RELATIVISTIC DENSITY FUNCTIONAL THEORY: FOUNDATIONS AND BASIC FORMALISM

Transcription

1 Chapter 0 RELATIVISTIC DENSITY FUNCTIONAL THEORY: FOUNDATIONS AND BASIC FORMALISM E. Engel a a Institut für Theoretische Physik J.W.Goethe Universität Frankfurt D Frankfurt/Main, Germany An overview of relativistic density functional theory (RDFT) is presented with special emphasis on its field theoretical foundations and the construction of relativistic density functionals. A summary of quantum electrodynamics (QED) for bound states provides the background for the discussion of the relativistic generalization of the Hohenberg-Kohn theorem and the effective single-particle equations of RDFT. In particular, the renormalization procedure of bound state QED is reviewed in some detail. Knowledge of this renormalization scheme is pertinent for a careful derivation of the RDFT concept which necessarily has to reflect all the features of QED, such as transverse and vacuum corrections. This aspect not only shows up in the existence proof of RDFT, but also leads to an extended form of the single-particle equations which includes radiative corrections. The need for renormalization is also evident in the construction of explicit functionals. In practice, on the other hand, radiative corrections are usually neglected in RDFT calculations. This neglect is formally introduced into RDFT via the no-pair approximation. Within this framework the main task is to find an appropriate approximation for the relativistic exchange-correlation energy functional. As explicit density functionals the relativistic local density approximation (RLDA) and the relativistic generalized gradient approximation (RGGA) are reviewed. Both their derivation from the properties of the relativistic homogeneous electron gas and a number of illustrative results are presented. In particular, it is shown that the RLDA does not provide an adequate description of the relativistic corrections in the case of atomic systems, while the RGGA

2 performs as well for heavy atoms as the nonrelativistic GGA does for light atoms. Finally, a new generation of relativistic density functionals is discussed in which, in addition to the four current, the effective single-particle spinors are used for the representation of the exchange-correlation functional. The most prominent example for such an implicit density functional is the exact exchange. The actual application of implicit functionals requires the solution of an integral equation for the exchange-correlation potential (Optimized Potential Method), which is also introduced. On this basis a selfconsistent treatment of the transverse exchange is possible, which allows a detailed investigation of the importance of transverse corrections.. Introduction In spite of the impressive progress which has been achieved with conventional ab-initio methods as the Configuration-Interaction or Coupled-Cluster schemes in recent years density functional theory (DFT) still represents the method of choice for the study of complex many-electron systems (for an overview of DFT see []). Today DFT covers an enormous variety of fields, ranging from atomic [,3], cluster [4,5] and surface physics [6,7] to the material sciences [8 0]. and theoretical biophysics [ 3]. Moreover, since the introduction of the generalized gradient approximation DFT has become an accepted method also for standard quantum chemical applications [4,5]. Given this tremendous success of nonrelativistic DFT the question for a relativistic extension (RDFT) arises quite naturally in view of the large number of problems in which relativistic effects play an important role (see e.g. Refs.[6,7]). As in the nonrelativistic case relativistic density functional methods have already been used for the study of many-electron systems prior to their formal derivation from some existence theorem. For instance, the atomic Dirac-Slater calculations of Liberman et al. [8] may be viewed as an early precursor of RDFT. Later Andersen [9,0] suggested to apply a relativistic form of the Kohn-Sham (KS) equations [] to solids. These methods have been given a rigorous foundation by Rajagopal and Callaway [], who formulated a relativistic generalization of the Hohenberg-Kohn (HK) theorem [3]. In this formalism the ground state four current density j µ plays the same role as the ground state density n j 0 does in the nonrelativistic context, i.e. j µ is the basic density variable which determines the many-particle ground state Φ uniquely (up to a Nevertheless this formalism is usually referred to as relativistic density functional theory, rather than currentdensity functional theory.

3 3 gauge transformation). Φ and consequently also the ground state energy E tot can thus be interpreted as unique functionals of j µ. Direct minimization of the functional E tot j with respect to j µ then allows the determination of the correct ground state four current and, upon insertion into E tot j, of the correct ground state energy. In this way the problem of solving the many-body problem has been transformed into the problem of finding an appropriate functional E tot j. The first attempts to model E tot j go back to the early days of quantum mechanics, in which Vallarta, Rosen and Jensen [4,5] suggested a purely density-dependent relativistic energy functional in the spirit of the Thomas- Fermi model [6,7]. However, an orbital-dependent treatment of the kinetic energy turned out to be mandatory for the reproduction of many basic quantum mechanical features. Such a treatment can be established via the noninteracting system which yields the same ground state four current as the interacting system of interest. This auxiliary noninteracting system (KS system) induces a suitable decomposition of the total energy of the interacting system: E tot is separated into the kinetic energy T s of the KS system, the coupling to the external fields, E ext, the direct (Hartree) energy E H and a remainder, the exchange-correlation (xc) energy functional E xc j, in which all complicated many-body aspects are absorbed. Minimization of E tot with respect to the single-particle states of the auxiliary system then yields the single-particle equations of RDFT, which have first been put forward by Rajagopal [8] as well as by MacDonald and Vosko [9]. These relativistic KS (RKS) equations have the form of the Dirac equation with a multiplicative, current-dependent four potential v s µ. In addition to the nuclear Coulomb potential, v s µ contains all electron-electron interaction effects in an effective fashion via the direct (Hartree) potential and an xc-potential originating from E xc j. This statement implies that not only the Coulomb interaction is included in E H and E xc but also the (retarded) Breit interaction. It thus points at the fact that a consistent and complete discussion of many-electron systems and consequently of RDFT must start from quantum electrodynamics (QED). RDFT necessarily has to reflect the various features of QED, both on the formal level and in the derivation of explicit functionals. The most important differences to the nonrelativistic situation arise from the presence of infinite zero point energies and ultraviolet divergencies. In addition, finite vacuum corrections (vacuum polarization, Casimir energy) show up in both fundamental quantities of RDFT, the four current and the total energy. These issues have to be dealt with by a suitable renormalization procedure which ultimately relies on the renormalization of the vacuum Greens functions of QED. The first attempt to take

