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Spectral Density Functional for Electronic Structure Calculations | 300 450, Study notes of History of Education

Material Type: Notes; Class: 300 - BIOLOGY & SOCIETY; Subject: EDUCATION; University: Rutgers University; Term: Unknown 2004;

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Download Spectral Density Functional for Electronic Structure Calculations | 300 450 and more Study notes History of Education in PDF only on Docsity! PHYSICAL REVIEW B 69, 245101 ~2004!Spectral density functionals for electronic structure calculations S. Y. Savrasov Department of Physics, New Jersey Institute of Technology, Newark, New Jersey 07102, USA G. Kotliar Department of Physics and Astronomy and Center for Condensed Matter Theory, Rutgers University, Piscataway, New Jersey 08854, USA ~Received 20 August 2003; revised manuscript received 5 February 2004; published 3 June 2004! We introduce a spectral density-functional theory which can be used to compute energetics and spectra of real strongly correlated materials using methods, algorithms, and computer programs of the electronic structure theory of solids. The approach considers the total free energy of a system as a functional of a local electronic Green function which is probed in the region of interest. Since we have a variety of notions of locality in our formulation, our method is manifestly basis-set dependent. However, it produces the exact total energy and local excitational spectrum provided that the exact functional is extremized. The self-energy of the theory appears as an auxiliary mass operator similar to the introduction of the ground-state Kohn-Sham potential in density-functional theory. It is automatically short ranged in the same region of Hilbert space which defines the local Green function. We exploit this property to find good approximations to the functional. For example, if electronic self-energy is known to be local in some portion of Hilbert space, a good approximation to the functional is provided by the corresponding local dynamical mean-field theory. A simplified implementation of the theory is described based on the linear muffin-tin orbital method widely used in electronic structure calculations. We demonstrate the power of the approach on the long standing problem of the anomalous volume expansion of metallic plutonium. DOI: 10.1103/PhysRevB.69.245101 PACS number~s!: 71.20.2b, 71.27.1a, 75.30.2mI. INTRODUCTION Strongly correlated electron systems display remarkably interesting and puzzling phenomena, such as high- temperature superconductivity, colossal magnetoresistance, heavy fermion behavior, huge volume expansions, and col- lapses to name a few. These properties need to be explored with modern theoretical methods. Unfortunately, the strongly correlated systems are complex materials with electrons oc- cupying active 3d , 4 f , or 5 f orbitals, ~and sometimes p orbitals as in many organic compounds and in Bucky-balls- based systems!. Here, the excitational spectra over a wide range of temperatures and frequencies cannot be described in terms of well-defined quasiparticles. Therefore, the design of computational methods and algorithms for quantitative de- scription of strongly correlated materials is a great intellec- tual challenge, and an enormous amount of work has ad- dressed this problem in the past.1–12 At the heart of the strong-correlation problem is the com- petition between localization and delocalization, i.e., be- tween the kinetic energy and the electron-electron interac- tions. When the overlap of the electron orbitals among themselves is large, a wavelike description of the electron is natural and sufficient. Fermi-liquid theory explains why in a wide range of energies systems, such as alkali and noble metals, electrons behave as weakly interacting fermions, i.e., they have a Fermi surface, linear specific heat and a constant magnetic susceptibility. The one-electron spectra form qua- siparticles and quasihole bands and the one-electron spectral functions show d functions like peaks corresponding to the one-electron excitations. We have powerful quantitative tech- niques such as the density-functional theory ~DFT! in the0163-1829/2004/69~24!/245101~24!/$22.50 69 2451local-density and generalized gradient approximation ~LDA and GGA!, for computing ground-state properties.1 These techniques can be successfully used as starting points for perturbative computation of one-electron spectra, for ex- ample using the GW method.2 They have also been success- fully used to compute the strength of the electron-phonon coupling and the resistivity of simple metals.13 When the electrons are very far apart, a real-space de- scription becomes valid. A solid is viewed as a regular array of atoms where each element binds an integer number of electrons. These atoms carry spin and orbital quantum num- bers giving rise to a natural spin and orbital degeneracy. Transport occurs with the creation of vacancies and doubly occupied sites. Atomic physics calculations together with perturbation theory around the atomic limit allow us to de- rive accurate spin-orbital Hamiltonians. The one-electron spectrum of the Mott insulators is composed of atomic exci- tations which are broaden to form bands that have no single- particle character. The one-electron Green functions show at least two polelike features known as the Hubbard bands,14 and the wave functions have an atomiclike character, and hence require a many-body description. The scientific frontier, one would like to explore, is a category of materials which falls in between the atomic and band limits. These systems require both a real-space and a momentum-space description. To treat these systems one needs a many-body technique which is able to treat Kohn- Sham bands and Hubbard bands on the same footing, and which is able to interpolate between well separated and well overlapping atomic orbitals. The solutions of many-body equations have to be carried out on the level of the Green functions which contain necessary information about the to-©2004 The American Physical Society01-1 S. Y. SAVRASOV AND G. KOTLIAR PHYSICAL REVIEW B 69, 245101 ~2004!tal energy and the spectrum of the solid. The development of such techniques has a long history in condensed matter physics. Studies of strongly correlated sys- tems have traditionally focused on model Hamiltonians using techniques such as diagrammatic methods,3 quantum Monte Carlo simulations,4 exact diagonalizations for finite-size clusters,5 density-matrix renormalization group methods,6 and so on. Model Hamiltonians are usually written for a given solid-state system based on physical grounds. In the electronic-structure community, the developments of LDA 1U ~Ref. 7! and self-interaction corrected8 methods , many- body perturbative approaches based on GW and its extensions,2 as well as time-dependent version of the density functional theory9 have been carried out. Some of these tech- niques are already much more complicated and time– consuming comparing to the standard LDA based algo- rithms, and the real exploration of materials is frequently performed by its simplified versions by utilizing such, e.g., approximations as plasmon–pole form for the dielectric function,15 omitting self–consistency within GW2 or assum- ing locality of the GW self-energy.16 In general, diagrammatic methods are most accurate if there is a small parameter in the calculation, say, the ratio of the on-site Coulomb interaction U to the band width W. This does not permit the exploration of real strongly correlated situations, i.e., when U/W;1. Systems near Mott transition is one of such examples, where strongly renormalized quasi- particles and atomiclike excitations exist simultaneously. In these situations, self-consistent methods based on the dy- namical mean–field based theory ~DMFT!,10 and its cluster generalizations such as dynamical cluster approximation,17 or cellular dynamical mean-field theory ~C-DMFT!,18,19 are the minimal many-body techniques which have to be em- ployed for exploring real materials. Thus, a combination of the DMFT based methods with the electronic-structure techniques is promising, because a realistic material-specific description where the strength of correlation effects is not known a priori can be achieved. This work is in its beginning stages of development but seems to have a success. The development was started20 by introducing so-called LDA1DMFT method and applying it to the photoemission spectrum of La12xSrxTiO3. Near Mott transition, this system shows a number of features incompat- ible with the one-electron description.21 The LDA11 method22 has been discussed, and the electronic structure of Fe has been shown to be in better agreement with experiment than the one based on LDA. The photoemission spectrum near the Mott transition in V2O3 has been studied, 23 as well as issues connected to the finite-temperature magnetism of Fe and Ni were explored.24 LDA1DMFT was recently gen- eralized to allow computations of optical properties of strongly correlated materials.25 Further combinations of the DMFT and GW methods have been proposed12,26,27 and a simplified implementation to Ni has been carried out.27 Sometimes the LDA1DMFT method11 omits full self- consistency. In this case the approach consists in deriving a model Hamiltonian with parameters such as the hopping in- tegrals and the Coulomb interaction matrix elements ex- tracted from a LDA calculation. Tight-binding fits to the24510LDA energy bands or angular momentum resolved LDA den- sities of states for the electrons