeZI ; i aretheLagrange factors forming the energy spectrum of singleparticlestates.The exchange correlation potential Vxc is a functional derivative Exc []Vxc(r) = .(1.6) (r) From(1.5)wecanndtheelectrondensity(r) andthetotalenergyofthe ground state of the system. AlthoughtheDFTisrigorouslyapplicableonlyforthegroundstate,and the exchangecorrelationenergyfunctionalat presentis onlyknown approx-imately,theimportanceofthistheorytopracticalapplicationscanhardlybe overestimated.It reduces the manyelectron problem to an essentially single-particle problem with the effective local potential 2ZI (r

)V (r) = + 2dr + Vxc(r).(1.7) |r RI | |r r

|I Obviously, (1.5) should be solved selfconsistently, since V (r) depends on the orbitals i(r) that we are seeking. Equations (1.25) are exact in so far as they dene exactly the electron den-sity and the total energy when an exact value of the functional Exc[] is given. Thus,thecentralissueinapplyingDFTisthewayinwhichthefunctional Exc[] is dened.It is convenient to introduce more general properties for the chargedensitycorrelationdeterminingExc.TheexactexpressionofExc[] for aninhomogeneouselectrongas may bewrittenas a Coulombinteraction betweentheelectronwithitssurroundingexchangecorrelationholeandthe charge density xc(r, r r) [46, 47]: 1

xc(r, r r) .(1.8)Exc[] = dr(r)dr2|r r

| In (1.8), xc is dened as 2 xc(r, r r) = (r

)d[g(r, r ; ) 1],(1.9) 0 where g(r, r

; ) is the pair correlation function; is the coupling constant. The Exc[] is independentof the actualshapeof the exchangecorrelation hole.Making the substitution R = r r is can be shown that [48] Exc[] = 4 dr(r)RdR xc(r, R)(1.10) and depends only on the spherical average of xc, 1 xc(r, R) = dxc(r, R).(1.11) 4 TLFeBOOK

6ELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS This means that the Coulomb energy depends only on the distance, not on the direction.Moreover, the hole charge density satises the sum rule [48] 4 R2dR xc(r, R) = 2.(1.12) Thisimpliesthattheexchangecorrelationholecorrespondstoanetcharge around the electron of one. 1.1.2Local Density Approximation In band calculations,usually certain approximationsfor the exchangecor-relationpotentialVxc(r) areused.Thesimplestandmostfrequentlyusedis the local density approximation (LDA), where xc(r, r r) has a form similar to that for a homogeneouselectron gas, but with the density at every point of the space replaced by the local value of the charge density,(r)for the actual system: 2 xc(r, r r) = (r)d[g0(|r r |, , (r)) 1],(1.13) 0 where g0(|r r

|, , (r)) isthepaircorrelationfunctionof a homogeneous electronsystem.Thisapproximationsatisesthesumrule(1.12),whichis one of its basic advantages.Substituting(1.13) into (1.8) we obtain the local density approximation [44]: Exc[] = (r)xc ()dr.(1.14) Here, xc is the contribution of exchange and correlation to the total energy (per electron) of a homogeneous interacting electron gas with the density (r). Thisapproximationcorrespondstosurroundingeveryelectronbyanexchange correlation hole and must, as expected, be quite good when (r) varies slowly. Calculations of xc by several techniques led to results which differed from one another by a few percent [49].Therefore, we may consider the quantity xc () to be reasonably well. An analytical expression for xc() was given by Hedin and Lundqvist [50].In the local density approximation, the effective potential (1.7) is 2ZI (r

)V (r) = + 2dr + xc(r),(1.15) |r RI | |r r

|I wherexc(r) istheexchangecorrelationpartofthechemicalpotentialofa homogeneous interacting electron gas with the local density (r), dxc() xc(r) = .(1.16) d TLFeBOOK 7THEORETICAL FRAMEWORK For spinpolarized systems, the local spin density approximation [45, 51] is used Exc[+, ] = (r)xc (+(r), (r))dr.(1.17) Here, xc (+, ) is the exchangecorrelationenergy per electron of a homo-geneoussystemwiththedensities+(r)and(r)forspinsupanddown, respectively. Note that the local density approximation and local spin density approxima-tioncontainnottingparameters.Furthermore,sincetheDFT hasnosmall parameter, a purely theoretical analysis of the accuracy of different approxima-tions is almost impossible.Thus, the applicationof any approximationto the exchangecorrelation potential in the real systems is most frequently validated by an agreement between the calculated and experimental data. Therearetwodifferenttypesofproblemsinquantummechanicalmany particlesystems:macroscopicmanyparticlesystemsandatomicsystemsor clustersofseveralatoms.MacroscopicsystemscontainN1023 particles andeffectsoccurringona N1 or N1/3 scalearenegligiblysmall.Atoms and clusters of N10 to 100 do not allow neglect of properties that scale with N1 and N1/3 .In addition,a strong change in electron density is observed on the boundary of a free atom or a cluster, while the electron density in metals on the atom periphery is a slowly varying function of the distance. For nite systems(atoms and clusters),the error in the total energycalcu-lated by the local density approximation is usually 5 to 8%.Even for a simple systemsuchasahydrogenatom,thetotalenergyiscalculatedto0.976 Ry insteadof1.0 Ry[52].Therefore,thecaseofnitemanyparticlesystems requires some other approach. Because metals are macroscopic manyparticlesystems, the applicationof the local density approximation yields sufciently good results for the ground state energy and electron density. TheDFTincludestheexchangeandcorrelationeffectsinamorenatural wayincomparisonwithHartree-Fock-Slatermethod.Here,theexchange correlation potential Vxc may be represented as Vxc(r) = (re)VGKS(r),(1.18) where VGKS is the GasparKohnSham potential, and re is given by 3

1/3 re(r) = (r).(1.19) 4 This parameter corresponds, in order of magnitude, to the ratio of the potential energy of particles to their average kinetic energy. In(1.18),theexchangeeffectsareincludedinVGKS,whileall correlation effects are containedin the factor (re) that dependson the electrondensity. TLFeBOOK8ELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS Wigner[53]suggestedthatthecorrelationenergyforintermediateelectron densitiescouldbeobtainedbyinterpolatingbetweenthelimitingvaluesof high and low densities of an electron gas: c = 0.88/re + 7.8.(1.20) With such c, we obtain W (re) =1 + [0.9604re (re + 5.85)/(re + 7.8)2].(1.21) Hedin and Lundqvist[50] used the results given in [54] to estimate c and obtained HL(re) = 1 + 0.0316re ln(1 + 24.3/re ).(1.22) Moreaccurateparametrizationformulasforc werederived[5557]by combiningrandomphaseapproximation(RPA)resultswiththettothe Greens-function Monte-Carlo results of Ceperly and Alder [58]. 1.2Modications of local density approximation Many calculations in the past decade have demonstrated that the local-spin-density approximation (LSDA) gives a good description of ground-state prop-ertiesofsolids.TheLSDAhasbecomethedef actotoolofrst-principles calculations in solid-state physics, and has contributed signicantly to the un-derstandingofmaterialpropertiesatthemicroscopiclevel.However,there are some systematic errors which have been observed when using the LSDA, such as the overestimation of cohesive energies for almost all elemental solids, and the related underestimation of lattice parameters in many cases. The LSDA also fails to correctly describe the properties of highly correlated systems, such as Mott insulators and certain f -band materials.Even for some "simple" cases the LSDA has been found wanting, for example the LSDA incorrectly predicts that for Fe the fcc structure has a lower total energy than the bcc structure. The early work of Hohenberg, Kohn, and Sham introduced the local-density approximation,butitalsopointedouttheneedformodicationsinsystems wherethedensityisnothomogeneous.OnemodicationsuggestedbyHo-henberg and Kohn [43] was the approximation

r + r

1 Exc = ELDA drdr

Kxc r r

, [(r)(r

)]2 ,(1.23) xc 4 2 wherethekernelKxc isrelatedtothedielectricfunctionofahomogeneous medium.This approximation is exact in the limit of weak density variations (r) = 0 + (r),(1.24) TLFeBOOK