4 these field theoretical aspects into account in the context of RDFT has been made in Ref.[30,3]. It has been verified that the standard HK argument which underlies the proof of the existence theorem of RDFT is compatible with the QED renormalization program. In addition, one finds radiative contributions to the single-particle equations of RDFT, which substantially complicate the structure of the RKS-equations on the exact level. In all practical calculations these radiative corrections have been neglected up to now. The issue of renormalization also has to be addressed in the derivation of approximations for E xc j. The most obvious route for the construction of explicit xc-functionals is the relativistic extension of the local density approximation (LDA) [8,9]. In the RLDA the density-dependence of the xc-energy of the relativistic homogeneous electron gas (RHEG) is utilized, which leads to a functional of the density only. Both the full exchange and the high-density limit of the correlation part of E xc have been evaluated [8,9,3]. However, applications soon indicated that the RLDA does not provide a realistic account of the relativistic xc-effects in atomic systems [33 36]. Thus, in practice the relativistic KS equations are usually combined with accurate nonrelativistic density functionals for E xc, as the generalized gradient approximation (GGA). No firstprinciples derivation of relativistic gradient corrections has been published so far. Recently, however, a semiempirical relativistic form of the GGA (RGGA) has been put forward [37 39] which extends the accuracy of the GGA for light atoms into the relativistic domain. The fact that in the RLDA the xc-functional only depends on the density, rather than on the full four current, has stimulated the search for alternative forms of RDFT which allow an explicit treatment of magnetization effects, i.e. the spin-degree of freedom. A formalism in which the spatial components of j µ are replaced by the magnetization density m has been suggested by MacDonald and Vosko [9] as well as Ramana and Rajagopal [40] (see also [4]). A corresponding m-dependent version of the RLDA has been derived for the exchange part of E xc [40,4 44]. The application of this formalism in its most general form suffers, however, from the intricate structure of its singleparticle equations, so that one usually restricts the orientation of the magnetization density to one direction (collinear m). Only recently first calculations with truly non-collinear magnetization densities have been performed [45,46]. While ground states with non-collinear m were found for a number of solids [47], non-collinearity turned out to be only of limited importance for open-shell atoms [46]. This seems to indicate that the standard collinear approach is sufficient for most applications. 4

5 One important reason for the failure of the RLDA in the case of atomic systems is its insufficient treatment of the electronic self-interaction [36]. This problem is resolved by a new generation of density functionals in which E xc is allowed to be orbital-dependent, extending the idea behind the kinetic energy T s to the xc-functional. For such implicit density functionals the corresponding xcpotential has to be evaluated with the optimized potential method (OPM), which has been introduced in the nonrelativistic context a long time ago [48,49] (for an overview see [50]). This method has attracted considerable interest after it had been realized that the OPM allows an exact treatment of the exchange in DFT [5 53], thus guaranteeing the complete cancellation of the self-interaction contained in the direct term. The OPM can also be formulated for the exact relativistic exchange [54], or its Coulomb limit [55,36,56] (i.e. neglecting the transverse interaction). In the meantime also first suggestions for orbital-dependent correlation functionals have been made [57 60], most of them being derived by perturbation theory on the basis of the auxiliary KS Hamiltonian [53,57]. This approach can be directly extended into the relativistic domain [54], including all transverse and vacuum corrections. Not only low order perturbative E c can be obtained in this way, but also resummed forms like the random phase approximation (RPA) [6]. The general applicability of this type of correlation functional, however, is not yet clear [6]. In its most general form the ROPM requires the solution of a set of four integral equations in order to determine the xc-components of v s µ. As a consequence the ROPM selfconsistency procedure is much more demanding than standard RKS-calculations. Even in the nonrelativistic case most applications thus either addressed spherical systems [63 66] or utilized the atomic sphere approximation [67,68]. Only few applications are available in which a spherical approximation is not exploited [69 7]. However, the computational demands of implicit functionals can be substantially reduced by resorting to a very efficient and accurate semi-analytical approximation to the OPM which has been introduced by Krieger, Li and Iafrate (KLI) [7]. This scheme is easily extended to the ROPM [56,54]. Applications of the KLI approximation within RDFT confirm the level of accuracy found in the nonrelativistic limit [73]. With this technique the use of implicit functionals represents a real alternative to the application of the RGGA. In this review an overview of the complete RDFT formalism is given. In Section the relevant aspects of the underlying quantum field theory, i.e. QED with an additional static external potential characterizing the nuclei, are summarized. In particular, the question of renormalization is addressed for both 5

