Analytical representation of the Local Field Correction of the Uniform Electron Gas within the Effective Static Approximation
AAnalytical representation of the Local Field Correction of the Uniform Electron Gaswithin the Effective Static Approximation
Tobias Dornheim,
1, 2, ∗ Zhandos A. Moldabekov,
1, 2 and Panagiotis Tolias Center for Advanced Systems Understanding (CASUS), D-02826 G¨orlitz, Germany Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany Space and Plasma Physics, Royal Institute of Technology, Stockholm, SE-100 44, Sweden
The description of electronic exchange–correlation effects is of paramount importance formany applications in physics, chemistry, and beyond. In a recent Letter, Dornheim et al. [ Phys. Rev. Lett. , 235001 (2020)] have presented the effective static approximation (ESA) to thelocal field correction (LFC), which allows for the highly accurate estimation of electronic propertiessuch as the interaction energy and the static structure factor. In the present work, we give ananalytical parametrization of the LFC within ESA that is valid for any wave number, and availablefor the entire range of densities (0 . ≤ r s ≤
20) and temperatures (0 ≤ θ ≤
4) that are relevant forapplications both in the ground state and in the warm dense matter regime. A short implementationin Python is provided, which can easily be incorporated into existing codes.In addition, we present an extensive analysis of the performance of ESA regarding the estimationof various quantities like the dynamic structure factor, static dielectric function, the electronicallyscreened ion-potential, and also stopping power in electronic medium. In summary, we find that theESA gives an excellent description of all these quantities in the warm dense matter regime, and onlybecomes inaccurate when the electrons start to form a strongly correlated electron liquid ( r s ∼ J. Chem. Phys. , 194104(2019)].
I. INTRODUCTION
The accurate description of many-electron systems is ofparamount importance for many applications in physics,quantum chemistry, material science, and related disci-plines [1, 2]. In this regard, the uniform electron gas(UEG) [3, 4], which is comprised of correlated electronsin a homogeneous, neutralizing positive background (alsoknown as ”jellium” or quantum one-component plasma),constitutes a fundamental model system. Indeed, our im-proved understanding of the UEG has facilitated manykey insights like the quasi-particle picture of collectiveexcitations [5] and the Bardeen-Cooper-Schrieffer theoryof superconductivity [6].In the ground state, many properties of the UEGhave been accurately determined on the basis of quan-tum Monte Carlo (QMC) simulations [7–19], which havesubsequently been used as input for various parametriza-tions [20–26]. These, in turn, have provided the basisof the possibly unrivaled success of density functionaltheory (DFT) regarding the description of real materi-als [27–29].Over the last decade or so, there has emerged a re-markable interest in warm dense matter (WDM)–an ex-otic state with high temperatures and extreme densities.In nature, these conditions occur in various astrophysi-cal objects such as giant planet interiors [30–32], browndwarfs [30, 33], and neutron star crusts [34]. On earth,WDM has been predicted to occur on the pathway to- ∗ [email protected] wards inertial confinement fusion [35], and is relevant forthe new field of hot-electron chemistry [36, 37].Consequently, WDM is nowadays routinely realized inlarge research facilities around the globe; see Ref. [38] fora recent review of different experimental techniques. Fur-ther, we mention that there have been many remarkableexperimental discoveries in this field, such as the obser-vation of diamond formation by Kraus et al. [39, 40], orthe measurement of plasmons in aluminum by Sperling et al. [41].At the same time, the theoretical description of WDMis notoriously difficult [42, 43] due to the complicated in-terplay of i) Coulomb coupling, ii) quantum degeneracyof the electrons, and iii) thermal excitations. Formally,these conditions are conveniently expressed by two char-acteristic parameters that are of the order of one simul-taneously: the density parameter (Wigner-Seitz radius) r s = r/a B , where r and a B are the average interparti-cle distance and Bohr radius, and the degeneracy tem-perature θ = k B T /E F , with E F being the usual Fermienergy [1, 44]. In particular, the high temperature rulesout ground state approaches and thermal DFT [45] sim-ulations, too, require as input an exchange–correlation(XC) functional that has been developed for finite tem-perature [46–49].This challenge has resulted in a substantial progress re-garding the development of electronic QMC simulationsat WDM conditions [50–68], which ultimately led to thefirst parametrizations of the XC-free energy f xc of theUEG [69, 70], allowing for thermal DFT calculations onthe level of the local density approximation (LDA). Atthe same time, DFT approaches are being developed thatdeal efficiently with the drastic increase in the basis size a r X i v : . [ phy s i c s . p l a s m - ph ] F e b for high temperatures [71–75], and even gradient correc-tions to the LDA have become available [49, 76].Of particular relevance for the further development ofWDM theory is the response of the electrons to an exter-nal perturbation as it is described by the dynamic den-sity response function χ ( q , ω ), see Eq. (1) below, where q and ω denote the wave vector and frequency. Such in-formation is vital for the interpretation of X-ray Thom-son scattering experiments (XRTS)–a standard methodof diagnostics for WDM which gives access to plasmaparameters such as the electronic temperature [77, 78].Furthermore, accurate knowledge of χ ( q , ω ) would al-low for the construction of advanced XC-functionals forDFT based on the adiabatic connection formula and thefluctuation-dissipation theorem, see Refs. [79–82] for de-tails, or as the incorporation as the dynamic XC-kernelin time-dependent DFT [83, 84]. Finally, we mentionthe calculation of energy-loss properties like the stop-ping power [85], the construction of effective ion-ion po-tentials [86–88], the description of electrical and thermalconductivities [89], and the incorporation of electronicexchange–correlation effects into other theories such asquantum hydrodynamics [90, 91] or average atom mod-els [92].Being motivated by these applications, Dornheim andco-workers have recently presented a number of investiga-tions of both the static and dynamic density response ofthe warm dense electron gas based on ab initio path inte-gral Monte Carlo (PIMC) [93] simulations [89, 94–99]. Inparticular, they have reported that often a static treat-ment of electronic XC-effects is sufficient for a highly ac-curate description of dynamic properties such as χ ( q , ω )or the dynamic structure factor (DSF) S ( q , ω ). Unfor-tunately, this static approximation (see Sec. II C below)leads to a substantial bias in frequency-averaged proper-ties like the interaction energy v [100].To overcome this limitation, Dornheim et al. [100]have presented the effective static approximation (ESA),which entails a frequency-averaged description of elec-tronic XC-effects by combining the neural-net represen-tation of the static local field correction (LFC) fromRef. [94] with a consistent limit for large wave vectorsbased on QMC data for the pair distribution functionevaluated at zero distance; see Ref. [101] for a recentinvestigation of this quantity. In particular, the ESAhas been shown to give highly accurate results for differ-ent electronic properties such as the interaction energyand the static structure factor (SSF) S ( q ) at the samecomputational cost as the random phase approximation(RPA). Furthermore, the value of the ESA for the inter-pretation of XRTS experiments has been demonstratedby re-evaluating the study of aluminum by Sperling etal. [41].The aim of the present work is two-fold: i) we introducean accurate analytical parametrization of the LFC withinESA, which exactly reproduces the correct limits at bothsmall and large wave numbers q = | q | and can be easilyincorporated into existing codes without relying on the neural net from Ref. [94]; a short Python implementationis freely available online [102]; ii) we further analyze theperformance of the ESA regarding the estimation of var-ious electronic properties such as S ( q, ω ) and χ ( q ) over alarge range of densities and temperatures.The paper is organized as follows: In Sec. II, we in-troduce the underlying theoretical background includingthe density response function, its relation to the dynamicstructure factor, and the basic idea of the ESA scheme.Sec. III is devoted to our new analytical parametrizationof the LFC within ESA (see Sec. III C for the final result),which is analyzed in the subsequent Sec. IV regarding theestimation of numerous electronic properties. The paperis concluded by a brief summary and outlook in Sec. V. II. THEORY
We assume Hartree atomic units throughout this work.
