Insights into the Electron-Electron Interaction from Quantum Monte Carlo Calculations
IInsights into the Electron-Electron Interaction from Quantum Monte CarloCalculations
Carl A. Kukkonen , ∗ and Kun Chen † Center for Computational Quantum Physics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010 (Dated: January 27, 2021)The effective electron-electron interaction in the electron gas depends on both the density andspin local field factors. Variational Diagrammatic Quantum Monte Carlo calculations of the spinlocal field factor are reported and used to quantitatively present the full spin-dependent, electron-electron interaction. Together with the charge local field factor from previous Diffusion QuantumMonte Carlo calculations, we obtain the complete form of the effective electron-electron interactionin the uniform three-dimensional electron gas. Very simple quadratic formulas are presented for thelocal field factors that quantitatively produce all of the response functions of the electron gas atmetallic densities.Exchange and correlation become increasingly important at low densities. At the compressibilitydivergence at rs = 5.25, both the direct (screened Coulomb) term and the charge-dependent exchangeterm in the electron-electron interaction at q=0 are separately divergent. However, due to largecancellations, their difference is finite, well behaved, and much smaller than either term separately.As a result, the spin contribution to the electron-electron interaction becomes an important factor.The static electron-electron interaction is repulsive as a function of density but is less repulsive forelectrons with parallel spins.The effect of allowing a deformable, rather than rigid, positive background is shown to be asquantitatively important as exchange and correlation. As a simple concrete example, the electron-electron interaction is calculated using the measured bulk modulus of the alkali metals with a linearphonon dispersion. The net electron-electron interaction in lithium is attractive for wave vectors0 − k F , which suggests superconductivity, and is mostly repulsive for the other alkali metals. PACS numbers:
I. INTRODUCTION
The electron-electron interaction is important for pair-ing and superconductivity, spin and magnetic phenom-ena, transport properties, and as an input for numeri-cal calculations. Many interesting phenomena occur inexotic materials, some with reduced dimensionality andunusual topologies, and different theoretical approacheshave been employed to explain experiment.This paper looks back to the well-studied three-dimensional electron gas to examine the quantitative ef-fects of exchange and correlation on the electron-electroninteraction to see if any insight may be gained for prob-lems of modern interest.The equation for the spin dependent effective electron-electron interaction in the three-dimensional electron gasis well-established in the mean field local approximation,and is given in terms of the local field factors that de-fine all of the response functions of the electron gas [1].Sum rules specify the local field factors at small and largewave vector q , and the difficult problem is to calculatethe intermediate wave vector dependence. The Quan-tum Monte Carlo method is considered to produce accu-rate results. The density local field factor was calculatedmany years ago by numerous methods including Diffu- ∗ Electronic address: [email protected] † Electronic address: kunchen@flatironinstitute.org sion Quantum Monte Carlo. Results for the spin localfield factor using the Variational Diagrammatic MonteCarlo (VDMC) method were first published in 2019[2]and additional results are presented here. The spin lo-cal field factor completes the specification of the effectiveelectron-electron interaction and has motivated this pa-per. The local field factors are also known as exchangeand correlation kernels in Time-Dependent Density Func-tional Theory.The equation for the electron-electron interaction itselfis indicative, but it is difficult to understand the relativeimportance of the terms until the actual local field factorsare used to show quantitative results.The numerically calculated values for the density andspin local field factors are discussed, and it is shown thatthey can approximated by very simple formulas that pro-duce all of the response functions of the electron gas.
II. ELECTRON-ELECTRON INTERACTION
Kukkonen and Overhauser[1] (KO) demonstrated thatthe standard self-consistent perturbation theory based onHartree-Fock theory and linear response theory could beused to calculate the effective many-body interaction V ee between two electrons in a simple metal, modeled by theelectron gas, in terms of the density local field factor a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n G + ( q, ω ) and spin local field factor G − ( q, ω ). V e(cid:126)σ ,e(cid:126)σ = 4 πe q (cid:32) (cid:0) ω − ω (cid:1) / (cid:0) ω − ω q (cid:1) (1 − G + Q ) [1 + (1 − G + ) Q ] − G Q − G + Q − G − Q − G − Q(cid:126)σ · (cid:126)σ (cid:19) (1)This is the interaction to be used for calculating matrixelements between two electrons with momenta k and k and spins σ and σ . For parallel spins, the wavefunctions must be properly anti-symmetric.The electron gas is characterized by the density pa-rameter r s , ( n = 4 π ( r s a ) / a is the Bohr radius). Q = v Π where v = 4 πe /q is the Coulomb potentialand Π ( q, ω ) is the Lindhard function. For convenience,we will not usually explicitly present the wave vector andfrequency dependence. We will also not explicitly use theword effective in discussing interactions. With the de-formable background, the standard phonon frequenciesand background (lattice) screening results were obtainedand are represented by the frequency dependence of thefirst term in Eq. (1). The intuitive physics conceptsbehind the electron-electron interaction in Eq. (1) arepresented in Ref. [1]. The frequencies ω q and ω referto the frequency response of the deformable backgroundwhich is discussed in Ref. [1] and Section VII of thispaper.The first calculations of the screened interaction inthe electron gas were the Thomas Fermi interaction andits quantum mechanical extension by Lindhard[3]. TheLindhard result is recovered by setting both G ’s equal tozero which results in V Lindhard = 4 πe q (1 + Q ) (2)In this approximation, the electron-electron, electron-test charge and test charge-test charge interactions areall the same. These latter interactions are discussed inAppendix B.The KO electron-electron interaction has been verifiedby many-body calculations [4], and has been extendedto multicarrier [5], spin polarized and two-dimensionalsystems[3]. The potential for superconductivity with-out phonons using the KO interaction was examined byTakada[6] and by Richardson and Ashcroft[7]. Takadafound superconductivity for a single carrier system andRichardson and Ashcroft did not. Richardson andAshcroft stated that an important difference was thateach group used different values of G + ( q ) and G − ( q ).Richardson and Ashcroft calculated their own G ’s in-cluding frequency dependence. Takada used frequencyindependent G ’s. The G ’s considered here are the staticlocal field factors.Connecting with Feynman diagrams, the frequency in-dependent factor in the first term can also