6 the ground state four current and the ground state energy. The details of the QED renormalization program, however, are relegated to the Appendices A-D. In Appendix A noninteracting fermions in a fixed external potential are considered, exhibiting the various stages of the quantization and the accompanying renormalization procedure. The case of interacting electrons is discussed in Appendix B, illustrating the renormalization of ultraviolet divergencies in the case of the QED vacuum. Appendix C is devoted to the relativistic homogeneous electron gas (RHEG), covering both the renormalization scheme for states with nonvanishing charge and the ground state energy functional of the RHEG. In addition, Appendix C provides some information on the response functions of the RHEG, which is then utilized in Appendix D to specify the renormalization procedure for (weakly) inhomogeneous systems, treating the external potential as a perturbation. Within this framework a proof can be given that there exists a one-to-one correspondence between the ground state and the renormalized ground state four current. This argument forms the basis for the discussion of the existence theorem of RDFT in Section 3.. The effective single-particle equations of RDFT are derived in Section 3.. The more practical variants of RDFT, utilizing the density and the magnetization density rather than the four current, are outlined in Section 3.3. This discussion also includes some remarks on the handling of the transverse interaction and of the negative energy states. The concept of relativistic implicit functionals is introduced in Section 3.4. Section 3 is concluded by making contact with nonrelativistic current-density functional theory. The presently available explicit approximations for the relativistic xc-energy functional are presented in Section 4. Both implicit functionals (as the exact exchange) and explicit density functionals (as the RLDA and RGGA) are discussed (on the basis of the information on the RHEG in Appendix C and that on the relativistic gradient expansion in Appendix E). Section 4 also contains a number of illustrative results obtained with the various functionals. However, no attempt is made to review the wealth of RDFT applications in quantum chemistry (see e.g.[74 88]) and condensed matter theory (see e.g.[89 00]) as well as the substantial literature on nonrelativistic xc-functionals (see e.g.[]). In this respect the reader is referred to the original literature. The review is concluded by a brief summary in Section 5. As is clear from this overview the present discussion focuses completely on the RKS scheme for stationary ground state problems of many-electron systems. Nevertheless it seems worthwhile to mention some further work on RDFT which is beyond this scope. A time-dependent generalization of the 6

7 RKS-equations has been introduced by Parpia and Johnson [0] and Rajagopal [0]. This method has been successfully applied to the photoionization of Hg and Xe [03,0] as well as to the evaluation of the polarizabilities of heavy closed-shell atoms [04] (using a direct time-dependent extension of the LDA for the xc-functional). A concept to deal with excited states in the framework of RDFT has been put forward by Nagy [05]. The derivation and first applications of relativistic extended Thomas-Fermi models may be found in Refs.[06 ]. Furthermore, an RDFT approach to meson field theory for hadronic matter (quantum hadrodynamics) [3] has been established by Speicher et al. [4]. This hadronic RDFT has been successfully applied to the description of nuclear ground states both within the extended Thomas-Fermi model [5 8] and within the KS scheme [9 ]. A corresponding formalism for finite temperature is also available [,3]. This Section is closed with a few remarks on units and notation. Throughout this work h is used. On the other hand, in order to allow direct access to both the nonrelativistic limit in which usually atomic units are applied ( h e m ) and the standard relativistic units ( h c m ) both e and m are kept in the formulae (e e ). Exceptions to this rule are the Appendices B and C in which keeping the speed of light would lead to expressions of excessive length. The space-time coordinates and metric are given by x x µ x 0 x ct r g µν g µν Greek indices run from 0 to 3, Latin indices from to 3. γ µ denotes the standard Dirac matrices, α µ γ 0 γ µ. The summation convention and the Feynman dagger notation p p µ γ µ are used throughout.. Field Theoretical Background The appropriate starting point for a fully relativistic description of the electronic structure of atoms, molecules, clusters and solids is QED. In a fully covariant QED-approach to these systems both the electrons and the nuclei would have to be treated as dynamical degrees of freedom (at least on a classical level in the case of the nuclei). However, in view of the large difference between the electron mass and the nuclear mass (in particular for heavy nuclei) the Born- Oppenheimer approximation is usually applied, at least for the discussion of ground state properties. The nuclei are thus treated as fixed external sources, 7

8 assuming them to be at rest in a common Lorentz frame. As external sources they may either be represented as classical charge distributions which interact with the quantized photon field or in the form of a classical potential which interacts with the electron field, both viewpoints being completely equivalent. In this contribution the second approach is chosen. The electrons thus interact with each other by the exchange of photons, while experiencing a static external potential. In order to keep the discussion as general as possible this potential is assumed to be of four vector form, V µ x, so that, in addition to Coulomb field of the nuclei, V µ may also represent external electromagnetic fields or nuclear magnetic moments. The system is thus characterized by the Lagrangian[4] int ext () e γ with e denoting the Lagrangian of noninteracting fermions, e x 4 ˆψ x ic mc ˆψ x ˆψ x ic γ being the Lagrangian of noninteracting photons, γ x 6π ˆF µν x ˆF µν x λ 8π µâµ x mc ˆψ x and int and ext providing the interaction between fermions and photons as well as between the fermions and the external potential, int x e ĵ µ x  µ x (4) ext x e ĵ µ x V µ x (5) (e e ). The operators ˆψ x and  µ x are the fermion and photon field operators, ˆF µν x is the electromagnetic field tensor, ˆF µν x µ  ν x ν  µ x (6) and ĵ µ x the fermionic current density, ĵ µ x ˆψ x γ µ ˆψ x (7) Both the fermionic Lagrangian and the electronic four current have been written in a form which ensures their correct behavior under charge conjugation [4] The vector bars on top of the partial derivatives indicate the direction in which the derivative has to be taken, i.e. in the second term of e the partial derivatives act on ˆψ x. 8 () (3)

