Short-range correlation energy of the relativistic homogeneous electron gas
aa r X i v : . [ phy s i c s . c h e m - ph ] F e b Short-range correlation energy of the relativistic homogeneous electron gas
Julien Paquier ∗ and Julien Toulouse , † Laboratoire de Chimie Th´eorique (LCT), Sorbonne Universit´e and CNRS, F-75005 Paris, France and Institut Universitaire de France, F-75005 Paris, France (Dated: February 15, 2021)We construct the complementary short-range correlation relativistic local-density-approximation functionalto be used in relativistic range-separated density-functional theory based on a Dirac-Coulomb Hamiltonian inthe no-pair approximation. For this, we perform relativistic random-phase-approximation calculations of thecorrelation energy of the relativistic homogeneous electron gas with a modified electron-electron interaction,we study the high-density behavior, and fit the results to a parametrized expression. The obtained functionalshould eventually be useful for electronic-structure calculations of strongly correlated systems containing heavyelements.
I. INTRODUCTION
Range-separated density-functional theory (RS-DFT) (see,e.g., Refs. 1 and 2) is an extension of Kohn-Sham density-functional theory (KS-DFT) [3] which allows one to rig-orously combines a multideterminant wave-function methodaccounting for the long-range part of the electron-electroninteraction with a complementary short-range density func-tional. RS-DFT can improve over usual Kohn-Sham density-functional approximations for the electronic-structure calcu-lations of strongly correlated systems (see, e.g., Refs. 4 and5) and / or systems involving weak intermolecular interactions(see, e.g., Refs. 6 and 7), while still enjoying a fast basis con-vergence [8].With the aim of describing compounds with heavy ele-ments which involve both strong correlation and relativistice ff ects, RS-DFT has been extended to a four-component rel-ativistic framework [9–12]. In this relativistic RS-DFT, theno-pair [13, 14] ground-state electronic energy of the Dirac-Coulomb Hamiltonian is written as [12] E = h Ψ + | ˆ T D + ˆ V ne + ˆ W lr ,µ ee | Ψ + i + ¯ E sr ,µ Hxc [ n Ψ + ] , (1)where ˆ T D is the kinetic + rest mass Dirac operator, ˆ V ne isthe nuclei-electron interaction operator, ˆ W lr ,µ ee is the electron-electron interaction operator associated with the long-rangepair potential w lr ,µ ee ( r ) = erf( µ r ) / r , and ¯ E sr ,µ Hxc [ n Ψ + ]is the corresponding complementary short-range relativisticHartree-exchange-correlation functional evaluated at the den-sity of Ψ + . The no-pair multideterminant wave function Ψ + isconstructed from positive-energy states only and can in princi-ple be obtained using a minmax principle [11, 12, 15–20]. Therange-separation parameter µ ∈ [0 , + ∞ [ controls the range ofthe separation. For µ =
0, the long-range interaction vanishesand no-pair relativistic KS-DFT (see, e.g., Refs. 21 and 22)is recovered. For µ → ∞ , the long-range interaction reducesto the full-range Coulomb interaction and no-pair relativisticwave-function theory (see, e.g., Refs. 19 and 23) is recovered.While any existing wave-function approximation can di-rectly be used for Ψ + , new approximations need to be devel-oped for the short-range relativistic functional ¯ E sr ,µ Hxc [ n ]. As ∗ [email protected] † [email protected] usual, this functional can be decomposed into Hartree, ex-change, and correlation contributions¯ E sr ,µ Hxc [ n ] = E sr ,µ H [ n ] + E sr ,µ x [ n ] + ¯ E sr ,µ c [ n ] . (2)The short-range Hartree functional is E sr ,µ H [ n ] = " n ( r ) n ( r ) w sr ,µ ee ( r )d r d r , (3)where w sr ,µ ee ( r ) = / r − w lr ,µ ee ( r ) is the short-range pairpotential. The short-range exchange