be writtenas Λ /(cid:15) – two vertex corrections divided by the dielec-tric function [8]. The deformable background (lattice) screens the first term, the Coulomb interaction, but notthe exchange and correlation and spin response termswhich arise from summing ladder diagrams.For a rigid lattice, the frequency dependence in the firstterm of Eq. (1) equals one, and the first term is repulsive.The second term is attractive. The spin dependent termis repulsive for opposite spins (singlet) and attractive forparallel spins (triplet). We initially consider the rigidbackground and discuss the deformable background insection VII below.Quantitative evaluation of the electron-electron inter-action is made in section V using the values of G + ( q )discussed in Appendix A and the new accurate values of G − ( q ) reported below. III. SPIN LOCAL FIELD FACTOR G − ( q ) A Quantum Monte Carlo method using renormalizedFeynman diagrams was developed by Chen and Haule[2].The resulting Variational Diagrammatic Monte Carlo(VDMC) method is a generic many-body solver that wastested on the electron gas. The VDMC method was usedto calculate the spin and density responses in the elec-tron gas. It is well suited for finite temperatures. Thedescription of the VDMC method, including grouping ofFeynman diagrams, numerical approach and high preci-sion results are given in Ref. [2].The data reported here are new VDMC calculationsof the static spin local field factor with higher accuracyfor densities r s = 1 − q = 0 − . k F , and addi-tional results for r s = 5. The calculation temperature is T = 0 . T F which is equivalent to T = 0. The methodprovides the highest accuracy calculations of the q = 0susceptibility, and our calculations at finite q also havehigh accuracy, but less than at q = 0. Typical error barsare shown with the data.Figure 1 shows the wave vector dependence of G − ( q )which demonstrates that it initially follows the quadraticbehavior required by the susceptibility sum rule. Thebehavior changes dramatically near q = 2 k F .The susceptibility enhancement at q = 0 was calcu-lated more accurately than the values at finite q . Theresults from Ref.[2] and the new result at r s = 5 arereported in Table I. r s χ/χ q = 0 for the three-dimensional electron gas calculated by Variational Diagram-matic Monte Carlo method. Uncertainty is indicated by thenumber in parentheses. In order to clearly see the small q behavior andthe effect of the susceptibility sum rule, the quantity G − ( q ) / ( q/q T F ) is plotted in Fig. 2. The Thomas Fermiscreening wave vector is defined by q T F = 4 k F /πa . The . . . . . q/k F . . . . . G ° ( q / k F ) r s = 1 r s = 2 r s = 3 r s = 4 r s = 5 FIG. 1: Spin local field factor G − ( q ) versus q/k F for r s =1 − spin exchange and correlation kernel for Time DependentDensity Function Theory is f spin xc = − πG − ( q ) / ( q/q T F ) . . . . . . q/k F . . . . . G ° ( q / k F ) / ( q / q T F ) r s = 1 r s = 2 r s = 3 r s = 4 r s = 5 FIG. 2: Spin local field factor divided by the wave vector di-vided by Thomas Fermi wave vector squared, G − ( q ) / ( q/q TF ) versus q/k F for r s = 1 −
5. Error bars are shown by shading.
Figure 2 shows that the spin local field factor G − ( q )follows the quadratic well and does not fall below thequadratic behavior until near 2 k F . In fact, for high den-sity r s = 1, G − ( q ) rises significantly above the quadraticbefore it falls below. The close adherence to the quadraticbehavior, suggests that in the metallic region r s = 2 − χ ( q ) χ ( q ) = 11 − G − Q , (3)and shown in Figure 3. . . . . . q/k F . . . . ¬ ( q / k F ) / ¬ ( q / k F ) r s = 1 r s = 2 r s = 3 r s = 4 r s = 5 FIG. 3: Susceptibility enhancement χ ( q ) /χ plotted versus q/k F for r s = 1 −
5. The data points use the actual valuesof G − ( q ) calculated here and reported in Figure 1. The solidlines are the simple quadratic function Eq. (5) set by thesusceptibility sum rule at q = 0. Error bars are shown byshading. Note that the Y axis starts at 1.0. The q = 0 value of the susceptibility enhancementis entirely set by the susceptibility sum rule. The en-hancement is modest because there is no divergence inthe susceptibility near the metallic region. The simplequadratic, which is the horizontal line in Fig. 2, fits thedata quite well and is adequate for the discussions in thispaper and for comparison with experiment. If higher ac-curacy is needed, the actual data shown in Figure 2 canbe used.The application of the VDMC method to the densitylocal field factor G + ( q ) is briefly discussed in AppendixA. IV. SIMPLE EXPRESSIONS FOR LOCALFIELD FACTORS G + ( q ) AND G − ( q ) The recommended simple quadratic forms for G + ( q )and G − ( q ) are: G + ( q ) = (cid:16) − κ κ (cid:17) (cid:18) qq T F (cid:19) (4) G − ( q ) = (cid:18) − χ χ (cid:19) (cid:18) qq T F (cid:19) . (5)These expressions are exact at small q and accuratelyrepresent the QMC data up to almost q = 2 k F for themetallic region r s = 2 − G + ( q ) is discussed in Ap-pendix A. Although these simple quadratic approxima-tions are not accurate beyond 2 k F , they are suitable forthe electron gas response functions which are cut off bythe Lindhard function above q = 2 k F . For any applica-tion that requires values of G at larger q , we recommendthe interpolation formula discussed in Appendix A for G + or the actual data above for G − .The simple quadratic approximation to G − ( q ) given inEq. (5) and used to calculate the susceptibility enhance-ment in Fig. 3 fits the VDMC data quite well. The fitis exact at q = 0, falls below by 2% at q = 1 . k F andslightly above at 2 k F . The average values are within1%. The susceptibility is the product of the enhance-ment times the Lindhard function and the Bohr mag-neton. The falloff of the Lindhard function above 2 k F makes this region unimportant for most applications.The recommended values of the compressibility aretaken from Perdew and Wang[9] and susceptibility ratiosare given in Table I. Both are plotted in Figure 4 and areaccurately fitted by quadratic interpolation formulas. -0.10.00.10.20.30.40.50.60.70.80.91.00 1 2 3 4 χ / χ κ / κ r s FIG. 4: Compressibility ratio κ /κ and susceptibility ratio χ /χ for the three-dimensional electron gas with a rigid uni-form positive background. The curves in Fig 4 are fits to the data in the metallicregion and fit the data to less than 0 . r s = 0. Since we are only interested in the metallic region r s = 1 −
5, the fitting curves were not required to havean intercept of 1 at q = 0, which results in a simpler andmore accurate equations in the metallic region.The compressibility and susceptibility ratios at q = 0are well fitted from r s = 1 − χ χ = 0 . − . r s + 0 . r s (6) κ κ = 1 . − . r s − . r s (7)With these G ’s and the compressibility and susceptibil-ity ratios, all of the response functions for the three- dimensional electron gas with a rigid background canbe quantitatively calculated. The same approach can beused for a spin polarized or two component electron gas.Figure 4 shows the well-known divergence and signchange of the compressibility at r s = 5 .
25. This causesthe vertex function and thus the dielectric function todiverge and become negative. The rigid uniform positivebackground prevents the overall model electron gas frombecoming unstable.