9 (for some details see Appendix A). Under the charge conjugation ˆ, which transforms electrons into positrons and vice versa, the four current (as a charge current rather than a probability current) must change its sign, 9 ˆ ĵ µ x ˆ ĵ µ x (8) On the other hand, in the Lagrangian the fermion charge only manifests itself in the coupling to V µ, ˆ V µ ˆ V µ (9) as an external potential which attracts electrons repels positrons, while the photon field behaves as ˆÂµ ˆ  µ. For the photon fields we have chosen to work in the covariant gauge [5], which relies on the Gupta-Bleuler indefinite metric quantization and leads to the gauge fixing term λ 8π µâµ x in the Lagrangian. For brevity explicit formulae will always be given for some particular gauge, i.e. a particular choice of λ. This choice primarily determines the form of one of the building blocks of the theory characterized by (), the propagator of noninteracting photons, D 0 µν x y i e c 0 0 T µ 0 x Âν 0 y 0 0 (0) where  µ 0 denotes the noninteracting photon field operator and 0 0 is the corresponding noninteracting vacuum. In Sections -4 Feynman gauge (λ ) will be used, i.e. the propagator Eq.(0). On the other hand, the renormalization procedure of QED presented in the Appendices is more easily discussed in Landau gauge (λ ), which leads to the form (03). The freedom to choose any of the covariant gauges for the photon field results from the gauge invariance of the Lagrangian (): A gauge transformation of the photon field,  µ x  µ x  µ x µ Λ x ; µ µ Λ x 0 () can be absorbed by an accompanying phase transformation of the fermion field ˆψ x ˆψ x exp ieλ x c ˆψ x ()

10 0 leaving the Lagrangian () invariant, ˆψ  ˆψ  (3) On the other hand, due to the choice of a particular Lorentz frame only static gauge transformations are admitted for the external potential in order to remain within the common rest frame of the nuclei, V µ x V µ x µ Λ x (4) ˆψ x exp ieλ x c ˆψ x (5) Λ x Ct λ x ; λ x 0 (6) ˆψ V ˆψ V (7) Not only the Lagrangian, but also the four current ĵ µ x, Eq.(7), is invariant under the transformations (),() and (4)-(6). For the external potential the Coulomb gauge i V i x V x 0 is applied consistently. An immediate consequence of the local gauge invariance of the Lagrangian is current conservation, µ ĵ µ x 0 and thus the conservation of the total charge, ˆQ d 3 x ĵ 0 x d 3 x (8) ˆψ x ˆψ x (9) Any eigenstate of the system characterized by () can therefore be classified with respect to its charge (but not particle number). Energy conservation can be directly deduced from the continuity equation for the energy momentum tensor ˆT µν, utilizing the framework of Noether s theorem [4,6]. For the Lagrangian () ˆT µν reads ˆT µν ic x 8 4π ˆψ x γ µ ν γ ν µ γ µ ν γ ν µ ˆF µρ x ˆF ν ρ x 4 gµν ˆF x λ ρ τ  τ x e ĵ µ x  ν x ĵ ν x  µ x e ĵ µ x V ν x ĵ ν x V µ x ˆψ x λ gµν ρ  ρ x g µν  ρ x g µρ  ν x g νρ  µ x (0)

11 The last line of Eq.(0) indicates that we are dealing with an open system: The source field breaks the symmetry of ˆT µν. The continuity equation for ˆT µν thus contains a source term which provides momentum to the system, µ ˆT µν x e ĵ µ x ν V µ x () However, as V µ x is time-independent one finds µ ˆT µ0 x 0 0 d 3 x ˆT 00 x 0 () which implies the conservation of energy in the rest frame of the sources and allows the identification of the Hamiltonian, Ĥ Ĥ e Ĥ γ Ĥ int Ĥ ext Ĥ hom Ĥ ext (3) Ĥ e x 0 d 3 x ˆψ x Ĥ γ x 0 d 3 x 8π icα β mc ˆψ x (4) 0  µ x 0  µ x  µ x  µ x (5) Ĥ int x 0 e d 3 x ĵ µ x  µ x (6) Ĥ ext x 0 e d 3 x ĵ µ x V µ x (7) (Feynman gauge has been chosen for brevity). The ground state Φ corresponding to Ĥ is nondegenerate in general, i.e. as long as V µ x does not exhibit some spatial symmetries. Not only continuous symmetries obviously depend on special forms of V µ (compare [3]), but also the discrete symmetries usually considered within QED [5]: For parity to be a good quantum number some reflection symmetry of the potential is required. Under charge conjugation the Hamiltonian shows the same transformation behavior as the Lagrangian, ˆĤ V µ ˆ Ĥ V µ. Finally, time reversal symmetry requires purely electrostatic potentials of the form V µ V 0 0. While the twofold degeneracy resulting for such potentials leads to an additional conserved quantum number, this does not cause any problems as the Fock space can be decomposed accordingly. In the subsequent discussion it will always be assumed that Φ is nondegenerate. Unfortunately, a straightforward application of the Hamiltonian (3) for the calculation of Φ is not possible: Without further prescriptions the theory based on the Lagrangian () is not well-defined but rather suffers from three types of