functional is E sr ,µ x [ n ] = h Φ + [ n ] | ˆ W sr ,µ ee | Φ + [ n ] i − E sr ,µ H [ n ] , (4)where Φ + [ n ] is the relativistic Kohn-Sham single-determinantwave function and ˆ W sr ,µ ee is the electron-electron interactionoperator associated with w sr ,µ ee ( r ). In Refs. 9 and 10, therelativistic short-range exchange and correlation functionals E sr ,µ x [ n ] and ¯ E sr ,µ c [ n ] were approximated by non-relativisticshort-range exchange and correlation functionals, which isa reasonable first approximation since for valence propertiesrelativistic e ff ects are usually dominated by the kinematic con-tribution and the induced change in the density (see, e.g.,Ref. 24). To go beyond this non-relativistic approximationand put relativistic RS-DFT on a firmer ground, we have con-structed for the short-range exchange functional E sr ,µ x [ n ] therelativistic local-density approximation (RLDA) in Ref. 11and approximations going beyond the RLDA in Ref. 12. Inthe present work, we turn to the short-range correlation func-tional ¯ E sr ,µ c [ n ] and we develop the RLDA for it.The complementary short-range correlation RLDA func-tional is defined as¯ E sr,RLDA ,µ c [ n ] = Z d r n ( r ) ¯ ǫ sr,RHEG ,µ c ( n ( r )) , (5)with ¯ ǫ sr,RHEG ,µ c ( n ) = ǫ RHEGc ( n ) − ǫ lr,RHEG ,µ c ( n ) , (6)where ǫ RHEGc ( n ) and ǫ lr,RHEG ,µ c ( n ) are the correlation ener-gies per particle of the relativistic homogeneous electron gas(RHEG) with full-range and long-range electron-electron in-teractions, respectively. We express each of these correlationenergies per particle as the correlation energy per particle ofthe corresponding non-relativistic homogeneous electron gas(HEG) multiplied by a relativistic correlation factor ǫ RHEGc ( n ) = ǫ HEGc ( n ) φ c ( n ) , (7) ǫ lr,RHEG ,µ c ( n ) = ǫ lr,HEG ,µ c ( n ) φ ˜ µ c ( n ) , (8)where we have introduced the scaled range-separation param-eter ˜ µ = µ/ k F (where k F = (3 π n ) / is the Fermi wave vec-tor), a natural adimensional parameter measuring the rangeof the interaction relative to the density. We must have φ lr , ˜ µ →∞ c ( n ) = φ c ( n ) since the long-range interaction reducesto the full-range one in this limit. Equations (7) and (8) allowone to use already existing parametrizations for ǫ HEGc ( n ) and ǫ lr,HEG ,µ c ( n ) [25, 26].The correlation energy per particle of the RHEG ǫ RHEGc ( n )was first estimated at the random-phase approximation (RPA)level by Ramana and Rajagopal [27] (see also Refs. 21, 28–31), and the corresponding relativistic correlation factor φ c ( n )was parametrized by Schmid et al. [32]. In the same spirit,we estimate in this work the relativistic long-range correlationfactor φ lr , ˜ µ c ( n ) at the RPA level, i.e. φ lr , ˜ µ c ( n ) ≈ φ lr,RRPA , ˜ µ c ( n ) = ǫ lr,RRPA , ˜ µ c ( n ) ǫ lr,RPA , ˜ µ c ( n ) , (9)where ǫ lr,RRPA , ˜ µ c ( n ) is the long-range relativistic random-phase-approximation (RRPA) correlation energy per particle of theRHEG and ǫ lr,RPA , ˜ µ c ( n ) is its non-relativistic analog. The useof the RPA appears consistent considering that relativistic ef-fects are most important in the high-density regime, for which the RPA provides a good approximation to the correlation en-ergy. Contrary to the RRPA calculations of Ramana and Ra-jagopal [27] which included the transverse contribution fromthe full quantum-electrodynamics (QED) photon propagatorand were performed within the no-sea approximation (i.e., in-cluding a renormalization contribution from negative-energystates) [31], here our RRPA calculations are limited to thelongitudinal component the interaction in the Coulomb gaugeand within the no-pair approximation. We do so for consis-tency since in the relativistic RS-DFT of Eq. (1), the long-range wave-function part is treated at the same level. The nu-merically calculated relativistic long-range correlation factor φ lr,RRPA , ˜ µ c ( n ) is then fitted to a parametrized expression impos-ing the correct high-density limit.Hartree atomic units (a.u.) are used throughout the paper. II. LONG-RANGE CORRELATION ENERGY FROMRANDOM-PHASE APPROXIMATIONA. Relativistic random-phase approximation