V. ELECTRON-ELECTRON INTERACTION:QUANTITATIVE RESULTS
Using G + ( q ) and G − ( q ), we plot the electron-electroninteraction. We plot each term in Eq. (1) separately toshow their relative importance. These are denoted V ee V ee V ee
3. The first term V ee V ee V ee V ee
3, thespin dependent term is positive (repulsive) for oppositespins (singlet) and negative (attractive) for parallel spins(triplet).These three terms are plotted together in Fig. 5 for r s = 2 and 5.The magnitude of the first term V ee q = 0 is( κ/κ ) /q T F which is divergent at the compressibility di-vergence. The results look “normal” at r s = 2 wherethe second and third terms are small corrections to V ee V ee r s = 2).At r s = 5, V ee V ee r s at the compressibility divergence approximatelyas 1 / (1 − r s / . V ee /(cid:15) ) V ext = Λ V et .This apparent divergence is one concern.The second term V ee V ee V ee
1. The fact that this term is large bringsinto question the use of perturbation theory and linearresponse. However, upon closer examination, V ee V ee V ee − V ee q = 0 is given as V ee − V ee κκ (cid:18) − ( κ/κ ) q T F (cid:19) = 2 − κ /κq T F . (8)For a rigid background, the net spin independent termsof the electron-electron interaction are completely well (a)(b) V ee ( q / ) π e ( - c m ) V ee ( q / ) π e ( - c m ) q/k F q/k F V ee V ee V ee V ee V ee V ee r s =2 r s =5 FIG. 5: The three terms in the equation for V ee ( q ) for r s = 2(a) and r s = 5 (b). The first term V ee V ee V ee πe . behaved and have no divergence at the compressibilitydivergence. This was also observed by KO[1]. The de-formable lattice will be discussed in Section VII below.Figure 6 shows the net electron-electron interaction forelectrons with parallel and opposite spins. This is theinteraction to be used for calculating matrix elementsand superconductivity. V ↑↓ ee = V ee − V ee V ee V ↑↑ ee = V ee − V ee − V ee V ee − V ee (a)(b) V Lindhard r s =2 V ee V ee ↑↓↑↑ V Lindhard V ee V ee V ee ( q / ) π e ( - c m ) V ee ( q / ) π e ( - c m ) q/k F q/k F r s =5 FIG. 6: The net electron-electron interactions V ee ( q ) for op-posite and parallel spins compared to the Lindhard potentialat r s = 2 (a) and 5 (b). A rigid background is assumed forthe electron gas. overall electron-electron interaction is smooth and has noremaining evidence of the large effects in V ee V ee r s = 2, V ee V ee r s = 5, the first two terms nearlycancel, and V ee
3, the spin dependent term, is relativelyimportant. The overall electron-electron interaction isconsiderably less repulsive for parallel spins.The values of the electron-electron interaction at q = 0are completely determined by the compressibility andsusceptibility sum rules. At large q (short distances), theelectron-electron interaction follows the Lindhard func-tion to the bare Coulomb interaction. The Lindhard in-teraction is shown for comparison.This quantitative evaluation of the electron-electroninteraction shows that with a rigid background, thestatic electron-electron interaction is well behaved andrepulsive throughout the metallic region. The electron-electron interaction (and the other interactions in theelectron gas discussed in Appendix B) are completelyspecified with simple equations provided in this paper.However, care must be taken with quantitative compari-son with experiment. The effective mass, renormalizationfactor z , core polarization and a deformable lattice canhave significant effects. Some of these were discussed byKukkonen and Wilkins[8]. The deformable lattice will bediscussed in section VII. VI. EFFECT OF THE COMPRESSIBILITY SUMRULE
We have emphasized that the compressibility and sus-ceptibility sum rules, which are derived from changes inthe total electron gas energy, completely determine thevarious interactions at q = 0. This is illustrated Fig. 7where we plot the ratio of the different interactions tothe Lindhard interaction at q = 0 versus r s . -0.20.00.20.40.60.81.01.21.41.61.82.02.22.40.0 1.0 2.0 3.0 4.0 5.0 6.0 V ( q = ) / V L i ndh a r d ( q = ) r s ↑↓↑↑ V ee V ee V et V tt FIG. 7: The ratio of the electron gas interactions at q = 0 tothe Lindhard (Thomas Fermi) interaction as a function of r s for a rigid background. The top curve is the electron-electroninteraction for electrons with opposite spins. The next curvedown is the electron- electron interaction for parallel spins.The electron-test charge interaction V et at q = 0 is equal tothe Thomas Fermi interaction at all r s . The bottom curve isthe test charge-test charge interaction V tt which always fallsbelow Thomas Fermi interaction and becomes negative at thecompressibility instability. Figure 7 shows that all of the interactions are the sameand equal to the Lindhard and Thomas Fermi interac-tions at high density near r s = 0. At lower density(larger r s ), the effects of exchange and correlation mani-fest themselves through the sum rules. Only the electron-electron interaction depends on spin and that is shownby the two curves for parallel and opposite spins.With a rigid uniform positive background, theelectron-electron and electron-test charge interactionsshow no unusual behavior as r s approaches the com-pressibility divergence at r s = 5 .
25. However, the testcharge-test charge interaction at q = 0 becomes negative(attractive) above the instability. This is due to the factthat the dielectric function has become negative. Thisinteraction is discussed in Appendix B. The sum rules dictate the q = 0 behavior. At large q ,all the interactions fall off as 1 /q which reflects the bareCoulomb interaction at short distances, and the wavevector dependence of the local field factors and the cutoff of the Lindhard function at 2 k F set the intermediate q behavior which has been discussed above. VII. DEFORMABLE BACKGROUND
KO considered a smooth but elastically deformablebackground which led to the frequency dependence inEq.(1). The result was verified in Ref.[3]. Introducing thedeformable background naturally results in the phononfrequencies[1]. ω q = N q M (cid:32) V bare ii − (cid:0) V bare ei (cid:1) v + (cid:0) V bare ei (cid:1) v(cid:15) (cid:33) ≡ ω + N q (cid:0) V bare ei (cid:1) M v(cid:15) (11)Where N is the density and M is the mass of the back-ground (ions), V bare ii is the bare ion-ion interaction, V bare ei is the bare electron-ion interaction, and (cid:15) is the dielectricfunction.It is instructive to rewrite the frequency dependence inequation (1) as ω − ω ω − ω q = 1 + ω q − ω ω − ω q . (12)This now multiplies the first term. The factor 1 allowsthe frequency independent part of the first term andsecond term to be formally combined which cancels thecompressibility divergence in the static interaction, andEq.