A. Density response and local field correction
The density response of an electron gas to an exter-nal harmonic perturbation [64] of wave-number q andfrequency ω is—within linear response theory—fully de-scribed by the dynamic density response function χ ( q, ω ).The latter is conveniently expressed as [1, 103] χ ( q, ω ) = χ ( q, ω )1 − πq [1 − G ( q, ω )] χ ( q, ω ) , (1)where χ ( q, ω ) denotes the density response functionof an ideal Fermi gas known from theory and the fullwave-number- and frequency-resolved information aboutexchange–correlation effects is contained in the dynamiclocal field correction G ( q, ω ). Hence, setting G ( q, ω ) = 0in Eq. (1) leads to the well known RPA which entails onlya mean-field description of the density response.Naturally, the computation of accurate data for G ( q, ω )constitutes a most formidable challenge, although first abinitio results have become available recently at least forparts of the WDM regime [89, 95–98].Let us next consider the static limit, i.e., χ ( q ) = lim ω → χ ( q, ω ) . (2)In this limit, accurate data for Eq. (1) have been pre-sented by Dornheim et al. [94, 99, 104] based on the re-lation [9] χ ( q ) = − n (cid:90) β d τ F ( q, τ ) , (3)with the imaginary-time density–density correlationfunction being defined as F ( q, τ ) = 1 N (cid:104) ρ ( q, τ ) ρ ( − q, (cid:105) . (4)We note that Eq. (4) is the usual intermediate scatteringfunction [77], but evaluated at an imaginary-time argu-ment τ ∈ [0 , β ]. In addition, we note that it is straight-forward to then use χ ( q ) to solve Eq. (1) for the staticlocal field correction G ( q ) = lim ω → G ( q, ω )= 1 − q π (cid:18) χ ( q ) − χ ( q ) (cid:19) . (5)Based on Eq. (5), Dornheim et al. [94] have obtained anextensive data set for G ( q ) for N p ∼
50 different density–temperature combinations. These data—together withthe parametrization of G ( q ; r s ) at zero temperature byCorradini et al. [25] based on ground-state QMC sim-ulations [10, 11]—was then used to train a deep neu-ral network that functions as an accurate representation G ( q ; r s , θ ) for 0 ≤ q ≤ q F , 0 . ≤ r s ≤
20 and 0 ≤ θ ≤ B. Fluctuation–dissipation theorem
The fluctuation–dissipation theorem [1] S ( q, ω ) = − Im χ ( q, ω ) πn (1 − e − βω ) (6)relates Eq. (1) to the dynamic structure factor S ( q, ω )and, thus, directly connects the LFC to different mate-rial properties. First and foremost, we mention that theDSF can be directly measured, e.g. with the XRTS tech-nique [77], which means that the accurate prediction of S ( q, ω ) from theory is of key importance for the diagnos-tics of state-of-the-art WDM experiments [78].The static structure factor is defined as the normaliza-tion of the DSF S ( q ) = (cid:90) ∞−∞ d ω S ( q, ω ) , (7)and thus entails an averaging over the full frequencyrange. We stress that this is in contrast to the staticdensity response function χ ( q ) introduced in the previ-ous section, which is defined as the limit of ω →
0. TheSSF, in turn, gives direct access to the interaction energyof the system, and for a uniform system it holds [4] v = 1 π (cid:90) ∞ d q [ S ( q ) − . (8)Finally, we mention the adiabatic connection formula [4,69, 70] f xc ( r s , θ ) = 1 r s (cid:90) r s d r s v ( r s , θ ) r s , (9)which implies that the free energy (and, equivalently thepartition function Z ) can be inferred if the dynamic den-sity response function—the only unknown part of whichis the dynamic LFC G ( q, ω )—of a system is known for allwave numbers and frequencies, and for different values of the coupling parameter r s . This idea is at the heart of theconstruction of advanced exchange–correlation function-als for DFT calculations within the ACFDT formulation;see, e.g., Refs. [79–82] for more details. C. The static approximation
Since the full frequency-dependence of G ( q, ω ) remainsto this date unknown for most parts of the WDM regime(and also in the ground-state), one might neglect dy-namic effects and simply substitute G ( q ) in Eq. (1). Thisleads to the dynamic density response function within the static approximation [89, 96], χ stat ( q, ω ) = χ ( q, ω )1 − πq [1 − G ( q )] χ ( q, ω ) , (10)which entails the frequency-dependence on an RPA level,but exchange-correlation effects are incorporated stati-cally. Indeed, it was recently shown that Eq. (10) allowsto obtain nearly exact results for χ ( q, ω ), S ( q, ω ), andrelated quantities for r s (cid:46) θ (cid:38) static approximation is problematic forquantities that require an integration over q , such asthe interaction energy v [100]. More specifically, it canbe shown that neglecting the frequency dependence inthe LFC (LFCs that are explicitly defined without a fre-quency dependence are hereafter denoted as G ( q )) leadsto the relation [105]lim q →∞ G ( q ) = 1 − g (0) , (11)where g (0) denotes the pair distribution function (PDF) g ( r ) evaluated at zero distance, sometimes also calledthe on-top PDF or contact probability. Yet, is has beenshown both in the ground state [106, 107] and at finitetemperature [94, 104] that the exact static limit of thedynamic LFC diverges towards either positive or nega-tive infinity in the q → ∞ limit. Eq. (11) thus impliesthat using G ( q ) as G ( q ) in Eq. (10) leads to a divergingon-top PDF, which is, of course, unphysical. This, too,is the reason for spurious contributions to wave-numberintegrated quantities like v at large q . D. The Effective Static Approximation
To overcome these limitation of the static approxima-tion , Dornheim et al. [100] have proposed to define aneffectively frequency-averaged theory that combines thegood performance of Eq. (10) for q (cid:46) q F with the con-sistent limit of G ( q ) from Eq. (11).More specifically, this so-called effective static approx-imation is constructed as [100] G ESA ( q ; r s , θ ) = G nn ( q ; r s , θ ) (1 − A ( x )) (12)+ (1 − g (0; r s , θ )) A ( x ) , with x = q/q F , and where G nn ( q ; t s , θ ) is the neural-net representation of PIMC data for the exact staticlimit G ( q ) = G ( q,
0) of the UEG [94], and g (0; r s , θ )denotes the on-top pair distribution function that wasparametrized in Ref. [100] on the basis of restricted PIMCdata by Brown et al. [53]. Further, A ( x ) denotes the ac-tivation function A ( x ) = A ( x, x m , η ) = 12 [1 + tanh ( η ( x − x m ))] (13)resulting in a smooth transition between G nn andEq. (11) for large q . Here the parameters x m and η canbe used to tune the position and width of the activation.In practice, the performance of the ESA only weakly de-pends on η and we always use η = 3 throughout thiswork. The appropriate choice of the position x m is lesstrivial and is discussed below.An example for the construction of the ESA is shownin Fig. 1 for the UEG at r s = 20 and θ = 1. In the toppanel, we show the wave-number dependence of the staticLFC G ( q ), with the green squares depicting exact PIMCdata for N = 66 taken from Ref. [104] and the blackdashed curve the neural-net representation from Ref. [94].Observe the positively increasing tail at large q from bothdata sets, which is consistent to the positive value of theexchange-correlation contribution to the kinetic energyat these conditions [106, 109].The solid red line corresponds to the ESA and is indis-tinguishable from the neural net for q (cid:46) q F . Further, itsmoothly goes over into Eq. (11) for larger q and attainsthis limit for q (cid:38) . q F . The purple dash-dotted curveshows the corresponding activation function A ( x ) [using x m = 3] on the right y -axis and illustrates the shape ofthe switchover between the two limits. As a reference,we have also included G ( q ) computed within the finite-temperature