9THEORETICAL FRAMEWORK where (r) 0, but the results for real systems are not encouraging.For free atoms the energy is innite, indicating that the sum rule (1.14) is not sat-ised. ThereareseveralalternativemethodstoimprovetheLSDapproximation described below. These include the approximations based on an exact equation forExc,thegradientcorrection,theself-interactioncorrection,theLDA+U method, and orbital polarization corrections. 1.2.1Approximations based on an exact equation for Exc The equation for the exchange-correlation energy (1.10) shows that the dif-ferences between the exact and the approximate exchange holes are largely due to the non-sphericalcomponentsof the hole.Since these do not contribute to Exc, total energies and total energy differences can be remarkably good, even in systems where the density distributionis far from uniform.In the LDA we assume that the exchange-correlationhole xc(r, r r

) depends only on the charge density at the electron.It would be more appropriate to assume [59, 60] that xc depends on a suitable average (r), xc(r, r r) = (r)d{gh [r r , , (r)] 1}.(1.25) Itispossibletochoosetheweightfunctionthatdetermines(r) sothatthe functionalreducestotheexactresultinthelimitofalmostconstantdensity. Approximation(1.25)satisesthesumrule(1.11).Somewhatdifferentpre-scriptions for the weight function have been proposed in [59, 60].The approx-imation gives improved results for total energies of atoms. Analternativeapproximationisobtainedifwekeeptheproperprefactor (r

) in Eq.1.9,leadingto the so-calledweighteddensity(WD) approxima-tion: xc(r , r r

) = (r

)G[| r r |, (r)],(1.26) where (r) ischosentosatisfythesumrule(1.12)[6062].Differentforms have been proposedfor G(r, ).Gunnarssonand Jones[48] proposean ana-lytical form of G(r, ).They assume that 5G(r, ) = C(){1 exp[()/ | r | ]},(1.27) where Cand are parameters to be determined.The functional G behaves as |r|5 for large distances,which is needed to obtain an image potential.For ahomogeneoussystemwithdensity,werequirethatthemodelfunctional shouldbothfulllthesumrulefor(r) = andgivetheexactexchange-correlation energy.This leads to two equations: drG(| r |, ) = 2,(1.28) TLFeBOOK 10ELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS 1 dr | r | G(| r |, ) = xc(),(1.29) which are sufcient to determine the two parameters C() and (). Thisfunctionalisexactinseverallimitingcases:(1)forahomogeneous system; (2) for one-electron systems, such as the hydrogen atom, where it gives an exact cancellation of the electron self-interactions; (3) for an atom, where it gives the correct behavior of the exchange-correlationenergy density far from the nucleus, xc = 1 ; (4) for far outside the surface, where it gives the image r potentialxc(z) = 1 .The LSDA gives qualitativelyincorrectanswers for 2z cases (3) and (4),and the cancellationin case (2) is only approximate.Since (2)issatised,thisapproximationprovidesa"self-interactioncorrection"in the sensethatwe shalldiscussbelow.Barstel et al.[63]and Przybylskiand Barstel [64,65] have used variationsof theWD approximationin studiesof Rh, Cu, and V. For Rh they found that the WD approximation correctly shifts unoccupiedbandsupward.For Cu theyobtainedanimproveddescriptionof the d-band and Fermi surface, while for Vthe error in the LSDA for the Fermi surface was substantially over corrected.For semiconductors it was found that thereiseitheronlylittle(Si,[66])orno(GaAs,[67])improvementoverthe LSDA for the band gap. 1.2.2Gradient correction An early attempt to improve the LSDA was the gradient expansion approx-imation(GEA)[43,44].Calculationsforatoms[68,69]andajelliumsur-face[70]showed,however,thattheGEA doesnotimprovetheLSDA ifthe abinitiocoefcientsofthegradientcorrection[68,71,72]areused.The errorsintheGEAwerestudiedbyLangrethandPerdew[70,46]andlater byPerdewandco-workers[7375].Itwasshownthatthesecondorderex-pansions of the exchange and correlation holes in gradients of the density are fairlyrealisticclosetotheelectron,butnotfaraway.Intheoriginalwork ofLangrethandco-workers[70,76]ageneralizedgradientapproximation (GGA)wasconstructedviacutoffofthespurioussmall-wave-vectorcontri-bution to the Fourier transform of the second order density gradient expansion fortheexchange-correlationholearoundanelectron.LaterPerdewandco-workersarguedthatthegradientexpansionscanbemademorerealisticvia real-spacecutoffschosentoenforceexactpropertiesrespectedbyzero-order or LSD terms but violated by the second-order expansions:The exchange hole is never positive,and integratesto -1,while the correlationhole integratesto zero. Numerous GGA schemeswere developedby LangrethandMehl [77],Hu and Langreth (LMH) [76], Becke [78], Engel and Vosko [79], and Perdew and co-workers (PW) [8083], the three most successful and popular ones are those TLFeBOOK 11THEORETICAL FRAMEWORK by Becke (B88) [78], Perdew and Wang (PW91) [82], and Perdew, Burke, and Ernzerhof (PBE) [83]. InGGAtheexchange-correlationfunctionaloftheelectronspindensities and takes the form EGGA[, ] = d3rf (, , , ) .(1.30) xc Because of the spin scaling relation the exchange part of the GGA functional can be written as 1 1 Ex [, ] = Ex [2] +Ex [2],(1.31) 2 2Exc =d3r unif ()Fx(s) ,(1.32) x wereunif () = 3kF /4istheexchangeenergyoftheuniformelectron x gas, kF = (32 )1/3 is the local Fermi wave vector, and s = ||/2kF is a reduced density gradient.The enhancement factor Fx(s) is given by 2F PBE(s) = 1 + /(1 + s/) ,(1.33) x where =0.21951 and =0.804. The PBE correlation energy EPBE =d3r {c (rs, ) + HPBE(rs , , t)} ,(1.34) c 1wherers =(3/4)1/3 ,=( )/,t=||/2ks ,= 2 [(1 + 1/2)2/3 + (1 )2/3], ks = (4kF /) , HPBE = 3 ln1 + 2 1 + At2 (1.35) t1 + At2 + A2t4 , A =[exp{unif /3 } 1]1 ,(1.36) c and=0.031091,=0.066725.Thereducedgradientssandtmeasurehow fast (r) is varying on the scales of the local Fermi wavelength 2/kF and the local Thomas-Fermi screening 1/ks , respectively. The GGA functionalswere tested in several cases,and were found to give improved resultsfor the ground-stateproperties.For atoms it was foundthat both total energies and removal energies are improved in the LMH functional comparedwiththeLSDA[77,76].ThePWfunctionalgivesafurtherim-provementin the total energyof atoms [80,81].The bindingenergiesof the rstrowdiatomicmoleculesarealsoimprovedbybothfunctionals[84,85]. TLFeBOOK12ELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS In a study of the band structure of V and Cu, Norman and Koelling [86] found that the LMH potential gave an improvement in the Fermi surface for V but not for Cu.The cohesiveenergy,the latticeparameters,and the bulk modulusof third-row elements have been calculated using the LMH, PW, and the gradient expansion functionals in [87].The PW functional was found to give somewhat better results than the LMH functional and both were found to typically remove half the errors in the LSD approximation,while the GEA gives worse results thanlocal-densityapproximation.ForFeGGAfunctionalscorrectlypredict a ferromagnetic bcc ground state, while the LSDA and the gradient expansion predict a nonmagnetic fcc ground state [8890].Also, the GGA corrects LSDA underestimation of the lattice constants of Li and Na. Large number of test calculations showed that GGA functionals yield great improvement over LSD in the descriptionof nite systems:they improve the total energies of atoms and the cohesive energy,equilibriumdistance,and vi-brational frequency of molecules [90, 91], but have mixed history of successes and failures for solids [9295, 91, 96, 97].This may be because the exchange-correlationhole can have a diffuse tail in a solid,but not in an atom or small molecule,where the densityitself is well localized.The generaltrend is that the GGA underestimatesthe bulk modulus and zone center transverseoptical phonon frequency [93, 96], corrects the binding energy [98, 96], and corrects or overcorrects, especially for semiconductor systems, the lattice constant [93 95, 91] compared to LDA. The GGA does not solve the problems encountered in the transition-metal monoxides FeO, CoO, and NiO [89]. The magnetic mo-ments and band structures obtained with the GGA for the oxides are essentially the same incorrect ones as obtained with the LSDA. Recently,a number of attempts have been made to extend the GGA by in-cluding higher order terms, in particular the Laplacian of the electron density, intotheexpansionoftheexchange-correlationhole[99,100].However,no extensive tests of the quality of these new potentials with application to solids have yet been made. 1.2.3Self-interaction correction In the density formalism each electron interacts with itself via the Coulomb electrostaticenergy.Thisnonphysicalinteractionwouldbecanceledexactly byacontributionfromtheexchange-correlationenergyintheexactformal-ism.IntheLSDapproximationthiscancellationisimperfect,butoftennu-mericallyrathergood.Theincorrecttreatmentoftheself-interactioninap-proximate functionalshas led a number of people to considerself-interaction corrected (SIC) functionals [101].Such a correction was studied in the context of theThomas-Fermi approximation[102],the Hartreeapproximation[103], the Hartree-Slater approximation [104], and the LSD approximation [105, 106, TLFeBOOK 13THEORETICAL FRAMEWORK 56].Within the LSD approximation the SIC functional takes the form [101] ESI C ] = ELSDocc xc [xc [] i d3rd3r i(|rr) ri