(1) can be rewritten as V e(cid:126)σ ,e(cid:126)σ = 4 πe q (cid:32) (cid:0) ω q − ω (cid:1) / (cid:0) ω − ω q (cid:1) (1 − G + Q ) [1 + (1 − G + ) Q ] +1 + (1 − G + ) G + Q − G + ) Q − G − Q − G − Q(cid:126)σ · (cid:126)σ (cid:19) (13)The new first term V ee − phonon is divergent at q = 0 atthe compressibility divergence, but the second two termshave no divergence. V ee − phonon represents the additional screening of theCoulomb interaction by the background (lattice). At ω =0, the numerator of the first term is negative which isthe expected result for static screening by the positivebackground. − ω q − ω ω q = − N q (cid:0) V bare ei (cid:1) (cid:46) ( M ve )( N q ) /M (cid:32) V bare ii − (cid:0) V bare ei (cid:1) v + (cid:0) V bare ei (cid:1) (cid:15) v (cid:33) (14)The screening depends on the stiffness of the backgroundrepresented by V bare ii and the properties of the electrongas. The rigid background is obtained when V bare ii (andthus ω q ) goes to infinity and this term goes zero.Another interesting limit occurs if all of the interac-tions including V b areii are Coulomb interactions. In thiscase − ω q − ω ω q = − V ee − phonon is large and negative (attractive) At q =0, the first term has the value set by the compressibilitysum rule as V ee − phonon ( q = 0) = − κκ V et ( q = 0)= − κκ V T F ( q = 0) (16)With all Coulomb interactions, this negative first term islarger than the other two terms combined and the overallelectron-electron interaction at q = 0 is attractive. Notethis term diverges at the compressibility divergence.An instructive intermediate case is to consider that V bare ei in the numerator of Eq.(14) is equal to the Coulombinteraction v = 4 πe /q , and to take ω q from experiment.The background (lattice) screening term can be rewrittenas V ee − phonon ( q ) = − ω q (cid:18) v ( (cid:15) (1 − G + Q )) (cid:19) N q M = − ω q V et ( q ) N q M (17)All of the effects of the deformable background are in ω q , the phonon frequencies which depend on the electrongas parameters as well as the bare ion-ion interaction.The electron test charge interaction appears because thelattice appears as a test charge to the electron gas. Theelectron test charge interaction is equal to the ThomasFermi and Lindhard interactions at q = 0, but differs atlarger q as shown in Appendix B.As the simplest example, we model the phonons by therelationship between ω q and the bulk modulus B whichis the inverse of the compressibility κ . ω q = Bq N M (18)where
N M is the mass density of the background, and (cid:112)
B/N M is the speed of sound. Using this relation-ship and the bulk modulus of the non-interacting elec-tron gas B = 1 / ( n V T F ( q = 0)), and the fact that theion density and the electron density are the same for themonovalent alkali metals, one obtains the equation forthe background screening contribution to the electron-electron interaction V ee − phonon ( q ) = − V et ( q )) V et (0) B B experiment (19) The measured bulk moduli, free electron values[10],and their ratios for the alkali metals are given in TableII. r s B B experiment B /B experiment Li 3.25 23.9 11 2.173Na 3.93 9.23 6.3 1.465K 4.86 3.19 3.1 1.029Rb 5.2 2.28 2.5 0.912Cs 5.62 1.54 1.6 0.963TABLE II: Free electron and experimentally measured bulkmoduli for the alkali metals. V ee ( q = ) / V L i ndh a r d ( q = ) r s -1.0-0.50.00.51.01.52.02.50.0 1.0 2.0 3.0 4.0Li Rb CsKNa 5.0 6.0 ↑↓↑↑ V ee V ee Rigid backgroundDeformable background
FIG. 8: Static ( ω = 0) electron-electron interaction at q = 0for the electron gas at the density of alkali metals (in unitsof the Lindhard (Thomas Fermi) interaction). The top twocurves are the electron-electron interaction for a rigid back-ground with opposite spins, and parallel spins. The data atthe lower right are the electron-electron interaction includ-ing the deformable background screening for the alkali metalsusing the measured bulk modulus for opposite and parallelspins. In Figure 8, the repulsive electron-electron interactionsat q = 0 for opposite and parallel spins in a rigid back-ground is plotted versus r s as in Figure 7. Also plotted isthe net electron-electron interaction including screeningby the deformable background using the experimentallymeasured bulk modulus for the alkali metals.The electron-electron interactions for a rigid back-ground are simply the second two terms of Eq. (13)evaluated at q = 0. These are completely specified bythe compressibility and susceptibility sum rules, and thevalues are given in Fig. 4. V ↑↓ ee (0) = (cid:34)(cid:16) − κ κ (cid:17) + χχ (cid:18) − χ χ (cid:19) (cid:35) V Lindhard (0)(20) -0.3-0.2-0.10.00.10.20.30.40.50.60.70.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 q/k F (a) V ee ( q / ) π e ( - c m ) ↑↓↑↑ V ee V ee ↑↓↑↑ V ee V ee r s = 3.25 Lithium Rigid backgroundDeformable background -0.2-0.10.00.10.20.30.40.50.60.70.80.9 q/k F (b) V ee ( q / ) π e ( - c m ) Deformable background r s = 3.93 Sodium Rigid background
FIG. 9: Electron-electron interaction at densities of lithium(a) and sodium (b) comparing the repulsive interaction calcu-lated for a rigid background to the net interaction when thedeformable background is included using the measured bulkmodulus. The top two curves are for the rigid lattice for op-posite and parallel spins. The bottom two curves subtract thedeformable background contribution from the top two curves.They represent the net electron-electron interaction. V ↑↑ ee (0) = (cid:34)(cid:16) − κ κ (cid:17) − χχ (cid:18) − χ χ (cid:19) (cid:35) V Lindhard (0)(21)The attractive background screening contribution to theelectron-electron interaction is given by Eq. (19) evalu-ated at q = 0. The electron-test charge interaction V et ( q )is given in Eq. (B3) in Appendix B, and is equal to theLindhard and Thomas Fermi potentials at q = 0. V et − phonon (0) = (cid:20) B B experiment (cid:21) V Lindhard (0) (22)This term is spin independent and is subtracted fromboth the opposite and parallel spin electron-electroninteractions for a rigid background and net electron-electron interaction is plotted at the r s values of the alkalimetals.Note that all of the electron-electron interactions at -10.0-5.00.05.010.015.00.0 0.5 1.0 1.5 2.0 (a) (b) ( r ) ( e V ) V ee ( r ) ( e V ) V ee r s = 3.25 Lithium Rigid backgroundDeformable background ↑↓↑↑ V ee V ee r /( r s a ) r /( r s a ) -5.00.05.010.0 ↑↓↑↑ V ee V ee Deformable background r s = 3.93 Sodium Rigid background