As already indicated, we want to determine the long-rangeRRPA correlation energy per particle of the RHEG withinthe no-pair approximation and for the longitudinal compo-nent the electron-electron interaction in the Coulomb gauge.With these approximations, the expression of ǫ lr,RRPA , ˜ µ c ( n ) isthe same as its non-relativistic counterpart (see, e.g., Refs. 33and 34) ǫ lr,RRPA , ˜ µ c ( n ) = − π n Z d q (2 π ) w lr , ˜ µ ( q ) Z ∞ d u Z d λ (cid:2) χ ( q , iu ) (cid:3) f lr , ˜ µ,λ H ( q )1 − χ ( q , iu ) f lr , ˜ µ,λ H ( q ) , (10)where λ is a coupling constant. In this expression, χ ( q , iu ) is the relativistic longitudinal non-interacting linear-response functionof the RHEG within the no-pair approximation at wave vector q = | q | and imaginary frequency iu (see Refs. 27, 31, and 35 andAppendix A) χ ( q , iu ) = − k F Z d ˜ k (2 π ) θ (1 − ˜ k ) √ ˜ k + ˜ c + q | ˜ k + ˜ q | + ˜ c ! − ˜ q √ ˜ k + ˜ c q | ˜ k + ˜ q | + ˜ c ˜ c q | ˜ k + ˜ q | + ˜ c − √ ˜ k + ˜ c ! ˜ u + ˜ c q | ˜ k + ˜ q | + ˜ c − √ ˜ k + ˜ c ! , (11)where we have introduced the adimensional variables ˜ k = k / k F , ˜ q = q / k F , ˜ u = u / k , and ˜ c = c / k F where c = . c is a natural adimensional parameter measuring the impor-tance of relativistic e ff ects (relativistic e ff ects are negligiblefor ˜ c ≫ c decreases). In Eq. (10), f lr , ˜ µ,λ H ( q ) is the long-range Hartree kernel at the coupling constant λ givenby the Fourier transform of the long-range interaction f lr , ˜ µ,λ H ( q ) = λ w lr , ˜ µ ( q ) = λ π ˜ q k exp " − ˜ q µ . (12)Performing the integrals in Eq. (10) over the angular variablesof q and over the coupling constant λ gives -0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 0 200 400 600 800 1000 1200a) ε clr,RPA, µ ε c l r , R P A , µ ( a . u . ) k F (a.u.) µ /k F ➞ ∞ µ /k F = 20 µ /k F = 2 µ /k F = 1 µ /k F = 0.5 µ /k F = 0.3 µ /k F = 0.2 µ /k F = 0.1 µ /k F = 0.05 µ /k F = 0.025 µ /k F = 0.01 µ /k F = 0.005 -1.6-1.4-1.2-1-0.8-0.6-0.4-0.2 0 0 200 400 600 800 1000 1200b) ε clr,RRPA, µ ε c l r , RR P A , µ ( a . u . ) k F (a.u.) µ /k F ➞ ∞ µ /k F = 20 µ /k F = 2 µ /k F = 1 µ /k F = 0.5 µ /k F = 0.3 µ /k F = 0.2 µ /k F = 0.1 µ /k F = 0.05 µ /k F = 0.025 µ /k F = 0.01 µ /k F = 0.005 FIG. 1. Non-relativistic (a) and relativistic (b) long-range RPA correlation energies per particle of the HEG. ǫ lr,RRPA , ˜ µ c ( n ) = − π Z ∞ d ˜ q Z ∞ d˜ u π exp " − ˜ q µ χ ( ˜ qk F , i ˜ uk ) + ˜ q k ln − π ˜ q k exp " − ˜ q µ χ ( ˜ qk F , i ˜ uk ) ! . (13)As in the non-relativistic case, the integral over ˜ q and ˜ u are performed numerically. However, contrary to the non-relativistic case, we also do numerically the integral over ˜ k inthe linear-response function in Eq. (11). In total, this gives afour-dimensional numerical integration that we calculate us-ing the software Wolfram Mathematica [36] with six digitsof accuracy. In the non-relativistic limit, i.e. ˜ c → ∞ , theintegral defining the linear-response function in Eq. (11) caneasily be done analytically and Eq. (11) reduces to the well-known non-relativistic Lindhard function [37]. However, forconsistency, we also use a four-dimensional numerical inte-gration with the same precision for ˜ c → ∞ to obtain thenon-relativistic RPA long-range correlation energy per parti-cle ǫ lr,RPA , ˜ µ c ( n ) = lim ˜ c →∞ ǫ lr,RRPA , ˜ µ c ( n ). We use 41 values of theFermi wave vector k F ranging from 0.005 to 1200 a.u. (corre-sponding to a range of Wigner-Seitz radius r s = [3 / (4 π n )] / from 384 to 0.0016 a.u.). The highest sampled density corre-sponds to more than twice the maximal core electronic den-sity of uranium, thus encompassing all chemically relevantelectronic densities. For the scaled range-separation param-eter ˜ µ = µ/ k F , we consider 25 di ff erent values ranging from0.005 to 20 a.u., in addition to the ˜ µ → ∞ limit giving the full-range RRPA and RPA correlation energies ǫ lr,RRPA , ˜ µ →∞ c ( n ) = ǫ RRPAc ( n ) and ǫ lr,RPA , ˜ µ →∞ c ( n ) = ǫ RPAc ( n ). Note that the speed oflight c is fixed to its physical value in our calculations, i.e. wedo not try to obtain the dependence on c of the RRPA corre-lation energy. For more details on the numerical calculations,see Ref. 38. B. Long-range correlation energy
We show in Fig. 1 the non-relativistic and relativistic long-range RPA correlation energies per particle as a function of k F for several values of ˜ µ . As regards the non-relativistic results, for ˜ µ → ∞ , we correctly reproduce the high-density expan-sion of the full-range RPA correlation energy per particle (see,e.g., Ref. 39) that we expressed here in terms of k F ǫ RPAc ( n ) = − − ln 2 π ln k F − . + O ln k F k F ! . (14)This is the usual weak-correlation limit where the correlationenergy per particle is negligible compared to the exchange en-ergy per particle which is linear in k F . We observe a similarlogarithmic behavior also for the long-range RPA correlationenergy per particle on our chosen range of k F for values of˜ µ larger than 0 . µ &
20 a.u., the long-range RPAcorrelation energy is nearly identical to the full-range RPAcorrelation energy.Turning now to the relativistic results, we observe a verydi ff erent behavior. Namely, for ˜ µ → ∞ , the full-range RRPAcorrelation energy per particle is linear with respect to k F ǫ RRPAc ( n ) ∼ k F →∞ − . k F , (15)which is in agreement with other RRPA calculations reportedin the literature [31, 32]. This is the ultra-relativistic limit,˜ c →
0, which is akin to a strong-correlation limit where boththe exchange and correlation energies per particle are linearwith respect to k F . A similar linear behavior is also observedfor the case of the long-range interaction. Again, for ˜ µ & C. Relativistic long-range correlation factor
We show in Fig. 2 the relativistic long-range correlation fac-tor φ lr,RRPA , ˜ µ c as a function of k F and ˜ µ . We observe that, for allvalues of ˜ µ and all relevant values of k F , the relativistic factor φ clr,RRPA, µ vs. k F φ c l r , RR P A , µ k F (a.u.) µ /k F ➞ ∞µ /k F = 20 µ /k F = 2 µ /k F = 1 µ /k F = 0.5 µ /k F = 0.3 µ /k F = 0.2 µ /k F = 0.1 µ /k F = 0.05 µ /k F = 0.025 µ /k F = 0.01 µ /k F = 0.005 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 0 1 2 3 4 5b) φ clr,RRPA, µ vs. µ /k F φ c l r , RR P A , µ µ /k F k F = 137.036k F = 100k F = 80k F = 60k F = 40 FIG. 2. Relativistic long-range correlation factor φ lr,RRPA , ˜ µ c ( n ) as a function of k F (a) and ˜ µ = µ/ k F (b). is greater than 1, i.e. relativistic e ff ects increase the magnitudeof the correlation energy. Moreover, φ lr,RRPA , ˜ µ c is an increasingfunction of k F , i.e. the relative relativistic e ff ects increase aswe increase the density.In Fig. 2 (a), it appears at first sight that φ lr,RRPA , ˜ µ is a mono-tonic decreasing function of ˜ µ , but the dependence on ˜ µ is infact more complicated and is plotted in Fig. 2 (b) for severalvalues of k F . For clarity, we show only values of k F lowerthan 200 a.u., but the behavior is similar for the whole rangeof Fermi wave vectors that we have considered. It