12 divergencies 3. These divergencies show up both in the Greens functions of the theory as well as in the expectation values characterizing physical observables. This is true in particular for the ground state energy and the ground state four current, which are the basic ingredients of any RDFT formalism. The physically consistent removal of these divergencies requires a renormalization of the fundamental parameters of the theory. This renormalization procedure, which is usually formulated in a perturbative framework, is described in detail in the Appendices A-D, addressing in particular the case of inhomogeneous systems. Here only a brief summary is given. The first type of divergency results from the presence of negative energy states, which lead to divergent vacuum expectation values (e.g. for the energy and the charge). This problem already exists for noninteracting fermions and is most easily resolved by explicit subtraction of the vacuum expectation values (or normal-ordering). For instance, if one considers noninteracting electrons subject to some external potential, corresponding to a Hamiltonian of the form Ĥ s Ĥ e Ĥ ext (8) the renormalized Hamiltonian Ĥ s R Ĥ s 0 s Ĥ s 0 s (9) leads to a finite ground state energy. In (9) 0 s represents the vacuum in the presence of the external potential. As a consequence, the prescription (9) introduces different reference energies for different external potentials, i.e. it ignores the energy difference between different vacua (Casimir energy [7]). This poses no problem as long as processes are considered in which V µ does not change (as e.g. the ionization of an electron). In the context of RDFT, however, a universal energy standard is required as one wants to compare different external potentials. A suitable definition for the renormalized Hamiltonian thus is Ĥ s R Ĥ s 0 0 Ĥ e 0 0 (30) where 0 0 is the homogeneous vacuum corresponding to the noninteracting fermion Hamiltonian Ĥ e. This universal choice of the energy zero, however, reintroduces a divergency into the expectation values of Ĥ s R : While for vanishing V µ all expectation values Ĥ s R remain finite, one encounters a divergency proportional to V µ for nonvanishing potential (see Appendix A). This singularity has exactly the same form as the second type of divergencies present in standard QED without external potential, the ultraviolet (UV) divergencies. 3 The discussion of infrared divergencies is not necessary for the present purpose and will be omitted.

13 The removal of UV divergencies represents the core of the renormalization program of QED [5]. In this procedure the divergencies are first suppressed at all intermediate steps by use of a suitable regularization and then absorbed into a redefinition of the fundamental parameters of the Lagrangian. This redefinition leads to additional contributions (counterterms) to the Greens functions when these are expressed in terms of the actual physical parameters. The precise form of the counterterms is controlled by a few unique, elementary requirements on the vacuum Greens functions (normalization conditions see Appendix B). As a consequence all vacuum Greens functions of QED without external potential are finite. However, as this procedure associates well-defined counterterms to each individual element (diagram) of the perturbation expansion, it also leads to finite Greens functions in the case of the relativistic homogeneous electron gas (RHEG): The individual perturbative contributions to the RHEG Greens functions can be split into vacuum and gas parts, the treatment of the former being defined by vacuum QED and the latter being finite without renormalization (Appendix C). Finally, using a perturbation expansion with respect to V µ, all Greens functions and observables corresponding to the inhomogeneous system () can be decomposed into contributions whose renormalization is uniquely defined by vacuum QED without external potential and finite remainders (Appendix D). The final renormalized ground state energy E tot and ground state four current j µ x are given by E tot Φ Ĥ Φ 0 Ĥ e Ĥ γ Ĥ int 0 E tot (3) j µ x Φ ĵ µ x Φ j µ x (3) where 0 denotes the vacuum of interacting QED without external potential, while E tot and j µ are the counterterms resulting from the program outlined above (explicit examples are given in the Appendices). It is sometimes advantageous to decompose the total counterterm E tot into an electron gas part E hom, which is independent of the external potential, and an inhomogeneity tot correction, E tot Etot hom E hom tot 3 E inhom tot (33) sums up exactly the counterterms required for the interacting RHEG. Eqs.(3) and (3) form a suitable starting point for RDFT, as one is now dealing with finite quantities only.

14 4 3. Foundations and Basic Formalism This Section is devoted to the discussion of the basic RDFT formalism. It starts with a summary of the relativistic generalization of the HK-theorem within the framework of QED [,30]. In Section 3. the resulting RKS-equations are outlined, emphasizing the presence of radiative corrections on the most rigorous level. In the case of implicit density functionals the RKS-equations have to be augmented by the ROPM integral equation [55,36,54] for the evaluation of v xc µ which is the subject of Section 3.3. In Section 3.4 various simplified forms of RDFT are discussed. In particular, the no-pair approximation is introduced and the role of the transverse interaction is illustrated. In addition, the relation between the four current version of RDFT and forms depending on the magnetization-density is analyzed. Finally, the weakly relativistic limit of RDFT is studied in Section 3.5, in order to make contact with nonrelativistic current density functional theory [8,9]. 3.. Existence Theorem The extension of the HK-theorem to relativistic systems was first formulated by Ragagopal and Callaway [] (see also [8,9]). Utilizing a QED-based Hamiltonian and four current, these authors applied the standard reductio ad absurdum of HK [3] to show that the ground state energy is a unique functional of the ground state four current. All questions related to zero-point energies, radiative corrections and, in particular, UV-divergencies were not addressed. However, the HK argument relies on the comparison of energy values, so that a rigorous proof within the framework of QED has to be based on renormalized ground state energies and four currents, i.e. on the quantities (3),(3). One thus has to make sure that the structure of the counterterms E tot and j µ is compatible with the logic of the HK argument. This compatibility can be explicitly proven within a perturbative framework [30]. In this proof the full machinery of the QED renormalization program is involved, which is therefore introduced step by step in the Appendices A-D. The discussion in this Section focuses on the actual reductio ad absurdum of HK, relying on the structure of the counterterms E tot and j µ provided in Appendix D. The argument of HK proceeds in two steps: In the first step the relation between external potentials and ground states is considered. Let us assume that two different potentials V µ and V µ yield the same (nondegenerate) ground state Φ, with the aim to find a statement which contradicts this assumption. One then has Ĥ Φ E Φ (34)