FIG. 10: V ee ( r ), the Fourier transform of V ee ( q ), the electron-electron interaction shown in Fig. 9, at the densities oflithium (a) and sodium (b). This figure compares the repul-sive interaction calculated for a rigid background to the netelectron-electron interaction when a deformable backgroundis included. The top two curves are for the rigid backgroundfor opposite and parallel spins. The bottom two curves in-clude the deformable background contribution. q = 0 are proportional to V Lindhard ( q = 0) = 1 /q T F = r s a / (2 . in units of 4 πe . The electron-electron in-teractions divided by V Lindhard ( q = 0) at each r s are plot-ted in Fig. 8 which shows the relative importance of ex-change and correlation and the deformable background.Figure 8 shows that the effects of screening by the de-formable background are as large as the effects of ex-change and correlation. The q = 0 value of the netelectron-electron for lithium is attractive for both paral-lel and opposite spins, while all of the other alkali metalsare repulsive. Lithium is the only alkali metal that ex-hibits superconductivity at ambient pressure. Althoughthe net electron-electron for lithium is more attractivefor parallel spins than opposite spins, this does not nec-essarily imply triplet pairing because the spatial part ofthe overall wave function must be anti-symmetric. Thisis discussed briefly after Fig. 10.The wave vector dependent electron-electron interac-tion for this simple model of lattice screening in the alkalimetals can be calculated using Eq. (13), Eq. (19) andEq. (B3). This assumes the linear phonon dispersion re-lation of Eq. (18). A linear phonon dispersion should bethe correct behavior at small q , but overestimates ω q as q approaches 2 k F (and a reciprocal lattice vector). Since ω q is in the denominator, a smaller value near 2 k F wouldimply an even more attractive potential. The results forlithium and sodium are shown in Fig. 9.The electron-electron interaction in a rigid backgroundis repulsive. The repulsion is less for parallel spins thanfor opposite spins. The deformable background at lowfrequencies contributes a negative term due the addi-tional screening by the deformable background/lattice.The background screening is due to the net Coulomb in-teraction and is independent of spin for a non-magnetizedelectron gas.The most interesting feature is that the net electron-electron interaction in lithium is attractive from q = 0 toabove 2 k F , for both parallel and opposite spins. A morecareful treatment of the background/phonons is needed,but this result is likely to be qualitatively correct. Thestrong attractive region in lithium may explain the sourceof the observed superconductivity.The other alkali metals with lower densities have morerepulsive electron-electron interactions and smaller ef-fects of the deformable lattice as measured by the bulkmodulus. The lattice screening is still a significant ef-fect. The repulsive interaction in the rigid lattice fallsoff more quickly with wave vector than the attractivecontribution from the deformable background. The re-sulting net electron-electron interaction has a minimumnear 1 . − k F . This minimum is slightly repulsive foropposite spins, and slightly attractive for parallel spinsfor all the alkali metals. The small attraction for parallelspins near 2 k F may lead to interesting physics. Sodiumis shown in Fig. 9. The curves for the other alkali metalsare similar.Figure 10 shows V ee ( r ), the Fourier transform of theelectron-electron interactions in Fig 9. The energy scaleis eV , and the Fermi energy for lithium is 4 . eV and forsodium it is 3 . eV . With a rigid lattice, the electron-electron interaction is repulsive as shown in the top twocurves of Figure 10. The classical turning points for scat-tering, where the repulsive potential is equal to the Fermienergy, are approximately 0 . r s for lithium and 0 . r s for sodium. Electron-electron scattering does not con-tribute to the electrical resistivity because momentumand charge are conserved in the scattering event (exceptfor a small effect due to umklapp scattering in a real-lattice as opposed to a uniform background). Electron-electron scattering does however contribute to the ther-mal resistivity.When the deformable background is included, the over-all interaction potential for lithium is attractive for bothparallel and opposite spins in the region 0 . − . r s witha depth comparable to the Fermi energy, with a repulsivecore at shorter distances. At larger distances, not shownin Fig 10, much smaller oscillations similar to Friedeloscillations are seen. The interaction or scattering oftwo electrons in the electron gas with an attractive po- tential due to the deformable background could includeresonances, virtual bound states or even bound states.We note that although the electron-electron attractionis stronger for parallel spins, the overall wave functionwhich must be anti-symmetric would imply that the spa-tial part is anti-symmetric (p-wave) and the probabilitythat the electron is found at small distances is lower.Thus it is likely that the opposite spin electrons with asymmetric (s-wave) wave function would sample more ofthe region of the attractive potential.For sodium, only the electron-electron interaction forparallel spins becomes significantly attractive.We present this Fourier transform real space calcula-tion to stimulate thinking within the single particle pic-ture of the electron gas– which has limitations. The usualthinking is in momentum and frequency space. Equation(13) gives the explicit frequency dependence. The lo-cal field factors used here are independent of frequency,but the frequency dependence of the Lindhard functionis known.The electron-electron interaction in this paper can beused in calculations of superconductivity. The simple in-clusion of the deformable lattice shows that lithium issubstantially different from the other alkali metals, andlithium is the only alkali that shows superconductivityat ambient pressure. Cesium exhibits superconductivityat high pressure. The net electron-electron interactionis sensitive to exchange and correlation and to the de-formable lattice. Richardson and Ashcroft[11] consideredsuperconductivity with the KO electron-electron interac-tion and phonons treated on an equal basis, and appliedthe theory to several metals including lithium. The con-siderable difficulties in comparison with experiment arediscussed in that paper.The region of largest static attraction between twoelectrons in lithium is at very short distances. The roleof this short range attraction in the dynamical theory ofsuperconductivity is not known to us.A strong word of caution is needed when comparingelectron gas calculations with experiment, particularlyat low density close to the compressibility divergence at r s = 5 .
25. Factors such as effective mass, core polar-ization and renormalization factor z have to be carefullyconsidered when comparing with experiment. These ef-fects were discussed in Kukkonen and Wilkins[8] and inRef[11]. A simple example is Cesium with r s = 5 . (cid:15) B that is fre-quency independent in the regions of interest. The cor-rect theory[8] is an electron gas with electrons of effectivemass m ∗ in a neutralizing uniform positive backgroundwith dielectric constant (cid:15) B . The result is obtained byscaling the known solutions for the electron gas with mass m and a non-polarizable background, but at a differentdensity r ∗ s = r s ( m ∗ /m(cid:15) B ) and evaluated at a scaled wavevector. For cesium, (cid:15) B = 1 .
