Relativistic density-functional theory based on effective quantum electrodynamics
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Relativistic density-functional theory based on effectivequantum electrodynamics
Julien Toulouse Laboratoire de Chimie Th´eorique (LCT), Sorbonne Universit´e and CNRS, F-75005 Paris,France Institut Universitaire de France, F-75005 Paris, France* [email protected] 18, 2021
Abstract
A relativistic density-functional theory based a Fock-space effective quantum-electrodynamics (QED) Hamiltonian using the Coulomb or Coulomb-Breit two-particle interaction is developed. This effective QED theory properly includesthe effects of vacuum polarization through the creation of electron-positron pairsbut does not include explicitly the photon degrees of freedom. It is thus a moretractable alternative to full QED for atomic and molecular calculations. Usingthe constrained-search formalism, a Kohn-Sham scheme is formulated in a quitesimilar way to non-relativistic density-functional theory, and some exact prop-erties of the involved density functionals are studied, namely charge-conjugationsymmetry and uniform coordinate scaling. The usual no-pair Kohn-Sham schemeis obtained as a well-defined approximation to this relativistic density-functionaltheory.
Contents N -negative-charge states 92.6 No-pair approximation 10 Conclusions 23A Charge-conjugation symmetry of the electron-positron Hamiltonian 24B Alternative definition of the electron-positron Hamiltonian 26References 29
The basic formulation of the relativistic extension of density-functional theory (DFT) wasfirst laid down by generalizing the Hohenberg-Kohn theorem [1] to a Hamiltonian based onquantum electrodynamics (QED) with the internal quantized electromagnetic field and an ex-ternal classical electromagnetic field [2–5]. These early works did not address the subtle issuesof QED renormalization. These issues were studied by Engel, Dreizler, and coworkers [6–10]who put relativistic (current) density-functional theory (RDFT) on more rigorous grounds.In their works, they confirmed the validity of the relativistic extension of the Hohenberg-Kohn theorem using a charge-conjugation-symmetric form of the QED Hamiltonian writtenwith commutators of field operators and appropriate renormalization counterterms. Eschrig et al. [11, 12] took another approach to RDFT based on Lieb’s Legendre transformation usinga normal-ordered QED Hamiltonian. Ohsaku et al. [13] proposed a local-density-matrix func-tional theory based on a QED Hamiltonian with an one-photon-propagator fermion-fermioninteraction. Despite these formal foundations of RDFT based on QED, in practice four-component RDFT is invariably applied in the Kohn-Sham (KS) scheme with a non-quantizedelectromagnetic field and in the no-pair approximation (i.e., neglecting contributions fromelectron-positron pairs) [14–21], most of the time using non-relativistic exchange-correlationdensity functionals.In this work, we examine an alternative RDFT based on a Fock-space effective QEDHamiltonian using the Coulomb or Coulomb-Breit two-particle interaction (see, e.g., Refs. [22–25]). This effective QED theory properly includes the effects of vacuum polarization throughthe creation of electron-positron pairs but does not include explicitly the photon degreesof freedom. It is thus a more tractable alternative to full QED for atomic and molecularcalculations. This so-called no-photon QED has been the subject of a number of detailedmathematical studies [26–31], which in particular established the soundness of this approachat the Hartree-Fock (HF) level. This is thus a good QED level to base a RDFT on. It isshown that we can develop indeed a RDFT formalism based on this effective QED theoryusing the constrained-search formalism [32, 33] in a quite similar way to non-relativistic DFT.The usual no-pair KS scheme is then obtained as a well-defined approximation to this RDFT.The paper is organized as follows. In Section 2, we expose the effective QED theory con-sidered in this work. We define the normal-ordered electron-positron Hamiltonian, we discusshow to define the polarized vacuum state and N -negative-charge states by a minimizationformulation, and we introduce the no-pair approximation in this approach. In Section 3, wedevelop a RDFT based on this effective QED theory. We describe the KS scheme in this ap-2roach, we give the expression of the Hartree, exchange, and correlation density functionals,we study some exact properties of these functionals, and discuss the local-density approxi-mation (LDA). Section 4 contains conclusions and perspectives. In the appendices, we provesome important and, to the best of my knowledge, seemingly unknown aspects of the ef-fective QED theory. First, in Appendix A, we show that the electron-positron Hamiltonianexpressed in terms of the normal ordering with respect to the free vacuum state has the correctcharge-conjugation symmetry. Second, in Appendix B, we show that the electron-positronHamiltonian based on normal ordering with respect to the free vacuum state is essentiallyequivalent to an electron-positron Hamiltonian based on commutators and anticommutatorsof Dirac field operators.In contrast to the quantum chemistry literature where often everything is formulated ina basis, here we prefer to use a real-space formalism which is more adapted to DFT. Hartreeatomic units (a.u.) are used throughout the paper. We consider the time-independent free Dirac equation D ( ~r ) ψ p ( ~r ) = ε p ψ p ( ~r ) , (1)with the 4 × D ( ~r ) = c ( ~ α · ~p ) + β mc , (2)where ~p = − i~ ∇ is the momentum operator, c = 137 .
036 a.u. is the speed of light, m = 1 a.u.is the electron mass, and ~ α and β are the 4 × ~ α = (cid:18) ~ σ ~ σ (cid:19) and β = (cid:18) I − I (cid:19) , (3)where ~ σ = ( σ x , σ y , σ z ) is the 3-dimensional vector of the 2 × and I are the 2 × { ψ p ( ~r ) } thatwe will assume as being discretized (by putting the system in a box with periodic boundaryconditions). This set can be partitioned into a set of positive-energy orbitals ( ε p >
0) and aset of negative-energy orbitals ( ε p < { ψ p ( ~r ) } = { ψ p ( ~r ) } p ∈ PS ∪ { ψ p ( ~r ) } p ∈ NS , where PSand NS designate the sets of “positive states” and “negative states”, respectively. The Diracfield is then quantized asˆ ψ ( ~r ) = X p ∈ PS ∪ NS ˆ a p ψ p ( ~r ) = X p ∈ PS ˆ b p ψ p ( ~r ) + X p ∈ NS ˆ d † p ψ p ( ~r ) , (4)where the sum has been decomposed in a contribution involving electron annihilation oper-ators ˆ b p ≡ ˆ a p for p ∈ PS and a second contribution involving positron creation operatorsˆ d † p ≡ ˆ a p for p ∈ NS. The annihilation and creation operators obey the usual fermionic anti-commutation rules { ˆ a p , ˆ a † q } = δ pq and { ˆ a p , ˆ a q } = { ˆ a † p , ˆ a † q } = 0 for p, q ∈ PS ∪ NS , (5)3 .2 Electron-positron Hamiltonian and the corresponding free vacuum state | i is defined such thatˆ b p | i = 0 for p ∈ PS and ˆ d p | i = 0 for p ∈ NS . (6) We then consider the normal-ordered electron-positron Hamiltonian in Fock space writtenwith this quantized Dirac field introduced in Refs. [22, 34] (see, also, Ref. [23]) that we canwrite as ˆ H = ˆ T D + ˆ W + ˆ V , (7)where the Dirac kinetic + rest mass operator ˆ T D , the two-particle interaction operator ˆ W ,and the external potential-energy interaction operator ˆ V are expressed as (using σ , ρ , τ , υ asspinor indices ranging from 1 to 4)ˆ T D = Z Tr[ D ( ~r )ˆ n ( ~r, ~r ′ )] ~r ′ = ~r d ~r ≡ X σρ Z [ D σρ ( ~r )ˆ n ,ρσ ( ~r, ~r ′ )] ~r ′ = ~r d ~r, (8)and ˆ W = 12 Z Z
Tr[ w ( ~r , ~r )ˆ n ( ~r , ~r )]d ~r d ~r ≡ X σρτυ Z Z w στρυ ( ~r , ~r )ˆ n ,ρυστ ( ~r , ~r ) d ~r d ~r , (9)and ˆ V = Z v ( ~r )ˆ n ( ~r ) d ~r, (10)where the one-particle density-matrix operator ˆ n ( ~r, ~r ′ ) and the pair density-matrix operatorˆ n ( ~r , ~r ) are defined using creation and annihilation Dirac field operators with normal order-ing N [ ... ] of the elementary creation and annihilation operators ˆ b † p , ˆ b p , ˆ d † p , ˆ d p with respect tothe free vacuum state | i ˆ n ,ρσ ( ~r, ~r ′ ) = N [ ˆ ψ † σ ( ~r ′ ) ˆ ψ ρ ( ~r )] , (11)ˆ n ,ρυστ ( ~r , ~r ) = N [ ˆ ψ † τ ( ~r ) ˆ ψ † σ ( ~r ) ˆ ψ ρ ( ~r ) ˆ ψ υ ( ~r )] , (12)and the (opposite) charge density operator isˆ n ( ~r ) = Tr[ˆ n ( ~r )] ≡ X σ ˆ n σσ ( ~r ) , (13)where n ( ~r ) = n ( ~r, ~r ). In Eq. (10) v ( ~r ) is a scalar external potential (for example, the Coulombpotential generated by the nuclei) and in Eq. (9) w ( ~r , ~r ) is a two-particle interaction matrixpotential which could be for example the Coulomb (C) + Breit (B) interaction w στρυ ( ~r , ~r ) = w C στρυ ( r ) + w B στρυ ( ~r ) , (14)4 .2 Electron-positron Hamiltonian with ~r = ~r − ~r and r = | ~r | , and w C στρυ ( r ) = w ( r ) δ σρ δ τυ , (15) w B στρυ ( ~r ) = − w ( r ) (cid:18) ~α σρ · ~α τυ + ( ~α σρ · ~r ) ( ~α τυ · ~r ) r (cid:19) , (16)where w ( r ) = 1 /r . The Coulomb-Breit interaction corresponds to the single-photon ex-change electron-electron scattering amplitude in QED evaluated with the zero-frequency limitof the photon propagator in the Coulomb electromagnetic gauge. More specifically, the in-stantaneous Coulomb interaction corresponds to the longitudinal component of the photonpropagator, whereas the Breit interaction is obtained from the zero-frequency transverse com-ponent of the photon propagator. The Breit interaction comprises the instantaneous magneticGaunt interaction, − w ( r ) ~α σρ · ~α τυ , and the remaining lowest-order retardation correction(see, e.g., Ref. [35]).The electron-positron Hamiltonian ˆ H does not commute separately with the electron andpositron number operators,ˆ N e = X p ∈ PS ˆ b † p ˆ b p and ˆ N p = X p ∈ NS ˆ d † p ˆ d p , (17)i.e., it does not conserve electron or positron numbers. However, the Hamiltonian ˆ H commuteswith the opposite charge operator (or electron-excess number operator)ˆ N = ˆ N e − ˆ N p , (18)i.e., it conserves charge. As a consequence, the eigenstates of the Hamiltonian ˆ H belongs tothe Fock space gathering together different particle-number sectors F = ( ∞ , ∞ ) M ( N e ,N p )=(0 , H ( N e ,N p ) , (19)where H ( N e ,N p ) is the Hilbert space for N e electrons and N p positrons. The Fock space canalso be decomposed into charge sectors F = ∞ M q = −∞ H q , (20)where H q is the Hilbert space for (opposite) charge q . For q ≥