version [105, 108] of the STLS approxima-tion [110], see the dotted blue curve. First and foremost,we note that STLS constitutes a purely static theory forthe LFC and, thus, exactly fulfills Eq. (11), i.e., it attainsa constant value in the limit of large wave numbers, al-though for significantly larger values of q . In addition,STLS is well known to violate the exact compressibilitysum-rule [108] (see Eq. (15) below) and deviates from theother curves even in the small- q limit. Finally, we notethat it does not reproduce the peak of both the neuralnet and ESA around q = 2 . q F .The bottom panel of Fig. 1 shows the correspond-ing results for the static structure factor S ( q ), with thegreen crosses again being the exact PIMC results fromRef. [104]. At this point, we feel that a note of cautionis pertinent. On the one hand, the PIMC method is lim-ited to simulations in the static limit, as dynamic sim-ulations are afflicted with an exponentially hard phaseproblem [111] in addition to the usual fermion sign prob-lem [112]. Therefore, PIMC results for both χ ( q, ω ) and G ( q, ω ) are only available for ω = 0. Yet, the PIMCmethod is also capable to give exact results for frequency-averaged quantities like S ( q ), as the frequency integra-tion is carried out in the imaginary time [93]. Thus, the r s =20, θ =1 G ( q ) A ( x ) q/q F PIMCSTLSESA1-g(0)StaticA(x,x m , η ) r s =20, θ =1 S ( q ) q/q F FIG. 1. Illustration of the effective static approximation(ESA) [100] for r s = 20 and θ = 1. Top panel: Static LFC.Green squares are exact PIMC data for G ( q ; r s , θ ) taken fromRef. [104], and dashed black line the neural net representa-tion from Ref. [94]. The solid red curve shows the frequency-averaged LFC G ( q ; r s , θ ) within ESA [Eq. (12)] and the dot-ted blue curve the same quantity within STLS [105, 108].The purple dash-dotted line shows the activation function A ( x, x m , η ) [for x m = η = 3, see Eq. (13)] and corresponds tothe right y -axis. Top panel: Static structure factor S ( q ) fromthe same methods, and in RPA (dash-dotted yellow). green squares do correspond to the results one would ob-tain if the correct, dynamic LFC G ( q, ω ) was insertedinto Eq. (1).This is in contrast to the black dashed curve, thathas been obtained on the basis of the static approxima-tion , Eq. (10), using as input the neural-net representa-tion [94] of the exact static limit G ( q ). Evidently, thestatic treatment of exchange–correlation effects is welljustified for q (cid:46) q F , but there appear systematic devia-tions for larger q ; see also the inset showing a magnifiedsegment around the maximum of S ( q ). In particular, S ( q ) does not decay to 1, and, while being small for eachindividual q , the error accumulates under the integral inEq. (8).The solid red curve has been obtained by inserting G ( q )within the ESA into Eq. (10). Plainly, the inclusion ofthe on-top PDF via Eq. (12) removes the spurious ef-fects from the static approximation , and the ESA curveis strikingly accurate over the entire q -range.The dotted blue curve has been computed using G ( q )within the STLS approximation. For small q , it, tooobeys the correct parabolic limit [59, 113], which is theconsequence of perfect screening in the UEG [1]. Forlarger q , there appear systematic deviations, and thecorrelation-induced peak of S ( q ) around q ∼ . q F isnot reproduced by this theory; see also Ref. [104] for anextensive analysis including even stronger values of thecoupling strength r s .Finally, the dash-dotted yellow curve has been com-puted within the RPA. Clearly, neglecting exchange–correlation effects in Eq. (1) leads to an insufficient de-scription of the SSF, and we find systematic deviationsof up to ∼ III. ANALYTICAL REPRESENTATION OF THEESAA. Choice of the activation function
The ESA as it has been defined in Eq. (12) has, inprinciple, two free parameters, which have to be de-fined/parametrized before an analytical representation of G ( q ; r s , θ ) can be introduced. More specifically, these arethe transition wave number x m and scaling parameter η from the activation function A ( x ; x m , η ); see Eq. (13). Scaling parameter η : We choose η ( r s , θ ) = 3 =const, as G ESA ( q ; r s , θ ) only weakly depends on this pa-rameter; see Ref. [100] for an example. Transition wave-number x m : The choice of areasonable wave-number of the transition between theneural-net and Eq. (11) is less trivial. What we need isa transition around x m ∼ . q F for θ (cid:46)
1, whereas itshould move to larger wave-number for higher tempera-tures. The dependence on the density parameter r s , onthe other hand, is less pronounced and can be neglected.We thus construct the function x m ( θ ) = A x + B x θ + C x θ , (14)with A x , B x , and C x being free parameters that we de-termine empirically. In particular, we find A x = 2 . B x = 0 .
31, and C x = 0 .
08. A graphical depiction ofEq. (14) is shown in Fig. 2An example for the impact of x m on both G ( q ) andthe corresponding SSF is shown in Fig. 3. The top panelshows the LFC, and we observe an overall similar trendas for θ = 1 depicted in Fig. 1. The main differences bothin the PIMC data and the neural net results for G ( q ) arei) the comparably reduced height of the maximum, ii)the increased width of the maximum regarding q , and iii) x m θ fi t FIG. 2. Temperature dependence of the transition wave-number x m from Eq. (14). the decreased slope of the positive tail at large wave num-bers. The red curve shows the ESA results for G ( q ) usingthe transition wave-number obtained from Eq. (14), i.e., x m ≈ .
58. In particular, the red curve reproduces thepeak structure of the exact static limit G ( q ), and subse-quently approaches the large- q limit from Eq. (11) [lightdotted grey line]. In contrast, the dash-dotted yellowand dashed-double-dotted purple lines are ESA resultsfor x m = 3 and x m = 2 .
5, respectively, and start to sig-nificantly deviate from G ( q ) before the peak. Finally, thedotted blue curve shows G ( q ) from STLS, and has beenincluded as a reference.Regarding S ( q ), the solid red curve shows the bestagreement to the PIMC data, whereas the static approx-imation again exhibits the spurious behaviour for large q , albeit less pronounced than for θ = 1 shown above.The ESA results for x m = 3, too, is in good agreementto the PIMC data, although there appears an unphysicalminimum around q = 3 q F . The ESA curve for x m = 2 . S ( q ) from the other data sets. Finally, the STLS curvedoes not provide an accurate description of the physi-cal behaviour and systematically deviates from the exactresults except in the limits of large and small q . B. Analytical representation
Let us start this discussion by introducing a suitablefunctional form for the q -dependence of G ESA when r s and θ are fixed. First and foremost, we note that ourparametrization is always constructed from Eq. (12),which means that the task at hand is to find an appro-priate representation of G nn ( q ; r s , θ ) that is sufficientlyaccurate in the wave-number regime where the neuralnet contributes to the ESA. The correct limit for large q ,on the other hand, is built in automatically.In addition, we would like to incorporate the exactlong-wavelength limit of the static LFC that is given by r s =20, θ =2 G ( q ) q/q F PIMCSTLSESA1-g(0)Staticx m =3x m =2.5 r s =20, θ =2 S ( q ) q/q F FIG. 3. Top panel: Static local field correction for r s = 20and θ = 2. Green squares are PIMC data for G ( q ) fromRef. [104], and dashed black line the neural-net representationfrom Ref. [94]. The solid red line shows G ( q ) within the ESAusing Eq. (14) [i.e., x m = 3 . x m = 3and x m = 2 .