27 and this re-normalizes r s from 5.62 to 4.44 which is below the compressibility di-0vergence. The core polarization corrections for lithiumand sodium are quite small. VIII. SUMMARY
Variational Diagrammatic Monte Carlo (VDMC) cal-culations of the wave vector dependent spin local fieldfactor (exchange and correlation kernel) have been pre-sented and utilized for all calculations. Using the densityand spin local field factors and explicit equations pre-sented in this paper, all of the response functions of thethree-dimensional electron gas can be easily and quan-titatively calculated. Exchange and correlation are fullyincluded within the self-consistent local field approxima-tion and the compressibility and susceptibility sum rulesat q = 0 are satisfied.The full spin dependent electron-electron interactionis calculated using these field factors. For a rigid back-ground, the electron-electron interaction shows no diver-gence at the compressibility divergence, is repulsive forall r s in the metallic region.Considering a deformable background, the ω = 0(static) screening by the background is shown to be veryimportant with an effect that is as large as exchange andcorrelation. A simple calculation shows that with a de-formable background modelled by using the measuredbulk modulus, the net electron-electron interaction in-cluding exchange and correlation is attractive (negative)in a large range of momentum and real space for lithiumwhich does exhibit superconductivity at ambient pres-sure, and is mostly positive (repulsive) for all the otheralkali metals.The compressibility divergence does not appear in theelectron-electron and electron-test charge interactions,but still shows up in the test charge-test charge inter-action as a divergence in the dielectric function and inthe screening by the deformable background.The quantitative spin dependent electron-electron in-teraction can be used in other calculations such as super-conductivity, and can be also used as a starting point forimproved numerical calculations. The self-consistent lo-cal field calculation of the KO interaction using Feynmandiagrams may lead to new techniques for identifying mas-sive cancellations of divergent diagrams and allow newperturbation techniques around the self-consistent solu-tion.More detailed numerical calculations of the density lo-cal field factor G + ( q ) from q = 0 to 3 k F would be wel-comed, as well as a physical explanation for the behavior. Acknowledgements
The authors are grateful to William Halperin, JanHerbst, Giovanni Vignale and David Ceperley for dis-cussions, and to Giovanni Onida and Massimiliano Cor-radini for providing the exact coefficients for their an- alytic function for G + ( q ) that fits the QMC data, andto Tori Hagiya for comparing his experimental data withour theory.C.A.K. acknowledges support from the US Social Secu-rity Administration. K.C. appreciates support from theSimons Foundation. Appendix A: DENSITY LOCAL FIELD FACTORG + ( q ) The density local field factor has been a subject of re-search for more than 60 years. G + ( q ) is needed to calcu-late the dielectric function and vertex correction. Thesequantities are sufficient to calculate all of the interactionsand response functions that do not depend on spin. Manyproposals have been made for the dielectric function and G + ( q ) particularly by Hubbard[12], Geldart[13] and col-laborators, and Singwi[14] and collaborators. When itwas realized that the compressibility sum rule was ofparamount importance at q = 0, the dielectric functionsand G ’s were manually adjusted to satisfy the sum ruleeven though the calculations themselves did not satisfythe sum rule. The calculations were used to interpolatethe wave vector dependence above q = 0. The behaviorof G + ( q ) at 2 k F and above has been the subject of con-siderable research and debate, but is not important forthe calculations of the static response properties of theelectron gas because the Lindhard function cuts off theresponse quickly above 2 k F .The Quantum Monte Carlo method was used by Mo-roni, Ceperley and Senatore[15] to calculate G + ( q ) from q = k F to q = 4 k F for r s = 2 , q = k F . Richardsonand Ashcroft[17] calculated the local field corrections atfinite frequencies, and their static results are similar, butdiffer somewhat in detail from the QMC results. Theyemphasized the importance of the sum rules. Richard-son and Ashcroft also provided an interpolation formula.Retrospective discussions of the local field factors weregiven by Simion and Giuliani[18] and by Hellal, Gasserand Issolah[19].We plot the Quantum Monte Carlo results of Ref. [12]for G + ( q ) together with the analytic interpolation func-tion Ref.[13], and the even simpler quadratic formula thatquite accurately reproduces the response functions of theelectron gas in Fig. 11. Although there are no data be-low q = k F , the QMC results above k F show that G + ( q )follows the quadratic required by the compressibility sumrule up until nearly 2 k F and then falls significantly be-low the initial quadratic. Theory predicts that the large q behavior will also be a quadratic but with a differentcoefficient. Other earlier versions of G + ( q ) [12] had a1 (a) q/k F . . . . . . G + ( q / k F ) QuadraticRef. 16QMC data (b) q/k F . . . . . G + ( q / k F ) QuadraticRef. 16QMC data
FIG. 11: Density local field function G + ( q ) plotted versus q/k F for r s = 2 (a) and r s = 5 (b). Data points are theQuantum Monte Carlo calculations from Ref. [12]. The solidcurves that fit the data are the analytic function from Ref.[13]. The quadratic Eq. (4) is the proposed simple approxi-mation to G + ( q ) for calculating the response functions. Errorbars are shown for all data points. If they are not evident,the error is smaller than the data point. much smaller value at 2 k F .Figure 11 emphasizes the large q behavior of G + ( q ).However the static response functions of the electron gasdepend mostly on the low q behavior, because the Lind-hard functions cuts off the effect of G + ( q ) quickly above q = 2 k F .At small q , the compressibility sum rule specifies that G + ( q →
0) = (cid:16) − κ κ (cid:17) (cid:18) qq T F (cid:19) (A1)To emphasize the low q behavior, G + ( q ) / ( q/q T F ) isplotted in Fig. 12 below, where the intercept at q = 0is (1 − κ /κ ). Note that the density exchange and corre-lation kernel needed for Time-Dependent Density Func-tional Theory is given by f xc = − πG + ( q ) / ( q/q T F ) . (a) q/k F . . . . G + ( q / k F ) / ( q / q T F ) QuadraticRef. 16QMC data (b) q/k F . . . . . . G + ( q / k F ) / ( q / q T F ) QuadraticRef. 16QMC data
FIG. 12: G + ( q ) / ( q/q TF ) , the density local field function di-vided by ( q/q TF ) is plotted versus q/k F for r s = 2 (a) and r s = 5 (b). Data points are the Quantum Monte Carlo calcu-lations from Ref. [15]. The solid curves that fit the data arethe analytic function from Ref. [16]. The straight line repre-sents the quadratic Eq. (4) that is the proposed simple ap-proximation to G + ( q ) for calculating the response functions.Error bars are shown for all data points. Note again that there are no Quantum Monte Carlodata below q = k F . The q = 0 values of the quadraticand the analytic function are set by the compressibil-ity sum rule. The compressibility is also calculated byMonte Carlo methods, and the q = 0 value is much moreaccurate than the q dependent data. The compressibil-ity sum rule dictates G + (0) and that the initial behaviorwill be quadratic, Eq. (A1), as represented by the con-stant horizontal line. The analytic interpolation formula(14) apparently fits the data above 1 . k F quite well, butmisses substantially the data at k F for r s = 2. Thisillustrates the problem of global curve fitting with an-alytical functions with a limited number of parameters.The simple quadratic does at least as good a job below2 k F and is substantially different at larger q , but theeffect at large q is not very important for the response2functions as will be shown below. Looking carefully atthe QMC data between k F and 2 k F , there is an intrigu-ing hint of structure in G + ( q ), with data both below andabove the quadratic. Further Quantum Monte Carlo cal-culations from q = 0 − k F would be informative. Thevertex function which embodies the effect of exchangeand correlation in the response functions is plotted inFig 13, where the simple quadratic is compared to theQMC data and the fitting function for this data. (a) q/k F . . . . / ( − G + Q ) QuadraticRef. 16QMC data (b) q/k F / ( − G + Q ) QuadraticRef. 16QMC data