which are believed to be correlated are performed. Constrained density-functional theory28 is used to find the screened on-site Coulomb U and exchange parameter J. This information is used in the down- folded model Hamiltonian with only active degrees of free- dom to explore the consequences of correlations. Such tech- nique is useful, since it allows us to study real materials already at the present stage of development. A more ambi- tious goal is to build a general method which treats all bands and all electrons on the same footing, determines both hop- pings and interactions internally using a fully self-consistent procedure, and accesses both energetics and spectra of cor- related materials. Several ideas to provide a theoretical underpinning to these efforts have been proposed. The effective action ap- proach to strongly correlated systems has been used to give realistic DMFT an exact functional formulation.29 Approxi- mations to the exact functional by performing truncations of the Baym-Kadanoff functional have been discussed.30 Simul- taneous treatment of the density and the local Green function in the functional formulation has been proposed.12 Total- energy calculations using LDA1DMFT have recently ap- peared in the literature.31–34 DMFT corrections have been calculated and added to the LDA total energy in order to explain the isostructural volume collapse transition in Ce.31 Fully self-consistent calculations of charge density, excita- tion spectrum and total energy of the d phase of metallic Plutonium have been carried out to address the problem of its anomalous volume expansion.32 The extensions of the method to compute phonon spectra of correlated systems with the applications to Mott insulators33 and high- temperature phases of Pu34 have been also recently devel- oped. In this paper we discuss the details of this unified ap- proach which computes both total energies and spectra of materials with strong correlations and present our applica- tions for Pu. We utilize the effective action free-energy ap- proach to strongly correlated systems29,30 and write down the functional of the local Green function. Thus, a spectral density-functional theory ~SDFT! is obtained. It can be used to explore strongly correlated materials from ab inito grounds provided useful approximations exist to the spectral density functional. One of such approximations is described here, which we refer to as a local dynamical mean-field ap- proximation. It is based on extended35 and cluster17–19 ver- sions of the dynamical mean-field theory introduced in con- nection with the model-Hamiltonian approach.10 Implementation of the theory can be carried out on the basis of the energy-dependent analog for the one-particle wave functions. These are useful for practical calculations in the same way as Kohn–Sham particles are used in density- functional based calculations. The spectral density-functional theory in its local dynamical mean-field approximation, re- quires a self-consistent solution of the Dyson equations coupled to the solution of the Anderson impurity model36 either on a single site10 or on a cluster.17,18 Since it is the most time consuming part of all DMFT algorithms, we are carrying out a simplified implementation of it based on a1-2 SPECTRAL DENSITY FUNCTIONALS FOR ELECTRONIC . . . PHYSICAL REVIEW B 69, 245101 ~2004!of a noninteracting system, potential energy, Hartree energy and exchange-correlation energy. The strategy consists in performing an expansion of the functional in powers of the charge of the electron.29,46,48,50,51 The lowest order term is the kinetic part of the action, and the energy associated with the external potential Vext . In the Baym Kadanoff Green func- tion theory this term has the form ~3!: KBK@G#5Tr ln G2Tr@G0 212G21#G . ~8! The G0(r,r8,iv) is the noninteracting Green function, which is given by G0 21~r,r8,iv!5d~r2r8!@ iv1m1¹22Vext~r!# , ~9! d~r2r8!5E dr9G021~r,r9,iv!G0~r9,r8,iv!, ~10! where m is a chemical potential. Note that since finite tem- perature formulation is adopted we did not obtain simply KBK@G#5Tr(2¹ 21Vext)G but also have got all entropy based contributions. Let us now turn to the density-functional theory. In prin- ciple, it does not have a closed formula to describe fully interacting kinetic energy as the density functional. However, it solves this problem by introducing a noninteracting part of the kinetic energy. It is described by its own Green function GKS(r,r8,iv), which is related to the Kohn–Sham ~KS! rep- resentation. An auxiliary set of noninteracting particles is introduced which is used to mimic the density of the system. These particles move in some effective one-particle Kohn- Sham potential Ve f f(r)5Vext(r)1Vint(r). This potential is chosen merely to reproduce the density and does not have any other physical meaning at this point. The Kohn–Sham Green function is defined in the entire space by the relation GKS 21(r,r8,iv)5G0 21(r,r8,iv)2Vint(r)d(r2r8), where Vint(r) is adjusted so that the density of the system r(r) can be found from GKS(r,r8,iv). Since the exact Green function G and the local Green function Gloc can be also used to find the density, we can write a general relationship: r~r!5T( iv GKS~r,r,iv!e iv01 5T( iv G~r,r,iv!eiv01 5T( iv Gloc~r,r,iv!e iv01, ~11! where the sum over iv assumes the summation on the Mat- subara axis at given temperature T. With the introduction of GKS the noninteracting kinetic portion of the action plus the energy related to Vext can be written in complete analogy with Eq. ~8! as follows KDFT@GKS#5Tr ln GKS2Tr@G0 212GKS 21#GKS . ~12! In order to describe the different contributions to the ther- modynamical potential in the spectral density-functional theory, we introduce a notion of the energy-dependent analog24510of Kohn-Sham representation. These auxiliary particles are interacting so that they will describe not only the density but also a local part of the Green function of the system, and will feel a frequency dependent potential. The latter is a field described by some effective mass operator Me f f(r,r8,iv) 5Vext(r)d(r2r8)1Mint(r,r8,iv). We now introduce an auxiliary Green function G(r,r8,iv) connected to our new ‘‘interacting Kohn–Sham’’ particles so that it is defined in the entire space by the relationship G 21(r,r8,iv) 5G0 21(r,r8,iv)2Mint(r,r8,iv). Thus, Mint(r,r8,iv) is a function which has the same range as the source that we introduce: it is adjusted until the auxiliary G(r,r8,iv) coin- cides with the local Green function inside the area restricted by u loc(r,r8), i.e., Gloc~r,r8,iv!5G~r,r8,iv!u loc~r,r8!. ~13! We illustrate the relationship between all introduced Green functions in Fig. 2. Note that G(r,r8,iv) also delivers the exact density of the system. With the help of G the kinetic term in the spectral density-functional theory can be repre- sented as follows KSDF@G#5Trln G2Tr@G0212G 21#G. ~14! Since GKS is a functional of r , DFT considers the density functional as the functional of Kohn-Sham wave functions, i.e., as GDFT@GKS# . Similarly, since G is a functional of Gloc , it is very useful to view the spectral density-functional GSDF as a functional of G: GSDF@G#5Tr ln G2Tr@G0212G 21#G1FSDF@Gloc# , ~15! where the unknown interaction part of the free energy FSDF@Gloc# is the functional of Gloc . If the Hartree term is explicitly extracted, this functional can be represented as FSDF@Gloc#5EH@r#1FSDF xc @Gloc# , ~16! FIG. 2. Relationship between various Green functions in spec- tral density-functional theory: exact Green function G, local Green function Gloc and auxiliary Green function G are the same in a certain region of space of our interest. They are all different outside this area, where the local Green function is zero by definiton.1-5 S. Y. SAVRASOV AND G. KOTLIAR PHYSICAL REVIEW B 69, 245101 ~2004!where EH@r# is the Hartree energy depending only on the density of the system, and where FSDF xc @Gloc# is the exchange-correlation part of the free energy. Note that the density of the system can be obtained via Gloc or G, there- fore the Hartree term can be also viewed as a functional of Gloc or G. Notice also, that since the kinetic energies ~8!, ~12!, ~14! are defined differently in all theories, the interac- tion energies FSDF@Gloc# , FBK@G# , FDFT@r# are also dif- ferent. The stationarity of the spectral density functional can be examined with respect to G: dGSDF dG~r,r8,iv! 50, ~17! similar to the stationarity conditions for GBK@G# and GDFT@GKS# dGBK dG~r,r8,iv! 50, ~18! dGDFT dGKS~r,r8,iv! 50. ~19! This leads to the equations for the corresponding Green func- tions in all theories: G 21~r,r8,iv!5G021~r,r8,iv!2Mint~r,r8,iv! ~20! as well as G21~r,r8,iv!5G0 21~r,r8,iv!2S int~r,r8,iv! ~21! GKS 21~r,r8,iv!5G0 21~r,r8,iv!2Vint~r!d~r2r8!. ~22! By using Eq. ~9! for G0 21 and by multiplying both parts by the corresponding Green functions we obtain familiar Dyson equations @2¹21Vext~r!2iv2m#G~r,r8,iv! 1E dr9Mint~r,r9,iv!G~r9,r8,iv!5d~r2r8! ~23! and @2¹21Vext~r!2iv2m#G~r,r8,iv! 1E dr9S int~r,r9,iv!G~r9,r8,iv!5d~r2r8!, ~24! @2¹21Vext~r!2iv2m#GKS~r,r8,iv! 1Vint~r!GKS~r,r8,iv!5d~r2r8!. ~25! The stationarity condition brings the definition of the auxil- iary mass operator Mint(r,r8,iv) which is the variational derivative of the interaction free energy with respect to the local Green function:24510Mint~r,r8,iv!5 dFSDF@Gloc# dG~r8,r,iv! 5 dFSDF@Gloc# dGloc~r8,r,iv! u loc~r,r8!. ~26! It plays the role of the effective self-energy which is short- ranged ~local! in the space. The corresponding expressions hold for the interaction parts of the exact self-energy of the electron S int(r,r8,iv) and