(| r

) +d3r i(r) xc(i(r, 0)(1.37) where i(r) is the charge density corresponding to the solution i of the SIC LSDequationandxc(, )istheexchange-correlation(xc) energyof ahomogeneous system with the spin densities and .The second term sub-tracts the nonphysical Coulomb interaction of an electron with itself as well as the correspondingLSD xc energy.The correspondingxc potential for orbital i with spin is [101] i(r

)V SI C d3 xc,i,(r) = V LSD ((r), (r))r | r r |V LSD (i(r), 0)(1.38) xc,xc, whereV LSD ((r), (r))istheLSDxcpotential.Animportantproperty xc, of the SIC potential is its orbital dependence.This leads to a state-dependent potential,andthesolutionsarethereforenotautomaticallyorthogonal.Itis therefore necessary to introduce Lagrange parameters to enforce the orthogo-nality occ2 + V (r) +V SI C (r)i = i i + ij j (1.39) i i =j where V (r) is the effective potential entering in a normal LSD calculation and ij are Lagrange parameters. TheSICremovesunphysicalself-interactionforoccupiedelectronstates and decreases occupied orbital energies.Calculations for atoms have been per-formedbyseveralauthors[105,106,56,107111].Theerrorsinthetotal exchange and correlationenergiesare much less than those obtained with the LSDapproximation.PerdewandZunger[56]alsoshowedthatthehighest occupied orbital energies of isolated atoms are in better agreement with exper-imental ionization energies. TheapplicationoftheLSDA-SIC tosolidshassevereproblemssincethe LSDA-SIC energy functional is not invariant under the unitary transformation oftheoccupiedorbitalsandonecanconstructmanysolutionsintheLSDA-SIC.IfwechooseBlochorbitals,theorbitalchargedensitiesvanishinthe innitevolumelimit.ThustheSICenergyisexactlyzeroforsuchorbitals. ThisdoesnotmeanthattheSICisinapplicableforsolidssincewecantake atomicorbitalsorconstructlocalizedWannierorbitalsthathaveniteSIC energies.Inmanycalculationsforsolids,theSICwasadaptedtolocalized TLFeBOOK14ELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS orbitals,which are selected under some physicalassumption.These methods have partly succeeded in providing improved electronic structures for wide-gap insulators[112, 113].The band gap was found to be substantiallybetter than in the LSD approximation.For LiCl the band gap is 10.6 (SIC), 6.0 (LSDA), and 9.4-9.9 eV (exp); for Ar it is 13.5 (SIC), 7.9 (X, = 2/3), and 14.2 eV (exp) [112, 113].In these systems the great improvement of the gap is related to an improvement of the eigenvalues for the corresponding free atoms. A longstanding problem in the DF formalism is the description of localiza-tion,forinstance,inaMottinsulatororinthe transitioninCe.The insulating,antiferromagnetictransition-metal(TM) oxideshavebeenstudied intensivelyforseveraldecades,becauseofthecontroversialnatureoftheir band gap.The LSD approximationascribescertainaspectsof the loss of the 3dcontributiontocohesion[114],butthebandgapsaremuchtoosmallor zero and the magnetic moments are in some cases also too small [115]. Recently,Svane andGunnarsson[116]andSzoteket al.[117]performed self-consistentcalculationsfortheTMmono-oxides(VO,CrO,MnO,FeO, CoO, NiO, and CuO) within the LSDA-SIC and obtained energy gaps and mag-neticmoments,whicharein goodagreementwith experiment.They didnot impose any physical assumption and chose the solutions from a comparison of the total energies.The selected solutions are composed from localized orbitals for transitionmetal d bandsand extendedBloch orbitalsfor oxygenp bands. In other words, the SIC is effective only for the transition metal d orbitals and the oxygen p orbitals are not affected directly by the SIC. The electronic struc-tures of TM mono-oxideshave also been calculatedby the LSDA-SIC [118]. Theauthorscarefullyexaminethecriteriontochooseorbitalsandtryboth solutionswithlocalizedandextendedoxygenporbitals.Thesolutionsare expressedaslinearcombinationsofmufn-tinorbitals(LMTO).Itisshown that the total energies of these solutions are strongly unaffected by the choices of exchange-correlationenergy functionals.Alternatively,if the solutionsare chosen so that all orbitals are localized as Wannier functions,the energy gaps areoverestimatedby1.5-3eV.Howeverinthesesolutions,therelativeposi-tions of occupied transition-metald bands and oxygen p bands are consistent with the analysis of photoemissionspectroscopyby the cluster conguration-interaction (CI) theory [119, 120]. Thetendencytoformananti-ferromagneticmomentintheLSDapprox-imationisseverelyunderestimatedinsomecases.Oneexampleistheone-dimensional Hubbard model, for which the exact solution is known [121].The bandgap,thetotalenergy,thelocalmoment,andthemomentumdistribu-tionaredescribedsubstantiallybetterbytheSICapproximationthanbythe LSD one [122].Another example is provided by the high-Tc superconductors, where the Stoner parameter Iis at least a factor 2-3 too small when using the LSDapproximation[123].SvaneandGunnarsson[124]performedcalcula-TLFeBOOK THEORETICAL FRAMEWORK15 2tionsforasimplemodelofLa2CuO4 whichincludestheimportantx2 yorbitalofCuandthe2porbitalsofoxygen,pointingtowardstheCuatoms. Itwasfoundthatthetendencytoantiferromagnetismisgreatlyenhancedin SIC,comparedwiththeLSDapproximation,andthattheexperimentalmo-ment may even be overestimatedby SIC. Recently the electronicstructureof La2CuO4 in the LSD-SIC approximationhas been calculated [125].The cor-rect antiferromagneticand semiconductinggroundstateis reproducedin this approximation.Good quantitative agreement with experiment is found for the Cu magnetic moment as well as for the energy gap and other electron excitation energies. 1.2.4LDA+Umethod AcrucialdifferencebetweentheLDAandtheexactdensityfunctionalis thatinthelatterthepotentialmustjumpdiscontinuouslyasthenumberof electronsNincreasesthroughintegervalues[126]andin theformer thepo-tential is a continuous function of N. The absence of the potential jump, which appears in the exact density functional, is the reason for the LDA failure in de-scribing the band gap of Mott insulators such as transition metal and rare-earth compounds.GunnarssonandSchonhammer[127]showedthatthedisconti-nuityintheone-electronpotentialcangivealargecontributiontotheband gap.The second important fact is that while LDA orbital energies,which are derivatives of the total energy E with respect to orbital occupation numbers ni (i = E/ni ), are often in rather poor agreement with experiment, the LDA total energy is usually quite good.A good example is a hydrogen atom where theLDAorbitalenergyis-0.54Ry(insteadof-1.0Ry)butthetotalenergy (-0.976 Ry) is quite close to -1.0 Ry [122]. Brandow [128] realized that param-eters of the nonmagneticLDA band structurein combinationwith on-site in-teractions among 3d electrons taken in a renormalized Hartree-Fock form pro-vide a very realistic electronic picture for various MottHubbard phenomena. Similarobservationsledtotheformulationoftheso-calledLDA+Umethod [129,130]inwhichanorbital-dependentcorrection,thatapproximatelyac-counts for strong electronic correlations in localized d or fshells, is added to theLDA potential.SimilartotheAndersonimpuritymodel[131],themain idea of the LDA+Umethod is to separate electrons into two subsystems lo-calized d or felectrons for which the strong Coulomb repulsionUshould be taken into account via a Hubbard-liketerm 1 U i =j ninj in a model Hamil-2 tonian and delocalizedconductionelectronswhich can be describedby using an orbital-independent one-electron potential. Hubbard [132, 133] was one of the rst to point out the importance,in the solid state, of Coulomb correlations which occur inside atoms. The many-body crystalwavefunctionhastoreducetomany-bodyatomicwavefunctionsas lattice spacing is increased.This limiting behavior is missed in the LDA/DFT. TLFeBOOK 16ELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS Thespectrumofexcitationsfortheshellofand-electronsystemisasetof many-bodylevels describingprocessesof removing and addingelectrons.In the simplied case, when every d electron has roughly the same kinetic energy d and Coulomb repulsion energy U, the total energy of the shell with n elec-tronsisgivenbyEn =dn + Un(n 1)/2andtheexcitationspectrumis given by n = En+1 En = d + Un. Letusconsiderdionasanopensystemwithauctuatingnumberofd electrons.ThecorrectformulafortheCoulombenergyofddinteractions asafunctionofthenumberofdelectronsNgivenbytheLDAshouldbe E=UN(N1)/2[130].IfwesubtractthisexpressionfromtheLDA totalenergyfunctionalandaddaHubbard-liketerm(neglectingforawhile exchange and non-sphericity) we will have the following functional: E= ELDA UN(N 1)/2 +1 2U ninj .(1.40) i=j The orbital energies i are derivatives of (1.40): E = LDA + U(1 i = ni) .(1.41) ni 2 This simple formula gives the shift of the LDA orbital energy U/2 for occu-pied orbitals (ni = 1) and +U/2 for unoccupiedorbitals (ni = 0).A similar formula is found for the orbital dependent potential Vi(r) = E/ni(r) where variation is taken not on the total charge density (r) but on the charge density of a particular i-th orbital ni(r): Vi(r) = V LDA(r) +U(1 ni) .(1.42) 2 Expression (1.42) restores the discontinuousbehaviorof the one-electronpo-tential of the exact density-functional theory. The functional (1.40) neglects exchange and non-sphericity of the Coulomb interaction.In the most general rotationallyinvariantform the LDA+Ufunc-tional is dened as [134, 135] n] = EL(S)DA n) Edc ( ELDA+U [(r), [(r)] + EU ( n) ,(1.43) where EL(S)DA [(r)] is the LSDA (or LDA as in Ref. [130]) functional of the total electron spin densities, EU ( n) is the electronelectron interaction energy of the localized electrons, and Edc( n) is the so-called double counting term which cancels approximatelythe part of an electron-electronenergy which is already included in ELDA .The last two terms are functions of the occupation matrix n dened using the local orbitals {lm }.Thematrix n=nm, m generallyconsistsofbothspin-diagonaland spin-non-diagonal terms. The latter can appear due to the spin-orbit interaction TLFeBOOK