FIG. 13: Vertex function Λ = 1 / (1 − G + Q ) plotted ver-sus q/k F for r s = 2 (a) and r s = 5 (b). Data points arethe Quantum Monte Carlo calculations from Ref. [15]. Thedotted curve that fits the data is the analytic function fromRef. [16]. The solid curve uses Eq. (4), the proposed simplequadratic approximation for calculating the response func-tions. Error bars are shown for all data points. If they arenot evident, the error is smaller than the data point. Notethat the y − axis starts at 1.0 for r s = 2, in order to emphasizesmall differences. The vertex function Λ = 1 / (1 − G + Q ) is required tocalculate the dielectric function, and the other interac-tions in the electron gas. Figure 13 shows the vertexfunction using the actual QMC data and the analyticfunction as well as the simple quadratic. The first point to note is that the q = 0 value of the vertex functionis exactly given by the compressibility sum rule and isequal to Λ(0) = κ/κ which diverges at the compress-ibility divergence as approximately 1 / (1 − r s / . r s = 2, the vertex enhancement is substantial, but it isnot near the compressibility divergence. At r s = 5, theelectron gas is very near the divergence and the vertexenhancement is very large. The error bar in the valueof G + ( q ) at q = k F reaches nearly a point of instabilityand its effect is magnified because it appears in the de-nominator. The vertex function is extremely sensitive tosmall changes at r s = 5.The simple quadratic function for G + ( q ) yields a vertexfunction that satisfies the compressibility sum rule andfits the vertex function derived from the QMC data aswell as the fitting formula of Ref. [13]. This is despite thelarge differences at large q , because the contributions atlarge q are cut off by the Lindhard function. The fittingfunction should be used for any calculations that dependon q substantially above 2 k F .The fact that the vertex function resulting from theanalytic function is larger than the quadratic for q greaterthan zero and less than k F is entirely due to the curvefitting, and there are no QMC data in this region.The simple quadratic approximation for G + ( q ) wassuggested by Taylor [20] 40 years ago and we concur. (a)(b) VDMCQuadratic r s = VDMCQuadratic r s = q/k F q/k F G + ( q / k F ) G + ( q / k F ) / ( q / q T F ) FIG. 14: Density local field factor G + ( q ) from VDMC calcula-tions for r s = 1 plotted versus q/k F (a), and G + ( q ) / ( q/q TF ) (b). Error bars are shown. χ /χ or κ /κ < . r s = 5. For the com-pressibility, this corresponds to r s = 2. As a test, we havecalculated the density local field factor G + ( q ) for r s = 1and 2, and the data are shown in Figures 14 and 15. At r s = 2, we compare the new VDMC calculations to QMCresults of Ref. [15] and the corresponding interpolationformula of Ref. [16].At r s = 1, Fig. 14 shows that the density local fieldfactor G + ( q ) has the same qualitative behavior as thespin local field factor G − ( q ) which are shown in Fig. 1and Fig. 2. The error bars are acceptable. Both showthat G rises above the quadratic at approximately 1 . kF and then falls below at 2 kF . (a)(b) r s = VDMCQMC r s = VDMCQMCRef. 14Quadratic q/k F q/k F G + ( q / k F ) / ( q / q T F ) G + ( q / k F ) FIG. 15: Density local field factor G + ( q ) from VDMC calcula-tions for r s = 1 plotted versus q/k F (a), and G + ( q ) / ( q/q TF ) (b). Error bars are shown. The new data for G + ( q ) in Fig. 15 demonstrates thelimitations of the current version of the VDMC approach.Ordinarily, we would not show data with such large er-ror bars. However, we want to compare with the QMCdata Ref. [15] and to provide new data below k F . TheVDMC data with error bars overlap the QMC data withits error bars except for two points near 2 k F , and eventhere, the agreement is quite close. The simple quadraticapproximation for G + ( q ) given in Eq. (4) represents theVDMC data quite well up to 2 k F . The interpolationcurve of Ref. [16] fits the data well above 2 k F . The data for G + ( q ) at r s = 2 are qualitatively similar to the datafor G − ( q ).The VDMC data for G − and the limited data for G + show that both of these local field factors are smoothfunctions of wave vector. Although the data rise slightlyabove the quadratic between 1.5 and 2 k F , there is no ev-idence of a large peak. We have not developed a physicalintuition for this behavior. Appendix B: TEST CHARGE-TEST CHARGE V tt AND ELECTRON-TEST CHARGE V et INTERACTIONS AND DENSITY RESPONSEFUNCTION
We use the same quadratic function for G + ( q ) to plotthe test charge test-charge and electron-test charge in-teractions at r s = 2 and 5, and compare them to theLindhard interaction. For the general reader, a simplephysically motivated derivation of these interactions andthe electron-electron interaction are in Ref. [1].The test charge-test charge interaction V tt is theCoulomb potential generated by a test charge plus theinduced screening cloud, and felt by another test charge.The dielectric function (cid:15) ( q, ω ) is defined by V tt = V ext (cid:15) (B1)and is written as (cid:15) = 1 + v Π − G + v Π = 1 + Λ Q, (B2)where Π is the Lindhard free electron response func-tion and v = 4 πe /q . Note that others may definethe response function with a minus sign. For conve-nience, we define Q = v Π and the vertex correctionΛ = 1 / (1 − G + Q ). The potentials are measured in unitsof 4 πe so that the Thomas Fermi and Lindhard poten-tials at q = 0 are simply 1 /q T F . Without exchange andcorrelation, G + = 0 and therefore Λ = 1 and the Lind-hard result is obtained.The electron-test charge interaction V et is simply thetest charge - test charge interaction multiplied by thevertex function. V et = Λ V tt = V ext − G + ) Q (B3)For these interactions and response functions of theelectron gas in the metallic region, the large q behaviorof the local field factor is not of significant importance inmost applications because the Lindhard function cuts offquickly above q = 2 k F .The q = 0 value of the interaction is set by the com-pressibility sum rule. V tt (0) = ( κ /κ ) /q T F is always lessthan the Lindhard or Thomas Fermi value. Both are vir-tually identical above 2 . k F . The q dependence of G + ( q )is only important between 0 and 2 k F .4 V tt ( q ) / π e ( - c m ) q/k F V tt V Lindhard (a) r s =2 V tt ( q ) / π e ( - c m ) q/k F V tt V Lindhard r s =5 (b) FIG. 16: V tt ( q ), the test charge-test charge or Coulomb in-teraction at r s = 2 and 5. The potential is measured in unitsof 4 πe . The dashed line is the Lindhard potential which isequal to the Thomas Fermi potential at q = 0 with the value V Lindhard (0) = 1 /q TF . V tt ( q ) in Fig 16 looks qualitatively like the Lindhardpotential at r s = 2, but is dramatically different at r s =5. This is due to the compressibility sum rule whichfixes the value at q = 0, and G + ( q ) interpolates between q = 0 and 2 k F . According to the compressibility sumrule, V tt ( q = 0) = 0 at the compressibility divergence at r s = 5 .