appearsthat, and for any value of k F , φ lr,RRPA , ˜ µ starts as an increasingfunction of ˜ µ until it reaches a maximum for a value ˜ µ max ( k F ),after which it becomes a decreasing function of ˜ µ convergingto its full-range interaction limit. The value of ˜ µ max ( k F ) is it-self an increasing function of k F , going from ˜ µ max (10) ≈ . µ max (1200) ≈ . φ lr,RRPA , ˜ µ increasesrapidly before ˜ µ max ( k F ), it decreases only slightly afterward.This behavior explains why in Fig. 2 (a) we observe that allcurves for ˜ µ higher than 1 appear to be superposed since thereis little variation of φ lr,RRPA , ˜ µ with respect to ˜ µ for these val-ues, and why we observe and a monotonic decreasing behav-ior with respect to ˜ µ only for lower values of ˜ µ . It appears thatfor ˜ µ → k F , i.e. the relativistic e ff ects disappear when onlythe very long-range part of the electron-electron interactionremains. In this limit, however, the long-range correlation en-ergy itself vanishes. III. PARAMETRIZATION
We now construct parametrizations of our numerical data.As building blocks for a parametrization of φ lr,RRPA , ˜ µ , we firstparametrize the high-density limits of the non-relativistic andrelativistic long-range correlation energies. A. High-density limit of the non-relativistic long-rangecorrelation energy
The parametrization of the high-density limit of thenon-relativistic correlation energy is done by combining aparametrization for large values of ˜ µ and a parametrizationfor small values of ˜ µ .For su ffi ciently large values of ˜ µ , the non-relativistic long-range RPA correlation energy per particle in the high-densitylimit follows a logarithmic behavior similar to the one of thenon-relativistic full-range RPA correlation energy per particle[see Eq. (14)], and we found that the dependence on ˜ µ can beapproximated by ǫ lr,RPA , ˜ µ, hd c ( n ) = − − ln 2 π ln k F + − . + + a ˜ µ a + a ˜ µ + a ˜ µ + a ˜ µ ! , (16)giving our first high-density (hd ) parametrization. The pa-rameters a = . a = . a = . a = . a = . ǫ lr,RPA , ˜ µ c ( n ) + [(1 − ln 2) /π ] ln k F at k F = µ ≥ . ffi ciently small values of ˜ µ , the high-density limitof the non-relativistic RPA long-range correlation energy perparticle can be approximated by the expression of Paziani etal. [26] ǫ lr,RPA , ˜ µ, hd c ( n ) = − π ln " + b x + b x + b x + b x + b x , (17)with x = µ √ r s = (3 √ π/ / ˜ µ √ k F and the parameters b = . b = . b = . b = . ) parametrization.We now combine these two high-density parametrizationsin a single parametrization by interpolating using the switch-ing function f ( ˜ µ ) = erf(3 ˜ µ ) ǫ lr,RPA , ˜ µ, hdc ( n ) = f ( ˜ µ ) ǫ lr,RPA , ˜ µ, hd c ( n ) + (1 − f ( ˜ µ )) ǫ lr,RPA , ˜ µ, hd c ( n ) . (18) -0.2 0 0.2 0.4 0.6 0.8 1 0 200 400 600 800 1000 1200a) ε clr,RPA, µ ,hd ( ε c l r , R P A , µ - ε c l r , R P A , µ , hd ) / ε c l r , R P A , µ ( % ) k F (a.u.) µ /k F ➞ ∞µ /k F = 20 µ /k F = 2 µ /k F = 1 µ /k F = 0.5 µ /k F = 0.3 µ /k F = 0.2 µ /k F = 0.1 µ /k F = 0.05 µ /k F = 0.025 0 1 2 3 4 5 6 0 200 400 600 800 1000 1200b) ε clr,RRPA, µ ,hd ( ε c l r , RR P A , µ - ε c l r , RR P A , µ , hd ) / ε c l r , RR P A , µ ( % ) k F (a.u.) µ /k F ➞ ∞µ /k F = 20 µ /k F = 2 µ /k F = 1 µ /k F = 0.5 µ /k F = 0.3 µ /k F = 0.2 µ /k F = 0.1 µ /k F = 0.05 µ /k F = 0.025 µ /k F = 0.01 µ /k F = 0.005 FIG. 3. Relative error of the high-density approximations for the non-relativistic (a) and relativistic (b) long-range RPA correlation energiesper particle [Eqs. (18) and (19)]. -0.8-0.6-0.4-0.2 0 0 100 200 300 400 500 µ =0 ε c ( a . u . ) k F (a.u.)Non-relativisticRelativistic FIG. 4. Non-relativistic and relativistic complementary short-rangecorrelation energies per particle of the HEG.