for the interaction part of the Kohn-Sham potential Vint(r). S int~r,r8,iv!5 dFBK@G# dG~r8,r,iv! , ~27! Vint~r!d~r2r8!5 dFDFT@r# dGKS~r8,r,iv! 5 dFDFT@r# dr~r! d~r2r8!. ~28! If the external potential is added to these quantities we obtain total effective self–energies/potentials of the SDF, BK, and DF theories: Me f f(r,r8,iv), Se f f(r,r8,iv), Ve f f(r), re- spectively. If the Hartree potential VH(r) is separated we obtain the exchange-correlation parts: Mxc(r,r8,iv), Sxc(r,r8,iv), Vxc(r). Note that strictly speaking the substitution of variables, GKS vs r , in the density functional as well as the substitution of variables, G vs Gloc , in the spectral density-functional is only possible under the assumption of the so-called V repre- sentability ~or M representability!, i.e., the existence of such effective potential ~mass operator! which can be used to con- struct the exact density ~local Green function! of the system via the noninteracting Kohn-Sham particles of the DFT or its energy-dependent generalization in SDFT. Note also that the effective mass operator of spectral density-functional theory is local by construction, i.e., it is nonzero only within the cluster area V loc restricted by u loc(r,r8). It is an auxiliary object which cannot be identi- fied with the exact self-energy of the electron Se f f(r,r8,iv). This is similar to the observation that the Kohn-Sham poten- tial of the DFT cannot be associated with the exact self- energy as well. Nevertheless, the SDFT always delivers local Green function and the total free energy exactly ~at least in principle! as long as the exact functional is used. In the limit when the exact self-energy of the electron is indeed localized within V loc , the SDF becomes the Baym–Kadanoff func- tional which delivers the full Green function of the system, i.e., we can immediately identify Me f f(r,r8,iv) with Se f f(r,r8,iv) and the poles of G(r,r8,iv) with exact poles of G(r,r8,iv) where the information about both k and en- ergy dependence as well as life time of the quasiparticles is contained. We thus see that, at least formally, increasing the size of V loc in the SDF theory leads to a complete descrip- tion of the many-body system, the situation quite different from the DFT which misses such scaling. From a conceptual point of view, the spectral density- functional approach constitutes a radical departure from the DFT philosophy. The saddle–point equation ~23! is the equa-1-6 SPECTRAL DENSITY FUNCTIONALS FOR ELECTRONIC . . . PHYSICAL REVIEW B 69, 245101 ~2004!tion for a continuous distribution of spectral weight and the obtained local spectral function Gloc can now be identified with the observable local ~roughly speaking, k integrated! one-electron spectrum. This is very different from the Kohn- Sham quasiparticles which are the poles of GKS not identifi- able rigorously with any one-electron excitations. While the SDFT approach is computationally more demanding than DFT, it is formulated in terms of observables and gives more information than DFT. On one side, spectral density functional can be viewed as approximation or truncation of the full Baym Kadanoff theory where FBK@G# is approximated by FSDF@Gloc# by restricting G to Gloc 30 and the kinetic functionals KBK@G# and KSDF@G# are thought to be the same. Such restriction will automatically generate a short-ranged self-energy in the theory. This is similar to the interpretation of DFT as approximation FBK@G#5FDFT@r# ,KBK@G#5KSDF@GKS# which would generate the DFT potential as the self-energy. However, SDFT can be thought as a separate theory whose manifestly local self-energy is an auxiliary operator intro- duced to reproduce the local part of the Green function of the system, exactly as the Kohn-Sham ground state potential is an auxiliary operator introduced to reproduce the density of the electrons in DFT. Spectral density-functional theory contains the exchange- correlation functional FSDF@Gloc# . An explicit expression for it involving a coupling constant l5e2 integration can be obtained in complete analogy with the Harris-Jones formula52 of density functional theory.51 One considers GSDF@G,l# at an arbitrary interaction l and expresses GSDF@G,e2#5GSDF@G,0#1E 0 e2 dl ]GSDF@G,l# ]l . ~29! Here the first term is simply the kinetic part KSDF@G# as given by Eq. ~14! which does not depend on l . The second part is thus the unknown functional FSDF@Gloc# . The deriva- tive with respect to the coupling constant in Eq. ~3! is given by the average ^c1(x)c1(x8)c(x)c(x8)&5Pl(x , x8,iv) 1^c1(x)c(x)&^c1(x8)c(x8)& where Pl(x , x8) is the density-density correlation function at a given interaction strength l computed in the presence of a source which is l dependent and chosen so that the local Greens function of the system is G. Since ^c1(x)c(x)&5r(r)d(t), we can ob- tain: FSDF@Gloc#5EH@r#1( iv E 0 e2 dl Pl~r,r8,iv! ur2r8u . ~30! Establishing the diagrammatic rules for the functional FSDF@Gloc# while possible, 29 is not as simple as for the functional FBK@G# . The latter is formally represented as a sum of two-particle diagrams constructed with G and vC . It is known that instead of expanding FBK@G# in powers of the bare interaction vC and G, the functional form can be ob- tained by introducing the dynamically screened Coulomb in- teraction W(r,r8,iv) as a variable.53 In the effective action formalism30 this can be done by introducing an auxiliary24510Bose variable coupled to the density, which transforms the original problem into a problem of electrons interacting with the Bose field. W is the connected correlation function of the Bose field. Our effective action is now a functional of G, W and of the expectation value of the Bose field. Since the latter couples linearly to the density it can be eliminated exactly, a step which generates the Hartree term. After this elimination, the functional takes the form GBK@G ,W#5Tr ln G2Tr@G0 212G21#G1FBK@G ,W# , ~31! FBK@G ,W#5EH@r#2 1 2 Tr ln W1 1 2 Tr@vC 212W21#W 1CBK@G ,W# . ~32! The entire theory is viewed as the functional of both G and W. Here, CBK@G ,W# is the sum of all two-particle diagrams constructed with G and W with the exclusion of the Hartree term, which is evaluated with the bare Coulomb interaction. An additional stationarity condition dGBK /dW50 leads to the equation for the screened Coulomb interaction W: W21~r,r8,iv!5vC 21~r2r8!2P~r,r8,iv!, ~33! where the function P(r,r8,iv)522dCBK /dW(r,r8,iv) is the exact interacting susceptibility of the system, which is already discussed in connection with representation ~30!. A similar theory is developed for the local quantities.30 and this generalization represents the ideas of extended dy- namical mean-field theory,35 now viewed as an exact theory, namely, one constructs an exact functional of the local Greens function and the local correlator of the Bose field coupled to the density which can be identified with the local part of the dynamically screened interaction. The real-space definition of it is the following: Wloc~r,r8,iv!5W~r,r8,iv!u loc~r,r8!, ~34! which is non-zero within a given cluster V loc . Note that formally this cluster can be different from the one considered to define the local Green function ~2! but we will not distin- guish between them for simplicity. An auxiliary interaction W(r,r8,iv) is introduced which is the same as the local part of the exact interaction within nonzero area of u loc(r,r8) Wloc~r,r8,iv!5W~r,r8,iv!u loc~r,r8!. ~35! The interaction part of the spectral density functional is rep- resented in the form similar to Eq. ~32!, FSDF@Gloc ,Wloc#5EH@r#2 1 2 Tr ln W1 12 Tr@vC212W 21#W 1CSDF@Gloc ,Wloc# ~36! and the spectral density functional is viewed as a functional GSDF@Gloc ,Wloc# or alternatively as a functional GSDF@G,W# . CSDF@Gloc ,Wloc# is formally not a sum of two-particle diagrams constructed with Gloc and Wloc , but in principle a more complicated diagrammatic expression can be derived. Alternatively, a more explicit expression in-1-7 S. Y. SAVRASOV AND G. KOTLIAR PHYSICAL REVIEW B 69, 245101 ~2004!electron local Green functions Gloc and the dynamically screened local interactions Wloc . Unfortunately, its full implementation is a very challenging project which so far has only been carried out at the level of model Hamiltonians.26 There are several simplifications which can be made, however. The screened Coulomb interaction W(r,r8,iv) can be treated on different levels of approxima- tions. In many cases used in practical calculations with the LDA1DMFT method, this interaction W is assumed to be static and parametrized by a set of some optimally screened on-site parameters, such as Hubbard U and exchange J. These parameters can be fixed by constrained density func- tional calculations, extracted from atomic spectra data or ad- justed to fit the experiment. Since the described theory can perform a search in a constrained space with fixed interaction W, this justifies the use of U and J as input numbers. A more refined approximation, can use a method such as GW to generate an energy-dependent W ~Ref. 56! which is then treated using extended DMFT.26 Alternatively we can envi- sion that W is already so short ranged that we can ignore the EDMFT self-consistency condition, and we treat W as Wf ix(x ,x8). This leads to performing a partial self- consistency with respect to the Green function only. The pro- cedure is reduced to solving Dyson equations ~20!, ~45! as well as to finding Mint via the solution of the impurity prob- lem. A full self-consistency can finally be restored by includ- ing a second loop to relax W. A methodological comment should be made in