0, we have H q = H ( q, ⊕H ( q +1 , ⊕ H ( q +2 , ⊕ · · · ⊕ H ( q + ∞ , ∞ ) , and for q ≤
0, we have H q = H (0 ,q ) ⊕ H (1 ,q +1) ⊕ H (2 ,q +2) ⊕· · · ⊕ H ( ∞ ,q + ∞ ) .Importantly, due to the fact that the electron-positron Hamiltonian in Eq. (7) is expressedin terms of the normal ordering with respect to the free vacuum state, it has the correct charge-conjugation symmetry, i.e. ˆ C ˆ H [ v ] ˆ C † = ˆ H [ − v ] where ˆ C is the charge-conjugation operator inFock space (see Appendix A). 5 .3 No-particle vacuum states By construction of the Hamiltonian ˆ H , the free vacuum state | i has a zero energy, i.e. E free0 = h | ˆ H | i = 0. However, this is generally not the lowest-energy vacuum state. We canconsider other no-particle vacuum states | ˜0 i (often referred to as polarized vacuum or dressedvacuum) parametrized as [23, 36] (see, also, Refs. [22, 34, 37, 38]) | ˜0 i = e ˆ κ | i , (21)where e ˆ κ performs an orbital rotation in Fock space (corresponding to a Bogoliubov trans-formation mixing electron annihilation operators ˆ b p and positron creation operators ˆ d † p [22])with the anti-Hermitian operator ˆ κ ˆ κ = X p,q ∈ PS ∪ NS κ pq ˆ a † p ˆ a q = X p,q ∈ PS κ pq ˆ b † p ˆ b q + X p ∈ PS X q ∈ NS κ pq ˆ b † p ˆ d † q + X p ∈ NS X q ∈ PS κ pq ˆ d p ˆ b q + X p,q ∈ NS κ pq ˆ d p ˆ d † q , (22)with the orbital rotation parameters κ pq ∈ C being the elements of an anti-Hermitian matrix κ . Note that the second term in the last expression of Eq. (22) creates electron-positron pairs.This generates new creation and annihilation operators related to the original ones via theunitary matrix U = e κ ˆ˜ a † p = e ˆ κ ˆ a † p e − ˆ κ = X q ∈ PS ∪ NS ˆ a † q U qp and ˆ˜ a p = e ˆ κ ˆ a p e − ˆ κ = X q ∈ PS ∪ NS ˆ a q U ∗ qp for p ∈ PS ∪ NS , (23)and corresponding new orbitals˜ ψ p ( ~r ) = X q ∈ PS ∪ NS ψ q ( ~r ) U qp for p ∈ PS ∪ NS , (24)such that the Dirac field operator in Eq. (4) can be rewritten asˆ ψ ( ~r ) = X p ∈ PS ∪ NS ˆ˜ a p ˜ ψ p ( ~r ) = X p ∈ PS ˆ˜ b p ˜ ψ p ( ~r ) + X p ∈ NS ˆ˜ d † p ˜ ψ p ( ~r ) , (25)with again ˆ˜ b p ≡ ˆ˜ a p for p ∈ PS and ˆ˜ d † p ≡ ˆ˜ a p for p ∈ NS. The new creation and annihilationoperators still obey the fermionic anticommutation rules in Eq. (5). Moreover, even thoughthis orbital rotation does not necessarily preserve the sign of the orbital energies, it doespreserve the charge, i.e. we have [ ˆ
N , ˆ˜ b † p ] = ˆ˜ b † p and [ ˆ N , ˆ˜ d † p ] = − ˆ˜ d † p . So the new creationoperators ˆ˜ b † p and ˆ˜ d † p can still be interpreted as creating electrons and positrons, respectively,and the partition into PS and NS should now be understood as a partition into positive andnegative opposite charge states. As expected, the new electron and positron annihilationoperators satisfy ˆ˜ b p | ˜0 i = 0 for p ∈ PS and ˆ˜ d p | ˜0 i = 0 for p ∈ NS . (26)The new vacuum state | ˜0 i contains electron-positron pairs associated with the original oper-ators ˆ b † p and ˆ d † p but does not contain any particle associated with the new operators ˆ˜ b † p andˆ˜ d † p . 6 .3 No-particle vacuum states We can then introduce a new one-particle density-matrix operator ˆ˜ n ( ~r, ~r ′ ) and a new pairdensity-matrix operator ˆ˜ n ( ~r , ~r ) defined using normal ordering ˜ N [ ... ] of the new elementarycreation and annihilation operators ˆ˜ b † p , ˆ˜ b p , ˆ˜ d † p , ˆ˜ d p with respect to the new vacuum state | ˜0 i ˆ˜ n ,ρσ ( ~r, ~r ′ ) = ˜ N [ ˆ ψ † σ ( ~r ′ ) ˆ ψ ρ ( ~r )] , (27)and ˆ˜ n ,ρυστ ( ~r , ~r ) = ˜ N [ ˆ ψ † τ ( ~r ) ˆ ψ † σ ( ~r ) ˆ ψ ρ ( ~r ) ˆ ψ υ ( ~r )] . (28)Using Wick’s theorem, the original one-particle density-matrix and pair density-matrix oper-ators in Eq. (11) and (12) can be rewritten as [22]ˆ n ,ρσ ( ~r, ~r ′ ) = ˆ˜ n ,ρσ ( ~r, ~r ′ ) + ˜ n vp1 ,ρσ ( ~r, ~r ′ ) , (29)andˆ n ,ρυστ ( ~r , ~r ) = ˆ˜ n ,ρυστ ( ~r , ~r ) + ˜ n vp1 ,υτ ( ~r , ~r )ˆ˜ n ,ρσ ( ~r , ~r ) + ˜ n vp1 ,ρσ ( ~r , ~r )ˆ˜ n ,υτ ( ~r , ~r ) − ˜ n vp1 ,υσ ( ~r , ~r )ˆ˜ n ,ρτ ( ~r , ~r ) − ˜ n vp1 ,ρτ ( ~r , ~r )ˆ˜ n ,υσ ( ~r , ~r ) + ˜ n vp2 ,ρυστ ( ~r , ~r ) , (30)where ˜ n vp1 ( ~r, ~r ′ ) is the vacuum-polarization (vp) one-particle density matrix˜ n vp1 ,ρσ ( ~r, ~r ′ ) = h ˜0 | ˆ n ,ρσ ( ~r, ~r ′ ) | ˜0 i = h ˜0 | ˆ ψ † σ ( ~r ′ ) ˆ ψ ρ ( ~r ) | ˜0 i − h | ˆ ψ † σ ( ~r ′ ) ˆ ψ ρ ( ~r ) | i = X p ∈ NS ˜ ψ ∗ p,σ ( ~r ′ ) ˜ ψ p,ρ ( ~r ) − X p ∈ NS ψ ∗ p,σ ( ~r ′ ) ψ p,ρ ( ~r ) , (31)and ˜ n vp2 ( ~r , ~r ) is the vacuum-polarization pair-density matrix˜ n vp2 ,ρυστ ( ~r , ~r ) = ˜ n vp1 ,υτ ( ~r , ~r )˜ n vp1 ,ρσ ( ~r , ~r ) − ˜ n vp1 ,ρτ ( ~r , ~r )˜ n vp1 ,υσ ( ~r , ~r ) . (32)The electron-positron Hamiltonian in Eq. (7) can then be rewritten as [22]ˆ H = ˆ˜ T D + ˆ˜ W + ˆ˜ V + ˆ˜ V vp + ˜ E , (33)with ˆ˜ T D = Z Tr[ D ( ~r )ˆ˜ n ( ~r, ~r ′ )] ~r ′ = ~r d ~r, (34)and ˆ˜ W = 12 Z Z
Tr[ w ( ~r , ~r )ˆ˜ n ( ~r , ~r )]d ~r d ~r , (35)and ˆ˜ V = Z v ( ~r )ˆ˜ n ( ~r ) d ~r, (36)7 .3 No-particle vacuum states with the new (opposite) charge density operatorˆ˜ n ( ~r ) = Tr[ˆ˜ n ( ~r )] , (37)where ˆ˜ n ( ~r ) = ˆ˜ n ( ~r, ~r ). In Eq. (33), the normal reordering with respect to the new vacuumstate | ˜0 i [Eqs. (29) and (30)] has generated two new terms: the vacuum-polarization potentialˆ˜ V vp and the new vacuum energy ˜ E . The vacuum-polarization potential [22] can be writtenas ˆ˜ V vp = ˆ˜ V vpH + ˆ˜ V vpx , (38)with a Hartree (or direct) contributionˆ˜ V vpH = Z Tr[˜ v vpH ( ~r )ˆ˜ n ( ~r )]d ~r ≡ X ρσ Z ˜ v vpH ,σρ ( ~r )ˆ˜ n ρσ ( ~r )d ~r, (39)where ˜ v vpH ,σρ ( ~r ) = P τυ R w στρυ ( ~r , ~r )˜ n vp υτ ( ~r )d ~r and ˜ n vp υτ ( ~r ) = ˜ n vp1 ,υτ ( ~r , ~r ), and an exchangecontribution ˆ˜ V vpx = Z Z
Tr[˜ v vpx ( ~r , ~r )ˆ˜ n ( ~r , ~r )]d ~r d ~r , (40)where ˜ v vpx ,τρ ( ~r , ~r ) = − P συ w στρυ ( ~r , ~r )˜ n vp1 ,υσ ( ~r , ~r ). Note that in the literature the name“vacuum polarization” is often restricted to the direct term in Eq. (39) whereas the exchangeterm in Eq. (40) is often designated as “self-energy” (see, e.g., Ref. [25]). Here, we adopt theterminology of Ref. [22] where vacuum polarization designates both terms. Finally, the newno-particle vacuum energy [22] can be written as˜ E = h ˜0 | ˆ H | ˜0 i = Z Tr[ D ( ~r )˜ n vp1 ( ~r, ~r ′ )] ~r ′ = ~r d ~r + Z v ( ~r )˜ n vp ( ~r ) d ~r + 12 Z Z
Tr[ w ( ~r , ~r )˜ n vp2 ( ~r , ~r )]d ~r d ~r . (41)Throughout the paper, | ˜0 i will refer to an arbitrary floating vacuum state, and { ˜ ψ p ( ~r ) } and ˜ E will refer to its associated orbitals and vacuum energy. The optimal HF vacuum stateis defined as the vacuum state minimizing ˜ E with respect to the orbital rotation parameters κ E HF0 = min κ ˜ E . (42)Clearly, if n vp1 ( ~r, ~r ′ ) = then ˜ E = 0, and thus E HF0 is necessarily negative. It can in factdiverges to −∞ due to infrared (IR) and ultraviolet (UV) divergences. The IR divergencesappear when taking the continuum limit of the sums in Eq. (31), but can simply be avoided byputting the system in a box with periodic boundary conditions and taking the thermodynamiclimit of quantities per volume unit (see, e.g., Refs. [11, 29, 30]), similarly to what is done forthe homogeneous electron gas. The UV divergences come from the unbound large-energy (orlarge index p ) limit of each sum in Eq. (31), even if we expect a cancellation of these UVdivergences to some extent between the two sums. A standard way of dealing with these UVdivergences is to introduce a fixed UV momentum cutoff and to remove the cutoff dependence8 .4 Correlated vacuum state via renormalization of the electron charge and mass in the Hamiltonian [26–31] (see alsoRef. [39]). In the rest of this work, we will simply assume that a proper renormalizationscheme is applied in order to keep everything finite.Finally, in Appendix B, we provide an alternative definition of the electron-positron Hamil-tonian based on commutators and anticommutators of Dirac field operators and we show that,after removing the vacuum energy, both Hamiltonians are equivalent to each other and alsoidentical to the effective QED Hamiltonian of Refs. [25, 40–44] [see Eq. (175)]. More generally, the vacuum state can be defined beyond the HF approximation as the lowest-energy state with zero charge, which will refer to as the correlated vacuum state | Ψ i ∈ H .In a full configuration-interaction approach, the correlated vacuum state can be parametrizedas a linear combination of states with arbitrary numbers of electron-positron pairs | Ψ i = c + X p ∈ PS X q ∈ NS c p q ˆ b † p ˆ d † q + X p ,p ∈ PS X q ,q ∈ NS c p q p q ˆ b † p ˆ d † q ˆ b † p ˆ d † q + X p ,p ,p ∈ PS X q ,q ,q ∈ NS c p q p q p q ˆ b † p ˆ d † q ˆ b † p ˆ d † q ˆ b † p ˆ d † q + · · · ! | i , (43)and minimizing the energy with respect to the coefficients leads to the correlated vacuumenergy E = h Ψ | ˆ H | Ψ i . Note that the particles inside this vacuum state cannot generally beabsorbed into an orbital rotation because of the two-particle interaction in the Hamiltonian.Therefore, the correlated vacuum state generally contains electron-positron pairs, in the sameway as the non-relativistic ground state contains excited Slater determinants that cannot beabsorbed into a redefinition of the orbitals. With the parametrization of the vacuum statein Eq. (43), there is no need to perform orbital rotations (i.e., orbital rotation parametersare redundant). The correlated vacuum state | Ψ i and correlated vacuum energy E includeall vacuum contributions (i.e., contributions from orbitals in the set NS) to all orders in thetwo-particle interaction. N -negative-charge states The ground-state energy for a net total amount of q = N negative charges (the equivalent of N electrons for the non-relativistic theory) is found as E N = min | Ψ i∈H N h Ψ | ˆ T D + ˆ W + ˆ V | Ψ i , (44)where | Ψ i is constrained to have a net total amount of N negative charges, i.e. R h Ψ | ˆ n ( ~r ) | Ψ i d ~r = N . Note that we will always tacitly assume that | Ψ i is constrained to be normalized to 1, i.e. h Ψ | Ψ i = 1. A state | Ψ i ∈ H N has the form | Ψ i = X p ,...,p N ∈ PS c p ...p N ˆ b † p · · · ˆ b † p N + X p ,...,p N ,p N +1 ∈ PS X q ∈ NS c p ...p N p N +1 q ˆ b † p · · · ˆ b † p N ˆ b † p N +1 ˆ d † q + X p ,...,p N ,p N +1 ,p N +2 ∈ PS X q ,q ∈ NS c p ...p N p N +1 q p N +2 q ˆ b † p · · · ˆ b † p N ˆ b † p N +1 ˆ d † q ˆ b † p N +2 ˆ d † q + · · · ! | i . (45)9 .6 No-pair approximation Again, vacuum contributions to all orders are included in the presence of N negative charges,and there is no need to perform orbital rotations. Obviously, in the special case N = 0, thisreduces to the correlated vacuum state in Eq. (43).Since the number of particles is not fixed for the Fock state | Ψ i in Eq. (45), there isno concept of N -particle wave function (depending on N space coordinates) associated withthe state | Ψ i . Thus, one cannot study for example the wave function at electron-electroncoalescence. However, one could study the small interparticle behavior of the pair-densitymatrix n ( ~r , ~r ) = h Ψ | ˆ n ( ~r , ~r ) | Ψ i , which should ultimately control the convergence rateof the energy with respect to the one-particle basis used to expand the orbitals. So far, asfar as we know, the electron-electron coalescence has been studied only for more approx-imate configuration-space-based relativistic theories where the concept of wave function isretained [45, 46]. How to extend in practice these studies to Fock-space approaches such asthe one of the present work is an open question.Finally, let us mention that we can allow for negative N to describe the case of N -positive-charge states, i.e. states with a majority of positrons. We will however normally think of N as being positive and write the equations accordingly. Finally, we consider the no-pair (np) approximation [47, 48]. In the context of the presenttheory, it is natural to first define what we will call here a “no-pair with vacuum-polarization”(npvp) approximation (see Ref. [22]) in which the ground-state energy for N electrons isexpressed as E npvp N = min | Ψ + i∈ ˜ H ( N, h Ψ + | ˆ T D + ˆ W + ˆ V | Ψ + i , (46)where the minimization is over normalized states in the set that we designate by ˜ H ( N, ≡ e ˆ κ H ( N, which is the set of states generated by all orbital rotations of N -electron states. Astate | Ψ + i ∈ ˜ H ( N, has the form | Ψ + i = e ˆ κ X p ,...,p N ∈ PS c p ...p N ˆ b † p · · · ˆ b † p N | i = X p ,...,p N ∈ PS c p ...p N ˆ˜ b † p · · · ˆ˜ b † p N | ˜0 i . (47)We can also write this state as | Ψ + i = ˆ˜ P + | Ψ i , (48)where | Ψ i is an arbitrary state constrained to have a net total amount of N negative charges,i.e. | Ψ i ∈ H N , and ˆ˜ P + is the projector onto the N -electron Hilbert space constructed fromthe set of electron creation operators { ˆ˜ b † p } associated with the floating vacuum state | ˜0 i . Theenergy is not only minimized with respect to | Ψ i but also with respect to the projector ˆ˜ P + byperforming orbital rotations between PS and NS orbitals. The optimal floating vacuum state | ˜0 i will of course depend on the number of electrons N considered. This npvp approximationthus restores the concept of the N -electron (4 N -component spinor) wave function, i.e.Ψ + ( ~r , ~r , ..., ~r N ) = X p ,...,p N ∈ PS c p ...p N ˜ ψ p ( ~r ) ∧ · · · ∧ ˜ ψ p N ( ~r N ) , (49)10here ∧ is the normalized antisymmetrized tensor product. In this approximation, the vacuumcontributions are taken into account at the mean-field level. Indeed, using the rewriting ofthe electron-positron Hamiltonian in Eq. (33), we have E npvp N = h Ψ + | ˆ˜ T D + ˆ˜ W + ˆ˜ V + ˆ˜ V vp | Ψ + i + ˜ E , (50)which includes the vacuum-polarization potential [Eq. (38)] and the vacuum energy [Eq. (41)].The common no-pair (np) approximation corresponds to additionally neglecting all vac-uum contributions E np N = min | Ψ + i∈ ˜ H ( N, h Ψ + | ˆ˜ T D + ˆ˜ W + ˆ˜ V | Ψ + i , (51)where we use now the Hamiltonian written with normal ordering with respect to the floatingvacuum state | ˜0 i . The no-pair approximation with optimized orbitals is analogous to thecomplete-active-space self-consistent-field method of quantum chemistry.Note that in Eq. (46) or (51) one can minimize with respect to the projector ˆ˜ P + since theFock-space normal-ordered electron-positron Hamiltonian is bounded from below. If one startsinstead with the configuration-space Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian,the same E np N can be obtained but using instead a minmax principle in which the energy ismaximized with respect to the projector (see Refs. [23, 49–51]). We now formulate a RDFT based on the electron-positron Hamiltonian in Eq. (7). Wewill consider the simplest case of functionals of only the (opposite) charge density n ( ~r ) = h Ψ | ˆ n ( ~r ) | Ψ i , which is appropriate for closed-shell systems. More generally, one could considerfunctionals depending also on the (opposite) charge current ~j ( ~r ) = h Ψ | ˆ ~j ( ~r ) | Ψ i with ˆ ~j ( ~r ) =Tr[ c~ α ˆ n ( ~r )]. Even more generally, one could consider functionals of the local density matrix n ( ~r ) = h Ψ | ˆ n ( ~r ) | Ψ i , as proposed in Ref. [13]. Using the constrained-search formalism [32, 33], we define the universal density functional F [ n ] for N -representable charge densities n ∈ D N , i.e. charge densities that come from astate | Ψ i ∈ H N , F [ n ] = min | Ψ i∈H N ( n ) h Ψ | ˆ T D + ˆ W | Ψ i = h Ψ[ n ] | ˆ T D + ˆ W | Ψ[ n ] i , (52)where H N ( n ) is the set of states | Ψ i ∈ H N constrained to yield the charge density n , and | Ψ[ n ] i designates a minimizing state. A N -representable charge density must of course containa net total amount of N negative charges, i.e. R n ( ~r )d ~r = N , but other than that the set of N -representable charge densities D N is a priori unknown. This is unlike the non-relativisticcase for which the mathematical set of N -representable densities is explicitly known [33]. The N -negative-charge ground-state energy can then be written as E N = min n ∈D N (cid:20) F [ n ] + Z v ( ~r ) n ( ~r ) d ~r (cid:21) . (53)11 .1 Kohn-Sham scheme Note that, in the special case N = 0 we obtain the correlated vacuum energy of Sec. 2.4. Also,as already indicated, we can allow for negative N to describe the case of N positive charges.To setup a KS scheme [52], we decompose F [ n ] as F [ n ] = T s [ n ] + E Hxc [ n ] , (54)where T s [ n ] is the non-interacting kinetic + rest-mass density functional T s [ n ] = min | Φ i∈ ˜ S ( N, ( n ) h Φ | ˆ T D | Φ i = h Φ[ n ] | ˆ T D | Φ[ n ] i , (55)where the minimization is over the set ˜ S ( N, ( n ) of single-determinant states | Φ i = ˆ˜ b † ˆ˜ b † · · · ˆ˜ b † N | ˜0 i with a fixed number of electrons N with respect to a floating vacuum state and yielding thecharge density n , and E Hxc [ n ] is the Hartree-exchange-correlation density functional. Theminimizing state (that we will assume unique up to a phase factor for simplicity) is the KSsingle-determinant state | Φ[ n ] i . Note that in Eq. (55) we have tacitly assumed that any N -representable charge density n can be represented by a single-determinant state | Φ i . Forthe non-relativistic theory, this can be proved to be true by explicitly constructing a singledeterminant yielding any given N -representable density [33,53,54]. This proof does not applyto the present relativistic theory due to the more complicated form of the charge density n ( ~r )which includes the vacuum-polarization contribution [see Eqs. (62) and (63)]. In fact, becauseof the vacuum-polarization contribution the charge density n ( ~r ) may not generally have thesame sign at all spatial points. This is particularly obvious for the case N = 0: the chargedensity integrates to zero R n ( ~r )d ~r = 0 and thus necessarily changes sign. Whether the proofsof Refs. [33,53,54] can be generalized to the relativistic case is an open question. We can thenwrite the ground-state energy as E N = min | Φ i∈ ˜ S ( N, h h Φ | ˆ T D + ˆ V | Φ i + E Hxc [ n | Φ i ] i , (56)where ˜ S ( N, is the set of single-determinant states with