15 Ĥ Φ E Φ 5 (35) where Ĥ denotes the Hamiltonian (3) in the Schrödinger picture and Ĥ is the corresponding Hamiltonian of the system with the potential V µ. The issue of renormalization need not be addressed at this point as no expectation values are taken. For the present purpose a suitable regularization is sufficient. Upon subtraction of both eigenvalue equations, e d 3 x ĵ ν x V ν x V ν x Φ E E Φ (36) one is left with the question whether the state on the left-hand side of Eq.(36) can be collinear with that on the right-hand side. If not, the desired contradiction would have been found. While the operator d 3 x ĵ ν V ν V ν in general does not commute with the Hamiltonian Ĥ, this does unfortunately not exclude the possibility of a single common eigenstate Φ. An obvious contradiction only arises for multiplicative potentials of the form V µ V 0 0. For this type of purely electrostatic potentials one can conclude that two potentials V 0 and V 0 lead to different ground states, as long as V 0 and V 0 differ by more than an additive constant (V 0 V 0 const as the total charge operator commutes with the Hamiltonian). One is thus led to the question whether the inclusion of a magnetic field can compensate the difference which results from two different electrostatic components V 0 and V 0, or whether two different magnetic fields can yield the same ground state. The answer to this question has not been finally settled to date. For nonrelativistic spin-density functional theory an argument has been given that two different magnetic fields can yield the same ground state under certain conditions [30,3]. While the nonrelativistic limit of the four current version of RDFT differs from the formalism used in [30,3] (compare Section 3.5), this result nevertheless suggests that a unique relation between V µ and the corresponding ground state does not exist in the relativistic case either. Fortunately, such a unique map between the space of four potentials and that of the corresponding ground states is not required for the existence of a ground state density functional. For the latter it is sufficient that the renormalized ground state four current j µ, Eq.(3), determines the ground state Φ uniquely (the second step of the HK argument). In order to prove this statement let us compare two weakly inhomogeneous systems obtained by perturbing an electron gas with density n 0 by two different external potentials V µ and V µ (the fact that the two systems must have the same average density just reflects the requirement that their charge has to be identical). The resulting ground states,

16 assumed to be nondegenerate, are denoted by Φ and Φ, the ground state four currents by j µ and j µ. The corresponding renormalized ground state energies are given by E tot Φ Ĥ hom e d 3 x ĵ µ V µ Φ E tot Φ Ĥ hom e d 3 x ĵ µ V µ Φ 0 Ĥ hom 0 E hom tot 0 Ĥ hom 0 E hom tot 6 E inhom tot V (37) E inhom tot V (38) where the counterterms have already been split into the electron gas component Etot hom and the inhomogeneity correction Etot inhom following (33). Neither Etot hom nor Etot inhom depends on the state under consideration. Within the perturbative approach summarized in the Appendices B-D one can show that these counterterms are completely determined by the external potential and the average density n 0. This is immediately clear for Etot hom, which only depends on n 0. On the other hand, Etot inhom can be written as (gauge invariant) functional of V µ via the response expansion (34). As long as Φ differs from Φ by more than a gauge transformation the state Φ represents some kind of excited state of the Hamiltonian with potential V µ. The renormalized energy E es associated with Φ in the unprimed system is given by E es Φ Ĥ hom e d 3 x ĵ µ V µ Φ 0 Ĥ hom 0 E hom tot E inhom tot V e d 3 x j µ V V µ V µ (39) with j µ as in (308) (of course, all individual components are understood to be regularized). On the one hand, the energy (39) is finite as can be seen by insertion of the counterterm required to extract the ground state energy E tot, E es Φ Ĥ hom e d 3 x ĵ µ V µ Φ 0 Ĥ hom 0 Etot hom Etot inhom V e d 3 x Φ ĵ µ Φ j µ V V µ V µ E tot e d 3 x j µ V µ V µ (40) On the other hand, the counterterms in (39) are unique. Ultimately, their form is determined by the normalization conditions for the Greens functions of vacuum QED and there is only one way to include these normalization conditions in expansions of the form (308) and (34): As long as the representation of the

17 counterterms relies on the ground state response functions of the RHEG, their precise form is defined by that potential V µ for which Φ is the ground state. As Φ is assumed to be nondegenerate, the energy associated with Φ in the unprimed system must be higher than the ground state energy, E tot E tot e d 3 x j µ V µ V µ (4) One can now interchange all primed and unprimed quantities in this argument to arrive at E tot E tot e d 3 x j µ V µ V µ (4) 7 Upon addition of (4) and (4), E tot E tot E tot E tot e d 3 x j µ j µ V µ V µ (43) one finally realizes that a contradiction arises if j µ j µ is assumed. We have thus shown that for states Φ and Φ which differ by more than a gauge transformation one also has j µ j µ, so that the ground state of any such system is uniquely determined by the ground state four current. On the other hand, if the two potentials only differ by a (static) gauge transformation, V µ x V µ x µ λ x λ x 0 the four currents obtained from (308) are identical due to the transversality of the response functions, Eq.(58). The same is then true for the counterterms in (308) and (34) and the inequality (43) becomes an equality. Consequently, there exists a one-to-one correspondence between the class of all ground states which just differ by gauge transformations and the associated ground state four current. In mathematical terms one can state that Φ is a unique functional of j µ once the gauge has been universally fixed, Φ Φ from V µ µ Λ j µ x Φ Φ j (44) The proof given relies on a perturbation expansion with respect to both the electron-electron interaction and V µ. The necessity for these expansions originates from the recursive nature of the renormalization scheme which proceeds order by order in the fine-structure constant and from the fact that the treatment of inhomogeneous systems has to be derived from the renormalization procedure for the homogeneous QED vacuum. Only in this framework is it possible