17THEORETICAL FRAMEWORK or a non-collinear magnetic order.Then, the second term in Eq. (1.43) can be written as [134, 136, 135]: 1 EU =2 (nm1 ,m2 Um1 m2 m3 m4 n m3 , m4 , ,{m} nm1 , m2 Um1 m4 m3 m2 n m3 ,m4 ) ,(1.44) where Um1 m2 m3 m4 are the matrix elements of the on-site Coulomb interaction which are given by 2lkUm1 m2 m3 m4 =am1 m2 m3 m4 F k ,(1.45) k=0 with F k being screened Slater integrals for a given l and 4 k a k =lm1|Ykq |lm2lm3|Y|lm4 .(1.46) m1 m2 m3 m4 kq2k + 1q=k The lm1|Ykq |lm2 angular integrals of a product of three spherical harmonics Ylm can be expressedin terms of Clebsch-Gordancoefcients and Eq. (1.46) becomes k Clm4a= m1 m2 +m3,m4 (Ckl00,l0)2Clm1 m1 m2 m3 m4km1 m2,lm2 km1 m2,lm3 .(1.47) ThematrixelementsUmmm mandUmm m m whichenterthosetermsin the sum in Eq. (1.44) which contain a product of the diagonal elements of the occupation matrix can be identied as pair Coulomb and exchange integrals =Umm, UmmmmUmm m