a fixed number of electrons N withrespect to a floating vacuum state. Note that, contrary to a general N -negative-charge state inEq. (45), we can associate a wave function to a single-determinant state, i.e. Φ( ~r , ~r , ..., ~r N ) =˜ ψ ( ~r ) ∧ · · · ∧ ˜ ψ N ( ~r N ).More explicitly, the expression of the energy in terms of the orbitals { ˜ ψ p } is E N [ { ˜ ψ p } ] = Z Tr[ D ( ~r ) n KS1 ( ~r, ~r ′ )] ~r ′ = ~r d ~r + Z v ( ~r ) n ( ~r ) d ~r + E Hxc [ n ] , (57)with the KS one-particle density matrix n KS1 ( ~r, ~r ′ ) = ˜ n KS1 ( ~r, ~r ′ ) + ˜ n vp1 ( ~r, ~r ′ ) , (58)which includes the contribution from the electronic occupied orbitals˜ n KS1 ( ~r, ~r ′ ) = N X i =1 ˜ ψ i ( ~r ) ˜ ψ † i ( ~r ′ ) , (59)and from the vacuum polarization [see Eq. (31)]˜ n vp1 ( ~r, ~r ′ ) = X p ∈ NS ˜ ψ p ( ~r ) ˜ ψ † p ( ~r ′ ) − X p ∈ NS ψ p ( ~r ) ψ † p ( ~r ′ ) , (60)12 .2 Hartree-exchange-correlation density functional and with the corresponding charge density n ( r ) = Tr[ n KS1 ( ~r, ~r )]. Taking the functional deriva-tive of E N [ { ψ p } ] with respect to ˜ ψ † p ( ~r ) with the orbital orthonormalization constraints, wearrive at the KS equations( D ( ~r ) + v ( ~r ) + v Hxc ( ~r )) ˜ ψ p ( ~r ) = ˜ ε p ˜ ψ p ( ~r ) , (61)where v Hxc ( ~r ) = δE Hxc [ n ] /δn ( ~r ) is the Hartree-exchange-correlation potential and ˜ ε p are theKS orbital energies. The KS equations must be solved self-consistently with the density n ( ~r ) = N X i =1 ˜ ψ † i ( ~r ) ˜ ψ i ( ~r ) + ˜ n vp ( ~r ) , (62)where the vacuum-polarization density is˜ n vp ( ~r ) = X p ∈ NS ˜ ψ † p ( ~r ) ˜ ψ p ( ~r ) − X p ∈ NS ψ † p ( ~r ) ψ p ( ~r )= 12 X p ∈ NS ˜ ψ † p ( ~r ) ˜ ψ p ( ~r ) − X p ∈ PS ˜ ψ † p ( ~r ) ˜ ψ p ( ~r ) ! , (63)where the last equality follows from Eqs. (167) and (171). Equations (61)-(63) have the sameform as for the KS scheme based on renormalized QED [7–10] except that we did not takeinto account any renormalization counterterms and that the present functional E Hxc [ n ] isassociated with the effective Coulomb or Coulomb+Breit two-particle interaction. The Hartree-exchange-correlation density functional E Hxc [ n ] can be decomposed as E Hxc [ n ] = E Hx [ n ] + E c [ n ] , (64)where E Hx [ n ] is the Hartree-exchange energy encompassing all first-order terms in the two-particle interaction E Hx [ n ] = h Φ[ n ] | ˆ W | Φ[ n ] i = 12 Z Z
Tr[ w ( ~r , ~r ) n KS2 ( ~r , ~r )]d ~r d ~r , (65)with the KS pair-density matrix n KS2 ( ~r , ~r ) = h Φ[ n ] | ˆ n ( ~r , ~r ) | Φ[ n ] i , and E c [ n ] is the corre-lation energy. The Hartree-exchange energy can be written more explicitly as E Hx [ n ] = ˜ E Hx [ n ] + ˜ E vpHx [ n ] , (66)where ˜ E Hx [ n ] is the main contribution˜ E Hx [ n ] = 12 Z Z
Tr[ w ( ~r , ~r )˜ n KS2 ( ~r , ~r )]d ~r d ~r , (67)depending on the part of the KS pair-density matrix coming from the electronic occupiedorbitals ˜ n KS2 ,ρυστ ( ~r , ~r ) = ˜ n KS1 ,υτ ( ~r , ~r )˜ n KS1 ,ρσ ( ~r , ~r ) − ˜ n KS1 ,ρτ ( ~r , ~r )˜ n KS1 ,υσ ( ~r , ~r ) , (68)13 .2 Hartree-exchange-correlation density functional and ˜ E vpHx [ n ] is the vacuum-polarization contribution˜ E vpHx [ n ] = Z Tr[˜ v vpH ( ~r )˜ n KS1 ( ~r, ~r )]d ~r + Z Z
Tr[˜ v vpx ( ~r , ~r )˜ n KS1 ( ~r , ~r )]d ~r d ~r + 12 Z Z
Tr[ w ( ~r , ~r )˜ n vp2 ( ~r , ~r )]d ~r d ~r , (69)where the vacuum-polarization potentials ˜ v vpH ( ~r ) and ˜ v vpx ( ~r , ~r ) were defined after Eqs. (39)and. (40), respectively, and the vacuum-polarization pair-density matrix ˜ n vp2 ( ~r , ~r ) was de-fined in Eq. (32).We can further decompose the functional E Hx [ n ] as E Hx [ n ] = E H [ n ] + E x [ n ] . (70)where the Hartree functional E H [ n ] collects all direct terms and the exchange functional E x [ n ]collects all exchange terms. The expression of the Hartree functional is E H [ n ] = ˜ E H [ n ] + ˜ E vpH [ n ] , (71)with ˜ E H [ n ] = 12 Z Z
Tr[ w ( ~r , ~r )˜ n KS2 , H ( ~r , ~r )]d ~r d ~r , (72)where ˜ n KS2 , H ( ~r , ~r ) is the Hartree contribution to ˜ n KS2 ( ~r , ~r ) [the first term in the right-handside of Eq. (68)], and˜ E vpH [ n ] = Z Tr[˜ v vpH ( ~r )˜ n KS1 ( ~r, ~r )]d ~r + 12 Z Z
Tr[ w ( ~r , ~r )˜ n vp2 , H ( ~r , ~r )]d ~r d ~r , (73)where ˜ n vp2 , H ( ~r , ~r ) is the Hartree contribution to ˜ n vp2 ( ~r , ~r ) [the first term in the right-handside of Eq. (32)]. Similarly, the expression of the exchange functional is E x [ n ] = ˜ E x [ n ] + ˜ E vpx [ n ] , (74)with ˜ E x [ n ] = 12 Z Z
Tr[ w ( ~r , ~r )˜ n KS2 , x ( ~r , ~r )]d ~r d ~r , (75)where ˜ n KS2 , x ( ~r , ~r ) is the exchange contribution to ˜ n KS2 ( ~r , ~r ) [the second term in the right-hand side of Eq. (68)], and˜ E vpx [ n ] = Z Z
Tr[˜ v vpx ( ~r , ~r )˜ n KS1 ( ~r , ~r )]d ~r d ~r + 12 Z Z
Tr[ w ( ~r , ~r )˜ n vp2 ,x ( ~r , ~r )]d ~r d ~r , (76)where ˜ n vp2 , x ( ~r , ~r ) is the exchange contribution to ˜ n vp2 ( ~r , ~r ) [the second term in the right-handside of Eq. (32)].The Hartree energy can also be more compactly written as a sum of Coulomb and Breitcontributions E H [ n ] = E CH [ n ] + E BH [ n ] , (77)14 .2 Hartree-exchange-correlation density functional where the Coulomb contribution has the same form as in non-relativistic DFT E CH [ n ] = 12 Z Z w ( r ) n ( ~r ) n ( ~r )d ~r d ~r , (78)involving the charge density n ( ~r ) [Eq. (62)], and the Breit contribution has the form E BH [ n ] = − c Z Z w ( r ) " ~j ( ~r ) · ~j ( ~r ) + ~j ( ~r ) · ~r ~j ( ~r ) · ~r r d ~r d ~r , (79)involving the KS charge current density ~j ( ~r ) ~j ( ~r ) = Tr[ c~ α n KS1 ( ~r, ~r )] = c N X i =1 ˜ ψ † i ( ~r ) ~ α ˜ ψ i ( ~r ) + ~ ˜ j vp ( ~r ) , (80)where ~ ˜ j vp ( ~r ) is the vacuum-polarization current density ~ ˜ j vp ( ~r ) = c X p ∈ NS ˜ ψ † p ( ~r ) ~ α ˜ ψ p ( ~r ) − X p ∈ NS ψ † p ( ~r ) ~ αψ p ( ~r ) . (81)Since we did not consider any vector potential in the KS equations [Eq. (61)], the KS Hamil-tonian has time-reversal symmetry and the KS orbitals { ˜ ψ p } come in degenerate Kramerspairs (see, e.g., Ref. [23]) with opposite current densities, and similarly for the orbitals { ψ p } of the free Dirac equation. It seems then reasonable to conclude that the vacuum-polarizationcurrent density ~ ˜ j vp ( ~r ) vanishes in the present context, glossing over the fact that the sumsin Eq. (81) are infinite. Moreover, for closed-shell systems, the contribution to the chargecurrent density ~j ( ~r ) coming from the occupied electronic states in Eq. (80) vanishes as well,and there is no Breit contribution to the Hartree energy. For open-shell systems, the chargecurrent density does not vanish and there is a Breit contribution to the Hartree energy. Sincethe charge current density ~j ( ~r ) is only an implicit functional of the charge density via the KSorbitals, the calculation of the Breit contribution to the Hartree potential would require touse the optimized-effective-potential method (see, e.g., Ref. [55]). A simpler alternative is toswitch to functionals depending also explicitly on the charge current density ~j ( ~r ).The correlation functional E c [ n ] is conveniently expressed with the adiabatic-connectionapproach [56–58] which can be straightforwardly generalized to the present relativistic theory.For this, we define an universal density functional similarly to Eq. (52) but depending on acoupling constant λ ∈ [0 , + ∞ [ F λ [ n ] = min | Ψ i∈H N ( n ) h Ψ | ˆ T D + λ ˆ W | Ψ i = h Ψ λ [ n ] | ˆ T D + λ ˆ W | Ψ λ [ n ] i , (82)where | Ψ λ [ n ] i denotes a minimizing state. This functional can be decomposed as F λ [ n ] = T s [ n ] + λE Hx [ n ] + E λ c [ n ] , (83)where the λ -dependent correlation contribution is E λ c [ n ] = h Ψ λ [ n ] | ˆ T D + λ ˆ W | Ψ λ [ n ] i − h Φ[ n ] | ˆ T D + λ ˆ W | Φ[ n ] i . (84)15 .3 No-pair approximation We will assume that F λ [ n ] is of class C as a function of λ for λ ∈ [0 ,
1] and that F λ =0 [ ρ ] = T s [ ρ ]. Taking the derivative of Eq. (84) with respect to λ and using the Hellmann-Feynmantheorem for the state | Ψ λ [ n ] i , we obtain ∂E λ c [ n ] ∂λ = h Ψ λ [ n ] | ˆ W | Ψ λ [ n ] i − h Φ[ n ] | ˆ W | Φ[ n ] i . (85)Integrating over λ from 0 to 1, and using E λ =1c [ n ] = E c [ n ] and E λ =0c [ n ] = 0, we arrive at theadiabatic-connection formula for the correlation functional E c [ n ] = Z d λ h Ψ λ [ n ] | ˆ W | Ψ λ [ n ] i − h Φ[ n ] | ˆ W | Φ[ n ] i = 12 Z d λ Z Z
Tr[ w ( ~r , ~r ) n λ , c ( ~r , ~r )]d ~r d ~r , (86)with the correlation contribution to the λ -dependent pair-density matrix n λ , c ( ~r , ~r ) = h Ψ λ [ n ] | ˆ n ( ~r , ~r ) | Ψ λ [ n ] i− n KS2 ( ~r , ~r ). More explicitly, the correlation functional has the expression E c [ n ] = ˜ E c [ n ] + ˜ E vpc [ n ] , (87)where ˜ E c [ n ] is the main contribution˜ E c [ n ] = 12 Z d λ Z Z
Tr[ w ( ~r , ~r )˜ n λ , c ( ~r , ~r )]d ~r d ~r , (88)with ˜ n λ , c ( ~r , ~r ) = h Ψ λ [ n ] | ˆ˜ n ( ~r , ~r ) | Ψ λ [ n ] i− ˜ n KS2 ( ~r , ~r ), and ˜ E vpc [ n ] is the vacuum-polarizationcontribution coming from the variation of the one-particle density matrix with λ ˜ E vpc [ n ] = Z d λ Z Tr[˜ v vpH ( ~r )˜ n λ , c ( ~r, ~r )]d ~r + Z d λ Z Z
Tr[˜ v vpx ( ~r , ~r )˜ n λ , c ( ~r , ~r )]d ~r d ~r , (89)with ˜ n λ , c ( ~r , ~r ) = h Ψ λ [ n ] | ˆ˜ n ( ~r , ~r ) | Ψ λ [ n ] i − ˜ n KS1 ( ~r , ~r ). Note that both ˜ n λ , c ( ~r , ~r ) and˜ n λ , c ( ~r , ~r ) include contributions from orbitals ˜ ψ p with p ∈ NS, which generate vacuum con-tributions to the correlation energy beyond first order in the two-particle interaction.