5. The dotted blue line shows G ( q ) from STLS,and the light grey line the analytical limit from Eq. (11).Bottom panel: Corresponding results for the static structurefactor S ( q ). the compressibility sum-rule [94, 108] (CSR)lim q → G ( q ; r s , θ ) = G CSR ( q ; r s , θ ) (15)= − q π ∂ ∂n ( nf xc ) . This is achieved by the ansatz G r s ,θ nn,fit ( q ) = G CSR ( q ; r s , θ ) (16) × (cid:20) α r s ,θ x + β r s ,θ √ x γ r s ,θ x + δ r s ,θ x . + G CSR ( q ; r s , θ ) (cid:21) , where x = q/q F is the reduced wave-number and thesuper-scripts in the four free parameters α r s ,θ , β r s ,θ , γ r s ,θ , and δ r s ,θ indicate that they are obtained for fixedvalues of θ and r s . We note that the G CSR ( q ; r s , θ ) term r s =5, θ =0 G ( q ) q/q F QMCCSRSTLSESA fi t1-g(0)Static r s =5, θ =3 G ( q ) q/q F PIMCCSRSTLSESA fi t1-g(0)Static FIG. 4. Static local field correction for r s = 5 and θ = 0 (top)and θ = 3 (bottom). Green squares are ground-state QMCdata from Ref. [11] (PIMC data for G ( q ) from Ref. [94]) for θ = 0 ( θ = 3), and dashed black lines the neural-net represen-tation from Ref. [94]. The solid red line shows G ( q ) within theESA using Eq. (14), and the light blue dash-dotted curve thecorresponding fit from Eq. (16). The dotted blue line shows G ( q ) from STLS, and the light grey line the analytical limitfrom Eq. (11). in the denominator of the square brackets compensatesthe equal pre-factor for large q .Two examples for the application of Eq. (16) are shownin Fig. 4, where the local field correction is shown for r s = 5 and θ = 0 (top) and θ = 3 (bottom). The redcurve shows G ( q ) within the ESA, and the light bluedash-dotted curve a fit to these data using Eq. (16) as afunctional form for θ and r s being constant. First andforemost, we note that the fit perfectly reproduces theESA, and no fitting error can be resolved with the nakedeye.The dash-dotted yellow curves show the CSR[Eq. (15)], which has been included into Eq. (16). In theground state, we indeed find good agreement betweenthe CSR, the QMC data, the neural net, and also theESA for q (cid:46) q F . This is somewhat changed for θ = 3,where the yellow curve exhibits more pronounced devia-tions from the PIMC data and all other curves. Still, wenote that the functional form from Eq. (16) is capableto accommodate this finding, and attains the small-wavenumber limit only for small q in this case.We thus conclude that Eq. (16) constitutes a suit-able basis for the desired analytical representation G ESA ( q ; r s , θ ). As a next step, we make Eq. (16) de-pendent on the density parameter r s . To achieve thisgoal, we parametrize the free parameters as: κ θ ( r s ) = a θκ + b θκ r s c θκ r s , (17)with κ ∈ { α, β, γ, δ } . Thus, the characterization of the r s -dependence for a single isotherm requires the determi-nation of 12 free parameters. This results in the isother-mic representation of the LFC of the form G θ nn,fit ( q ; r s ) = G CSR ( q ; r s , θ ) (18) × (cid:20) α θ ( r s ) x + β θ ( r s ) √ x γ θ ( r s ) x + δ θ ( r s ) x . + G CSR ( q ; r s , θ ) (cid:21) , This isothermic representation is illustrated in Fig. 5,where we show the full r s -dependence of the four free pa-rameter α − δ (clockwise) for θ = 0 (green), θ = 1 (red),and θ = 4 (black). The symbols have been obtained byfitting Eq. (16) to ESA data for G ( q ) for constant valuesof r s and θ . The solid lines have been subsequently ob-tained by fitting the representation of Eq. (17) to thesedata over the entire r s -range. The resulting curves areindeed smooth and qualitatively capture the main trendsfrom the data points. Finally, the dashed curves havebeen computed by fitting Eq. (18) to ESA data over theentire r s -range, but for constant values of θ . Interest-ingly, this final optimization step results in qualitativechange of the description of all four parameters for θ = 4,but only mildly changes the results for both θ = 1 and θ = 0.Let us for now postpone the discussion of the dottedcurve in Fig. 5, and consider Fig. 6 instead. In particular,we show the results of the isothermic fitting procedure for θ = 1 (top) and θ = 4 (bottom), with the red, green, andblack curves corresponding to different data sets for r s =0 . r s = 2, and r s = 5, respectively. More specifically,the solid lines show the ESA reference data for G ( q ),and the dashed curves have been obtained by fitting thedata points for α − δ shown in Fig. 5 via Eq. (17). For θ = 1, this simple procedure alone leads to an excellentrepresentation of G θ ( q ; r s ). The dotted curve has beenobtained by performing the full isothermic fits, i.e., byfitting Eq. (18) to ESA data over the entire r s -range,but with θ being constant. Indeed, we find only minordeviations between the dashed and the dotted curve.For θ = 4, on the other hand, the simple representa-tion of the fit parameters from Eq. (16) results in a sub-stantially less accurate representation of G θ ESA ( q ; r s ), andthe systematic error is most pronounced at high density, r s = 0 .
7. This shortcoming can be remedied by perform-ing the full isothermic fit of the entire q - r s -dependence,and the dotted curves are in excellent agreement to theoriginal ESA data everywhere. We thus conclude that the functional form of Eq. (18) constitutes an adequaterepresentation of G θ ESA ( q ; r s ). C. Final representation of G ESA ( q ; r s , θ ) The final step is then given by the construction of ananalytical representation of the full r s - θ - q -dependence byexpressing the parameters a θκ , b θκ , and c θκ in Eq. (17) asa function of θ , f κ ( θ ) = a f + b f θ + c f θ . . (19)This results in three free parameters for each of the 12coefficients required for the characterization of the r s -dependence, i.e., a total of 36 parameters that have to bedetermined by the fitting procedure.The full three-dimensional fit-function is then given by G nn,fit ( q ; r s , θ ) = G CSR ( q ; r s , θ ) (20) × (cid:20) α ( r s , θ ) x + β ( r s , θ ) √ x γ ( r s , θ ) x + δ ( r s , θ ) x . + G CSR ( q ; r s , θ ) (cid:21) , where the functions κ ( r s , θ ) [with κ ∈ { α, β, γ, δ } ] aregiven by κ ( r s , θ ) = a κ ( θ ) + b κ ( θ ) r s c κ ( θ ) r s , (21)and the θ -dependent coefficients follow Eq. (19).Our final analytical representation of the LFC withinthe effective static approximation immediately followsfrom plugging Eq. (20) into Eq. (12), G ESA,fit ( q ; r s , θ ) = G nn,fit ( q ; r s , θ ) (1 − A ( x )) (22)+ (1 − g (0; r s , θ )) A ( x ) . The thus fitted coefficients are given in Tab. I, and acorresponding python implementation is freely availableonline [102].The resulting analytical representation G ESA ( q ; r s , θ )is illustrated in Fig. 7, where we compare it (dashed lines)to the original ESA data at r s = 5 (top) and r s = 2,i.e., two metallic densities that are of high interest in thecontext of WDM research.More specifically, r s = 5 corresponds to a strongly cou-pled system, where an accurate treatment of electronicexchange–correlation effects is paramount [114]. Theseconditions can be realized experimentally in hydrogenjets [115] and evaporation experiments [47, 114, 116, 117].The green, red, black, and blue curves show results for θ = 0, θ = 1, θ = 2, and θ = 4, respectively, and we findthat our new analytical representation of G ESA ( q ; r s , θ )is in excellent agreement to the ESA input data every-where.The bottom panel corresponds to r s = 2, which is rel-evant e.g. for the investigation of aluminum [41, 118].Here, too, we find excellent agreement between the fittedfunction and the ESA input data for θ = 0 and θ = 4, -1-0.5 0 0.5 0 5 10 15 20 α r s θ =0 θ =1 θ =4 fi tr s - fi tr s - θ - fi t -0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0 5 10 15 20 β r s -4-3.5-3-2.5-2-1.5-1-0.5 0 0.5 1 0 5 10 15 20 δ r s -1.5-1-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 5 10 15 20 γ r s FIG. 5. Dependence of the fit parameters α − δ (clockwise) from Eq. (16 on the density parameter r s for θ = 0 (green), θ = 1(red), and θ = 4 (black). The symbols have been obtained by fitting Eq. (16) to ESA data for individual constant values ofboth r s and θ , and the solid lines have been fitted to these data using Eq. (17) as a functional form. The dashed lines havebeen obtained from isothermal fits over the full r s -range using Eq. (18), and the dotted curves from the final fit over the full r s - θ - q -dependence, see Eq. (20).TABLE I. Fit parameters for the analytic parametrization of G ESA ( q ; r s , θ ) from Eq. (20). For each of the coefficients a α , b α , . . . , c γ , we give the three free parameters from Eq. (19), a f , b f , and c f . A short python implementation is freelyavailable online [102]. a α . − . . α b α − . . − . c α . . − . a β − . . − . β b β . − . . c β . . − . a γ − . . . γ b γ . − . − . c γ . . − . a δ . − . − . δ b δ − . − . . c δ . . − . while small, yet significant deviations appear at interme-diate wave numbers for θ = 2 and θ = 1. Still, it is im-portant to note that these deviations do not exceed thestatistical uncertainty of the original PIMC input datafor G ( q ) on which the neural net from Ref. [94] and theESA are based. We thus conclude that our analytical representation of G ESA ( q ; r s , θ ) provides a highly accurate description ofelectronic–exchange correlation effects over the entire rel-evant parameter range. The application of this represen-tation for the computation of other material propertieslike the static structure factor S ( q ), interaction energy v , θ =1 G ( q ) q/q F r s =0.7r s =2r s =5 ESA fi tr s - fi t 0 0.2 0.4 0.6 0.8 1 1.2 0 1 2 3 4 5 6 θ =4 G ( q ) q/q F r s =0.7r s =2r s =5 ESA fi tr s - fi t FIG. 6. Illustration of the isothermic fit function of the localfield correction G ESA ( q ) for θ = 1 (top) and θ = 4 (bottom).The red, green, and black curves depict different results for r s = 0 . r s = 2, and r s = 5, respectively. Solid: ESA;dashed: fitted r s -dependence of the individual coefficients α − δ from Eq. (16) according to Eq. (17); dotted: full isothermicfits of G θ ESA ( q ) via Eq. (18). or dielectric function (cid:15) ( q ) is discussed in detail in Sec. IV. IV. RESULTSA. The static local field correction