order to make contact with the literature of cluster extensions of single site DMFT within model Hamiltonians. We adopted a less restrictive notion of locality by defining an effective ac- tion of the one-particle Green function ~and of the interac- tion! whose arguments are in nearby unit cells. This main- tains the full translation invariance of the lattice. At the level of the exact effective action , this is an exact construction, and its extremization will lead to portions of the exact Greens function which obeys causality. Note however that it has been proved recently19 that generating approximations to the exact functional by restricting the Baym–Kadanoff func- tional to nonlocal Green’s functions leads to violations of causality. For this reason, we propose to use techniques such as CDMFT which are manifestly causal for the purpose of realizing approximations to the local Greens functions. Our final general comment concerns the optimal choice of local representation or, precisely, the definition of the local Green function. This is because the local dynamical mean- field approximation is likely to be accurate only if we know in which portion of the Hilbert space the real electronic self- energy is well localized. Unfortunately, this is not known a priori, and in principle, only a full cluster DMFT calcula- tion is capable to provide us some hints in attempts to answer this question. However, considerable empirical evidence can be used as a guide for choosing a basis for DMFT calcula- tions, and we discuss these issues in the following sections. C. Choice of local representation We have already pointed out that spectral density- functional theory is a basis set dependent theory since it245101probes the Green function locally in a certain region deter- mined by a choice of basis functions in the Hilbert space. Provided the calculation is exact, the free energy of the sys- tem and the local spectral density in that Hilbert space will be recovered regardless the choice of it. We have developed the theory assuming that the basis in the Hilbert space is indeed the real space which gives us the choice ~2! for the local Green function, i.e., the part of the real Green function restricted by u loc(r,r8). While this is most natural choice for the purpose of formulating locality in r and r8 variables, it is also very useful to discuss a more general choice, connected to some space of orbitals xj(r) which can be used to repre- sent all the relevant quantities in our calculation. As we have in mind to utilize sophisticated basis sets of modern elec- tronic structure calculations, we will sometimes waive the orthogonality condition and will introduce the overlap matrix Ojj85^xjuxj8& especially in cases when we discuss a prac- tical implementation of the method. We note that the space xj(r) can in principle be inter- preted as the reciprocal space plane wave representation xj(r)5e i(k1G)r,j5k1G with k being the Brillouin zone vector and G being the reciprocal lattice vector. Thus the Green function can be probed in the region of the reciprocal space. It can be interpreted as the real space representation if xj(r)5d(j2r) where the sums over j are interpreted as the integrals over the volume, and the locality in this basis is precisely exploited in Eq. ~2!. A tremendous transparency of the theory will also arrive if we interpret the orbital space $xj% as a general nonorthogonal tight-binding basis set when index j combines the angular momentum index lm , and the unit cell index R, i.e., xj(r)5x lm(r2R)5xa(r2R). Note that we can add additional degrees of freedom to the index a such, for example, as multiple k basis sets of the linear muffin-tin orbital based methods, Gaussian decay constants in the Gaussian orbital based methods, and so on. If more than one atom per unit cell is considered, index a should be supplemented by the atomic basis position within the unit cell, which is currently omitted for simplicity. For spin un- restricted calculations a accumulates the spin index s and the orbital space is extended to account for the eigenvectors of the Pauli matrix. Let us now introduce the representation for the exact Green function in the localized orbital representation G~r,r8,iv!5( ab ( k xa k~r!Gab~k,iv!xb k*~r8! 5( ab ( RR8 xa~r2R!Gab~R2R8,iv! 3xb*~r82R8!. ~48! Assuming the single-site impurity case, we can separate local and nonlocal parts Gloc(r,r8,iv)1Gnon2loc(r,r8,iv) as fol- lows Gloc~r,r8,iv!5( ab Gloc ,ab~ iv!( R xa~r2R!xb*~r82R! 5( ab Gloc ,ab~ iv!( k xa k~r!xb k*~r8!, ~49!-10 SPECTRAL DENSITY FUNCTIONALS FOR ELECTRONIC . . . PHYSICAL REVIEW B 69, 245101 ~2004!where we denoted the site-diagonal matrix elements dRR8Gab(R2R8,iv) as Gloc ,ab(iv). Note that this defini- tion is different from the real-space definition ~2!. For ex- ample, Eq. ~2! contains the information about the density of the system. The formula ~49! does not describe the density since RÞR8 elements of the matrix Gab(R2R8,iv) are thrown away. The locality of Eq. ~49! is controlled exclu- sively by the decay of the orbitals xa(r) as a function of r, not by u loc(r,r8) The local part of the Green function, Gloc(r,r8,iv), which is just defined with respect to the Hilbert space $xa% can be found by developing the corresponding spectral density-functional theory. Since the basis set is assumed to be fixed, the matrix elements Gloc ,ab(iv) appear only as variables of the functional. As above, we introduce an aux- iliary Green function Gab(k,iv) to deal with kinetic energy counterpart. Stationarity yields the matrix equation: G0,ab 21 ~k,iv!5G ab21~k,iv!1Mint ,ab~ iv!, ~50! where the noninteracting Green function ~9! is the matrix of noninteracting one-electron Hamiltonian: G0,ab 21 ~k,iv!5^xa k uiv1m1¹22Vextuxb k&. ~51! The self-energy Mint ,ab(iv) is the derivative dFSDF@Gloc ,ab(iv)#/dGloc ,ab(iv) and takes automatically the k independent form. While formally exact, this theory would have at least one undesired feature since, for example, the density of the sys- tem can no longer be found from the definition ~49! of Gloc(r,r8,iv). As a result, the Hartree energy cannot be sim- ply recovered. If treated exactly FSDF@Gloc ,ab(iv)# should contain the Hartree part. However, we see that the theory delivers k independent Mint ,ab(iv) including the Hartree term. There seems to be a paradox since modern electronic structure methods calculate the matrix element of the Hartree potential within a given basis exactly, i.e., ^xa k uVHuxb k&. The k dependence is trivial here and is connected to the known k dependence of the basis functions used in the calculation. Therefore, while formulating the spectral density-functional theory for electronic structure calculation, we need to keep in mind that in many cases, the k dependence is factorizable and can be brought into the theory without a problem. This warns us that the choice of the local Green function has to be done with care so that useful approximations to the func- tional can be worked out. It also shows that in many cases the k dependence is encoded into the orbitals. It is not that nontrivial k dependence of the self-energy operator, which is connected to the fact that Mint(r,r8,iv) may be long-range, i.e., decay slowly when r departs from r8. It may very well be proportional to d(r2r8) such as the LDA potential and still deliver the k dependence. It turn out that the desired k dependence with the choice of the Green function after Eq. ~49! can be quickly reinstated if we add the density of the system as another variable to the functional. This is clear since the density is a particular case of the local Green function in Eq. ~2! taken at r5r8 and summed over iv . Therefore combination of definition ~49! and r is another, third possibility of defining Gloc . This will245101allow treatment of all local Hartree-like potentials without a problem. Moreover, as we discuss below, this may allow to design better approximations to the functional since the Hil- bert space treatment of locality is more powerful: it may allow us to treat more long-ranged self-energies than the ones restricted by u loc(r,r8), and the basis sets can be opti- mally adjusted to specific self-energies exactly as the basis sets used in electronic structure calculations are tailored to the LDA potential. We have noted earlier that the mass operator Mint(r,r8,iv) is an auxiliary object of the spectral density- functional theory. It has the same meaning as the DFT Kohn–Sham potential: it is local operator that needs to be added to the noninteracting Green function in order to repro- duce the local Green function of the system, as the DFT potential is added to the noninteracting Green function to reproduce the density of the system. Roughly speaking, SDFT provides the exact energy and exact one-electron den- sity of states which is advantageous compared to the DFT which provides the energy and the density only. However, we obtain the full k dependent one-particle spectra as the poles of auxiliary Green function G(r,r8,z). Can these poles be interpreted as the exact k dependent one-electron excita- tions? This question is similar to the question of the DFT: can the Kohn-Sham spectra be interpreted as the physical one-electron excitations? To answer both questions we need to know something about exact self-energy of the electron. If it is energy-independent, totally local, i.e., proportional to d(r2r8) and well-approximated by the DFT potential, the Kohn-Sham spectra represent real one-electron excitations. The exact SDFT waives most of the restrictions: if the real self-energy is localized within the area Rloc , the exact SDFT calculation with the cluster V loc including Rloc will find the exact k dependent spectrum. If we pick V loc larger than Rloc , the SDFT equations themselves will choose physical localization area for the self-energy during our self- consistent calculation. However, these statements become approximate if we utilize the local dynamical mean-field ap- proximation instead of extremizing the exact functional. Even if the real self-energy of the electron is sufficiently short ranged, this approximation will introduce some error in the calculation, the situation similar to LDA within DFT. However, the local dynamical mean-field theory does not necessarily have to be formulated in real space. The assump- tion of localization for self-energy can be done in some por- tion of the Hilbert space. In that portion of the Hilbert space the cluster impurity model needs to be solved. The choice of the appropriate Hilbert space, such, e.g., as atomiclike tight-binding basis set is crucial to obtain an eco- nomical solution of the impurity model. Let us for simplicity discuss the problem of optimal basis in some orthogonal tight-binding ~Wannier-like! representation for the electronic self-energy S~r,r8,iv!5( ab ( k xa k~r!Sab~k,iv!xb k*~r8! 