In the npvp approximation introduced in Eq. (46), the universal density functional becomes F npvp [ n ] = min | Ψ + i∈ ˜ H ( N, ( n ) h Ψ + | ˆ T D + ˆ W | Ψ + i (90)where ˜ H ( N, ( n ) is the set of states in ˜ H ( N, yielding the charge density n . In this approxima-tion, the definition of T s [ n ] in Eq. (55) is left unchanged and consequently the KS determinantstate | Φ[ n ] i and the Hartree and exchange functionals E H [ n ] and E x [ n ] are also left unchanged.We thus have the decomposition F npvp [ n ] = T s [ n ] + E Hx [ n ] + E npvpc [ n ] , (91)16 .3 No-pair approximation where E npvpc [ n ] is the new correlation functional in this approximation. In this npvp KSscheme, the ground-state energy is then obtained as E npvp N = min | Φ i∈ ˜ S ( N, h h Φ | ˆ T D + ˆ V | Φ i + E Hx [ n | Φ i ] + E npvpc [ n | Φ i ] i . (92)Hence, this approximation affects only the correlation functional, namely E npvpc [ n ] has thesame expression than E c [ n ] but in Eqs. (88) and (89) ˜ n λ , c ( ~r , ~r ) and ˜ n λ , c ( ~r , ~r ) are nowcalculated with a state | Ψ λ + [ n ] i ∈ ˜ H ( N, ( n ) and thus do not contain any contributions comingfrom orbitals ˜ ψ p with p ∈ NS. However, vacuum contributions are still included at themean-field level with the potentials ˜ v vpH ( ~r ) and ˜ v vpx ( ~r , ~r ).In the most common no-pair approximation of Eq. (51), the universal functional is definedas F np [ n ] = min | Ψ + i∈ ˜ H ( N, ( n ) h Ψ + | ˆ˜ T D + ˆ˜ W | Ψ + i , (93)where we use now the operators written with normal ordering with respect to the floatingvacuum state | ˜0 i , and the non-interacting kinetic + rest-mass density functional is defined as T nps [ n ] = min | Φ i∈ ˜ S ( N, ( n ) h Φ | ˆ˜ T D | Φ i = h Φ np [ n ] | ˆ˜ T D | Φ np [ n ] i , (94)where | Φ np [ n ] i is the KS determinant state in this approximation (again, assumed to be uniqueup to a phase factor for simplicity). The functional F np [ n ] can then be decomposed as F np [ n ] = T nps [ n ] + E npHx [ n ] + E npc [ n ] , (95)where E npHx [ n ] is the no-pair Hartree-exchange functional E npHx [ n ] = h Φ np [ n ] | ˆ˜ W | Φ np [ n ] i = 12 Z Z
Tr[ w ( ~r , ~r )˜ n KS , np2 ( ~r , ~r )]d ~r d ~r , (96)with the no-pair KS pair-density matrix ˜ n KS , np2 ( ~r , ~r ) = h Φ np [ n ] | ˆ˜ n ( ~r , ~r ) | Φ np [ n ] i (which, asbefore, can be trivially separated into Hartree and exchange contributions), and E npc [ n ] is theno-pair correlation functional E npc [ n ] = 12 Z d λ Z Z
Tr[ w ( ~r , ~r )˜ n λ, np2 , c ( ~r , ~r )]d ~r d ~r , (97)with ˜ n λ, np2 , c ( ~r , ~r ) = h Ψ λ + [ n ] | ˆ˜ n ( ~r , ~r ) | Ψ λ + [ n ] i − ˜ n KS , np2 ( ~r , ~r ) and | Ψ λ + [ n ] i is a λ -dependentno-pair minimizing state for the charge density n . Finally, the no-pair ground-state energy isobtained as E np N = min | Φ i∈ ˜ S ( N, h h Φ | ˆ˜ T D + ˆ˜ V | Φ i + E npHx [ n | Φ i ] + E npc [ n | Φ i ] i , (98)and the no-pair charge density is simply n ( ~r ) = P Ni =1 ˜ ψ † i ( ~r ) ˜ ψ i ( ~r ).This constitutes a no-pair KS RDFT with well-defined universal exchange and correla-tion functionals E npx [ n ] and E npc [ n ]. This contrasts with the RDFT based on the relativistic17 .4 Exact properties of the density functionals extension of the Hohenberg-Kohn theorem of Refs. [7–10] for which the no-pair approxima-tion is only introduced a posteriori without giving an unambiguous definition of the involvedfunctionals. Indeed, the no-pair approximation involves the projector ˆ˜ P + onto the subspaceof electronic states [Eq. (48)] which depends on the separation of the orbitals into PS and NSsets, and therefore depends on the potential used to generate these orbitals. If the projectoris applied to the Hamiltonian, the whole resulting projected Hamiltonian is thus dependenton this potential, and one cannot isolate, as normally done in DFT, an universal part of theHamiltonian, and one thus cannot define universal density functionals. In the present work,instead of thinking of the projector ˆ˜ P + as being applied to the Hamiltonian, we equivalentlythink of the projector as being applied to the state, i.e. | Ψ + i = ˆ˜ P + | Ψ i , and optimize theprojector simultaneously with the state | Ψ i . In this way, we can introduce universal den-sity functionals, similarly to non-relativistic DFT, defined such that for a given density aconstrained-search optimization in Eq. (93) or (94) of the projected state | Ψ + i determinesalone the optimal projector without the need of pre-choosing a particular potential, at leastfor systems for which orbitals can be unambiguously separated into PS and NS sets. Thesame view can be taken in the configuration-space approach using a minmax principle [51]. Charge-conjugation symmetry
A state | Ψ[ n ] i in Eq. (52) yields the charge density n and minimizes h Ψ | ˆ T D + ˆ W | Ψ i . Thecharge-conjugated state ˆ C | Ψ[ n ] i , where ˆ C is the charge-conjugation operator in Fock space(see Appendix A), yields the charge density − n since h Ψ[ n ] | ˆ C † ˆ n ( ~r ) ˆ C | Ψ[ n ] i = −h Ψ[ n ] | ˆ n ( ~r ) | Ψ[ n ] i = − n ( ~r ) , (99)where we have used the antisymmetry of the density operator under charge conjugation,ˆ C † ˆ n ( ~r ) ˆ C = − ˆ n ( ~r ) [Eq. (143)]. Moreover, the charge-conjugated state ˆ C | Ψ[ n ] i minimizes h Ψ | ˆ T D + ˆ W | Ψ i since h Ψ[ n ] | ˆ C † ( ˆ T D + ˆ W ) ˆ C | Ψ[ n ] i = h Ψ[ n ] | ˆ T D + ˆ W | Ψ[ n ] i , (100)since both ˆ T D and ˆ W are symmetric under charge conjugation [Eqs. (142) and (147)]. Wethus conclude that ˆ C | Ψ[ n ] i = | Ψ[ − n ] i , (101)and that the universal density functional is symmetric under charge conjugation F [ n ] = F [ − n ] . (102)Similarly, the KS determinant state in Eq. (55) transforms asˆ C | Φ[ n ] i = | Φ[ − n ] i , (103)and the functionals T s [ n ], E H [ n ], E x [ n ], and E c [ n ] are all symmetric under charge conjugation T s [ n ] = T s [ − n ] , (104)18 .4 Exact properties of the density functionals E H [ n ] = E H [ − n ] , (105) E x [ n ] = E x [ − n ] , (106) E c [ n ] = E c [ − n ] . (107)In other words, these functionals must be even functionals of the charge density. Consequently,their functional derivatives with respect to n ( ~r ) must be odd functionals of the charge density.This is particularly obvious for the Coulomb contribution to the Hartree energy in Eq. (78). Uniform coordinate scaling relations
In non-relativistic DFT, the uniform coordinate scaling relations [59–61] are important con-straints on the density functionals. We show how to generalize them for the present RDFT.Since there is generally no concept of wave function in the present relativistic theory, wecannot define coordinate scaling on wave functions, as normally done. Instead, we must workin Fock space and we thus define an unitary uniform coordinate scaling operator ˆ S γ whichtransforms the Dirac field operator asˆ S † γ ˆ ψ ( ~r ) ˆ S γ = γ / ˆ ψ ( γ~r ) , (108)where γ ∈ ]0 , + ∞ [ is a scaling factor, and similarly for the separate electron and positron fieldoperators in Eq. (136), i.e. ˆ S † γ ˆ ψ + ( ~r ) ˆ S γ = γ / ˆ ψ + ( γ~r ) and ˆ S † γ ˆ ψ − ( ~r ) ˆ S γ = γ / ˆ ψ − ( γ~r ). Theone-particle density-matrix and density operators transform asˆ S † γ ˆ n ( ~r, ~r ′ ) ˆ S γ = γ ˆ n ( γ~r, γ~r ′ ) , (109)and ˆ S † γ ˆ n ( ~r ) ˆ S γ = γ ˆ n ( γ~r ) , (110)while the pair density-matrix operator transforms asˆ S † γ ˆ n ( ~r , ~r ) ˆ S γ = γ ˆ n ( γ~r , γ~r ) . (111)Since the scaling relations involve scaling the speed of light c , we will explicitly indicatein this section the dependence on c . A state | Ψ λ,c [ n ] i in Eq. (82) for any coupling constant λ and speed of light c yields the charge density n and minimizes h Ψ | ˆ T c D + λ ˆ W | Ψ i . The scaledstate | Ψ λ,cγ [ n ] i = ˆ S γ | Ψ λ,c [ n ] i , (112)yields the scaled charge density [see Eq. (110)] n γ ( ~r ) = γ n ( γ~r ) , (113)and minimizes h Ψ | ˆ T cγ D + λγ ˆ W | Ψ i since h Ψ λ,cγ [ n ] | ˆ T cγ D + λγ ˆ W | Ψ λ,cγ [ n ] i = γ h Ψ λ,c [ n ] | ˆ T c D + λ ˆ W | Ψ λ,c [ n ] i , (114)19 .4 Exact properties of the density functionals where we have used Eqs. (109) and (111). We thus conclude that the scaled state | Ψ λ,cγ [ n ] i atcoupling constant λ and speed of light c corresponds to the state at scaled density n γ , scaledcoupling constant λγ , and scaled speed of light cγ | Ψ λ,cγ [ n ] i = | Ψ λγ,cγ [ n γ ] i , (115)or, equivalently, | Ψ λ/γ,c/γγ [ n ] i = | Ψ λ,c [ n γ ] i , (116)and that the universal density functional satisfies the scaling relation F λγ,cγ [ n γ ] = γ F λ,c [ n ] , (117)or, equivalently, F λ,c [ n γ ] = γ F λ/γ,c/γ [ n ] . (118)At λ = 0, we find the scaling relation of the KS single-determinant state | Φ c/γγ [ n ] i = | Φ c [ n γ ] i , (119)which directly leads to the scaling relation for the non-interacting kinetic density functional T c s [ n γ ] = γ T c/γ s [ n ] , (120)and for the Hartree and exchange density functionals E c H [ n γ ] = γE c/γ H [ n ] and E c x [ n γ ] = γE c/γ x [ n ] . (121)The correlation density functional has the same scaling as F λ,c [ n ] E λ,c c [ n γ ] = γ E λ/γ,c/γ c [ n ] , (122)and, in particular, for λ = 1 E c c [ n γ ] = γ E /γ,c/γ c [ n ] . (123)These scaling relations imply that the low-density limit ( γ →