Let us begin the investigation of the results that can beobtained within the ESA by briefly recapitulating a fewimportant properties of G ESA ( q ; r s , θ ) itself. To this end,we show the LFC in the θ - q -plane for r s = 20 (top) and r s = 5 (bottom) in Fig. 8. More specifically, the dashedblack lines show the neural-net results for G ( q ) fromRef. [94], and the solid red lines the corresponding datafor our analytical representation of G ESA ( q ; r s , θ ). Firstand foremost, we note that the temperature dependenceis qualitatively similar for both values of the density pa-rameter; a more detailed analysis of the r s -dependence ofthe LFC is presented in Fig. 9 below. As usual, G ( q ) ex-hibits a non-constant behaviour for large wave numbers, r s =5 G ( q ) q/q F θ =0 θ =1 θ =2 θ =4 ESAq-r s - θ fi t 0 0.2 0.4 0.6 0.8 1 1.2 0 1 2 3 4 5 6 7 r s =2 G ( q ) q/q F θ =0 θ =1 θ =2 θ =4 ESAq-r s - θ fi t FIG. 7. Analytical representation of G ESA ( q ; r s , θ ): Shownare ESA results (solid lines) and our final analytical represen-tation, Eq. (22). whereas the ESA converges towards Eq. (11). In addi-tion, our parametrization nicely reproduces the neural-net for x < x m ( θ ), which further illustrates the highquality of the representation. Finally, we find that theexact static limit of the LFC, too, becomes increasinglyflat at large q for high temperatures, which can be seenparticularly well for r s = 20. In fact, simultaneously con-sidering large values of r s and θ brings us to the classicallimit, where G ( q ) converges towards one for large wavenumbers [119], lim q →∞ G classical ( q ) = 1 . (23)Moreover, the ESA and G ( q ) converge in this regime asthe static structure factor can always be computed fromthe static LFC only via the exact relation [119, 120] S classical ( q ) = 11 − πq [ G classical ( q ) − βn . (24)In other words, the spurious effects due to the static ap-proximation and the need for the ESA in WDM appli-cations are a direct consequence of quantum effects onelectronic exchange–correlation effects, which only van-ish in the classical limit.0
0 1 2 3 4 0 1 2 3 4 5 6 0.5 1 1.5 2G(q;r s , θ ) r s =20 ESAStatic θ q/q F
0 1 2 3 4 0 1 2 3 4 5 6 0.5 1 1.5 2G(q;r s , θ ) r s =5 ESAStatic θ q/q F FIG. 8. The local field correction in the θ - q -plane: The solidred and dashed black curves show our analytical representa-tion of G ESA ( q ; r s , θ ) and the neural-net representation of theexact static limit G ( q ) from Ref. [94] for r s = 20 (top) and r s = 5 (bottom). Let us next consider the dependence of the LFC onthe density parameter r s , which is shown in Fig. 9 for θ = 1. For strong coupling, we observe a positive tail inthe neural-net results for G ( q ) which begins at smallervalues of x = q/q F for larger r s . Between r s = 2 and r s = 1, i.e., in the middle of the WDM regime, this be-haviour changes and we find instead a negative slope,which ultimately even leads to negative values of G ( q ).From a physical perspective, the long wave-number limitis dominated by single-particle effects and the sign of theslope follows from the exchange–correlation contributionto the kinetic energy K [106, 107], which changes its signat these conditions [101, 109].The ESA, on the other hand, is invariant to this effect s , θ ) θ =1 ESAStaticr s q/q F FIG. 9. The local field correction in the r s - q -plane: Thesolid red and dashed black curves show our analytical repre-sentation of G ESA ( q ; r s , θ ) and the neural-net representationof the exact static limit G ( q ) from Ref. [94] for θ = 1. Notethe logarithmic scale of the r s -axis. and, as usually, attains the consistent limit for G ( q ) givenby Eq. (11) for all values of r s .As a further motivation for our ESA scheme, we con-sider an effective local field correction G invert ( q ), which,by definition, exactly reproduces QMC data for S ( q )where they are available. More specifically, such a quan-tity can be defined as G invert ( q ) = min G (cid:16)(cid:12)(cid:12)(cid:12) S G ( q ) − S ( q ) (cid:12)(cid:12)(cid:12)(cid:17) , (25)where S G ( q ) denotes the SSF computed with respect tosome trial static LFC G . In practice, we solve Eq. (25)by scanning over a dense G -grid for each q -point andsearch for the minimum deviation in the SSF. In thisway, we have effectively inverted S ( q ) for the LFC G ,even though the relation between the two quantities is notstraightforward when quantum mechanical effects cannotbe neglected.The results for this procedure are depicted in Fig. 10,where we show different LFCs at r s = 20. The top andcenter panels corresponds to θ = 0 and θ = 1, and both G ( q ) and G ESA ( q ) exhibit the familiar behaviour that hasbeen discussed in the context of Fig. 1 above. The yellowtriangles show the inverted results for Eq. (25) and are inremarkably good agreement to both G ( q ) and G ESA ( q )for q (cid:46) q F . For larger q , G invert ( q ) follows G ESA ( q )and attains the same finite limit instead of diverging likethe exact static limit of the LFC. In fact, the curves canhardly be distinguished within the given level of accuracy(in particular at θ = 0), which further substantiates thesimple construction of the ESA, Eq. (12).1 r s =20, θ =0 G ( q ) q/q F G invert G classical ESA1-g(0)Static r s =20, θ =1 G ( q ) q/q F PIMCG invert G classical ESA1-g(0)Static r s =20, θ =4 G ( q ) q/q F PIMCG invert G classical ESA1-g(0)Static
FIG. 10. Inverted local field correction at finite temperature:The for r s = 20 at θ = 0 (top), θ = 1 (center) and θ =4 (bottom). Green squares and black dashed line: PIMCresults for G ( q ) from Ref. [104] and corresponding neural-netresults [94]. Solid red and dotted grey: ESA and large- q limit,Eq. (11). Yellow triangles: inverted LFC G classical ( q ), seeEq. (25). Purple diamonds: LFC from the classical relationEq. (26). Let us briefly postpone the discussion of the purple di-amonds and instead consider the bottom panel of Fig. 10showing results for θ = 4. At these conditions, G ( q ) and G ESA ( q ) only start to noticeably deviate for q (cid:38) q F , andthe PIMC data, too, appear to remain nearly constantfor large q . In addition, the black dashed curve is only reliable for q ≤ q F as data for larger wave numbers hadnot been included into the training of the neural net, seeRef. [94] for details.Unsurprisingly, the inverted data for G inverted ( q )closely follow G ESA ( q ) over the entire q -range, and bothESA and the static approximation give highly accurateresults for S ( q ) and v .Let us next more closely examine the connection be-tween the ESA and the classical limit, where G ( q ) is suf-ficient to compute exact results for S ( q ), see Eq. (24)above. In particular, Eq. (24) can be straightforwardlysolved for G ( q ), which gives the relation G classical ( q ) = 1 − q π (cid:18) S ( q ) − (cid:19) βn , (26)which, too, is exact in the classical limit.At the same time, it is interesting to evaluate Eq. (26)for a quantum system to gauge the impact of quantumeffects on exchange–correlation effects at different wavenumbers q . The results are depicted by the purple di-amonds in Fig. 10. In the ground state, i.e., β → ∞ ,it holds G classical ( q ) = 1 for all q , as the second termis proportional to T and, hence, vanishes. For θ = 1, G classical ( q ) does depend on q , but is still qualitativelywrong over the entire depicted wave-number range. Inparticular, it strongly violates the compressibility sum-rule Eq. (15) and does not even decay to zero in thelimit of small q . Finally, G classical ( q ) does more closelyresemble the other curves at θ = 4, but still substantiallydeviates everywhere. We thus conclude that quantum ef-fects are paramount even at θ = 4 and r s = 20, and canonly be neglected at significantly higher temperatures. B. The static structure factor
The next quantity to be investigated with the ESAscheme is the static structure factor S ( q ), which we showin Fig. 11. The left column corresponds to r s = 20 and,thus, constitutes the most challenging case for the ESAdue to the dominant character of exchange–correlationeffects at these conditions.Let us start with the top panel, showing results forthe ground state. The green squares are state-of-the-artdiffusion Monte Carlo results by Spink et al. [17] and con-stitute the gold standard for benchmarks. The solid redcurve has been obtained using G ESA ( q ) and is in remark-able agreement for all q , even in the vicinity of the peakof S ( q ) around q ≈ . q F . In contrast, the blue dottedSTLS curve does not capture this feature and exhibitspronounced systematic deviations except in the limits ofsmall and large wave numbers.The center panel in the left column has been obtainedfor θ = 0 .