The use of the fourth power of the error function allows fora steep enough switching and using 3 ˜ µ as the argument putsthe transition between the two parts around ˜ µ = . µ . In particular, for ˜ µ → ∞ , it correctlyreduces to the full-range behavior in Eq. (14).The relative error of this high-density approximation ǫ lr,RPA , ˜ µ, hdc ( n ) is plotted in Fig. 3 (a). For k F &
400 a.u. and˜ µ ≥ .
025 a.u., the high-density approximation gives a rel-ative error of less than 0 . µ (notshown), the maximal relative error increases up to around 3%but the error is made on very small values of the correlationenergy. B. High-density limit of the relativistic long-range correlationenergy
In the high-density limit, the relativistic long-range RRPAcorrelation energy is linear in k F for all values of ˜ µ and it iswell approximated by ǫ lr,RRPA , ˜ µ, hdc ( n ) = − . × − + c ˜ µ + c ˜ µ + c ˜ µ + c ˜ µ + c ˜ µ + c ˜ µ + c ˜ µ + c ˜ µ + c ˜ µ ! / ˜ c , (19)where the parameters c = . c = . c = . c = − . c = . c = . c = . c = . c = .
956 have beenobtained by fitting at k F = µ considered in this work. For ˜ µ → ∞ , Eq. (19) correctlyreduces to the full-range behavior in Eq. (15).The relative error of this high-density approximation ǫ lr,RRPA , ˜ µ, hdc ( n ) is plotted in Fig. 3 (b). For ˜ µ → ∞ , the rela-tive error gets below 1% for k F & µ decreases,the high-density regime is reached for smaller values of k F ,e.g. for ˜ µ = .
005 a.u. we obtain 1% accuracy for k F & C. Parametrization of the relativistic long-range correlationfactor
Having found parametrizations for the high-density limit ofthe non-relativistic and relativistic long-range RPA correlationenergies per particle, we now use these expressions to build aPad´e-like expression for the relativistic long-range correlationfactor φ lr,RRPA , ˜ µ c . We found that it is accurately represented by TABLE I. Parameters for the relativistic long-range correlation factor φ lr,RRPA , ˜ µ c [Eq. (20)]. i a , i a , i a , i b , i b , i b , i × − × − × − - - -2 7.04721 × − × − × − × − -2.40333 × − × − × − - 5.62594 × − × − × − × − × − - - -5 - 3.07852 5.32685 × − - 7.56679 × − × − φ lr,RRPA , ˜ µ c ( n ) = + a , + a , ˜ µ a , + ˜ µ / ˜ c + a , + a , ˜ µ + a , ˜ µ a , + a , ˜ µ + ˜ µ / ˜ c + a , + a , ˜ µ + a , ˜ µ a , + a , ˜ µ + ˜ µ / ˜ c − ǫ lr,RRPA , ˜ µ, hdc ( n ) / ˜ c + a , + b , ˜ µ a , + ˜ µ / ˜ c + a , + b , ˜ µ + b , ˜ µ a , + b , ˜ µ + ˜ µ / ˜ c + a , + b , ˜ µ + b , ˜ µ a , + b , ˜ µ + ˜ µ / ˜ c − ǫ lr,RPA , ˜ µ, hdc ( n ) / ˜ c . (20)The choice of using the opposite of the high-density correla-tion energies as coe ffi cients of 1 / ˜ c terms ensures that thesecoe ffi cients are positive and reduces the risk of introducingpoles within the parametrization. The parameters are givenin Table I. They have been found by fitting to the numericalvalues of ǫ lr,RRPA , ˜ µ c /ǫ lr,RPA , ˜ µ c using all values of k F and ˜ µ con-sidered in this work. The maximal absolute error is less than0 .