m =Jmm. The averaging of the matrices Ummand Umm Jmmover all possible pairs of m, m denes the averaged Coulomb Uand exchange Jintegrals which enter theexpressionforEdc .UsingthepropertiesofClebsch-Gordancoefcients one can show that 1 U=Umm= F 0 ,(1.48) (2l + 1)2 mm1 U J= (Umm Jmm )2l(2l + 1)mm1 2l=F 0 (Cl0 ,(1.49) n0,l0)2F k 2l k=2 TLFeBOOK18ELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS where the primed sum is over m = m. Equations (1.48) and (1.49) allow us to establishthe followingrelationbetweenthe averageexchangeintegralJand Slater integrals: J= 1 2l 2l k=2 (Cl0 n0,l0)2F k ,(1.50) or explicitly J= 1 14 (F 2 + F 4)forl = 2 ,(1.51) J= 1 6435 (286F 2 + 195F 4 + 250F 6 )forl = 3 .(1.52) The meaning of Uhas been carefully discussedby Herring [137].In, e.g., a 3d electron system with n 3d electrons per atom, Uis dened as the energy cost for the reaction 2(dn) dn+1 + dn1 ,(1.53) i.e., the energycost for moving a 3d electronbetween two atoms which both initially had n 3d electrons.It should be emphasized that Uis a renormalized quantitywhichcontainstheeffectsofscreeningbyfast4sand4pelectrons. The number of these delocalizedelectronson an atom with n+1 3d electrons decreaseswhereastheirnumberonanatomwithn-13delectronsincreases. ThescreeningreducestheenergycostforthereactiongivenbyEq.(1.53). It is worth notingthat becauseof the screeningthe value of Uin L(S)DA+U calculations is signicantly smaller then the bare U used in the Hubbard model [132, 133]. Inprinciple,thescreenedCoulombUandexchangeJintegralscanbe determinedfromsupercellLSDAcalculationsusingSlaterstransitionstate technique [138] or from constrained LSDA calculations [139141].Then, the LDA+Umethod becomes parameter free.However, in some cases,as for in-stance for bcc iron [138],the value of Uobtainedfrom such calculationsap-pearstobe overestimated.Alternatively,the valueof UestimatedfromPES and BIS experiments can be used.Because of the difculties with unambigu-ous determination of Uit can be considered as a parameter of the model. Then itsvaluecanbeadjustedsotoachievethebestagreementoftheresultsof LDA+UcalculationswithPESoropticalspectra.Whiletheuseofanad-justable parameter is generally considered an anathema among rst principles practitioners, the LDA+U approach does offer a plausible and practical method to approximatelytreatstronglycorrelatedorbitalsin solids.It hasbeenfond thatmanypropertiesevaluatedwiththeLDA+Umethodarenotsensitiveto smallvariationsofthevalueofUaroundsomeoptimalvalue.Indeed,the optimal value of Udetermined empirically is ofter very close to the value ob-tained from supercell or constrained density functional calculations. TLFeBOOK 19THEORETICAL FRAMEWORK In order to calculate the matrix elements Um1 m2 m3 m4 dened by Eq. (1.45) one needs to know not only F 0, which can be identied with U , but also higher order SlatersintegralsF 2 , F 4 for d as well as F 6 for felectrons.Once the screenedexchangeintegralJhasbeendeterminedfromconstrainedLSDA calculation the knowledge of the ratio F 4/F 2 (and F 6/F 4 for felectrons) is sufcientforcalculationoftheSlaterintegralsusingtherelation(1.50).De Groot et al. [142] tabulated F 2 and F 4 for all 3d ions.The ratio F 4/F 2 for all ions is between 0.62 and 0.63.By substituting these values into Eq. (1.51) one obtains screenedF 2 and F 4 .Alternatively,one can calculate unscreenedF n using their denition nr where l(r) is the radial wave function of the localized elections and r< (r>) isthesmaller(larger)ofrandr .Then,F n canberenormalizedkeeping theirratiosxedsoastosatisfytherelation(1.50).Itisworthmentioning, thatinsolidsthehigherorderSlatersintegralsarescreenedmostlydueto anangularrearrangementofdelocalizedelectronswhereasthescreeningof F 0 =Uinvolvesaradialchargeredistributionandachargetransferfrom neighboring sites.Because of this, F 0 is screened much more effectively than other F n. In 3d metals oxides, for example, Uis reduced from the bare value of about 20 eV to 6-8 eV [129]. The third term in Eq. (1.43) is necessary in order to avoid the double count-ing of the Coulomb and exchange interactionswhich are included both in the L(S)DA energy functional and in EU . Following the arguments of Czy zyk and Sawatzky [143] one can dene 1 1 Edc =UN(N 1) J N (N 1) ,(1.55) 2 2 whereN isthenumberoflocalizedelectronswiththespingivenbya partial trace of the occupation matrix N = m nm,m, N= N + N, andUandJareaveragedon-siteCoulombandexchangeintegrals,respectively. Theexpressions(1.55)forEdc and(1.44)forEU substitutedtogetherwith ELSDA[]intothefunctional(1.43)denethetotalenergyfunctionalofthe LSDA+Umethod.Inthisapproach,theexchangesplittingofmajorityand minority spin states is governed mainly by the LSDA part of the effective one electron potential, whereas the EU Edc part is responsible for the Coulomb repulsion between the localized electrons and non spherical corrections to the exchange interaction. TLFeBOOK 20ELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS Anisimov et al.[130] introducedthe LDA+Umethod in which the double counting term in the form 1 1 Edc =UN(N 1) JN (N 2)(1.56) 2 4is used in conjunction with the LDA total energy functional ELDA[].In con-trasttotheLSDA+Uapproach,intheLDA+Utheexchangesplittingofthe localized shell is provided by the EU Edc part of the functional (1.43). Most oftheresultspresentedinthisbookwereobtainedusingtheLSDA+Uver-sion of the LDA+Umethod.However, since both versions of the L(S)DA+U approachgivesimilarresultsifthevalueoftheaverageexchangeintegralJ is determined from constrained LSDA calculations, we use the more common LDA+U abbreviation in what follows. After the total energy functional has been dened, the effective Kohn-Sham equations of the L(S)DA+U method can be obtained by minimizing (1.43) with respect to the wave function i (r): E j fj j |j i (r) i (r) = 0 ,(1.57) wherej areLagrangemultipliers,fj istheoccupationofthestate,andthe electron density is given by 2(r) = fi|i(r)| .(1.58) i The minimization leads to the set of equations [2 + V LDA(r)]i(r) +V1 m1 ,2m2 n1 m1 ,2m2 = ii(r) , 1 m1 ,2 m2 i (1.59) where the orbital dependent potential V1 m1 ,2 m2 is given by the derivative of EU Edc with respect to the elements of the occupation matrix V1 m1 ,2 m2 =1 2 Um1 m2 m3 m4 nm3 ,m4 m3 m4 Um1 m4 m3 m2 n2 m3 ,1m4 (1.60) m3 m4 1 1 1 2 [U (N 2) +J(N1 2)] . Thevariationn1 m1 ,2m2 / denestheprojectionofthewavefunction i i ontotheYlm subspace.Itsexplicitformdependsonthechoiceofthe basis functionsof a particularband structuremethod.The correspondingex-pressions for the LMTO method will be given in the Appendix A whereas the expressions for the LAPW method can be found in Ref. [144]. TLFeBOOK