5, and the green squares are finite- T PIMCdata taken from Dornheim et al. [104]. Again, the ESAgives a very good description of S ( q ), although the peakheight is somewhat overestimated. Still, the description2 r s =20, θ =0 S ( q ) q/q F r s =5, θ =0 S ( q ) q/q F r s =20, θ =0.5 S ( q ) q/q F r s =5, θ =1 S ( q ) q/q F r s =20, θ =4 S ( q ) q/q F r s =5, θ =4 S ( q ) q/q F FIG. 11. Static structure factor at r s = 20 (left) and r s = 5 (right) at different θ . Green squares: T = 0 QMC data [17] andfinite- T PIMC data [104]; Solid red: ESA; dotted blue: STLS [105, 108, 110]; dash-dotted yellow: RPA. is strikingly improved compared to the STLS approxima-tion.Lastly, the bottom panel has been obtained for θ = 4,where ESA cannot be distinguished from the PIMC ref-erence data within the given Monte Carlo error bars.STLS, too, is quite accurate in this regime, althoughthere remain systematic deviations at intermediate q . Finally, we mention the dash-dotted yellow curve in allthree panels, that have been obtained within RPA. Ev-idently, this mean field description is unsuitable at suchlow densities even at relatively high values of the reducedtemperature θ .The right column of Fig. 11 has been obtained for adensity that is of prime interest to WDM research, r s =35. Again, the top panel corresponds to the ground-stateand shows relatively good agreement between diffusionMonte Carlo, ESA, and STLS, although the latter doesnot capture the small correlation induced peak in S ( q ).The RPA, on the other hand, remains inaccurate despitethe reduced coupling strength compared to the left panel.At θ = 1 (center panel), the situation is quite similar,with the ESA being nearly indistinguishable to the PIMCdata over the entire q -range, whereas STLS is too largefor small and too small for large wave numbers.Finally, the bottom panel corresponds to θ = 4. Here,too, only the ESA is capable to reproduce the PIMCdata, whereas STLS and in particular RPA exhibit sys-tematic errors. C. Interaction energy
The next important quantity to be investigated in thiswork is the interaction energy v , which, in the caseof a uniform electron gas, is simply given by a one-dimensional integral over the static structure factor S ( q )[see Eq. (8)] that we evaluate numerically. The resultsare shown in Fig. 12, where we depict the θ -dependenceof v for four relevant values of the density parameter r s .More specifically, the top left panel corresponds to r s = 2, i.e., a metallic density that is typical for WDMexperiments using various materials, and we plot the rela-tive deviation in v compared to the accurate parametriza-tion of the UEG by Groth et al. [69]. At these conditions,both the ESA (solid red) and the static approximation (dashed grey) are very accurate over the entire θ -range,with a maximum deviation of ∆ v/v ∼ v , with a maximum deviation of ∼ r s = 5, a relatively sparse density that can be realized e.g.in experiments with hydrogen jets, see above. First andforemost, we note that both the ESA and STLS providea remarkably good description of the interaction energy,and the systematic error never exceeds 2%. Somewhatsurprisingly, STLS even gives slightly more accurate darafor small values of θ compared to ESA. Yet, this is dueto a fortunate cancellation of errors in S ( q ) under theintegral in Eq. (8) [ S ( q ) is too large for small q and toosmall for large q , which roughly balances out] [4, 100],since the static structure factor S ( q ) is comparativelymuch better in ESA than in STLS, cf. Fig. 11. In ad-dition, we note that the static approximation performssubstantially worse for low temperatures, which is due tothe unphysically slow convergence of S ( q ) towards 1 forlarge q , see Secs. II C and II D above.The bottom left panel shows the same analysis for r s =10, and even for this strong coupling strength that con-stitutes the boundary of the electron liquid regime [96],the error in ESA does not exceed 2%. In addition, theSTLS exhibits a comparable accuracy in v , whereas the static approximation fails at low θ as it is expected. Finally, the bottom right panel shows results for verystrong coupling, r s = 20. Overall, the ESA gives themost accurate data for v of all depicted approximations,and is particularly good both at large temperature and inthe ground state. In contrast, the STLS approximationfor G ( q ) results in a relatively constant relative deviationof ∼ − static approximation cannotreasonably used for this values of the density parameter. D. Density response function
This section is devoted to a discussion of the suitabilityof frequency-averaged LFCs for the determination of theexact static limit of the density response function χ ( q ).In this case, the previously discussed static approxima-tion , i.e., using the neural-net representation of G ( q, q limit of frequency-independent theories G ( q ) given by Eq. (11) is spurious.On the other hand, we might expect that the impact ofthe LFC decreases for large q , such that G ESA ( q ) and G ( q ) could potentially give similar results.To resolve this question, we show χ ( q ) in Fig. 13 forthree representative values of the density parameter r s ,with the green, red, and black sets of curves correspond-ing to θ = 4, θ = 1, and θ = 0, respectively. Let us startwith the top panel showing results for a metallic den-sity, r s = 2, with the dotted, dashed, and solid curvescorresponding to ESA, the exact static limit, and STLS,respectively. Firstly, we note that all three curves exhibitthe correct parabolic shape for small wave-numbers [113],lim q → χ ( q ) = − πq . (27)In particular, Eq. (27) is a direct consequence of the4 π/q pre-factor in front of the LFC in Eqs. (1) and (10),which means that its impact vanishes for small q . Withincreasing wave numbers, χ ( q ) exhibits a broad peakaround q ≈ . q F , which is also well reproduced by allcurves. Moreover, the ESA is virtually indistinguishablefrom the exact result for all three temperatures, whereasSTLS noticeably deviates, in particular at θ = 0.The center panel shows the same analysis for r s =5. As discussed above, the increased coupling strengthmeans that the impact of the LFC is more pronounced inthis case, and the STLS curve substantially deviates atintermediate wave numbers, except for the highest tem-perature θ = 4. In stark contrast, the ESA is in excellentagreement to the exact curve everywhere, and we findonly minor deviations for 2 q F (cid:46) q (cid:46) q F . In this sense,the ESA combines the best from two worlds, by givingexcellent results both for frequency-averaged quantitieslike S ( q ), and really static properties like χ ( q,
0) over theentire WDM regime.This nice feature of the ESA is only lost when enter-ing the strongly coupled electron liquid regime, as it isdemonstrated in the bottom panel of Fig. 13 for r s = 20.In this case, the static density response function is more4 r s =2 Δ v / v [ % ] θ ESASTLSStatic -2 0 0 1 2 3 4 r s =5 Δ v / v [ % ] θ ESASTLSStatic -6-4-2 0 0 1 2 3 4 r s =10 Δ v / v [ % ] θ ESASTLSStatic -10-8-6-4-2 0 2 4 0 1 2 3 4 r s =20 Δ v / v [ % ] θ ESASTLSStatic
FIG. 12. Relative deviation in the interaction energy v [see Eq. (8)] compared to the parametrization by Groth et al. [69]. Solidred circles: ESA; dotted blue diamonds: STLS [105, 108, 110]; dashed grey crosses: static approximation using the neural-netrepresentation from Ref. [94]. sharply peaked at low temperature and exhibits a non-trivial shape that is difficult to resolve. Therefore, theSTLS approximation is not capable to give a reasonabledescription of either the peak position or the shape, seeRef. [104] for a more extensive analysis on this point in-cluding even larger values of the density parameter r s .The ESA, on the other hand, is strikingly accurate forboth θ = 4 and θ = 1, but substantially deviates fromthe exact curve for 2 q F (cid:46) q (cid:46) q F in the ground state. E. Dielectric function
The dynamic dielectric function (cid:15) ( q, ω ) is defined as (cid:15) ( q, ω ) = 1 − χ ( q, ω ) q π + χ ( q, ω ) , (28)and is important in both classical and quantum electro-dynamics, in particular for the description of plasma os-cillations [98, 121, 122]. Since a more detailed analysis ofthis quantity has been presented elsewhere [89, 98], herewe restrict ourselves to a brief discussion of ESA resultsfor the static limit of Eq. (28), (cid:15) ( q ).The results are shown in Fig. 14, where the left panelshows the dielectric function for r s = 5 and θ = 1. Re-markably, we find substantial disagreement between thedifferent results for small wave numbers q , which is instriking contrast to linear response properties like χ ( q )and also the SSF S ( q ). For the latter quantities, the impact of the LFC vanishes for small q as it has beenexplained above, such that even the mean-field descrip-tion within the RPA becomes exact in this limit. Thedielectric function, on the other hand, always divergesfor small q , and this divergence is connected to the CSRfor the static LFC [Eq. (15)] [89, 108],lim q → (cid:15) ( q ) = − πχ ( q ) q [1 + 4 πCχ ( q )] , (29)where C is the pre-factor to the parabola in Eq. (15), C = − π ∂ ∂n ( nf xc ) . (30)In principle, exact knowledge of the static LFC as it isencoded in the neural-net representation from Ref. [94]gives access to the exact static dielectric function de-picted in Fig. 14. Yet, while the exact relation Eq. (15)was indeed incorporated into the training procedure ofthe neural net, it was not strictly enforced and, thus, isonly fulfilled by the static (grey dashed) curve with afinite accuracy. Therefore, this curve violates Eq. (29)and attains a finite value in the limit of q →