0) corresponds to the non-relativistic limit ( c → ∞ ), while the high-density limit ( γ → ∞ ) corresponds to the ultra-relativistic limit ( m → m is the electron mass).In the low-density limit, we indeed recover the well-known behaviors of the non-relativisticdensity functionals. After removing the rest-mass energy of N electrons, N mc , the non-interacting kinetic-energy functional scales quadratically as γ → T c s [ n γ ] − N mc ∼ γ → γ T NRs [ n ] , (124)where T NRs [ n ] = lim c →∞ ( T c s [ n ] − N mc ) is the non-relativistic (NR) non-interacting kinetic-energy functional. The Hartree and exchange functionals scale linearly as γ → E c H [ n γ ] ∼ γ → γE NRH [ n ] and E c x [ n γ ] ∼ γ → γE NRx [ n ] , (125)20 .4 Exact properties of the density functionals where E NRH [ n ] = lim c →∞ E c H [ n ] = E CH [ n ] [Eq. (78)] and E NRx [ n ] = lim c →∞ E c x [ n ] are the non-relativistic Hartree and exchange functionals. The correlation functional also scales linearlyas γ → E c c [ n γ ] ∼ γ → γW NR,SCEc [ n ] , (126)where W NR,SCEc [ n ] = lim λ →∞ E NR, λ c [ n ] /λ is the non-relativistic strictly-correlated-electron(SCE) correlation functional [62–65] obtained from the non-relativistic correlation functionalalong the adiabatic connection E NR, λ c [ n ] = lim c →∞ E c,λ c [ n ] [see Eq. (84)] in the limit of infinitecoupling constant λ → ∞ . The low-density limit is also called the strong-interaction limitsince in this limit the Hartree, exchange, and correlation energies dominate over the non-interacting kinetic energy.The high-density limit of the relativistic density functionals is more exotic. The non-interacting kinetic-energy functional scales linearly as γ → ∞ T c s [ n γ ] ∼ γ →∞ γT c, URs [ n ] , (127)where T c, URs [ n ] = lim m → T c s [ n ] is the ultra-relativistic (UR) non-interacting kinetic-energyfunctional obtained by letting the electron mass going to zero in the Dirac operator [Eq. (2)].This is in contrast with the quadratic scaling of the non-relativistic kinetic-energy functional,i.e. T NRs [ n γ ] = γ T NRs [ n ]. The Hartree and exchange functionals also scale linearly as γ → ∞ E c H [ n γ ] ∼ γ →∞ γE c, URH [ n ] and E c x [ n γ ] ∼ γ →∞ γE c, URx [ n ] , (128)where E c, URH [ n ] = lim m → E c H [ n ] and E c, URx [ n ] = lim m → E c x [ n ] are the ultra-relativistic Hartreeand exchange functionals. This is similar to the linear scaling of the non-relativistic Hartreeand exchange functionals E NRH [ n γ ] = γE NRH [ n ] and E NRx [ n γ ] = γE NRx [ n ]. Finally, the correla-tion functional scales linearly as γ → ∞ E c c [ n γ ] ∼ γ →∞ γE c, URc [ n ] , (129)where E c, URc [ n ] = lim m → E c c [ n ] is the ultra-relativistic correlation functional. This is againin contrast with the non-relativistic case where the correlation functional goes to a constantas γ → ∞ , for a KS Hamiltonian with a non-degenerate ground state, lim γ →∞ E NRc [ n γ ] = E NR,GL2c [ n ], where E NR,GL2c [ n ] is the second-order G¨orling-Levy (GL2) correlation energy [66,67]. Hence, in the relativistic case, the high-density limit is no longer a weak-interaction orweak-correlation limit since T c s [ n γ ], E c H [ n γ ], E c x [ n γ ], and E c c [ n γ ] all scale linearly in γ . In par-ticular, the divergence of the relativistic correlation functional in the high-density limit hasimportant implications for relativistic functional development. Indeed, many non-relativisticcorrelation functionals, such as the Perdew-Burke-Ernzerhof (PBE) one [68], have been de-signed to saturate in the high-density limit. Hence, these non-relativistic correlation func-tionals should be rethought so as to satisfy Eq. (129).The same scaling relations apply in the no-pair approximation, as well as in the npvpvariant of Eq. (90). In the configuration-space approach of the no-pair approximation, thesescaling relations could be obtained using the minmax principle (see Ref. [51]).In the non-relativistic theory, the high-density limit is realized in atomic ions in the limit oflarge nuclear charge, Z → ∞ , at fixed electron number N (see Refs. [69, 70]). In a relativistic21 .5 Local-density approximation setting, the relation between the high-density limit and the large nuclear-charge limit is morecomplicated due to the scaling of the speed of light [49]. However, we note that numericalstudies show that relativistic no-pair and beyond-no-pair correlation energies (calculated withrespect to HF) of two-electron atoms diverge as Z increases [49, 71], which is in line with thedivergence of E c c [ n γ ] as γ → ∞ [Eq. (129)].Finally, for γ = λ , the scaling relation in Eq. (122) gives an expression for the correlationfunctional along the adiabatic connection at coupling constant λE λ,c c [ n ] = λ E c/λ c [ n /λ ] , (130)which could be useful for analyzing approximate correlation functionals and for developing arelativistic extension of the multideterminant KS scheme of Refs. [72, 73]. The LDA is usually the first approximation considered in DFT. In the present relativistictheory, the LDA exchange-correlation functional may be written as E LDAxc [ n ] = Z | n ( ~r ) | ǫ RHEGxc ( | n ( ~r ) | )d ~r, (131)where ǫ RHEGxc ( n ) is the exchange-correlation energy per particle of the relativistic homogeneouselectron gas (RHEG) of constant charge density n ∈ [0 , + ∞ [. We have used the absolute valueof the charge density in order to satisfy charge-conjugation symmetry [Eqs. (106) and (107)].Since the RHEG has a spatially constant charge density, its KS potential v + v Hxc inEq. (61) must necessarily be a spatial constant as well. Since the KS potential does notdepend on spinor indices either (contrary to the HF potential), the KS orbitals of the RHEGare thus simply the eigenfunctions of the free Dirac equation. In other words, due to trans-lational symmetry, the KS vacuum state | ˜0 i of the RHEG is equal to the free vacuum state | i . Consequently, the vacuum-polarization one-particle density matrix in Eq. (60) vanishesfor the RHEG and the LDA exchange functional does not contain any vacuum-polarizationcontribution, i.e. E LDAx [ n ] = ˜ E LDAx [ n ] [Eq. (75)] or ˜ E vp,LDAx [ n ] = 0 [Eq. (76)]. Similarly, forthe LDA correlation functional, we have E LDAc [ n ] = ˜ E LDAc [ n ] [Eq. (88)] or ˜ E vp,LDAc [ n ] = 0[Eq. (89)], but E LDAc [ n ] still contains vacuum contributions via the correlation pair-densitymatrix ˜ n λ , c ( ~r , ~r ) of the RHEG.Moreover, for the same reason, the KS orbitals of the RHEG obtained in the no-pairapproximation [Eq. (94)] are also necessarily the eigenfunctions of the free Dirac equation, andthus the no-pair approximation has no impact on the LDA exchange functional, i.e. E LDAx [ n ] = E np,LDAx [ n ]. By contrast, the no-pair approximation or its npvp variant [Eq. (91)] do have animpact of the LDA correlation functional, i.e. E LDAc [ n ] = E npvp,LDAc [ n ] = E np,LDAc [ n ], sincethe vacuum contributions are now suppressed from ˜ n λ , c ( ~r , ~r ).The exchange energy per particle of the RHEG for the Coulomb interaction of Eq. (15)is [4, 74] (see, also, Ref. [50]) ǫ RHEG,Cx ( n ) = − k F π "
56 + 13 ˜ c + 23 p c arcsinh (cid:18) c (cid:19) − c ! ln (cid:18) c (cid:19) − (cid:18)p c − ˜ c arcsinh (cid:18) c (cid:19)(cid:19) , (132)22here k F = (3 π n ) / is the Fermi wave vector and ˜ c = mc/k F is a relativistic parameter.The exchange energy per particle for the Breit interaction of Eq. (16) has a similar form [75](see, also, Ref. [50]) ǫ RHEG,Bx ( n ) = 3 k F π " − (cid:16) c (cid:17) − ˜ c − c ) + ln (cid:0) c (cid:1) !! +2 (cid:18)p c − ˜ c arcsinh (cid:18) c (cid:19)(cid:19) . (133)Note that these expressions are valid for an arbitrary speed of light c . The dependence onthe adimensional parameter ˜ c is then necessary for the LDA exchange functional to satisfythe scaling relation of Eq. (121). Note that the Breit exchange energy per particle is anapproximation to the exchange energy per particle obtained with the transverse componentof the full QED photon propagator [3, 4, 74]. The exchange energy per particle obtained withthe full QED photon propagator has in fact a simpler expression than the Coulomb-Breit one,thanks to the cancellation of many terms between the Coulomb and transverse components, ǫ QEDx ( n ) = − k F π " − (cid:18)p c − ˜ c arcsinh (cid:18) c (cid:19)(cid:19) . (134)The Coulomb-Breit exchange energy per particle is a good approximation to the exchangeenergy per particle obtained with the full QED photon propagator for k F . c [50]. In anycase, the LDA exchange functional corresponding to the present RDFT is given by Eqs. (132)and (133), and not by Eq. (134).Contrary to the case of exchange, the correlation energy per particle of the RHEG cannotbe calculated analytically. It has been estimated numerically at the level of the relativisticrandom-phase approximation, using either the no-sea approximation (which includes parts ofthe vacuum contributions) or the no-pair approximation, and the full QED photon propagatoror the Coulomb-Breit interaction [76,77] (see also Refs. [7–9,14,78–80]). However, to the bestof my knowledge, these calculations were done for the fixed physical value of the speed of light.Therefore, we do not have the dependence on c and we cannot apply the scaling relation ofEq. (123) or (130). More work seems necessary to construct the LDA correlation functionalincluding the dependence on c with or without the no-pair approximation. In this work, we have examine a RDFT based on an effective QED without the photondegrees of freedom. The formalism is appealing since it is simpler than RDFT based on fullQED. We have used this formalism to unambiguously define density functionals in the no-pairapproximation, thus making a closer contact with calculations done in practice, and to studysome exact properties of the involved functionals, namely charge-conjugation symmetry anduniform coordinate scaling. The formalism has also the advantage to be easily extended tomultideterminant KS schemes which combine wave-function methods with density functionalsbased on a decomposition the electron-electron interaction (see, e.g., Refs. [72, 81, 82]).In possible future works on the present RDFT, one may study whether this approachcan be made mathematically rigorous, one may develop density-functional approximations23or this approach, one may examine the extension to functionals of the charge current densityor of the one-particle density matrix, and one may implement this approach for example forcalculations of vacuum-polarization effects in heavy atoms.