5( ab ( RR8 xa~r2R!Sab~R2R8,iv! 3xb*~r82R8!. ~52!-11 S. Y. SAVRASOV AND G. KOTLIAR PHYSICAL REVIEW B 69, 245101 ~2004!We can separate our orbital space $xa% onto the subsets de- scribing light $xA% and heavy $xa% electrons. Assuming ei- ther off-diagonal terms between them are small or we work with exact Wannier functions, the self-energy S(r,r8,iv) can be separated onto contributions from the light, SL(r,r8,iv), and from the heavy, SH(r,r8,iv), electrons. Sab(k,iv) is expected to be k dependent but largely v in- dependent for the light block, i.e, SL(r,r8,iv) 5(AB(kxA k(r)SAB(k)xB k*(r8). The k dependency here should be well-described by LDA-like approximations, therefore we expect SL(r,r8,iv);Ve f f(r)d(r2r8). A dif- ferent situation is expected for the heavy block where we would rely on the result SH~r,r8,iv!;Ve f f~r!d~r2r8!1( ab xa~r!Sab8 ~ iv!xb*~r8!. ~53! The first term here gives the k dependence coming from an LDA-like potential. It describes the dispersion in the heavy band. The second term is the energy dependent correction where site-diagonal approximation R5R8 is imposed. What is the best choice of the basis to use in connection with Sab8 (iv) in Eq. ~53!? Here the decay of the orbitals xa(r) as a function of r is now entirely in charge of the self-energy range. In light of the spectral density-functional theory, the answer is the following: the local dynamical mean-field ap- proximation would work best for such basis xa(r) whose range approximately corresponds to a self-energy localiza- tion Rloc of the real electron. Even though Rloc is not known a priori, something can be learned about its value based on a substantial empirical evidence. It is, for example, known that LDA energy bands when comparing to experiments at first place miss the energy dependent Sab8 (iv) like corrections. This is the case for bandwidths in transition metals ~and also in simple metals!, the energy gaps of semiconductors, etc. It is also known that many-body based theories work best for massively downfolded model Hamiltonians where only ac- tive low-energy degrees of freedom at the region around the Fermi level EF remain. The many-body Hamiltonian Ĥ5( ab ( RR8 haRbR8 (0) @caR 1 cbR81H.c.# 1 ( abgd ( RR8R9R- Vabgd RR8R9R-caR 1 cbR8 1 cdR-cgR9 ~54! with Vabgd RR8R9R- 5E drdr8xaR* ~r!xbR8* ~r8!vC~r2r8!xgR9~r!xdR-~r8! assumes the one-electron Hamiltonian haRbR8 (0) is obtained as a fit to the bands near EF . This can always be done by long-ranged Wannier functions. It is also clear that the cor- relation effects are important at first place for the partially occupied bands since only these bring various configura- tional interactions in the many-body electronic wave func-245101tions. For example, the well-known one-band Hamiltonian for CuO2 plane of high-Tc materials considers an antibond- ing combination of Cux22y2 and Ox ,y orbitals which crosses EF . Also, the calculations based on the LDA1DMFT method usually obtain reliable results when treating only the bands crossing the Fermi level as the correlated one-electron states. This is, for example, the case of Pu or our25 and previous57 calculation for LaTiO3 where t2g three-band Hamiltonian is considered. All this implies that the range for Sab8 (iv) term in Eq. ~53! should correspond to the properly constructed Wannier orbitals, which is fairly long ranged. What happen if we instead utilize mostly localized represen- tation which, for example, can be achieved by tight binding fits to the energy bands at higher energy scale? For the case of CuO2 this would correspond to a three-band Hamiltonian with Cux22y2 and Ox ,y orbitals treated separately. For LaTiO3 system this is a Hamiltonian derived from Tit2g and Op or- bitals. The answer here can be given as a practical matter of most economic way to solve the impurity problem: provided Cu and O levels are well separated, provided both ap- proaches use properly downfolded for each case Coulomb interaction matrix elements Vabgd RR8R9R- , and provided correla- tions are treated on all orbitals, the final answer should be similar regardless the choice of the basis. A faster algorithm will be obtained by treating the one-band Hamiltonian with antibonding Cux22y22Ox ,y orbital. If indeed the self-energy is localized on the scale of the distance between Cu and O, it is clear where the inefficiency of the three-band model ap- pears: the second term in Eq. ~53! needs to be extended within the cluster involving both Cu and nearest O sites and should involve both Cu and O centered orbitals simply to reach the cluster boundary. In the one-band case this is en- coded into the decay of the properly constructed Wannier state. The preceding discussion is merely a conjecture. It does not imply that the localization range for the real self-energy of correlated electron at given frequency v is directly pro- portional to the size of Wannier states located in the vicinity of v1m . It may very well be that in many cases this range is restricted by a single atom only ~atomic sphere of Cu in the example above!. Clearly more experience can be gained by studying a correlation between the decay of the Coulomb matrix element VRRR8R8 as a function of R2R8 and the ob- tained matrix S(R2R8,v) using a suitable cluster DMFT technique. These works are currently being performed and will be reported elsewhere.58 The given discussion however warns that in general the best choice of the basis for single- site dynamical mean-field treatment may not be the case of mostly localized representation. In this respect the area re- stricted by u loc(r,r8) which is used to formulate SDFT in the real space may need to be extended up to a cluster. However, alternative formulation with the choice of local Green func- tion after Eq. ~49! may be more economical since a single- site approximation may still deliver good results. As we have argued, such spectral density-functional theory will also need the density of the system to complete the definition of local Green function. The local dynamical mean-field approxima- tion can be applied to the interaction functional FSDF which-12 SPECTRAL DENSITY FUNCTIONALS FOR ELECTRONIC . . . PHYSICAL REVIEW B 69, 245101 ~2004!Therefore, the LDA1U method can be viewed as an ap- proximation ~Hartree–Fock approximation! to the spectral density functional within LDA1DMFT. The correct interaction energy among the correlated elec- trons can be written down explicitly using the Hartree–Fock approximation. In our language the LDA1DMFT interaction energy functional ~55! is rewritten in the form FLDA1U@r ,nab#5EH@r#1Fxc LDA@r#1F̃U@nab#2FDC@ n̄c# , ~65! where the functional form F̃U@nab# is known explicitly: F̃U@nab#5 1 2 (abcd ~Uacbd2Uacdb!nabncd . ~66! Here, indexes a ,b ,c ,d involve fixed angular momentum l of the heavy orbitals and run over magnetic m and spin s quan- tum numbers. The on-site Coulomb interaction matrix Uabcd is the on-site Coulomb interaction matrix element Va5ab5bg5cd5d RRRR appeared in Eq. ~54! which is again taken for the subblock of the heavy orbitals. Note that sometimes Uabcd is defined as Va5ab5cg5bd5d RRRR . The double counting term FDC@nab# needs to be intro- duced since both the L~S!DA and U terms account for the same interaction energy between the correlated orbitals. This includes in first place the Hartree part. However, the precise form of the double counting is unclear due to nonlinear na- ture of the LDA exchange–correlation energy. In practice, it was proposed7 that the form for FDC is FDC@ n̄c#5 1 2 Ūn̄c~ n̄c21 !2 1 2 J̄@ n̄c ↑~ n̄c ↑21 !1 n̄c ↓~ n̄c ↓21 !