18 to explicitly extract the structure of the required counterterms, which is the first important ingredient for establishing the inequality (43). On the other hand, even if one accepts the fact that the QED perturbation series may be an asymptotic expansion, the discussion is formally valid to all orders. In addition, the case of finite systems is covered, at least in principle, by the limit n 0 0. This limit is particularly transparent for noninteracting systems as the -loop counterterms (3) and (36) are independent of n 0 and are thus directly applicable to arbitrary inhomogeneous systems, as e.g. the KS system. The second important ingredient of the inequality (43) is a minimum principle for the ground state energy (3). While the Ritz variational principle is well established in the nonrelativistic context, no mathematically rigorous proof of a minimum principle for the renormalized ground state energies seems to be available. On the other hand, it is exactly the requirement that the energy spectrum must have a lower bound which is the main motivation for the first step of the renormalization program, the elimination of the divergent zero-point energy. This is not only true for the standard QED without external potential, but also for electrons subject to some V µ (in this case the Furry picture can be utilized see Appendix A). Moreover, within the perturbative approach outlined in Appendices B-D UV-divergencies can be handled in a unique fashion. This scheme directly leads to a finite ground state energy, but also provides a unique answer for the renormalization of the energy expectation value of excited states. Consequently, the mere assumption of a nondegenerate ground state Φ implies that all other states lead to energies higher than the energy of Φ. In other words: If the renormalized energies would not reflect the minimum principle for the energy which is observed in nature, this would question the renormalization program, rather than the minimum principle. Fixing the gauge once and for all, the relation (44) allows to understand all ground state observables as unique functionals of j µ. The most important functional of this type is the ground state energy itself, E tot j Φ j Ĥ Φ j 0 Ĥ e Ĥ γ Ĥ int 0 E hom tot 8 E inhom tot (45) This energy functional contains not only all relativistic kinetic effects for both electrons and photons but also all radiative effects. Utilizing once again the energy minimum principle, one may then formulate the basic variational principle of RDFT as δ δ j ν r E tot j µ d 3 x j 0 x 0 (46)

19 The subsidiary condition implies charge conservation and all quantities involved are supposed to be fully renormalized. Solution of (46) with the exact functional E tot j yields the exact ground state four current j µ and, upon insertion of j µ into E tot j, the exact ground state energy. Thus Eq.(46) opens the possibility to calculate two important quantities of the system without explicit knowledge of the ground state. As it stands the functional (45) is well defined for all those j µ which result as ground state four currents from some external potential V µ. Strictly speaking, this does not yet guarantee the existence of the functional derivative δ E tot j δ j µ on the set of ground state four currents, which is a prerequisite for the applicability of the variational equation (46). The question of the existence of the functional E tot j for a sufficiently dense set of j µ in the neighbourhood of any ground state four current, the so-called interacting v-representability, has not been investigated in the relativistic situation. However, it seems quite plausible that the statements found in the nonrelativistic context also apply to the functional (45). One would thus expect that difficulties only arise for δ -type potentials V µ, so that any discretization of space ensures v-representability (for details see []). As the nuclear potentials of actual interest are much less singular than the δ -distribution and any solution of (46) is based on some kind of discretization, v-representability should not pose any problem in practice. Moreover, the v-representability problem may be resolved already on the formal level by a redefinition of the energy functional in the spirit the Levy-Lieb constrained search [3,33]. In view of the difficulties associated with the renormalization procedure one may ask whether it is possible to base RDFT on an approximate relativistic many-body approach, as e.g. the Dirac-Coulomb (DC) Hamiltonian, Ĥ DC Ĥ e 0 Ĥ ext 0 Ĥ e e (47) Ĥ e e e d 3 r d 3 r 9 ˆψ 0 r ˆψ 0 r ˆψ 0 r ˆψ 0 r (48) r r or its Dirac-Coulomb-Breit (DCB) extension, from the very outset. In this case the no-pair approximation (np) plays the role of the renormalization scheme, Ĥ DC np ˆΛ ĤDC ˆΛ ĵ µ np ˆΛ ĵµ ˆΛ (49) where ˆΛ is a projection operator onto positive energy states. However, the no-pair approximation can be unambiguously specified only within some welldefined single-particle scheme. Even in this case ˆΛ depends on the actual

20 single-particle potential and thus on the external potential, ˆΛ V µ. As a consequence, Ĥnp DC is a nonlinear functional of V µ, which does not allow the usual reductio ad absurdum of the HK-proof. In addition, the no-pair approximation introduces a gauge dependence of the ground state energy [54], so that an unambiguous comparison of two ground state energies is only possible if one neglects the Breit interaction and restricts oneself to an external potential of the form V µ V 0 0. It thus appears that the existence theorem of RDFT has to be based on the field theoretical formalism. The no-pair approximation, which is used in most applications, is much more easily introduced at a later stage, i.e. in the context of the single-particle equations of RDFT. 3.. Relativistic Kohn-Sham Equations The relativistic variant of the HK theorem guarantees the formal existence of a density functional description of relativistic systems but does not give any hint how to construct the crucial functional E tot j. Explicit approximations to E tot j can be derived by a variety of methods. The most important approach starts with a study of the homogeneous electron gas, for which the energy functional is a simple function of the gas density. This functional can then be extended in a systematic fashion by inclusion of inhomogeneity corrections which depend on the gradients of the density. If this approach is utilized for the complete energy functional one ends up with relativistic (extended) Thomas Fermi models [4, 5,08]. As in the nonrelativistic context, however, these models have found very limited use due to the fact that they omit important quantum mechanical properties: They neither reproduce atomic shell structure nor do they lead to molecular binding. As is obvious from these fundamental deficiencies which are not related to the electron-electron interaction, the Thomas Fermi models suffer from their description of the kinetic energy part of E tot j. This problem is resolved by the KS-scheme, which allows an exact treatment of the kinetic energy of noninteracting particles. The starting point for this scheme is the assumption that there exists a noninteracting, relativistic system with exactly the same ground state four current j µ r as the interacting system one is actually interested in. The question whether such a noninteracting system always exists, usually termed noninteracting v-representability, has not been examined in the relativistic case. One would, however, expect analogous statements as in the nonrelativistic situation [], so that this assumption should not be a serious restriction. As the auxiliary system is noninteracting its basic Hamiltonian is of the type (73), its ground state is given by (63) and its vacuum four current has the form (76). 0