25, and becomes negative at larger r s . It mustturn positive again and match 1 /q at q beyond 2 kF .We don’t have a physical intuition for this“overscreening”behavior resulting from a negative dielectric function.Near the compressibility divergence, the vertex cor-rection and thus the dielectric function and V tt are ex-tremely sensitive to small changes, and pressure may bean interesting variable. When applying this formula toreal metals, the effective mass and core polarization willre-normalize the equations to make the effective r ∗ s lowerthan the actual physical r s . Another Ward identity spec-ifying the renormalization factor z must also be consid- -1.00.01.02.03.04.00.0 0.5 1.0 1.5 2.0 r /( r s a ) r /( r s a ) ( r ) ( e V ) V tt ( r ) ( e V ) V tt V tt V Lindhard r s =2 V tt V Lindhard r s =5 (a)(b) FIG. 17: The test charge-test charge interactions for r s = 2(a) and 5(b) have been numerically Fourier transformed andare plotted versus distance. The Fourier transform of theLindhard function is shown for comparison. The data areshown from r/ ( r s a ) = 0 . − . ered.The Fourier transforms of the test charge-test chargepotentials are shown in Fig. 17 and Fig. 18 compared tothe Lindhard potential.Although it is not shown in Fig. 17, V tt and V Lindhard converge at small distances r/ ( r s a ) < . q > k F ) and become equal to the bare inter-action at even smaller distances. At intermediate dis-tances (derived from intermediate q ), V tt is less repul-sive than V Lindhard . At large distances the oscillations in V tt are larger than the Friedel oscillations in V Lindhard .This is dramatically different for r s = 5 where there isa broad attractive region around the test charge from r/ ( r s a ) = 0 . .
4. The oscillations at larger distancesalso have a larger amplitude.The lattice spacing of alkali metals is approximately1.1 times r s , and the diameter of the core electrons isabout 0.5 times r s . The attractive minimum for r s = 5located at r/ ( r s a ) = 0 . . eV compared5 -2.0-1.00.01.02.02.0 3.0 4.0 5.0 6.0 7.0 8.0 -7.0-6.0-5.0-4.0-3.0-2.0-1.00.01.02.03.02.0 3.0 4.0 5.0 6.0 7.0 8.0 V tt V Lindhard V tt V Lindhard ( r ) ( - e V ) V tt ( r ) ( - e V ) V tt r s =2 r s =5 (a)(b) r /( r s a ) r /( r s a ) FIG. 18: The test charge-test charge interactions for r s = 2(a)and 5(b) have been numerically Fourier transformed and areplotted versus distance. The Fourier transform of the Lind-hard function is shown for comparison. The data are shownfrom r/ ( r s a ) = 2 . − to a cohesive energy of rubidium of 0 . eV . This attrac-tion can be part of the explanation for the contractionof the inter-atomic spacing in liquid rubidium that is ob-served in x-ray scattering experiments as a function ofpressure and temperature[21].The electron-test charge interaction V et is shown inFig. 19. The electron-test charge interaction at q = 0is equal to 1 /q T F which is the same as the Lindhardand Thomas- Fermi interactions (obtained by setting G + = 0). The effect of the vertex correction in the nu-merator is canceled by the vertex correction in the di-electric function at q = 0. The effect of exchange andcorrelation only occurs between zero and about 2 . k F .For r s = 2, the effects of exchange and correlation aresmall. The effects are larger for r s = 5.A recent paper[22] calculated the static density re-sponse function of lithium from a Kramers-Kronig trans-formation of the dynamic structure factor measured byinelastic electron scattering. Figure 20 was prepared byT. Hagiya, the lead experimental author, using the for- V e t ( q ) / π e ( - c m ) q/k F V et V Lindhard r s =2 (a) V e t ( q ) / π e ( - c m ) q/k F V et V Lindhard r s =5 (b) FIG. 19: The electron-test charge interaction V et ( q ) at r s = 2and 5. The dashed curve is the Lindhard potential. mula for X ( q ) including exchange and correlation pro-vided by us compared to the Random Phase Approxima-tion.The density response function is given by: χ ( q ) = Π (1 − G + Q )(1 + Q/ (1 − G + Q )) = λ Π ε . (B4)The RPA response function is obtained by setting G + = 1.The theory has no adjustable parameters. The ex-perimentalists point out that their data is not accurateenough to definitively distinguish between the responsefunctions using exchange and correlation and the RPA.Nevertheless, the experimental results are very impres-sive as is the data analysis.6 FIG. 20: Measured static density response function forlithium from Ref. [22] compared to theoretical value withno adjustable parameters. The top curve includes exchangeand correlation using G + ( q ) and the bottom curve uses theRandom Phase Approximation which is obtained by setting G + ( q ) = 1.[1] C. A. Kukkonen and A. Overhauser, Physical Review B , 550 (1979).[2] K. Chen and K. Haule, Nature Communications , 1(2019).[3] G. Giuliani and G. Vignale,Quantum theory of the electron liquid (CambridgeUniversity Press, 2005).[4] G. Vignale and K. Singwi, Physical Review B , 2156(1985).[5] G. Vignale and K. S. Singwi, Physical Review B , 2729(1985).[6] Y. Takada, Physical Review B , 5202 (1993).[7] C. Richardson and N. Ashcroft, Physical Review B ,R764 (1996).[8] C. A. Kukkonen and J. W. Wilkins, Physical Review B , 6075 (1979).[9] J. P. Perdew and Y. Wang, Physical Review B , 079904(2018).[10] N. W. Ashcroft and N. D. Mermin, Solid state physics(New York: Holt, Rinehart and Winston,, 1976).[11] C. Richardson and N. Ashcroft, Physical Review B ,15130 (1997).[12] J. Hubbard, Proceedings of the Royal Society of London.Series A. Mathematical and Physical Sciences , 336 (1958).[13] D. Geldart and S. Vosko, Canadian Journal of Physics , 2137 (1966).[14] P. Vashishta and K. Singwi, Physical Review B , 875(1972).[15] S. Moroni, D. M. Ceperley, and G. Senatore, PhysicalReview Letters , 689 (1995).[16] M. Corradini, R. Del Sole, G. Onida, and M. Palummo,Physical Review B , 14569 (1998).[17] C. Richardson and N. Ashcroft, Physical Review B ,8170 (1994).[18] G. E. Simion and G. F. Giuliani, Physical Review B ,035131 (2008).[19] S. Hellal, J.-G. Gasser, and A. Issolah, Physical ReviewB , 094204 (2003).[20] R. Taylor, Journal of Physics F: Metal Physics , 1699(1978).[21] K. Matsuda, K. Tamura, and M. Inui, Physical ReviewLetters , 096401 (2007).[22] T. Hagiya, K. Matsuda, N. Hiraoka, Y. Kajihara,K. Kimura, and M. Inui, Physical Review B102