0, which isunphysical.Our new analytical representation of G ESA ( q ), in con-trast, exactly incorporates the CSR, which means thatthe solid red curve exhibits the correct asymptotic be-haviour (depicted as the dash-dotted green curve). Fi-nally, the dotted blue curve has been obtained on thebasis of the approximate G STLS ( q ), and starkly deviates5 -0.07-0.06-0.05-0.04-0.03-0.02-0.01 0 0 1 2 3 4 5 r s =2 χ q/q F θ =4 θ =1 θ =0 ESAStaticSTLS -0.025-0.02-0.015-0.01-0.005 0 0 1 2 3 4 5 r s =5 χ q/q F θ =4 θ =1 θ =0 ESAStaticSTLS -0.006-0.005-0.004-0.003-0.002-0.001 0 0 1 2 3 4 5 r s =20 χ q/q F θ =4 θ =1 θ =0 ESAStaticSTLS FIG. 13. Static density response function χ ( q ) for r s = 2(top), r s = 5, and r s = 20 (bottom). The dotted, solid, anddashed lines have been obtained by inserting into Eq. (10) G ESA ( q ), G STLS ( q ), and the neural-net representation of G ( q,
0) from Ref. [94], respectively. Green curves: θ = 4;red: θ = 1; black: θ = 0. from the exact asymptotic limit. Indeed, the violationof the CSR is a well-known shortcoming of the STLS ap-proach [108], which has ultimately led to the developmentof the approach by Vashista and Singwi [123, 124].The right panel of Fig. 14 shows the correspondingdata for the inverse dielectric function (cid:15) − ( q ). Here the static and ESA curves are in excellent agreement over theentire q -range, which, again, highlights the value of theanalytical parametrization which is capable to accurately describe both (cid:15) ( q ) and (cid:15) − ( q ) at the same time.Let us conclude this section with an example at strongcoupling, r s = 20 and θ = 1, depicted in Fig. 15. Firstly,we note that here the ESA and CSR curves for (cid:15) ( q ) di-verge towards negative infinity, which is the result ofa negative compressibility at these conditions, see alsoRefs. [89, 108]. For completeness, we note that this is anecessary, but not sufficient condition for instability [1],and, thus, not problematic. The STLS curve, too, di-verges towards negative infinity, although with a substan-tially different slope. Finally, the static curve becomesincreasingly inaccurate for small q and again attains afinite value for q = 0.Regarding the inverse dielectric function (right panel),the negative compressibility is reflected by a nontrivialshape of this quantity, with a minimum around q ≈ . q F .Here, too, we note that ESA and the static curve arein excellent agreement everywhere, whereas the STLSapproximation gives a substantially wrong prediction ofboth the location and the depth of the minimum in (cid:15) − ( q ). F. Dynamic structure factor
The final property of the UEG to be investigated inthis work is the dynamic structure factor S ( q, ω ), whichis shown in Fig. 16 for θ = 1. The left panel corre-sponds to the usual metallic density, r s = 2, and the dot-ted green curves are ab initio PIMC results taken fromRef. [96] that have been obtained by stochastically sam-pling the dynamic LFC G ( q, ω ). In addition, the solidred and dashed black curves have been obtained by us-ing the ESA and the static approximation , and are in vir-tually perfect agreement to the PIMC data everywhere.This illustrates that a static description of the LFC isfully sufficient to describe the dynamic density responseof electrons at these conditions, see also Refs. [89, 95–97]for more details.The right panel corresponds to a stronger couplingstrength, r s = 10, which is located at the margins ofthe electron liquid regime. While the ESA and staticapproximation here, too, basically give the same results,both curves exhibit systematic deviations towards the ex-act PIMC data. This is a direct consequence of the in-creased impact of the frequency-dependence of electronicexchange–correlation effects expressed via the dynamicLFC at these conditions [96].Interestingly, the impact of the dynamic LFC onlymanifests in a pronounced way in the shape of S ( q, ω ),whereas its normalization [i.e., the SSF, see Eq. (7)] ishardly affected. This is demonstrated in Fig. 17, wherewe show the corresponding S ( q ) for the same conditions.For example, for both q = 1 . q F and q = 1 . q F , theshape of the PIMC data for S ( q, ω ) significantly deviatesfrom the other curves, whereas the SSF is nearly perfectlyreproduced by both the ESA and the static approxima-tion .6 r s =5, θ =1 ε q/q F ESAstaticCSRSTLS r s =5, θ =1 ε - q/q F ESAstaticCSRSTLS
FIG. 14. Left: Static dielectric function (cid:15) ( q ) for r s = 5 and θ = 1. Solid red: ESA; dashed grey: exact static limit usingthe neural-net from Ref. [94]; dash-dotted green: CSR, Eq. (15); dotted-blue: STLS [105, 108, 110]. Right: Same data for theinverse dielectric function (cid:15) − ( q ). -200-100 0 100 200 300 0 0.5 1 r s =20, θ =1 ε q/q F ESAstaticCSRSTLS -1-0.5 0 0.5 1 0 1 2 3 4 r s =20, θ =1 ε - q/q F ESAstaticCSRSTLS
FIG. 15. Left: Static dielectric function (cid:15) ( q ) for r s = 20 and θ = 1. Solid red: ESA; dashed grey: exact static limit usingthe neural-net from Ref. [94]; dash-dotted green: CSR, Eq. (15); dotted-blue: STLS [105, 108, 110]. Right: Same data for theinverse dielectric function (cid:15) − ( q ). For larger q , the results for the SSF of G ( q ) and G ESA ( q ) do start to deviate, but this has no pronouncedimpact on S ( q, ω ) itself.We thus conclude that both the usual static approxi-mation and our new ESA scheme [100] are equally wellsuited for the description of dynamic properties at WDMconditions, but are not suited for a qualitative descriptionof the dynamic density response of the strongly coupledelectron liquid regime, for which a fully dynamic localfield correction has been shown to be indispensable. G. Test charge screening.
According to linear response theory, the screened po-tential of an ion (with charge Ze ) can be computed usingthe static dielectric function as [88, 125]:Φ( r ) = (cid:90) d q (2 π ) πZeq e i q · r (cid:15) ( q ) , (31)which is valid for the weak electron-ion coupling. Thelatter condition is satisfied at large distances from theion [126].As discussed in Sec. IV E above, the violation of the ex-act limit Eq. (29) leads to the unphysical behavior of the7
0 1 2 3 0 3 6 9 12 15 0.1 0.2S(q, ω ) r s =2, θ =1 ESAstaticPIMCq/q F ω / ω p
0 1 2 3 0 1 2 3 4 5 6 7 0.5S(q, ω ) r s =10, θ =1 ESAstaticPIMCq/q F ω / ω p FIG. 16. Dynamic structure factor of the uniform electron gas at θ = 1 for r s = 2 (left) and r s = 10 (right). Solid red:ESA; dashed black: static approximation ; dotted green: ab initio reconstructed PIMC results using a stochastically sampleddynamic LFC, taken from Ref. [96]. r s =10, θ =1 S ( q ) q/q F FIG. 17. Static structure factor of the UEG for r s = 10 and θ = 1 (cf. right panel of Fig. 16). Green squares: PIMC datataken from Ref. [96]; solid red: ESA; dashed black: static ap-proximation ; dotted blue: STLS [105, 108, 110]; dash-dottedyellow: RPA. static dielectric function computed using the neural-netrepresentation of the LFC from Ref. [94]. This results inincomplete screening when the corresponding static di- electric function is used to compute the screened poten-tial. To illustrate this, we show the screened ion potential(with Z = 1) for r s = 2, θ = 0 . θ = 1 . ∼ /r asymptotic behavior at large dis-tances. In contrast, the screened potential obtained us-ing the analytical representation G ESA ( q ; r s , θ ) correctlyreproduces complete screening like RPA based data,with a Yukawa type exponential screening at large dis-tances [126]. Finally, we note that electronic exchange–correlation effects, taken into account by using the LFC,lead to a stronger screening of the ion potential comparedto the RPA result [88, 126, 127]. H. Stopping power
A further example for the application of the LFC isthe calculation of the stopping power, i.e. the mean en-ergy loss of a projectile (an ion) per unit path length,and related quantities such as the penetration length,straggling rate etc. These energy dissipation character-istics are of paramount importance for such applications8
FIG. 18. Screened ion potential at r s = 2, θ = 0 . θ = 1. Solid red: the data computed using the analyticalrepresentation of the ESA Eq. (22); dashed black: static ap-proximation computed using the neural-net representation ofthe static local field correction from Ref. [94]; dotted blue:RPA result; dashed grey line shows ∼ /r behavior of theneural-net representation based data at large distances. as ICF and laboratory astrophysics [128, 129]. A linearresponse expression based on the dynamic dielectric func-tion that describes the stopping power for a low-Z pro-jectile when the ion–electron coupling is weak [130, 131]is given by [130]: S ( v ) = 2 Z e πv (cid:90) ∞ d kk (cid:90) kv d ω ω Im (cid:20) − (cid:15) ( k, ω ) (cid:21) , (32)where v is the ion velocity.Recently, using Eq. (32), the neural-net representationof the LFC [94] was used to study the ion energy-losscharacteristics and friction in a free-electron gas at warmdense matter conditions [85]. Therefore, it is required tocheck whether the discussed unphysical behavior of cer-tain quantities based on the neural-net representation ofthe LFC [94] also manifests in the stopping power. Thecomparison of the ESA (22) based data for the stoppingpower to the results obtained using the neural-net repre-sentation of the LFC [94] is shown in Fig. 19 for r s = 2, θ = 0 . θ = 1 .