21THEORETICAL FRAMEWORK Astheoutputfromself-consistentLDA+Ucalculationsoneobtainsthe bandstructure,renormalizedduetothecorrelationeffects,andtheoccupa-tionmatrix nofthelocalizedelectrons.Inordertosimplifytheanalysisof the LDA+Uresults it is helpful to make a transformation to a new set of local orbitals |li =|lmdm,i ,(1.61) m where the matrix dm,i diagonalizes the occupation matrix, i.e. dm,i nm, m d m ,j = ij ni .(1.62) mmInthisrepresentationtheoccupationmatrixisdiagonalandtheeigenvalues ni have the meaningof orbitaloccupationnumbers for the |li local orbitals. Then, the energy of the Coulomb interaction EU [ n] can be written as 1 EU = (Uij Jij )ninj ,(1.63) 2 i,j where Uij and Jij are Coulomb and exchange matrix elements calculated be-tween the new local orbitals |li. The functions|li are the partnersof the irreduciblerepresentationsof the localsymmetrysubgroupofanatomicsitewithcorrelatedelectrons.Ifthe localsymmetryis sufcientlyhigh,so that all theirreduciblerepresentations are inequivalent, the transformation matrix does not depend on the occupation of the local orbitals and can be constructed using group theoretical techniques. Forexample,ifthespin-orbitcouplingisneglectedfora3dioninacubic environment |li are the well known eg and t2g orbitals. On the other hand, in the relativistic atomic limit, when both the crystal-eldsplittingofthelocalizedstatesandtheirhybridizationwithdelocalized bandsaremuchsmallerthanthespin-orbitsplitting,theoccupationmatrix isdiagonalinthe|jmj representationwherej=l 1/2andmj arethe total angular momentum of a localized electron and its projection, respectively. Then,the indexi is a short-cutforj, mj pairsandtheelementsof dm,i are given by the corresponding Clebsch-Gordan coefcients Cjmj . lm, 1 2 AsonecanseefromEq.(1.45)thematrixelementsUm1 m2 m3 m4 contain both F 0 =U , which provides the splittingof the localizedstates into lower and upper Hubbard subbands,and the terms proportionalto F n with n> 0, whichareresponsibleforangularcorrelationswithinthelocalizedshell.In the case when Uis effectively screened and U e = U Jbecomes small, the latter terms give the dominant contribution to Um1 m2 m3 m4 . Looking ahead and comparing Eq. (1.45) to Eqs. (1.72)(1.74), that dene the so-called orbital po-larization correctionsto LSDA described in section 1.2.5, one can notice that TLFeBOOK22ELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS theLDA+UmethodincludestheOPcorrectionsbetweenthelocalizedelec-trons but without making the assumptionthat the occupationmatrix is diago-nal in spin indices.Thus, the OP corrections can be considered as the limiting caseofthemoregeneralLDA+Uapproach[134].Inthefollowingwewill refertocalculationsperformedusingtheLDA+UmethodwithU e =0 as LDA+U (OP) calculations. The most important property of the LDA+Ufunctional is the discontinuity ofthepotentialandthemaximumoccupiedorbitalenergyasthenumberof electrons increases through an integer value, the absence of which is the main deciency of the local-density approximation compared with the exact density functional [126] as far as band gaps are concerned. ItshouldbementionedthatwhereastheSICequationsarederivedwithin theframeworkof homogeneous-electron-gastheoryandthemethodis there-fore a logical extension of LDA, this is obviously not the case for the LDA+U method. The latter method has the same deciencies as the mean-eld (Hartree-Fock) method.The orbital-dependent one-electron potential in Eq. (1.60) is in the form of a projectionoperator.This means that the LDA+Umethod is es-sentiallydependentonthechoiceofthesetofthelocalizedorbitalsinthis operator.ThatisaconsequenceofthebasicAnderson-model-likeideology [131] of the LDA+Uapproach.That is, the separationof the total variational spaceintoa localizedd- (f -) orbitalsubspace,withthe Coulombinteraction betweenthemtreatedwithaHubbard-typetermintheHamiltonian,andthe subspaceof all other statesfor which the local densityapproximationfor the Coulomb interaction is regarded as sufcient.The imprecision of the choice of the localized orbitals is not as crucial as might be expected.The d (f ) orbitals for which Coulomb correlation effects are important are indeed well localized in space and retain their atomic characterin a solid.The experienceof using the LDA+Uapproximation in various electronic structure calculation schemes shows thatthe resultsare notsensitiveto theparticularformof the localized orbitals. TheLDA+Umethodwasprovedtobeaveryefcientandreliabletool in calculatingthe electronicstructureof systems where the Coulomb interac-tion is strong enough to cause localizationof the electrons.It works not only for nearly core-like4forbitals of rare-earthions, where the separation of the electronic states in the subspaces of the slow localized orbitals and fast itiner-antonesis valid,butalsoforsuchsystemsastransitionmetal oxides,where 3dorbitalshybridizequitestronglywithoxygen2porbitals[129].Inspite ofthefactthattheLDA+Umethodisamean-eldapproximationwhichis in general insufcient for the descriptionof the metal-insulatortransitionand stronglycorrelatedmetals,insomecases,suchasthemetal-insulatortransi-tion in FeSi and LaCoO3, LDA+Ucalculationsgave valuable information by providing insight into the nature of these transitions [145]. TLFeBOOK23THEORETICAL FRAMEWORK UsingtheLDA+Umethoditwasfound[129]thatalllate-3d-transition-metal monoxides, as well as the parent compounds of the high-Tc compounds, arelarge-gapmagneticinsulatorsofthecharge-transfertype.Further,the methodcorrectlypredictsthatLiNiO2 isalow-spinferromagnetandNiSa local-momentp-typemetal.Themethodwasalsosuccessfullyappliedto the calculationof the photoemission(X-ray photoemissionspectroscopy)and bremsstrahlung isochromatic spectra of NiO [130]. The advantage of theLDA+U method is the ability to treat simultaneously delocalized conduction band elec-tronsandlocalizedelectronsinthesamecomputationalscheme.Forsucha method it is important to be sure that the relative energy positions of these two typesofbandsarereproducedcorrectly.TheexampleofGdgivesuscon-denceinthis [146].Gd is usuallypresentedasan examplewherethe LSDA gives the correct electronic structure due to the spin-polarization splitting of the occupied and unoccupied 4fbands (in all other rare-earth metals LSDA gives anunphysical4fpeakattheFermienergy).IntheLSDA,theenergysep-arationbetween4fbandsisnotonlystronglyunderestimated(theexchange splitting is only 5 eV instead of the experimental value of 12 eV) but also the unoccupied 4f band is very close to the Fermi energy thus strongly inuencing the Fermi surface and magnetic ground-state properties (in the LSDA calcula-tion the antiferromagnetic state is lower in total energy than the ferromagnetic oneincontradictiontotheexperiment).The applicationofLDA+Umethod to Gd gives good agreement between calculated and experimentalspectra not onlyfortheseparationbetween4fbandsbutalsoforthepositionofthe4f peaks relative to the Fermi energy [146]. 1.2.5Orbital polarization correction Many magneticmetals possessa considerableorbital magneticmoment in addition to the spin magnetic moment.In LSDA the exchange correlation po-tential does only depend on the spin density and an induced spin moment wold correspondtothegaininexchangeenergyimpliesbyHundsrstrulefor atoms. To obtain an orbital moment the spin-orbit interaction must be included in the Hamiltonian.However the so calculatedorbital moment is found to be toosmalltoaccountfortheexperimentallyobservedorbitalmoments.This is already noticeable for the ferromagnetictransition metals where the orbital moments are very small, but the effect is much more drastic in actinide inter-metallic ferromagnets.What is lacking in LSDA is clearly that there is nothing in the theory (by its construction) which would account for Hunds second rule (maximizetheorbitalmoment).Inordertocorrectforthisseveraldifferent orbital splitting theories have been developed [147151]. The problem is not simple because the appropriatedensity functional must be nonlocal.The authors of [149] suggest the following approximate method whichyieldsanenergyfunctionandthuseigenvalueshiftratherthan TLFeBOOK 24ELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS themoreaccuratebutunknownfunctionalandcorrespondingpotential.The ground state of the atomic conguration can be obtained from a vector model [152,153]involvinginteractionoftypesisj andlilj (si andli arethespin and angular momenta for the i-th electron in the atomic conguration, respec-tively).Byreplacingtheinteractionenergyoftheformsisj withthe z zmean-eldapproximation(i si )(j sj ),oneobtainsanenergy 1 IM z 4 s (Ms = 2sz is spin magnetizationand I Stoner exchangeparameter)which is the Stoner expression for the spin-polarization energy [154, 155].Hence the Stonerenergyisafunctionofthespinandthespin-upand-downbandsare split rigidly.For the orbitalpolarizationwe followan analogousroute [149]. Wereplace 1 lilj ,whichoccursintheenergyofthegroundstateofan 2

atom as a functionof occupationnumber,with 1 2 (i liz )(j ljz ) and obtain a, so-called, orbital polarization energy 1 Eop = RL2 (1.64) 2 z , where Lz is the total angular momentum, R is the appropriate Racah parame-ter,eitherE3 (f -states)or B(d-states).From this energyan associatedshift of one-electron eigenvalue is obtained Eopme = = RLz me.(1.65) nme Thus,inanopenfshellatom,thespin-uporspin-downfmanifoldsare splitintosevenequidistantlevelswhen thetotalorbitalsmomentis nonzero. E3 now plays role analogous to the Stoner parameter I. The possibility of orbital polarization given by the energy correction in Eq. (1.64)has been usedfor some applicationsandhas to some extentbeensuc-cessfulinthesensethatabetterdescriptionoftheorbitalpartofthetotal magnetic moment is obtained if this splitting is included in the self-consistent band structurecalculation.This theorywas used to describethe localization-delocalizationtransitionin Ce,Pr,and Nd [149].Applicationto Fe,Co,and Ni,transition.metalmonoxidesandvariouscompoundshasalsoimproved agreement between theory and experiment [156159]. Another scheme suggestedby Severin et al [150, 151].The scheme based upon a generalization of Slaters expression for the spin polarization energy of an open atomic shell [160]. In Hartree-Fock theory the exchange energy is written Exc =1 lm, l m | g | l m

, lmnlmnl

m slm,sl m ,(1.66) 2 lm,l m TLFeBOOK 25THEORETICAL FRAMEWORK where g is the Coulomb interaction,lm labels the orbitals and in an extended system, nlm which are local occupation numbers are in general nonintegral. ThisexpressionisevaluatedintermsofSlaterintegralsthroughamultipole expansion of the Coulomb interaction.The term lm = l

m is included in the sum but for integraloccupationnumbers the rst term in the expansion,con-taining F 0, exactly cancels the spherical part of the direct Coulomb interaction [160,161].Slaterthereforeremovedthiscontributionbutevaluatedhigher multipolecontributionstotheexchangeinteractionusingnonintegralorbital occupation numbers.The spin polarization part of Eq. (1.66) is [150] 1 Es sp Vll mlml ,(1.67)= 4 l,l where ml is the l-th partial spin moment, the exchange integrals Vllare given in terms of radial Slater exchange integrals Gk byll ll

k Vll=Gk (1.68) 000 ll , k where (...) is a Wigner 3j symbol, and Gll= Fll when l = l

. When l = l = 0Eqs. (1.67) and (1.68) reduce to Es = 1 F 0m2 and F 0 = JHF A = U . Moresp 4 llkgenerally, but with F 0 F k for k > 0 the 3j symbol 000 = 1/2l + 1and JHF A = U/(2l + 1), whereUis the full Coulomb integral. We should remind here that LSDA moment formation is driven by the spin polarizationenergy,ELSDA .Ina muchused,andoftenquitegoodapproxi-sp mation [162] 1 Es sp Jll mlml ,(1.69)= 4 l,l where Jllare the LSDA exchange interactions between shells l and l . We wish to separate in (1.67) the contributionswhich would correspond to Hunds rst and second rules, since the former is well accounted for in LSDA. 1Thisisachievedbywritingmlm =ml + mlm. where ml = 2l+1 mlm m is the average spin moment for a shell whereas mlm represents the deviation ofthemomentsfromtheaverageandthusrepresentstheenergygaindueto orbital polarization.Since the average moment does not depend on m this part of the sum is easily evaluated and the spin polarization energy becomes 1

