Comparison of local density functionals based on electron gas and finite systems
CComparison of local density functionals based on electron gas and finite systems
M. T. Entwistle, M. Casula, and R. W. Godby Department of Physics, University of York, and European TheoreticalSpectroscopy Facility, Heslington, York YO10 5DD, United Kingdom Institut de Min´eralogie, de Physique des Mat´eriaux et de Cosmochimie (IMPMC),Sorbonne Universit´e, CNRS UMR 7590, IRD UMR 206, MNHN, 4 Place Jussieu, 75252 Paris, France (Dated: June 27, 2018)A widely used approximation to the exchange-correlation functional in density functional theoryis the local density approximation (LDA), typically derived from the properties of the homogeneouselectron gas (HEG). We previously introduced a set of alternative LDAs constructed from one-dimensional systems of one, two, and three electrons that resemble the HEG within a finite region.We now construct a HEG-based LDA appropriate for spinless electrons in one dimension and findthat it is remarkably similar to the finite LDAs. As expected, all LDAs are inadequate in low-density systems where correlation is strong. However, exploring the small but significant differencesbetween the functionals, we find that the finite LDAs give better densities and energies in high-density exchange-dominated systems, arising partly from a better description of the self-interactioncorrection.
I. INTRODUCTION
Density functional theory (DFT) is the most popu-lar method to calculate the ground-state properties ofmany-electron systems . In the widely employed Kohn-Sham (KS) formalism of DFT, the real system of inter-acting electrons is mapped onto a fictitious system ofnoninteracting electrons moving in an effective local po-tential, with both systems having the same electron den-sity. While in principle an exact theory, in practice theaccuracy of DFT calculations is constrained by our abil-ity to approximate the exchange-correlation (xc) part ofthe KS functional, whose exact form is unknown. Identi-fying properties of the exact xc functional that are miss-ing in commonly used approximations is vital for furtherdevelopments.A widely used approximation is the local densityapproximation (LDA) which assumes that the true xcfunctional is solely dependent on the electron density ateach point in the system. LDAs are traditionally de-rived from knowledge of the xc energy of the homoge-neous electron gas (HEG), a model system where theexchange energy is known analytically and the correla-tion energy is usually calculated using quantum MonteCarlo simulations. LDAs have been hugely successful inmany cases , however, their validity breaks down in anumber of important situations , particularly whenthere is strong correlation. They are known to miss outsome critical features that are present in the exact xc po-tential, such as the cancellation of the spurious electronself-interaction , or the Coulomb-type − /r decay ofthe xc potential far from a finite system , instead fol-lowing an incorrect exponential decay . They also failto capture the derivative discontinuity , the discon-tinuous nature of the derivative of the xc energy withrespect to electron number N , at integer N .In a previous paper , we introduced a set of LDAswhich, in contrast to the traditional HEG LDA, wereconstructed from systems of one, two, and three electrons which resembled the HEG within a finite region. Illus-trating our approach in one dimension (1D), we foundthat the three LDAs were remarkably similar to one an-other. In this paper, we construct a 1D HEG LDAthrough suitable diffusion Monte Carlo (DMC) tech-niques, along with a revised set of LDAs constructed fromfinite systems. We compare the finite and HEG LDAswith one another to demonstrate that local approxima-tions constructed from finite systems are a viable alter-native, and explore the nature of any differences betweenthem.In order to test the LDAs, we employ our iDEA code which solves the many-electron Schr¨odinger equation ex-actly for model finite systems to determine the exact,fully-correlated, many-electron wave function. Using thisto obtain the exact electron density , we then utilize ourreverse engineering algorithm to find the exact KS sys-tem. In our calculations we use spinless electrons to moreclosely approach the nature of exchange and correlationin many-electron systems, which interact via the appro-priately softened Coulomb repulsion ( | x − x | + 1) − . II. SET OF LDASA. LDAs from finite systems