Acknowledgements
I thank Christian Brouder, Julien Paquier, and Trond Saue for discussions and/or commentson the manuscript.
A Charge-conjugation symmetry of the electron-positron Hamil-tonian
Under charge conjugation, the Dirac field operator transforms as (see, e.g., Refs. [9,42,44,83])ˆ C ˆ ψ ( ~r ) ˆ C † = C ˆ ψ † T ( ~r ) , (135)with the unitary charge-conjugation symmetry operator in Fock space ˆ C , the unitary ma-trix C = − i α y β defined up to an unimportant phase factor, and T designating the matrixtransposition. If we decompose the Dirac field operator into free electron and positron fieldcontributions ˆ ψ ( ~r ) = ˆ ψ + ( ~r ) + ˆ ψ − ( ~r ) , (136)with ˆ ψ + ( ~r ) = P p ∈ PS ˆ b p ψ p ( ~r ) and ˆ ψ − ( ~r ) = P p ∈ NS ˆ d † p ψ p ( ~r ) in which { ψ p ( ~r ) } is the set ofeigenfunctions of the free Dirac equation, then charge conjugation interchanges these contri-butions as ˆ C ˆ ψ + ( ~r ) ˆ C † = C ˆ ψ † T − ( ~r ) , (137)ˆ C ˆ ψ − ( ~r ) ˆ C † = C ˆ ψ † T+ ( ~r ) . (138)Let us stress that Eqs. (137) and (138) are only valid when using the orbitals of the freeDirac equation { ψ p ( ~r ) } and not arbitrary orbitals { ˜ ψ p ( ~r ) } . These equations allow us to findthe transformation under charge conjugation of the electron-positron Hamiltonian in Eq. (7)expressed with normal ordering with respect to the free vacuum state.In terms of the free electron and positron field operators, the one-particle density-matrixoperator in Eq. (11) has the expressionˆ n ,ρσ ( ~r, ~r ′ ) = ˆ ψ † + ,σ ( ~r ′ ) ˆ ψ + ,ρ ( ~r ) + ˆ ψ † + ,σ ( ~r ′ ) ˆ ψ − ,ρ ( ~r ) + ˆ ψ †− ,σ ( ~r ′ ) ˆ ψ + ,ρ ( ~r ) − ˆ ψ − ,ρ ( ~r ) ˆ ψ †− ,σ ( ~r ′ ) , (139)which becomes under charge conjugationˆ C ˆ n ,ρσ ( ~r, ~r ′ ) ˆ C † = X ρ ′ σ ′ C ρρ ′ [ ˆ ψ − ,σ ′ ( ~r ′ ) ˆ ψ †− ,ρ ′ ( ~r ) + ˆ ψ − ,σ ′ ( ~r ′ ) ˆ ψ † + ,ρ ′ ( ~r )+ ˆ ψ + ,σ ′ ( ~r ′ ) ˆ ψ †− ,ρ ′ ( ~r ) − ˆ ψ † + ,ρ ′ ( ~r ) ˆ ψ + ,σ ′ ( ~r ′ )] C † σ ′ σ = − X ρ ′ σ ′ C ρρ ′ ˆ n ,σ ′ ρ ′ ( ~r ′ , ~r ) C † σ ′ σ , (140)24r, in matrix form, ˆ C ˆ n ( ~r, ~r ′ ) ˆ C † = − C ˆ n T1 ( ~r ′ , ~r ) C † . (141)From this, we deduce that the Dirac kinetic + rest mass operator ˆ T D in Eq. (8) is symmetricunder charge conjugationˆ C ˆ T D ˆ C † = − Z Tr[ D ( ~r ) C ˆ n T1 ( ~r ′ , ~r ) C † ] ~r ′ = ~r d ~r = − Z Tr[ C † D ( ~r ) C ˆ n T1 ( ~r ′ , ~r )] ~r ′ = ~r d ~r = Z Tr[ D ( ~r )ˆ n ( ~r ′ , ~r )] ~r ′ = ~r d ~r = ˆ T D , (142)where we have used C † D ( ~r ) C = − D ∗ ( ~r ) = − D T ( ~r ). Moreover, from Eq. (141), we find theexpected antisymmetry of the charge density operator under charge conjugationˆ C ˆ n ( ~r ) ˆ C † = − ˆ n ( ~r ) , (143)which immediately shows that the external potential operator ˆ V in Eq. (10) is also antisym-metric ˆ C ˆ V ˆ C † = − ˆ V . (144)A similar calculation gives the transformation of the pair density-matrix operator inEq. (12) under charge conjugationˆ C ˆ n ,ρυστ ( ~r , ~r ) ˆ C † = X ρ ′ υ ′ τ ′ σ ′ C ρρ ′ C υυ ′ ˆ n ,τ ′ σ ′ υ ′ ρ ′ ( ~r , ~r ) C † τ ′ τ C † σ ′ σ , (145)or, in matrix notation,ˆ C ˆ n ( ~r , ~r ) ˆ C † = ( C ⊗ C )ˆ n T2 ( ~r , ~r )( C ⊗ C ) † , (146)where ⊗ is the matrix tensor product. This shows that the two-particle interaction operatorˆ W in Eq. (10) is symmetric under charge conjugationˆ C ˆ W ˆ C † = 12 Z Z
Tr[ w ( ~r , ~r )( C ⊗ C )ˆ n T2 ( ~r , ~r )( C ⊗ C ) † ]d ~r d ~r = 12 Z Z
Tr[( C ⊗ C ) † w ( ~r , ~r )( C ⊗ C )ˆ n T2 ( ~r , ~r )]d ~r d ~r = 12 Z Z
Tr[ w ( ~r , ~r )ˆ n ( ~r , ~r )]d ~r d ~r = ˆ W , (147)where we have used ( C ⊗ C ) † w ( ~r , ~r )( C ⊗ C ) = w ( ~r , ~r ) = w T ( ~r , ~r ) and w ( ~r , ~r ) = w ( ~r , ~r ).In conclusion, we thus have found the expected transformation of the electron-positronHamiltonian under charge conjugationˆ C ˆ H [ v ] ˆ C † = ˆ H [ − v ] . (148)25 Alternative definition of the electron-positron Hamiltonian
As an alternative to the definition of the electron-positron Hamiltonian based on normalordering with respect to the free vacuum state in Eq. (7), an electron-positron Hamiltonianbased on commutators and anticommutators (which we indicate by using the superscript c)of Dirac field operators can be defined asˆ H c = ˆ T cD + ˆ W c + ˆ V c , (149)with ˆ T cD = Z Tr[ D ( ~r )ˆ n c1 ( ~r, ~r ′ )] ~r ′ = ~r d ~r, (150)and ˆ W c = 12 Z Z
Tr[ w ( ~r , ~r )ˆ n c2 ( ~r , ~r )]d ~r d ~r , (151)and ˆ V c = Z v ( ~r )ˆ n c ( ~r ) d ~r. (152)In these expressions, ˆ n c1 ( ~r, ~r ′ ) is an one-particle density matrix operator defined as a commu-tator of Dirac field operatorsˆ n c1 ,ρσ ( ~r, ~r ′ ) = 12 h ˆ ψ † σ ( ~r ′ ) , ˆ ψ ρ ( ~r ) i , (153)ˆ n c ( ~r ) = Tr[ˆ n c1 ( ~r, ~r )] is the associated (opposite) charge density operator, and similarly ˆ n c2 ( ~r , ~r )is a pair density-matrix operator defined as an anticommutator of products of Dirac field op-erators ˆ n c2 ,ρυστ ( ~r , ~r ) = 12 n ˆ ψ † τ ( ~r ) ˆ ψ † σ ( ~r ) , ˆ ψ ρ ( ~r ) ˆ ψ υ ( ~r ) o . (154)Whereas the commutator form in Eq. (153) is well known in the literature (see, e.g.,Refs. [9, 25]), the anticommutator form in Eq. (154) is, to the best of my knowledge, originalto the present work. The commutator and the anticommutator in these definitions imposethe correct transformation under charge conjugation without having to use normal orderingwith respect to the free vacuum state. Indeed, using Eq. (135), it is straightforward to seethat ˆ n c1 ( ~r, ~r ′ ) correctly transforms as in Eq. (141)ˆ C ˆ n c1 ( ~r, ~r ′ ) ˆ C † = − C ˆ n cT1 ( ~r ′ , ~r ) C † , (155)and, similarly, ˆ n c2 ( ~r , ~r ) correctly transforms as in Eq. (146)ˆ C ˆ n c2 ( ~r , ~r ) ˆ C † = ( C ⊗ C )ˆ n cT2 ( ~r , ~r )( C ⊗ C ) † . (156)Using Wick’s theorem, we can express ˆ n c1 ( ~r, ~r ′ ) in terms of the one-particle density-matrixoperator ˆ˜ n ( ~r, ~r ′ ) defined with normal ordering with respect to the alternative no-particlevacuum state | ˜0 i in Eq. (27)ˆ n c1 ,ρσ ( ~r, ~r ′ ) = ˆ˜ n ,ρσ ( ~r, ~r ′ ) + ˜ n c , vp1 ,ρσ ( ~r, ~r ′ ) , (157)26ith the associated vacuum-polarization one-particle density matrix˜ n c , vp1 ,ρσ ( ~r, ~r ′ ) = h ˜0 | ˆ n c1 ,ρσ ( ~r, ~r ′ ) | ˜0 i = 12 (cid:16) h ˜0 | ˆ ψ † σ ( ~r ′ ) ˆ ψ ρ ( ~r ) | ˜0 i − h ˜0 | ˆ ψ ρ ( ~r ) ˆ ψ † σ ( ~r ′ ) | ˜0 i (cid:17) = 12 X p ∈ NS ˜ ψ ∗ p,σ ( ~r ′ ) ˜ ψ p,ρ ( ~r ) − X p ∈ PS ˜ ψ ∗ p,σ ( ~r ′ ) ˜ ψ p,ρ ( ~r ) ! . (158)Similarly, we can express ˆ n c2 ( ~r , ~r ) in terms of the pair density-matrix operator ˆ˜ n ( ~r , ~r )defined with normal