0. From Fig. 19 we see that the ESAand the neural-net representation based results for thestopping power are in agreement with a high accuracy.Additionally, a comparison to the RPA based data showsthat electronic exchange-correlation effects are significantat projectile velocities v (cid:46) v F . We refer an interested FIG. 19. Stopping power at r s = 2, θ = 0 . θ = 1.Solidred: data computed using the analytical representation of theESA Eq. (22); dashed black: static approximation computedusing the neural-net representation of the static local fieldcorrection from Ref. [94]; dotted blue: RPA result. The lower x axis corresponds to v/v th and the upper x axis to v/v F , with v th and v F being the thermal velocity and Fermi velocity ofelectrons, respectively. reader to Ref. [85] for a more detailed study in a widerparameter range. V. SUMMARY AND DISCUSSIONA. Summary
The first main achievement of this work is the con-struction of an accurate analytical representation of the effective static approximation for the local field correction G ESA ( q ; r s , θ ) covering all wave-numbers and the entirerelevant range of densities (0 . ≤ r s ≤
20) and temper-atures (0 ≤ θ ≤ G invert ( q ) that,when being inserted into Eq. (1), exactly reproduces thestatic structure factor S ( q ) known from QMC calculationboth in the ground state and at finite temperature. Re-markably, G invert ( q ) almost exactly follows G ESA ( q ) forall wave numbers, which further substantiates the qual-ity of the relatively simple idea behind the ESA. As it isexpected, the latter gives very accurate results both for S ( q ) and the interaction energy v , in particular at metal-lic densities where we find relative deviations to PIMC9data not exceeding 1%.A further point of interest is the utility of the ESAregarding the estimation of the static density responsefunction χ ( q ) and the directly related dielectric func-tion (cid:15) ( q ). More specifically, the neural-net representa-tion of the exact static LFC G ( q ; r s , θ ) should give ex-act result for this quantities, whereas the definition of G ESA ( q ; r s , θ ) as a frequency-averaged LFC could poten-tially introduce a bias in this limit. Yet, we find that theESA gives virtually exact results over the entire WDMregime (even in the ground-state), whereas said bias onlymanifests in χ ( q ) for the strongly coupled electron liq-uid regime, r s = 20. In addition, the exact incorpora-tion of the CSR for small q in our parametrization of G ESA ( q ; r s , θ ) means that the present results for the di-electric function (cid:15) ( q ) are even superior to the correspond-ing prediction by the neural net, where the CSR is onlyfulfilled approximately, i.e., with finite accuracy. In par-ticular, the ESA gives the correct divergence behaviourof (cid:15) ( q ) in the limit of small q , whereas the neural-net pre-dicts a finite value for q = 0, which is unphysical [1, 89].A third item of our analysis is the application of theESA for the estimation of the dynamic structure fac-tor S ( q, ω ), where we find no difference to the usual static approximation [89, 95, 96]. More specifically,both G ( q ; r s , θ ) and G ESA ( q ; r s , θ ) are highly accurateat WDM densities, but cannot reproduce the nontrivialshape of S ( q, ω ) associated with the predicted incipientexcitonic mode [26, 132] in the electron liquid regime.Furthermore, we have compared our parametrizationof G ESA ( q ; r s , θ ) and the neural-net representation of G ( q ; r s , θ ) regarding the construction of an electronicallyscreened ionic potential Φ( r ). While the resulting poten-tials are in excellent agreement for small to intermediatedistances r , the aforementioned inaccuracies of the neu-ral net at small q lead to a spuriously slow convergenceof Φ( r ) at large ionic separations r .Finally, the stopping power calculation results showthat the ESA and the neural-net representation of theLFC are equivalent for this application. Therefore, boththe presented analytical fit formula for the ESA andthe neural-net representation of the LFC can be usedto study ion energy-loss in WDM and hot dense matter. B. Discussion and outlook
The ESA scheme has been shown to give a highly reli-able description of electronic XC-effects and, in our opin-ion, constitutes the method of choice for many applica-tions both in the context of WDM research and solidstate physics in the ground state.Due to its definition as a frequency-averaged LFC, theESA is particularly suited for the construction of ad-vanced XC-functionals for DFT simulations based on the adiabatic connection and the fluctuation dissipation the-orem [79–82]. This is a highly desirable project, as thepredictive capability of DFT for WDM calculations isstill limited [118].Secondly, we mention the interpretation of XRTSexperiments [77, 78] within the Chihara decomposi-tion [133] where electronic correlations are often treatedinsufficiently. In this regard, the remarkable degree ofaccuracy provided by both ESA and the static approxi-mation , and the promising results for aluminum shownin Ref. [100] give us hope that an improved description ofXRTS signals can be achieved with hardly any additionaleffort.Thirdly, the ESA can be used to incorporate electronicXC-effects into many effective theories in a straightfor-ward way. Here examples include quantum hydrodynam-ics [90, 91, 134], average atom models [92], electronicallyscreened ionic potentials [127, 135, 136], and dynamicelectronic phase-field crystal methods [137].Finally, we mention the value of the LFC in generaland the ESA in particular for the estimation of a mul-titude of material properties like the electronic stoppingpower [85], thermal and electrical conductivities [89], andenergy relaxation rates [138–140].From a theoretical perspective, the main open chal-lenge is given by the estimation of the full frequency-dependence of the LFC G ( q, ω ), which is currently onlypossible for certain parameters [89, 95, 96]. One waytowards this goal would be the development of newfermionic QMC approaches at finite temperature, to es-timate the imaginary-time density–density correlationfunction F ( q, τ )–the crucial ingredient for the reconstruc-tion of both S ( q, ω ) and G ( q, ω ). Here the phaselessauxiliary-field QMC method constitutes a promising can-didate [66].A second topic for future research is given by thecomparison of G ESA ( q ; r s , θ ) to different dielectric the-ories [105, 108, 124, 141–143], in particular the recentscheme by Tanaka [141] and the frequency-dependentversion of STLS [144–146]. ACKNOWLEDGMENTS
We thank Jan Vorberger for helpful comments. Thiswork was partly funded by the Center for AdvancedSystems Understanding (CASUS) which is financed byGermany’s Federal Ministry of Education and Research(BMBF) and by the Saxon Ministry for Science, Cultureand Tourism (SMWK) with tax funds on the basis ofthe budget approved by the Saxon State Parliament. Wegratefully acknowledge CPU-time at the NorddeutscherVerbund f¨ur Hoch- und H¨ochstleistungsrechnen (HLRN)under grant shp00026 and on a Bull Cluster at the Centerfor Information Services and High Performace Comput-ing (ZIH) at Technische Universit¨at Dresden.0 [1] G. Giuliani and G. Vignale,
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