In Ref. 29 we chose a set of finite locally homogeneoussystems in order to mimic the HEG, which we referredto as “slabs” (Fig. 1). We generated sets of one-electron(1 e ), two-electron (2 e ), and three-electron (3 e ) slab sys-tems over a typical density range (up to 0.6 a.u.) and ineach case calculated the exact xc energy E xc . From thiswe parametrized the xc energy density ε xc = E xc /N interms of the electron density of the plateau region of theslabs, repeating for the 1 e , 2 e , and 3 e set.To approximate the xc energy of an inhomogeneoussystem, the LDA focuses on the local electron density at a r X i v : . [ c ond - m a t . o t h e r] J un
10 0 10 20 30 x (a.u.) . . . . . . . n ( a . u . ) x (a.u.) V e x t ( a . u . ) x (a.u.) V e x t ( a . u . ) FIG. 1. The exact many-body electron density (solid lines)for a selection of the two-electron slab systems. The densityis locally homogeneous across a plateau region and decaysexponentially at the edges. Inset: the external potential fora typical two-electron slab system (middle density in mainfigure). each point in the system: E LDAxc [ n ] = Z n ( x ) ε xc ( n ) dx, (1)where in a conventional LDA ε xc ( n ) is the xc energy den-sity of a HEG of density n . This approximation becomesexact in the limit of the HEG, and so it is a reasonablerequirement for the finite LDAs to become exact in thelimit of the slab systems. Due to the initial parametriza-tion of ε xc ( n ) focusing on the plateau regions of the slabs(i.e., ignoring the inhomogeneous regions at the edges),we used a refinement process in order to fulfill this re-quirement.The refined form for the xc energy density in thethree finite LDAs has now been increased from the four-parameter fit in Ref. 29 to a seven-parameter fit in thispaper: ε xc ( n ) = ( A + Bn + Cn + Dn + En + F n ) n G , (2)where the optimal parameters for each LDA are given inTable I. The xc potential V xc is defined as the functionalderivative of the xc energy which in the LDA reduces toa simple form : V LDAxc ( n ) = ε xc ( n ( x )) + n ( x ) dε xc dn (cid:12)(cid:12)(cid:12)(cid:12) n ( x ) . (3) B. HEG exchange functional
In Ref. 29 we solved the Hartree-Fock equations tofind the exact exchange energy density ε x for a fully TABLE I. Optimal fit parameters for ε xc ( n ) in the finiteLDAs. The last two rows contain the mean absolute error(MAE) and root-mean-square error (RMSE) of the fits. ε xc ( n )is graphed in Sec. II D below. Parameter 1 e value 2 e value 3 e value A − . − . − . B C − . − . − . D E − . − . − . F G × − × − × − MAE 1.3 × − × − × − RMSE 1.9 × − × − × − spin-polarized [ ζ = 1 where ζ ≡ ( N ↑ − N ↓ ) /N ] 1DHEG of density n consisting of an infinite number ofelectrons interacting via the softened Coulomb repulsion u ( x − x ) = ( | x − x | + 1) − : ε x ( n ) = − π n Z πn − πn dk Z πn − πn dk u ( k − k ) , (4)where the Fourier transform of u ( x − x ) is integrated overthe plane defined by the Fermi wave vector k F = πn .Solving Eq. (4) for the range of densities we used inthe finite LDAs, we parametrized ε x ( n ). Once again, wehave increased our fit from four parameters to seven pa-rameters, as in Eq. (2) above . The optimal parametersare given in Table II. The ε x ( n ) curve is shown in theinset of Fig. 2. TABLE II. Optimal fit parameters for ε x ( n ) in the HEG LDA.The last two rows contain the mean absolute error (MAE) androot-mean-square error (RMSE) of the fit. Parameter Value A − . B C − . D E − . F G × − MAE 6.5 × − RMSE 7.2 × − C. HEG correlation functional