1 Es = Vll mlmlVlm,l m mlmml

m ,(1.70) sp 4 4 l,l lm,l

mwhere Vlm,l

m= [C(lm; l m

)]2Gk ll ,(1.71) k TLFeBOOK 26ELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS C(lm; l

m

) is a Gaunt coefcient. It is known from atomic calculationsthat Slater integrals are too large due to neglect of screening effects.In atomic theory it is customary to scale them down. Since LSDA works so well and due to the formally equivalence between Vlland the exchange integrals obtained in LSDA, Jll , it can be proposed the scaling procedure Vll Jll , whence 1

1 Esp = Jll mlmlV lm,l m mlmml

m ,(1.72) 4 4 l,l lm,l

mwhere V lm,l

m= dlm,l m Jll ,(1.73) and

[C(lm, l

m

)]2Gk ll dlm,l

m= k

ll

k (1.74) .Gk 000ll k ThespinpolarizationenergyobtainedfromEq.(1.72)dependsuponthe occupation of the individual orbitals, and the energies of the orbitals are shifted by amounts given by 1 = Jll (ml+dlm,l

m ml m ).(1.75) lm 2 l mEquation(1.75)reducestoLSDAwhenml m=0,sothesecondterm describes the orbital splitting. The latter approximation produces an improvement, bringing both total mag-netic moment and individual contributions to the magnetic moment into better agreement with experiment for US [150]. 1.3Excitations in crystals The Landau theory of the Fermi liquid is very useful in studying the collec-tive excitations and other physical properties of manyfermion systems [163]. We review the salient features relevant for the topics of this book. 1.3.1Landau Theory of the Fermi Liquid TheLandautheorywasinitiallydevelopedfor 3He,anisotropicquantum Fermi liquid.However, the conduction electrons in metals may also be consid-ered to be such a system since their degeneracy temperature is 104 105 K. AccordingtoLandau,theenergyspectrumofaFermiliquidissimilarto that of an ideal Fermi gas.This is valid for small deviations of the distribution TLFeBOOK27THEORETICAL FRAMEWORK functionn(k) =1/ exp[(k )/T ] + 1fromitsequilibriumdistribution n0(k): n(k) = n(k) n0(k).Thetheoryproceedsfromtheassumption that the classication of energy levels remains unchanged in a gradual increase ofinteractionbetweenparticles,i.e.,inatransitionfromagastoaliquid. Here,thegas particlesarerepresentedaselementaryexcitationsbehavingas quasiparticles of energies and momentum k.The concept of an elementary excitationarisesfromthequantummechanicaldescriptionofthecollective motion of particles and, in this case it may be identied with a real particle in a selfconsistent eld of surrounding particles. The dening characteristics of the Fermi liquid are the quasiparticle disper-sion law = (k) dened as a variational derivative of the total energy E of a system with respect to the distribution function E = (k),(1.76) n(k) and the correlation function f (k, k

) giving the quasiparticleenergy variation (k) through the variation of the particle distribution over the k space: (k)= 2 E = f (k, k

).(1.77) n(k

)n(k)n(k ) When the interaction between quasiparticles is taken into account, the exci-tation energy of the system can be written 1

W=(k)n(k) +f (k, k

)n(k)n(k ).(1.78) 2 kk k TheLandautheorywasoriginallydevelopedforquasiparticleswithshort range interactions,and may seem inappropriatefor describing the conduction electronsinmetalswhicharesubjecttolongrangeCoulombforces.Silin showed[164],however,thatthescreeningofelectronsinmetalsreducethe effectiverangeofthisforce.Themovingelectron,whichrepelsitsneigh-boring electrons,can also be seen as an electron surroundedby an exchange correlation hole of positive charge.With these considerations, it is not surpris-ing that the Landau theory has provided a successful theoretical framework for metals. TheLandautheoryisalsovalidforaperiodicpotential.Theelementary excitations are still dened as quasiparticles with energy (k) and interaction f (k, k

). However, due to the anisotropy of the lattice, (k) depends on the di-rection of k, andf (k, k

) depends on both k and k directions.Although Lan-daus function f (k, k

) is unknown for most metals, it is not essential since this functionisoftennotincorporatedwithintheformulasthatrelateexperimen-tallymeasuredquantitiestovaluesderivedfromtheelectrondispersionlaw. TLFeBOOK

28ELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS However, the Landau function can not be completely ignored since f (k, k

) is directly included in high frequency, nonlinear and other properties of metals. The Landau theory provides a framework to describe various collective mo-tionsofliquids,i.e.,primaryacoustic,zeroacoustic,plasmons,spinwaves, etc., and offers a simple means to interpret different results of the manybody theory.In our case, the most important feature of the Landau theory is that the excitationspectrumofanymacroscopicmanyfermionsystemhasageneral character.Weak excitationsof such a system are of a oneparticletype (with thequasiparticlesrepresentingtheparticles).The interactionbetweenquasi-particlesisofimportanceonlywhenthenumberofquasiparticlesissmall. Thus,atsufcientlylowtemperaturestheinteractionbetweenquasiparticles maybeignoredandtheseparticlescanbeconsideredtobeinanidealgas state. The Quasiparticle Description in Density Functional Theory.The DFT, rigorouslyspeaking,is applicableonlyto the descriptionof propertiesof the ground state, for example, the total energy and the electron density distribution. Tocalculatethesystemexcitationenergy,weneedtomodifythistheoryby introducingtheoccupationnumbers,ni foreachofthestatesanddetermine the electron density [165, 166] N (r) =i=1 ni|i(r)|2 ,(1.79) and the functional N

Tniti ,(1.80) i=1 where ti = i (2)idr = i i (VC + Vxc)idr;(1.81) VC(r) U/(r) is the Coulomb potential; Vxc is the exchangecorrelation potential.Then,forthegivensetofni,theselfconsistentsolutionofthe (1.5,1.79) will give the energy of the system E T

+ U [] + Exc[],(1.82) wheretheCoulombU []andexchangecorrelationenergiesdependonthe occupation numbers. Ingeneral,Eisnotthetotalenergyofthesystem,sinceT []

=T

fora random set of occupation numbers ni.Only when the ni are in a FermiDirac distribution(atzerotemperature),(1.79)isequivalentto(1.4)andEisthe totalenergyofthesystemE.Thisisbecausetheinnitesimaltransferofa TLFeBOOK

29THEORETICAL FRAMEWORK chargefromtheregionofoccupiedstatestoabovetheFermilevelresultsin an increasein E, since the region of free states is always above the occupied levels. Only when all the lower states are occupied

E is at minimum and equals to the total energy of the system E. Let us differentiate

E over the occupation numbers

E tj=ti +njnij ni + (VC + Vxc) |i|2 +nj |i|2 dr.(1.83) nij Now, using (1.81) we obtain

E j (2 + VC + Vxc)j dr + c.c.,(1.84) ni = i +nj j ni where c.c. is the complex conjugate of the preceding expression in parentheses. From (1.5), the last component in (1.84) becomes nj j |j |2dr = 0.(1.85) nij Thus, we obtain nally [165, 166]

E i =,(1.86) ni whichistrueonlyiftheenergyofthesystemEisacontinuousfunctionof the occupation numbers and can be differentiated with respect to them.This is the differencebetweenthe DFT andthe HartreeFockmethodswhere the ni areeitherzerooronesinceeverysingle-determinantwavefunctioniseither unoccupied or lled with one electron only. Let us consider the relationshipof the oneparticleenergy spectrumto the system excitation spectrum.Let the occupation numbers ni change by a value i due to some excitation of the system.We introducethe intermediate occu- i = ni + i (01).The energy of this intermediatepation numbers nstate is

E(). The excited system energy W[166] is then 1

E W=EN +1 EN =d = 0

1

1E =i i 0 n

( ) id =i i()d.(1.87) i 0 TLFeBOOK30ELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS Thisformulacanbewrittenin amore instructiveform.By expandingi() into a series in and allowing for the dependence of these values on the occu-pation numbers ni, we obtain [166] W= i ii(0) + 1 2 ij fij ij + .(1.88) The values fij = i nj = j ni 2