4% for the smallest values of ˜ µ considered. In the specialcase ˜ µ → ∞ , we obtain the full-range relativistic correlationfactor φ lr,RRPA , ˜ µ →∞ c ( n ) = φ RRPAc ( n ), with a maximal absolute er-ror less than 0 . c , and not for an arbitrary value of c . For more details on thefit, see Ref. 38. D. Complementary short-range correlation energy perparticle
From Eqs. (6)-(8), we finally obtain our approximation forthe complementary short-range correlation energy per particleof the RHEG¯ ǫ sr,RHEG ,µ c ( n ) ≈ ǫ HEGc ( n ) φ RRPAc ( n ) − ǫ lr,HEG ,µ c ( n ) φ lr , ˜ µ, RRPAc ( n ) , (21)in which we use the Perdew-Wang-92 parametrization for ǫ HEGc ( n ) [25] and the parametrization of Paziani et al. [26] for ǫ lr,HEG ,µ c ( n ).In the limit µ =
0, this short-range correlation energy perparticle reduces to the full-range correlation energy per parti-cle, i.e. ¯ ǫ sr,RHEG ,µ = ( n ) = ǫ RHEGc ( n ). In Fig. 4, we compare ourobtained ǫ RHEGc ( n ) with its non-relativistic analog ǫ HEGc ( n ). Asalready indicated, relativistic e ff ects increase the magnitude ofthe correlation energy for large densities and turn the logarith-mic dependence with respect to k F into a linear dependence.We plot in Fig. 5 the relativistic and non-relativistic com-plementary short-range correlation energies per particle as afunction of ˜ µ for several values of k F . For k F =
100 a.u.,we already see the impact of the relativistic e ff ects for small values of ˜ µ . For k F =
550 a.u., the relativistic e ff ects are im-portant for all relevant values of ˜ µ . Note that the wiggling be-havior with respect to ˜ µ observed on the graphs for k F = ǫ lr,HEG ,µ c ( n ). This is not so surprising sincesuch high densities were not considered in the construction ofthe parametrization of Ref. 26. This calls perhaps for a re-finement of this parametrization. For high enough densities,however, the possible refinement of ǫ lr,HEG ,µ c ( n ) is secondaryin comparison to the relativistic e ff ects. IV. CONCLUSION
From RRPA calculations on the RHEG, we have con-structed the complementary short-range correlation RLDAfunctional to be used in relativistic RS-DFT based on a Dirac-Coulomb Hamiltonian in the no-pair approximation. Thisshort-range correlation RLDA functional could be tested onatomic and molecular systems, and will most likely servea starting point for building more sophisticated relativisticshort-range correlation functionals, e.g. depending on thedensity gradient or on the on-top pair density as already donefor the short-range exchange functional [12]. We believe thatthe present work helps to establish relativistic RS-DFT ona firm ground and will eventually be useful for electronic-structure calculations of strongly correlated systems contain-ing heavy elements.
Appendix A: Non-interacting linear-response function of theRHEG in the no-pair approximation
The non-interacting one-electron Green function of theRHEG in the no-pair approximation at wave vector k = | k | -0.08-0.07-0.06-0.05-0.04-0.03-0.02-0.01 0 0 0.5 1 1.5 2 2.5 3a) k F = 1 a.u. ε cs r , µ ( a . u . ) µ /k F Non-relativisticRelativistic-0.25-0.2-0.15-0.1-0.05 0 0 0.5 1 1.5 2 2.5 3b) k F = 100 a.u. ε cs r , µ ( a . u . ) µ /k F Non-relativisticRelativistic-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 0 0.5 1 1.5 2 2.5c) k F = 550 a.u. ε cs r , µ ( a . u . ) µ /k F Non-relativisticRelativistic
FIG. 5. Non-relativistic and relativistic complementary short-rangecorrelation energy per particle of the HEG for k F = k F =
100 a.u. (b), and k F =