# . ~67! where Ū5@1/(2l11)2#(abUabab , J̄5Ū2@1/2l(2l 11)#(ab(Uabab2Uabba) and where n̄c s5(aPlcnaadsas , n̄c5 n̄c ↑1 n̄c ↓ . Some other forms of the double countings have also been discussed in Ref. 61. The minimization of the functional GLDA1U@r ,nab# is now performed. The self-energy correction in Eq. ~58! ap- pears as the orbital dependent correction M̃ab2VabDC : M̃ab5 dF̃U dnab 5( cd ~Uacbd2Uacdb!ncd , ~68! Vab DC5 dFDC dnab 5dabF ŪS n̄c2 12 D2 J̄S n̄cs2 12 D G . ~69! While the correction is static, it is best viewed as the Hartree- Fock approximation to the self-energy Mab(iv) within the LDA1DMFT method. Note that such interpretation allows us to utilize double counting forms within LDA1DMFT as M̃(r,r8,i`) or M̃(r,r8,i0). Note also that the solution of the impurity problem collapses in the LDA1U method since the self-energy is known analytically by formula ~68!. From a practical point of view, despite the great success of the LDA1U theory in predicting materials properties of correlated solids7 there are obvious problems with this ap- proach when applied to metals or to systems where the or-245101bital symmetries are not broken. They stem from the well- known deficiencies of the Hartree–Fock approximation. The most noticeable is that it only describes spectra of magneti- cally ordered systems which have Hubbard bands. We have however argued that a correct treatment of the electronic structure of strongly correlated systems has to treat both Hubbard bands and quasiparticle bands on the same footing. Another problem occurs in the paramagnetic phase of Mott insulators: in the absence of any broken symmetry the LDA1U method reduces to the LDA, and the gap collapses. In systems such as NiO where the gap is of the order of eV, but the Neel temperature is a few hundred Kelvin, it is un- physical to assume that the gap and the magnetic ordering are related. For this reason the LDA1U predicts magnetic order in cases that it is not observed, as, e.g., in the case of Pu.62 F. Local GW approximation We now discuss the relaxation of the screened Coulomb interaction W(r,r8,iv) which appeared in the spectral density-functional formulation of the problem. Both LDA 1DMFT and LDA1U methods parametrize the interaction W with optimally screened set of parameters, such, e.g., as the matrix Uabcd appeared in Eq. ~66!. This matrix is sup- posed to be given by an external calculation such, e.g., as the constrained LDA method.28 To determine this interaction self-consistently an additional self-consistency loop de- scribed by the Eqs. ~33! and ~46! has to be switched on together with calculation of the local susceptibility P(r,r8,iv) by the impurity solver. This brings a truly self- consistent ab initio method without input parameters and the double counting problems. A simplified version of this method has been recently proposed12,16 which is known as a local version of the GW method ~LGW!. Within the spectral density-functional theory, this approximation appears as approximation to the functional CSDF@Gloc ,Wloc# taken in the form CLGW@Gloc ,Wloc#52 1 2 TrGlocWlocGloc . ~70! As a result, the susceptibility P(r,r8,iv) is approximated by the product of two local Green functions, i.e., P 522dCLGW /dWloc5GlocGloc , and the exchange- correlation part of our mass operator is approximated by the local GW diagram, i.e., Mxc5dCLGW /dGloc 52GlocWloc . Thus, the impurity model is solved and the procedure can be made self-consistent: For a given Mint and P, the Dyson equations ~20!, ~37! for G and W are solved. Then, the local quantities Gloc , Wloc are generated and used to find new Mint and P thus avoiding the computation of the bath Green function G0 after Eq. ~45!, and the interaction V, after Eq. ~46!. Note that since the local GW approximation Eq. ~70! is relatively cheap from computational point of view, its imple- mentation on a cluster and for all orbitals should not be a problem. The results of the single-site approximation for the local quantities have been developed independently and re--15 S. Y. SAVRASOV AND G. KOTLIAR PHYSICAL REVIEW B 69, 245101 ~2004!ported in the literature.16 The cluster extension is currently being performed and the results will be reported elsewhere.63 Note finally that the local GW approximation is not the only one which can be implemented as the simplified impu- rity solver. For example, another popular approximation known as the fluctuational exchange approximation ~FLEX! can be worked out along the same lines. Note also that the combination of the DMFT and full GW diagram has been recently proposed12,27 and a simplified implementation for Ni,27 and for a model Hamiltonian26 have been carried out. This procedure incorporates full k dependence of the self- energy known diagrammatically within GW together with the additional local DMFT diagrams. III. CALCULATION OF LOCAL GREEN FUNCTION The solution of the Dyson equations described in the pre- ceding section for a given strongly correlated material re- quires the calculation of the local Green function during the iterations towards self-consistency. This is very similar to the procedure in the density-functional theory, when the charge density is computed. A big advantage of DFT is the use of Kohn–Sham orbitals which reduces the Eq. ~22! for the Kohn–Sham Green function to a set of one-particle Schröd- inger’s like equations for the wave functions. As a result the kinetic-energy contribution is calculated directly and the evaluation of the total energy of a solid is not a problem. Here, a similar algorithm will be described for the energy- dependent Dyson equation, the solution in terms of the linear-muffin-tin orbital basis set will be discussed, and the formula for evaluating the total energy will be given. A. Energy Resolved One-Particle Representation We introduced the auxiliary Green function G(r,r8,iv) to deal with the kinetic part of the action in SDFT. It satisfies to the Dyson Eq. ~9!. Let us now introduce the representation of generalized energy-dependent one-particle states G~r,r8,iv!5( kj ckjv R ~r!ckjv L ~r8! iv1m2Ekjv , ~71! G 21~r,r8,iv!5( kj ckjv R ~r!~ iv1m2Ekjv!ckjv L ~r8!, ~72! where the left ckjv L (r) and right ckjv R (r) states satisfy to the following Dyson equations: @2¹21Vext~r!1VH~r!#ckjv R ~r! 1E Mxc~r,r8,iv!ckjvR ~r8!dr85EkjvckjvR ~r! ~73! @2¹21Vext~r!1VH~r!#ckjv L ~r! 1E ckjvL ~r8!Mxc~r8,r,iv!dr85EkjvckjvL ~r! ~74! @we dropped the imaginary unit for simplicity in the notation ckjv(r) which shall be thought as a shortened version of245101ckj(r,iv)]. These equations should be considered as the ei- genvalue problems with complex non-Hermitian self-energy. As a result, the eigenvalues Ekjv @a shortened form for Ekj(iv)] being the same for both equations are complex in general. The explicit dependency on the frequency iv in both eigenvectors and eigenvalues comes from the self- energy. Note that left and right eigenfunctions are orthonor- mal E drckjvL ~r!ckj8vR ~r!Äd j j8 ~75! and can be used to evaluate the charge density of a given system using the Matsubara sum and the integral over the k space: r~r!5T( iv ( kj gkjvckjv L ~r!ckjv R ~r!eiv0 1 , ~76! where gkjv5 1 iv1m2Ekjv . ~77! We have cast the notation of spectral density theory in a form similar to DFT. The function gkjv is the Green function in the orthogonal left/right representation which plays a role of a ‘‘frequency-dependent occupation number.’’ It needs to be pointed out that the frequency-dependent energy bands Ekjv represent an auxiliary set of complex ei- genvalues. These are not the true poles of the exact one- electron Green function G(r,r8,z) considered at complex z plane. However, they are designed to reproduce the local spectral density of the system. Note also that these bands Ekjz are not the true poles of the auxiliary Green function G(r,r8,z). The latter ones still need to be located by solving a nonlinear equation corresponding to the singularities in the expression ~71! after analytic continuation to real frequency. For a one-band case this equation is simply: z1m2Ekz 50, whose solution delivers the quasiparticle dispersion Zk . General knowledge of the poles positions Zkj will allow us to write an alternative expression for G which is similar to Eq. ~71!, but with the eigenvectors found at Zkj thus carrying out no auxiliary frequency dependence. These poles are the real one-electron excitational spectra in case G is a good approxi- mation to G. However, the use of Eq. ~71! is advantageous, since it avoids additional search of poles and allows direct evaluation of the local spectral and charge densities the sys- tem. The energy-dependent representation allows us to obtain a very compact expression for the total energy. As we have argued, the entropy terms are more difficult to evaluate. However, they are generally small as long as we stay at low temperatures. The pure kinetic part of the free energy ex- pressed via @see, Eq. ~39!#-16 SPECTRAL DENSITY FUNCTIONALS FOR ELECTRONIC . . . PHYSICAL REVIEW B 69, 245101 ~2004!Tr ln G2TrMe f fG5T( iv eiv0 1E drdr8ln G~r,r8,iv! 2T( iv E drdr8Me f f~r,r8,iv! 3G~r8,r,iv! ~78! needs to be separated onto the energy and entropy terms. Both contributions can be evaluated without a problem, but in light of neglecting the entropy correction in the interaction part, we concentrate on evaluating the kinetic energy only: T( iv eiv0 1E dr@~2¹r2!G~r,r8,iv!#r5r8 5T( iv eiv0 1 ( kj ^ckjv L u2¹2uckjv R & iv1m2Ekjv . ~79! The SDFT total energy formula is now arrived by utilizing the relationship Ekjv5^ckjv L u2¹21Me f f uckjvR &5^ckjvL u2¹21Vext1VH1MxcuckjvR &: ESDF5T( iv eiv0 1 ( kj gkjvEkjv 2T( iv E drdr8Me f f~r,r8,iv!G~r8,r,iv!1 1E drVext~r!r~r!1 12E drVH~r!r~r! 1 1 2 T( iv E drdr8Mxc~r,r8,iv!Gloc~r8,r,iv!. ~80! If the self-energy is considered as input to the iteration while the Green function is the output, near stationary point, it should have a convergency faster than the convergency in the Green function. It is instructive to consider the noninteractive limit when the self-energy represents a local energy-independent poten- tial, say, the ground-state Kohn Sham potential of the density-functional theory. This provides an important test for our many-body calculation. It is trivial to see that in the DFT limit, we obtain the Kohn–Sham eigenfunctions ckjv R ~r!→ckj~r!, ~81! ckjv L ~r!