E ninj (1.89) can be related to the effective twoparticle interaction of quasiparticles existing inthestatesiandj.Thenexttermsoftheexpansionwillberelatedtothe three-particle, fourparticle interaction and so on. Therelationshipbetweeni andthesystemexcitationenergiesbecomes complicated because real excitations involve discrete electrons.Therefore, the excitation energy is not given by the rst derivative of the total energy but by the complete Taylor series (1.88).Slater [42] showed numericalexamplesof the convergence of this expansion. Note that (1.88)is identicalto the expressionpostulatedby Landau(1.78) foraFermiliquid,andthefunctionfij istheanalogoftheLandaufunction f(k, k

). For a oneparticle excitation from state i to j 1 W= [j () i()]dj (0.5) i(0.5).(1.90) 0 Inthisformula,theintegraliscalculatedwitha linearapproximationforthe integrand.The result is equivalent to the transition state method developed by Slater by another way [42] and used by him for the calculation of excited states in the approximation. In conclusion we should mention that Landau theory is applicable to Fermi systemswithweakinteractionsbetweenquasi-particles.Thetheoryfailsto describesystemswithstrongcorrelationssuchasoccurforexcitationwithin highlylocalizedorbitals.Forsuchsystemsthemostsuitablemathematical method is the Greens functions approach initially developed in quantum eld theory. 1.3.2 Greens Functions of Electrons in Metals and Elementary Excitations By denition, the Greens function for a macroscopic system is [41]

(x)

G(x, x

) = iT

+(x

),(1.91) TLFeBOOK

31THEORETICAL FRAMEWORK where x is a combination of the time t and the position r; . . . denotes aver-agingoverthe groundstateofa system;Tis thechronologicalproduct,that is,operatorsfollowingit shouldbelocatedfromthe rightto the leftin order of increasingtime t, t

;

are the Heisenbergoperators.For convenience,we omit in (1.91), the dependence on spin. The Greens function in the r, E representation, i.e., the Fourier image with respect to (t t

), G(r, r

; E), is the solution of the Dyson equation [41]

; E)

G(r

(2 + E)

G(r, r

; E) dr

(r, r , r

; E) = (r r

).(1.92) Here,(r, r

; E)isthemassorselfenergyoperatorthatdescribestheex-changeandcorrelationeffects.Thisnon-Hermitianoperatorisnonlocaland dependsonenergy.Itisaneffectivepotentialthatallowsfortheinteraction with all other particles of the system.Obviously, the mass operator in a crystal possesses the translational symmetry of the lattice (r + a, r + a; E) = (r, r

; E),(1.93) where a is a lattice vector of the crystal. Close to the

G-function pole, the r.h.s.of (1.92) may be omitted so that we get a homogeneousintegrodifferentialequationwith the eigenvaluesdening the excitation energy spectrum of a system [41, 45] 2k(r, E) + dr

(r, r

; E)k(r , E) = kk(r, E).(1.94) Here,thefunctionsk(r, E)aresimilartotheBlochwavefunctionsofan electron in a periodic eld.For metals, (1.94) replaces the usual Schrdinger equation.However, in contrast to the latter, the energy eigenvalues in (1.94) are in the general case, complex, because the mass operator (r, r

; E) is complex. From (1.92,1.94), we obtain for the Greens function k(r, E)+ k (r

, E)

(1.95)G(r, r ; E) =E k(E) + i0signE. k The quantity k(E) is the change in the system energy when a particle is added to it.If we attribute this change to one quasiparticle, k(E) may be dened asin (1.76) as the energy of the quasiparticle,in strict conformity with Landaus theory.For states close to the Fermi energy [in (1.95) the energy is referenced to the Fermi level, ], the function

G(r, r

; E) has a pole at E= k(E).Thus, the Greens function poles determine the spectrum of elementary excitations in a multifermionsystem.In general,the energyof a quasiparticleis complex duetointeractionwiththeotherquasiparticles.This ismanifestedasa shift of the Greens function pole into the complex region, expressed by the term i0 TLFeBOOK32ELECTRONIC STRUCTURE AND MO PROPERTIES OF SOLIDS in the denominator (1.95).The complex nature of energy levels, according to general rules of quantum mechanics [167], causes the life time of an excited statetobenite(1/|Im |).ThequantityIm denesthewidthofan energylevel.Generallyspeaking,thedescriptionofamultifermionsystem intermsof quasiparticlesastheelementaryexcitationsofa systemis anap-proximation whose accuracy decreases with an increase in | |. However,oneshouldnottakethisliterally.Sinceenergyisadimensionalquantity,to decide on the suitabilityof the conceptof quasiparticlesfor a given case it is required that the lifetime of an excited state be compared with the characteristic relaxation times of the system. In a small vicinity about the Fermi level, Im k(E) 0 like Im 0. Inthis case, Re k(E) is obtained as a solution of (1.94) with real (r, r

; E) and k(E). Since the self-energy(r, r

; E) is dened by the groundstateof the sys-tem,accordingtotheHohenbergKohntheorem,itisalsoafunctionofthe electrondensity.Proceedingfromthis,ShamandKohnsuggestedusingthe following form for (r, r

; E) [45]: (r, r

; E) = VC(r)(r r

) +0(r r

; E VC(r0); (r0 )),(1.96) where0 istheenergyofanelectrongaswithzeroCoulombpotentialand electrondensity;VC(r0)istheelectrostaticHartreepotentialatthepoint r0 =(r + r

)/2.Moreover,in(1.96)we haveseparatedthelocalpartfrom the self-energy in the form of a Hartree (Coulomb) potential.This operation enables us to isolate the shortrange interactions into 0. If, now we use the following approximation for the eigenfunction k k(r, E) = A(k) exp[ip(r)r](1.97) andignorethedependenceofAandtheelectronmomentumonr,then,as shown in [50], (1.94), which denes the quasiparticleenergy,will be reduced to an equation of the form of (1.5) [2 + VC(r) + xc((r), E)]k (r, E) = kk(r, E).(1.98) With E=, xc((r), )xc((r)),we come to the main DFT equation thatholdsinthelocaldensityapproximation.Inboth(1.98)and(1.15),the exchangecorrelation potential xc is the same. The authors of [50] have tabulatedxc relative to Efor different re. It hasbeenshownthat,withinaband,xc variesinsignicantly(5%)withE. Therefore,fornotveryhighexcitationenergies,a simpleapproximationcan be used xc((r), E)xc .(1.99) This resultcorrespondsto theneglectof interactionbetweenquasiparticles, that is, to the case when the Landau function f(k, k

) =0. TLFeBOOK33THEORETICAL FRAMEWORK Thus,withsomecaveats,onemayconcludethatbandcalculationsbased on the DFT theory, which adequately allows for the exchange and correlation effects,enableusto calculatetheenergyof elementaryexcitationsinmany electron systems.We do not claim that the argument leading to (1.99) is rigor-ous, since the theory does not incorporate a small parameter. The applicability of the oneparticle approach requires that the energy bands be broad.In an atomic or local orbital basis this means there should be a strong overlap of the wave function centered at different points and also, the Coulomb interaction energy between electrons in local orbitals should be much less than the band width of a given atom.This will depend on the particular parameters of the system being considered.The approximation (1.99) will be poor in the case of low electron density in the conduction band of semiconductors,where exciton,polaronandothereffectsareimportant,andincrystalswithnarrow bands,e.g.,thef -bandinrareearthelements.In thelattercase,inaddition to strongCoulombcorrelations,thevelocityof anelectroninthe bandstate, determined by the derivative of the energy with respect to |k|, is so small that thenucleihaveenoughtimetoshiftand,hence,tostabilizetheexcitation. Therefore, the electronphononinteractionbecomes important.When a one particle approach is suitable (wide energy bands), the velocity of the electron is so high that it can travel a considerable distance during the period required for the nuclei to go out of equilibrium and, hence, the electronphonon interaction may be neglected. 1.3.3The GWapproximation Whilewehaveprovidedsomejusticationsforusingastraightforward LSDAbandstructureapproachtoevaluateexcitationenergies,anumberof materials with inter