550 a.u. (c). and frequency ω is G ( k , ω ) = X s ∈{↑ , ↓} ψ k , s ψ † k , s " θ ( k − k F ) ω − ε k + i + + θ ( k F − k ) ω − ε k − i + , (A1)where ψ k , s are the four-component spinors associated with thepositive-energy solutions of the non-interacting Dirac equa-tion ψ k , s = s ε k + c ε k ϕ sc ( σ · k ) ε k + c ϕ s , (A2)where σ is the vector composed of the three Pauli matrices, ε k = √ k c + c are the one-electron energies, and ϕ s are thetwo-component spinors ϕ ↑ = and ϕ ↓ = . (A3)Note that the Green function G ( k , ω ) is a 4 × q = | q | and frequency q is χ ( q , q ) = Z d k (2 π ) Z + ∞−∞ d ω π i Tr (cid:2) G ( k , ω ) G ( k + q , ω + q ) (cid:3) , (A4)which, after calculating the trace of the products of spinors(see, e.g., Ref. 11) and calculating the integral over ω by con-tour integration, gives χ ( q , q ) = Z d k (2 π ) + k · ( k + q ) c + c ε k ε | k + q | ! θ ( | k + q | − k F ) θ ( k F − k ) " − q + ε | k + q | − ε k − i + + q + ε k − ε | k + q | + i + . (A5)Evaluating the linear-response function at imaginary frequency q = iu , and after simplifying, we find χ ( q , iu ) = − Z d k (2 π ) θ ( k F − k ) + k · ( k + q ) c + c ε k ε | k + q | ! ε | k + q | − ε k ) u + ( ε | k + q | − ε k ) , (A6)which can also be written as χ ( q , iu ) = − Z d k (2 π ) θ ( k F − k ) h(cid:0) ε k + ε | k + q | (cid:1) − q c i (cid:0) ε | k + q | − ε k (cid:1) ε k ε | k + q | h u + ( ε | k + q | − ε k ) i . (A7)This expression is equal, up to a trivial sign convention, to thefirst term of the longitudinal non-interacting linear-responsefunction given by Ramana and Rajagopal [27] [Eq. (6) ofRef. 27]. The expression determined in their work is notwithin the no-pair approximation but within the no-sea ap-proximation, and thus their expression includes a renormaliza-tion term coming from the negative-energy states. The no-pairlongitudinal non-interacting linear-response function of theRHEG was also calculated by Facco Bonetti et al. [31], who gave a closed-form expression for real frequencies [Eq. (A1)of Ref. 31]. However, to the best of our knowledge, their ex-pression cannot be straightforwardly used for imaginary fre-quencies. We prefer then to use Eq. (A7) in order to work withimaginary frequencies. After introducing adimensional vari-ables and simplifying, Eq. (A7) leads to Eq. (11) and we per-form the integral numerically. For more details on the deriva-tion of Eq. (A7), see Ref. 38. [1] A. Savin, in Recent Developments of Modern Density Functional Theory ,edited by J. M. Seminario (Elsevier, Amsterdam, 1996) pp.327–357.[2] J. Toulouse, F. Colonna, and A. Savin,Phys. Rev. A , 062505 (2004).[3] W. Kohn and L. J. Sham, Phys. Rev. , A1133 (1965).[4] E. D. Hedegård, J. Toulouse, and H. J. A. Jensen,J. Chem. Phys. , 214103 (2018).[5] A. Fert´e, E. Giner, and J. Toulouse,J. Chem. Phys. , 084103 (2019).[6] D. E. Taylor, J. G. ´Angy´an, G. Galli, C. Zhang, F. Gygi, K. Hi-rao, J. W. Song, K. Rahul, O. A. von Lilienfeld, R. Podeszwa,I. W. Bulik, T. M. Henderson, G. E. Scuseria, J. Toulouse,R. Peverati, D. G. Truhlar, and K. Szalewicz, J. Chem. Phys. , 124105 (2016).[7] C. Kalai, B. Mussard, and J. Toulouse,J. Chem. Phys. , 074102 (2019).[8] O. Franck, B. Mussard, E. Luppi, and J. Toulouse, J. Chem.Phys. , 074107 (2015).[9] O. Kullie and T. Saue, Chem. Phys. , 54 (2012).[10] A. Shee, S. Knecht, and T. Saue, Phys. Chem. Chem. Phys. ,10978 (2015).[11] J. Paquier and J. Toulouse, J. Chem. Phys. , 174110 (2018).[12] J. Paquier, E. Giner, and J. Toulouse,J. Chem. Phys. , 214106 (2020).[13] J. Sucher, Phys. Rev. A , 348 (1980).[14] M. H. Mittleman, Phys. Rev. A , 1167 (1981).[15] J. D. Talman, Phys. Rev. Lett. , 1091 (1986).[16] S. N. Datta and G. Devaiah, Pramana , 387 (1988).[17] M. Grisemer and H. Siedentop, J. London Math. Soc. , 490(1999).[18] J. Dolbeault, M. J. Esteban, and E. S´er´e, J. Funct. Anal. ,208 (2000).[19] T. Saue and L. Visscher, in Theoretical Chemistry and Physics of Heavy and Superheavy Elements ,edited by S. Wilson and U. Kaldor (Kluwer, Dordrecht, 2003)pp. 211–267.[20] A. Almoukhalalati, S. Knecht, H. J. Aa. Jensen, K. G. Dyall, and T. Saue, J. Chem. Phys. , 074104 (2016).[21] E. Engel, in
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