21 As discussed in Appendix A both the energy and the four current need to be renormalized. On the one hand, the subtraction of the appropriate zero-point energy is required, Eq.(79). On the other hand, the lowest order UV-divergencies have to be eliminated by (3) and (36). The ground state four current j µ of both the auxiliary system and the interacting system is thus given by j µ r k j v µ r Θ k Θ k φ k r α µ φ k r j µ v r (50) φ k r α µ φ k r ε k mc 0 for ε k mc for mc ε k ε F 0 for ε F ε k mc ε k φ k r α µ φ k r j µ 0 r (5) (5) where φ k denotes the single-particle spinors of the auxiliary system and ε F represents the Fermi level which separates occupied from unoccupied auxiliary states. In (50) the total current has been decomposed into a vacuum jµ 0 contribution j v µ and the contribution of the actual electronic states. r is the lowest order counterterm which keeps the vacuum polarization current j v µ UVfinite, as specified in Eq.(3) by use of dimensional regularization (with the total RKS-potential on the right-hand side). In the next step one decomposes the ground state energy functional (45), E tot T s E ext E H E xc (53) where the counterterms for E tot given in Eq.(3) are understood to be included in the individual energy components. The latter are defined as follows: The noninteracting kinetic energy functional T s, i.e. the kinetic energy of the auxiliary system, is given by T s T s v Θ k d 3 r φ r k icα β mc φ k r T s v (54) k d 3 r φ k r icα β mc φ k r ε k mc mc ε k φ k r icα β mc φ k r 0 0 Ĥ e 0 0 Ts inhom (55)

22 with the counterterms as in (79) and (39). The interaction energy with the external potential term reads E ext j e d 3 r j µ r V µ r (56) where the counterterm (38) has already been absorbed into the renormalized current j µ. The (direct) Hartree energy E H is defined in a covariant fashion, utilizing the free photon propagator D 0 µν, Eq.(0), E H j d 3 x d 4 y j µ x D 0 µν x y jν y (57) so that it includes all direct matrix elements of the order e. E H can be split into a Coulomb contribution EH C and a transverse component ET H which reflects the presence of the magnetic interaction in D 0 µν, E H j E C H E T H (58) EH C e d 3 r d 3 n r n r r r r (59) EH T e c d 3 r d 3 j r j r r r r (60) where the standard notation j µ r n r j r c (6) has been introduced. Finally, the xc-energy functional contains all remaining contributions to E tot j, i.e. (53) defines E xc. As the HK argument is also valid for noninteracting systems for which E tot T s E ext one concludes that T s is a density functional, T s j, in spite of its explicit orbital-dependent form. With this information and E ext and E H being obvious density functionals E xc is also a functional of j µ, E xc j. Given the fact that the ground state of the auxiliary system and thus its ingredients φ k are uniquely determined by the four current of the interacting system, the variational principle may then be exploited to minimize E tot with respect to the φ k, rather than j µ (a more careful argument can be given along the lines of Ref.[]). This minimization leads to the relativistic KS-equations icα β mc α µ v µ s r φ k r ε k φ k r (6)

23 with the multiplicative KS potential v µ s consisting of the sum of V µ, the Hartree potential v µ H and the xc-potential vµ xc, v µ s r ev µ r v µ H r v µ xc v µ H r e d 3 r v µ xc r δ E xc j δ j µ r 3 r (63) j µ r (64) r r (65) Eqs.(50),(6)-(65) have to be solved selfconsistently in order to obtain the exact j µ r of the interacting system. As a matter of principle, however, the solution does not provide direct information on the ground state itself. Nevertheless one would expect the Slater determinant Φ s, Eq.(63), obtained from the N lowest auxiliary orbitals φ k to be a reasonable approximation to the actual Φ. For completeness we finally note an alternative form of the total energy, obtained by use of (6), E tot mc ε k ε F ε k E H j E xc j d 3 r v µ xc r j µ r ε k mc ε k mc ε k ε k 0 0 Ĥ e 0 0 Ts inhom (66) which is often applied in practical calculations. The last three terms represent the energy shift induced in the vacuum by the presence of the RKS-potential, i.e. the Casimir energy of the KS-system [7]. It seems worthwhile to emphasize the fact that the KS vacuum 0 s results from a selfconsistency procedure and thus changes during the iterative solution of (6). In addition to these obvious vacuum contributions, the terms of the first line include vacuum corrections via the current (50) and the functional dependence of E xc and v xc µ on jµ. This fact is most easily illustrated by the exchange component of E xc. The exact exchange is defined as that contribution to E xc which is linear in e within a perturbation expansion with respect to the auxiliary KS Hamiltonian [54]. This leads to E x : d 3 x d 4 yd 0 µν x y tr d 3 x d 4 yd 0 µν x y tr Ex hom Ex inhom G s x y γ ν G s y x γ µ G 0 v x y γ ν G 0 v y x γ µ (67)