→ckj* ~r!, ~82! Ekjv→Ekj , ~83! and the one-electron energy bands are no longer frequency dependent. The sum over Matsubara frequencies in the ex- pression for the charge density ~76! can be performed ana- lytically using the expression for the Fermi-Diraq occupation numbers:245101f kj5 1 e (Ekj2m)/T11 5T( iv eiv0 1 iv1m2Ekj ~84! and the formula ~76! collapses to the standard expression for the density of noninteracting fermions. The total-energy ex- pression ~80! is converted back to the DFT expression for the total energy since the eigenvalue Ekjv becomes the DFT band structure Ekj , and the summation over Matsubara fre- quencies T( ive iv01gkjv gives according to Eq. ~84! the Fermi-Diraq occupation number f kj . The standard DFT ex- pression is recovered: EDFT5( kj f kjEkj2E drVe f f~r!r~r!1E drVext~r!r~r! 1 1 2E drVH~r!r~r!1Exc@r# , ~85! where Ekj5^ckju2¹21Ve f f uckj&5^ckju2¹21Vext1VH 1Vxcuckj&. B. Use of linear muffin–tin orbitals The next problem is to solve the Dyson equation for the eigenvalues. The sophisticated basis sets developed to solve the one-electron Schrödinger equation can be directly used in this case. We utilize the LMTO method described exten- sively in the past literature42–44 as it provides a minimal atom-centered local orbital basis set ideally suited for the electronic structure calculation. Within the LMTO basis, the full Green function is represented as a sum G~r,r8,iv!5( k ( ab xa k~r!Gab~k,iv!xbk*~r8! ~86! and, as we have argued in the preceding section, the matrix Gab(k,iv) needs to be considered as a variable in the spec- tral density functional. The stationarity yields the equation for the Green function Gab~k,iv!5@~ iv1m!Ô~k!2 ĥ (0)~k!2Mint~k,iv!#ab21 , ~87! where the overlap matrix Oab(k)5^xa k uxb k&, the noninteract- ing Hamiltonian matrix hab (0)(k)5^xa k u2¹21Vext(r)uxb k& and the self–energy formally comes as a matrix element Mint ,ab~k,iv!5E drdr8xak*~r!Mint~r,r8,iv!xbk~r8! ~88! over the LMTOs. Again, it is worth to point out that the self-energy here depends on k via the orbitals even if the single-impurity case is considered. In calculations performed on a cluster, the self-energy will also pick its nontrivial k dependence coming from the nearest sites. While formally valid, the present approach is not very efficient since the Green function G(r,r8,iv) has to be evaluated via Eq. ~86!. This is the k integral which has poles in a complex frequency plane, and integrating singular func- tions need to be performed with care. In this respect, we-17 S. Y. SAVRASOV AND G. KOTLIAR PHYSICAL REVIEW B 69, 245101 ~2004!centers of interest constructed with the spherical part of the LDA potential. As the entire procedure is variational slight modifications brought by the nonlocal self-energies after Eq. ~61! and ignored in the present study should not lead to significant modifications of the obtained total energies and the one-electron spectra as long as TB-LMTO basis is thought to be complete within a given energy window. The second step is a construction of the charge density using the obtained local Green function. In this regard, the formula ~60! was used which takes into account the modification of the one-electron densities of states Nl(E) brought by the correlations. As, the redistribution of the spectral weight for the f electrons involves the feedback on the remaining s ,p ,d electrons, change in the densities of states Nl(E) appears in all l channels. The final step is the self-consistent evaluation of the total energy using the formula ~80!. In the LDA 1DMFT approximation using a fixed basis set all compli- cated integrals appeared in Eq. ~80! can be reduced to con- volutions between various matrices. Since the interaction functional has the form of Eq. ~55!, we subtract from the LDA the average interaction energy of the f electrons in the form ~67! of the double counting term and then add im- proved estimates of these quantities using the self-consistent solution of the impurity model. This results in a simplified expression for the total energy: Etot5T( kj ( iv gkjvekjv2E Ve f f~r!r~r!dr 2T( iv ( ab @Mint ,ab~ iv!2VabDC#Gloc ,ba~ iv! 1E Vext~r!r~r!dr1EH@r#1ExcLDA@r# 1 1 2 T( iv ( ab Mint ,ab~ iv!Gloc ,ba~ iv!2FDC@ n̄c# , ~103! where Vab DC is given by Eq. ~69! and FDC@ n̄c# is given by Eq. ~67!. Since the dynamical mean-field theory requires the solu- tion of the Anderson impurity model for the multiorbital f shell of Pu, we have developed a method which, inspired by the success of the iterative perturbation theory,10 interpolates the self-energy between small and large frequencies. At low frequencies, the exact value of the self-energy and its slope is extracted from the Friedel sum rule and from a slave-boson mean-field treatment.37–39 This approach is accurate as it has been shown recently to give the exact critical value of U in the large degeneracy limit at half-filling78 At high frequen- cies the self-energy behavior can be computed based on high-frequency moments expansions.14,40 The result of inter- polation can be encoded into a simple rational form for the self-energy.41 In practical calculations for Pu, we used a two- pole approximation: S~ iv!5S~ i`!1 ( n51,2 Wn iv2Pn , ~104!245101where the unknown four coefficients Wn and Pn are deter- mined to satisfy known conditions in the low- and high- frequency limits. We found that this kind of self-energy fits quantum Monte Carlo ~QMC! data in large region of param- eters, such as U and doping, and where this comparison is at all possible ~small degeneracy and high temperature!. Thus, our approach interpolates between four major lim- its: small and large iv’s valid for any U as well as small and large U8s ~band vs atom! valid for any iv . The analytical continuation to the real frequency axis is not a problem with the present method and avoids the use of the Pade79 and maximum entropy4 based techniques. Complete details of our method can be found in Ref. 41. Here we only mention technicalities connected to the f electrons of Pu where we deal with the impurity Green functions which are the matri- ces 14314. However, for the relativistic f level in cubic symmetry, the matrices can be reduced to 535 with four nonzero off-diagonal elements. The solution of such impurity problem is still a formidable numerical problem. We there- fore make some simplifications. First, the off-diagonal ele- ments are in general small and will be neglected. We are left with the 5 f 5/2 state split into 2 levels which are twofold (G7), and fourfold (G8) degenerate, and with the 5 f 7/2 state split into three levels which are twofold (G6), twofold (G7), and fourfold (G8) degenerate. Second, since in Pu the inter- multiplet spin-orbit splitting is much larger than the intra- multiplet crystal field splitting ~.5:1!, we reduce the prob- lem of solving Anderson impurity model ~AIM! for the lev- els separately by treating the 5 f 5/2 G7 and G8 levels as one sixfold degenerate level, and the 5 f 7/2 G6 , G7 and G8 levels as another eightfold degenerate level. We first study in detail the total energy as a function of the parameter U and give our predictions for the volumes in a , d , and « phases. We then discuss the one-electron spectra in both a and d phases and compare our results with the pho- toemission experiment.68 Since our method does not yet al- low us to treat complicated lattices, we perform our calcula- tions for simple fcc and bcc structures and report only a simplified study of the a phase which formally has 16 atoms per unit cell. A. Calculation of volume To illustrate the importance of correlations, we discuss the results of our total energy calculations for various strengths of the on-site Coulomb interaction U. Figure 5 reports our theoretical predictions. First, the total energy as a function of volume of the fcc lattice is computed. The temperature is fixed at 600 K, i.e. in the vicinity of the region where the d phase is stable. U50 GGA curve indicates a minimum at V/V050.7. This volume is in fact close to the volume of the a phase. Certainly, we expect that correlations should be less important for the compressed lattice in general, but there is no sign whatever of the d phase in the U50 calculation. The total energy curve is dramatically different for U larger than 0. The details depend sensitively on the actual value of U. The behavior at U54 eV shows the possibility of a double minimum; it is actually realized for a slightly smaller value of U. We find that for U53.8 eV, the minimum occurs near-20 SPECTRAL DENSITY FUNCTIONALS FOR ELECTRONIC . . . PHYSICAL REVIEW B 69, 245101 ~2004!V/Vd50.80 which corresponds to the volume of the a phase if we allow for monoclinic distortions and a volume- dependent U. When U increases by 0.2 eV the minimum occurs at V/Vd51.05 which corresponds to the volume of the d phase, in close agreement with experiment. Since the energies are so similar, we may expect that as temperature decreases, the lattice undergoes a phase transition from the d phase to the a phase with the remarkable decrease of the volume by 25%. We repeated our calculations for the bcc structure using the temperature T5900 K where the e phase is stable. Fig. 5 shows these results for U54 eV with a location of the mini- mum at around V/Vd51.03. While the theory has a residual inaccuracy in determining the d and e phase volumes by a few percent, a hint of volume decrease with the d→e tran- sition is clearly reproduced. Thus, our first-principles calcu- lations reproduce the main features of the experimental phase diagram of Pu. Note that the values of U;4 eV which are needed in our simulation to describe the a→d transition, are in good agreement with the values of on-site Coulomb repulsion be- tween f electrons estimated