ordering with respect to the vacuum state | ˜0 i in Eq. (28)ˆ n c2 ,ρυστ ( ~r , ~r ) = ˆ˜ n ,ρυστ ( ~r , ~r ) + ˜ n c , vp1 ,υτ ( ~r , ~r )ˆ˜ n ,ρσ ( ~r , ~r ) + ˜ n c , vp1 ,ρσ ( ~r , ~r )ˆ˜ n ,υτ ( ~r , ~r ) − ˜ n c , vp1 ,υσ ( ~r , ~r )ˆ˜ n ,ρτ ( ~r , ~r ) − ˜ n c , vp1 ,ρτ ( ~r , ~r )ˆ˜ n ,υσ ( ~r , ~r ) + ˜ n c , vp2 ,ρυστ ( ~r , ~r ) , (159)with the associated vacuum-polarization pair density matrix˜ n c , vp2 ,ρυστ ( ~r , ~r ) = h ˜0 | ˆ n c2 ,ρυστ ( ~r , ~r ) | ˜0 i = 12 (cid:16) h ˜0 | ˆ ψ † τ ( ~r ) ˆ ψ υ ( ~r ) | ˜0 ih ˜0 | ˆ ψ † σ ( ~r ) ˆ ψ ρ ( ~r ) | ˜0 i − h ˜0 | ˆ ψ † τ ( ~r ) ˆ ψ ρ ( ~r ) | ˜0 ih ˜0 | ˆ ψ † σ ( ~r ) ˆ ψ υ ( ~r ) | ˜0 i + h ˜0 | ˆ ψ υ ( ~r ) ˆ ψ † τ ( ~r ) | ˜0 ih ˜0 | ˆ ψ ρ ( ~r ) ˆ ψ † σ ( ~r ) | ˜0 i − h ˜0 | ˆ ψ ρ ( ~r ) ˆ ψ † τ ( ~r ) | ˜0 ih ˜0 | ˆ ψ υ ( ~r ) ˆ ψ † σ ( ~r ) | ˜0 i (cid:17) = 12 X p,q ∈ NS ˜ ψ ∗ p,τ ( ~r ) ˜ ψ p,υ ( ~r ) ˜ ψ ∗ q,σ ( ~r ) ˜ ψ q,ρ ( ~r ) − X p,q ∈ NS ˜ ψ ∗ p,τ ( ~r ) ˜ ψ p,ρ ( ~r ) ˜ ψ ∗ q,σ ( ~r ) ˜ ψ q,υ ( ~r )+ X p,q ∈ PS ˜ ψ ∗ p,τ ( ~r ) ˜ ψ p,υ ( ~r ) ˜ ψ ∗ q,σ ( ~r ) ˜ ψ q,ρ ( ~r ) − X p,q ∈ PS ˜ ψ ∗ p,τ ( ~r ) ˜ ψ p,ρ ( ~r ) ˜ ψ ∗ q,σ ( ~r ) ˜ ψ q,υ ( ~r ) ! . (160)Similarly to what was done in Eq. (33), the electron-positron Hamiltonian in Eq. (149)can then be rewritten as ˆ H c = ˆ˜ T D + ˆ˜ W + ˆ˜ V + ˆ˜ V vp + ˜ E c0 , (161)where ˆ˜ T D , ˆ˜ W , and ˆ˜ V have been already defined in Eqs. (34)-(36), and ˆ˜ V vp and ˜ E c0 are thevacuum-polarization potential and no-particle vacuum energy associated with this Hamilto-nian. Similarly to Eq. (38), the vacuum-polarization potential can be written asˆ˜ V vp = ˆ˜ V vpd + ˆ˜ V vpx , (162)with a direct contribution ˆ˜ V vpd = Z Tr[˜ v c , vpd ( ~r )ˆ˜ n ( ~r )]d ~r , (163)where ˜ v c , vpd ,σρ ( ~r ) = P τυ R w στρυ ( ~r , ~r )˜ n c , vp υτ ( ~r )d ~r and ˜ n c , vp υτ ( ~r ) = ˜ n c , vp1 ,υτ ( ~r , ~r ), and an ex-change contribution ˆ˜ V vpx = Z Z
Tr[˜ v c , vpx ( ~r , ~r )ˆ˜ n ( ~r , ~r )]d ~r d ~r , (164)27here ˜ v c , vpx ,τρ ( ~r , ~r ) = − P συ w στρυ ( ~r , ~r )˜ n c , vp1 ,υσ ( ~r , ~r ). Finally, the associated no-particle vac-uum energy can be written as˜ E c0 = h ˜0 | ˆ H c | ˜0 i = Z Tr[ D ( ~r )˜ n c , vp1 ( ~r, ~r ′ )] ~r ′ = ~r d ~r + Z v ( ~r )˜ n c , vp ( ~r ) d ~r +12 Z Z
Tr[ w ( ~r , ~r )˜ n c , vp2 ( ~r , ~r )]d ~r d ~r . (165)As suggested by the fact that we used the same notation, it turns out that both the directand exchange contributions to the vacuum-polarization potential in Eq. (162) are identical tothe ones introduced in Eq. (38). This can be shown as follows. First, using the fact that theorbital rotation in Eq. (24) leaves invariant the following sum over orbitals X p ∈ PS ˜ ψ ∗ p,σ ( ~r ′ ) ˜ ψ p,ρ ( ~r ) + X p ∈ NS ˜ ψ ∗ p,σ ( ~r ′ ) ˜ ψ p,ρ ( ~r ) = X p ∈ PS ψ ∗ p,σ ( ~r ′ ) ψ p,ρ ( ~r ) + X p ∈ NS ψ ∗ p,σ ( ~r ′ ) ψ p,ρ ( ~r ) , (166)the vacuum-polarization one-particle density matrix in Eq. (158) can be expressed in termsof the vacuum-polarization one-particle density matrix introduced in Eq. (31) as˜ n c , vp1 ,ρσ ( ~r, ~r ′ ) = ˜ n vp1 ,ρσ ( ~r, ~r ′ ) + n c , vp1 ,ρσ ( ~r, ~r ′ ) , (167)where we have introduced n c , vp1 ,ρσ ( ~r, ~r ′ ) = 12 X p ∈ NS ψ ∗ p,σ ( ~r ′ ) ψ p,ρ ( ~r ) − X p ∈ PS ψ ∗ p,σ ( ~r ′ ) ψ p,ρ ( ~r ) ! , (168)which is the vacuum-polarization one-particle density matrix associated with the operator inEq. (153) but over the free vacuum state, i.e. n c , vp1 ( ~r, ~r ′ ) = h | ˆ n c1 ( ~r, ~r ′ ) | i . Using charge-conjugation symmetry on the set of eigenfunctions { ψ p ( ~r ) } of the free Dirac equation, wehave n c , vp1 ,ρσ ( ~r, ~r ′ ) = 12 X p ∈ NS ψ ∗ p,σ ( ~r ′ ) ψ p,ρ ( ~r ) − X p ∈ NS X ρ ′ σ ′ C ρρ ′ ψ p,σ ′ ( ~r ′ ) ψ ∗ p,ρ ′ ( ~r ) C † σ ′ σ ! , (169)or, in matrix form, n c , vp1 ( ~r, ~r ′ ) = n c , vp1 , − ( ~r, ~r ′ ) − Cn c , vp T1 , − ( ~r ′ , ~r ) C † , (170)where n c , vp1 , − ,ρσ ( ~r, ~r ′ ) = (1 / P p ∈ NS ψ ∗ p,σ ( ~r ′ ) ψ p,ρ ( ~r ). We then immediately see that the densityassociated with n c , vp1 ( ~r, ~r ′ ) vanishes n c , vp ( ~r ) = Tr[ n c , vp1 ( ~r, ~r )] = 0 , (171)i.e., the free electron vacuum density and the free positron vacuum density are identical, asalready known [25]. Now, using C † α C = α T , it can be checked that X τυ w στρυ ( ~r , ~r ) n c , vp υτ ( ~r ) = 0 , (172)28 EFERENCES and therefore the contribution of n c , vp1 ( ~r, ~r ′ ) to the direct vacuum-polarization potential inEq. (163) vanishes. Finally, even tough ˆ˜ n ( ~r , ~r ) does not satisfy charge-conjugation symme-try in the sense of Eq. (141), it does satisfy the following relationˆ˜ n ( ~r , ~r ) = C ˆ˜ n T1 ( ~r , ~r ) C † , (173)and, together with the symmetry properties of w στρυ ( ~r , ~r ), it can be used to check that Z Z X τρσυ w στρυ ( ~r , ~r ) n c , vp1 ,υσ ( ~r , ~r )ˆ˜ n ,ρτ ( ~r , ~r )d ~r d ~r = 0 , (174)and therefore the contribution of n c , vp1 ( ~r, ~r ′ ) to the exchange vacuum-polarization potential inEq. (164) vanishes as well. This establishes the equivalence between the vacuum-polarizationpotential in Eq. (38) and in Eq. (162).The no-particle vacuum energies ˜ E in Eq. (41) and ˜ E c0 in Eq. (165) are different how-ever. In particular, in comparison to the situation for ˜ E discussed after Eq. (41), the UVdivergences are more serious for ˜ E c0 since the sums in Eq. (165) tend to give cumulative neg-ative energies rather than cancelling energies. For this reason, we prefer to work with theelectron-positron Hamiltonian ˆ H in Eq. (7). The form of the electron-positron Hamiltonianˆ H c in Eq. (149) remains useful however to establish links with the literature. In particular,by writing explicitly ˆ H c in Eq. (161) in terms of elementary creation and annihilation op-erators corresponding to the orbital basis { ˜ ψ p ( ~r ) } , and after removing the vacuum energy˜ E c0 , it can be checked that one exactly recovers the effective QED (eQED) Hamiltonian ofRefs. [25, 40–44]. So we have ˆ H eQED = ˆ H c − ˜ E c0 = ˆ H − ˜ E , (175)where ˆ H eQED is the Hamiltonian in Eq. (46) of Ref. [25]. Whereas this eQED Hamiltonianwas obtained in Ref. [25] via a “charge-conjugated contraction” of the fermion operators, hereit is obtained via the commutator and anticommutator in Eqs. (153) and (154), or equivalentlyvia the normal ordering with respect to the free vacuum state in Eqs. (11) and (12). References [1] P. Hohenberg and W. Kohn,
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