by atomic spectral data,80 con- strained density-functional studies,81 and our previous LDA 1U studies.62 The double-well behavior in the total-energy curve is un- precedented in LDA or GGA based calculations but it is a natural consequence of the proximity to a Mott transition. Indeed, recent studies of model Hamiltonian systems10,82 in the vicinity of the Mott transition show that two DMFT so- lutions which differ in their spectral distributions can coexist. It is very natural that allowing the density to relax in these conditions can give rise to the double minima as seen in Fig. 5. B. Calculation of spectra We now report our calculated spectral density of states for the fcc structure using the volume V/Vd50.8 and V/Vd 51.05 corresponding to our theoretical studies of a and d phases. To compare the results of the dynamical mean-field FIG. 5. Total energy as a function of volume in Pu for different values of U calculated using the LDA1DMFT approach. Data for the fcc lattice are computed at T5600 K, while data for the bcc lattice are given for T5900 K.245101calculations with the LDA method as well as with the experi- ment, we discuss the results presented in Fig. 6. Figure 6~a! shows the density of states calculated using LDA1DMFT method in the vicinity of the Fermi level. Solid black line corresponds to the d phase and solid gray line corresponds to the a phase. We predict the appearance of a strong quasipar- ticle peak near the Fermi level which exists in the both phases. Also, the lower and upper Hubbard bands can be clearly distinguished in this plot. The width of the quasipar- ticle peak in the a phase is found to be larger by 30 per cent compared to the width in the d phase. This indicates that the low-temperature phase is more metallic, i.e., it has larger spectral weight in the quasiparticle peak and smaller weight in the Hubbard bands. Recent advances have allowed the experimental determination of these spectra, and our calcu- lations are consistent with these measurements.68 Figure 6~b! shows the measured photoemission spectrum for d ~black line! and a ~gray line! Pu. We can clearly see a strong qua- siparticle peak. Also a smaller peak located at 0.8 eV for the d phase can be found. We interpret it as the lower Hubbard band. The result of the local-density approximation is shown on Fig. 6~a! by dashed line. The LDA produces two peaks near the Fermi level corresponding to 5 f 5/2 and 5 f 7/2 states sepa- rated by the spin-orbit coupling. The Fermi level falls into the dip between these states and cannot reproduce the fea- tures seen in photoemission. We should also mention that LDA1U fails completely62,71 to reproduce the intensity of the f states near the Fermi level as it pushes the f band 2–3 eV below the Fermi energy. This is the picture expected from the static Hartree-Fock theory such as the LDA1U. Only full inclusion of the dynamic effects within the DMFT allows to account for both the quasiparticle resonance and the Hub- bard satellites which explains all features of the photoemis- sion spectrum in d Pu. The calculated by LDA1DMFT densities of states at EF equal to 7 st./@eV*cell# are consistent with the measured val- FIG. 6. ~a! Comparison between calculated densities of states using the LDA1DMFT approach for fcc Pu: the data for V/Vd 51.05, U54.0 eV ~black line!, the data for V/Vd50.80, U 53.8 eV ~gray line! which correspond to the volumes of the d and a phases, respectively. The result of the GGA calculation ~dotted line! at V/Vd51(U50) is also given. ~b! Measured photemission spectrum of d ~black line! and a ~gray line! Pu at the scale from 21.0 to 0.4 eV ~after Ref. 68!.-21 S. Y. SAVRASOV AND G. KOTLIAR PHYSICAL REVIEW B 69, 245101 ~2004!ues of the linear specific-heat coefficient. We still find a re- sidual discrepancy by about factor of 2 due to either inaccu- racies of the present calculation or due to the electron– phonon interactions. However, these values represent an improvement as compared to the LDA calculations which appear to be five times smaller. Similar inaccuracy has been seen in the LDA1U calculation.62 A simple physical explanation drawn from these studies suggests that in the d phase the f electrons are slightly on the localized side of the interaction-driven localization- delocalization transition with a sharp and narrow Kondo-like peak and well-defined upper and lower Hubbard bands. It therefore has the largest volume as has been found by previ- ous LDA1U calculations62,71 which take into account Hub- bard bands only. The low-temperature a phase is more me- tallic, i.e. it has larger spectral weight in the quasiparticle peak and smaller weight in the Hubbard bands. It will there- fore have a much smaller volume that is eventually repro- duced by LDA/GGA calculations which neglect both Cou- lomb renormalizations of quasiparticles and atomic multiplet structure. The delicate balance of the energies of the two minima may be the key to understanding the anomalous properties of Pu such as the great sensitivity to small amounts of impurities ~which intuitively would raise the en- ergy of the less symmetric monoclinic structure, thus stabi- lizing the d phase to lower temperature! and the negative thermal expansion. Notice however, that the a phase is not a weakly correlated phase: it is just slightly displaced towards the delocalized side of the localization–delocalization transi- tion, relative to the d phase. This is a radical new viewpoint in the theoretical literature on Pu, which has traditionally regarded the a phase as well understood within LDA. How- ever, the correlation viewpoint is consistent with a series of anomalous transport properties in the a phase reminiscent of heavy electron systems. For example, the resistivity of a –Pu around room temperature is anomalously large, temperature independent and above the Mott limit67 ~the maximum resis- tivity allowed to the conventional metal!. Strong correlation anomalies are also evident in the thermoelectric power.83 V. CONCLUSION In conclusion, this work describes a first-principles method for calculating the electronic structure of materials where many-body correlation effects between the electrons are not small and cannot be neglected. It allows simultaneous evaluation of the total free energy and the local electronic spectral density. The approach is based on the effective ac- tion functional formulation of the free energy and is viewed as spectral density-functional theory. An approximate form of the functional exploits a local dynamical mean-field245101theory of strongly correlated systems accurate in the situa- tions when the self-energy is short ranged in a certain portion of space. The localization is defined with reference to some basis in Hilbert space. It does not necessarily imply localiza- tion in real space and is treated using a general basis set following the ideology of the cellular dynamical mean-field theory. Further approximations of the theory, such as LDA 1DMFT and local GW are discussed. Implementation of the method is described in terms of the energy-dependent one- particle states expanded via the linear muffin-tin orbitals. Ap- plication of the method in its LDA1DMFT form is given to study the anomalous volume expansion in metallic Pluto- nium. We obtain equilibrium volume of the d phase in good agreement with experiment with no magnetic order imposed in the calculation. The calculated one-electron densities of states are consistent with the results of the photoemission. Our most recent studies34 of the lattice dynamical properties of Pu address the problem of the d→« transitions and show good agreement with experiment.84 Alternative developments of the LDA1DMFT approach by several groups around the world discuss other applica- tions of the dynamical mean-field theory in electronic- structure calculations. The results obtained are promising. Volume collapse transitions, materials near the Mott transi- tion, systems with itinerant and local moments, as well as many other exciting problems are beginning to be explored using these methods. ACKNOWLEDGMENTS The authors would like to thank for many enlightening discussions the participants of the research school on Real- istic Theories of Correlated Electron Materials, Kavli Insti- tute for Theoretical Physics, fall 2002, where the part of this work was carried out. Many helpful discussions with the participants of weekly condensed matter seminar at Rutgers University are also acknowledged. Conversations with A. Georges, A Lichtenistein, N. E. Zein related to various as- pects of dynamical mean field and GW theories are much appreciated. The authors are indebted to E. Abrahams, A. Arko, J. Joyce, H. Ledbetter, A. Migliori, and J. Thompson for discussing the issues related to the work on Pu. The work was supported by the NSF DMR Grants No. 0096462, 02382188, 0312478, 0342290, US DOE division of Basic Energy Sciences Grant No. DE-FG02-99ER45761, and by Los Alamos National Laboratory Subcontract No. 44047- 001-0237. The authors also acknowledge support from the Computational Material Science network operated by US DOE. Kavli Institute for Theoretical Physics is supported by NSF Grant No. PHY99-07949.1 For a review, see, e.g., Theory of the Inhomogeneous Electron Gas, edited by S. Lundqvist and S. H. March ~Plenum, New York, 1983!. 2 For a review, see, e.g., F. Aryasetiawan and O. Gunnarsson, Rep.Prog. Phys. 61, 237 ~1998!. 3 See, e.g., N.E. Bickers and D.J. Scalapino, Ann. Phys. ~N.Y.! 193, 206 ~1989!. 4 For a review, see, e.g., M. Jarrell and J.E. Gubernatis, Phys. Rep.-22
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