We use the lattice regularized diffusion Monte Carlo(LRDMC) algorithm to compute the ground-state en-ergy of the fully spin-polarized HEG over a wide rangeof densities, much higher than the 0.6 a.u. limit used inthe finite LDAs. This is in order to ensure the resultantparametrization of the correlation energy density ε c re-duces to the known high-density and low-density limits.We determine ε c by subtracting the kinetic energy and ε x contributions from the total energy.To parametrize the correlation energy density we usea fit of the form : ε c ( r s ) = − A RPA r s + Er Br s + Cr + Dr ln(1 + αr s + βr ) α , (5)where r s is the Wigner-Seitz radius and is related to thedensity (in 1D) by 2 r s = 1 /n . The optimal parameters(with estimated errors) are given in Table III. The fitapplied to the data is shown in Fig. 2. TABLE III. Optimal fit parameters with estimated errors inparentheses for ε c ( r s ) in the HEG LDA. The last two rowscontain the mean absolute error (MAE) and root-mean-squareerror (RMSE) of the fit. Note: A RPA has been determinedfrom the high-density limit for ε c (in which the random phaseapproximation (RPA) is exact ), which is exactly fulfilledby our fit, and hence has no associated error. Parameter Value A RPA × − B × − C × − D × − E × − α β × − RMSE 1.3 × − . . . . n (a.u.) . . . . . " c ( a . u . ) fitLRDMC . . . . n (a.u.) . . . . " x ( a . u . ) FIG. 2. The ε c (with associated error bars) for a set of HEGsover the density range used in the finite LDAs. The fit applied(solid blue) becomes exact in the known high-density and low-density limits. Inset: The ε x curve in the HEG LDA. The high-density limit (infinitely-weak correlation) ofthe parametrization is: ε c ( r s →
0) = − A RPA r , (6) and its low-density limit (infinitely-strong correlation) is: ε c ( r s → ∞ ) = − EαD ln( r s ) r s . (7)Therefore, the parametric form in Eq. (5) correctly re-produces the expected behavior of the correlation energydensity in the high-density limit [ ε c ∝ r ] and low-density limit [ ε c ∝ ln( r s ) /r s ]. D. Comparison of 1 e , 2 e , 3 e and HEG LDAs Summing together the HEG exchange and correlationparametric fits, we can now compare the HEG LDA thatwe have developed against the three finite LDAs. Thestriking similarity between the four ε xc curves can beseen in Fig. 3(a). While very similar in the low-densityrange, there are some differences between them. Theseare highlighted in Fig. 3(b) which, using the 1 e LDA as areference, plots its difference with the remaining LDAs.There is a competing balance between exchange and cor-relation. At low densities, these differences can be mainlyattributed to ε c , which is entirely absent in the 1 e LDA,and increases in magnitude as we progress to 2 e to 3 e toHEG (Fig. 4). As we move to higher densities in whichthe magnitude of ε c decreases, and the magnitude of ε x increases, the order of the four ε xc curves reverses. Theyincreasingly separate as we move to higher densities withthe 1 e LDA, which consists entirely of self-interactioncorrection, giving the largest magnitude for ε xc . By plot-ting the difference between the 1 e LDA (where correla-tion is absent) and the exchange part of the HEG LDA(i.e., removing the correlation term), it can be seen thatthe 1 e LDA yields a larger exchange energy density thanthe HEG LDA at all densities (Fig. 5).The refinement process used in the construction of thefinite LDAs focused on giving the correct E xc in the limitof the slab systems, but did not ensure that the correct V xc , and by extension electron density, were reproduced(a property of HEG LDAs). We find that the finite LDAsare completely inadequate at reproducing the densities ofthe slab systems. We compare the exact V xc against n and find that there is a high nonlocal dependence on n ,implying that no local density functional can accuratelyreproduce V xc and hence n for the slab systems. In lightof this, the success of the finite LDAs reported below isall the more surprising. III. TESTING THE LDAS
In the previous section we observed the close similaritybetween the four LDAs. In this section we apply themto a range of model systems in order to identify thedifferences between them. . . . . n (a.u.) . . . . " x c ( a . u . ) e LDA e LDA e LDAHEG LDA . . . . n (a.u.) . . . . " x c ( a . u . ) e e LDA e e LDAHEG e LDA . . . n (a.u.) . . . . " x c ( a . u . ) . . . n (a.u.) . . . . " x c ( a . u . ) . . . n (a.u.) . . . . " x c ( a . u . ) FIG. 3. (a) The ε xc curves in the 1 e (dashed red), 2 e (solidgreen), 3 e (dotted blue) and HEG (dotted-dashed black)LDAs. Inset: Close-up of the four curves at higher densities.The similarity between them is striking, with a clear progres-sion from 1 e to 2 e to 3 e to HEG. (b) The 1 e LDA is used asa reference here. Plotted is its difference ( δε xc = ε xc − ε e xc )with the 2 e (solid green), 3 e (dotted blue) and HEG (dotted-dashed black) LDAs. A. Weakly correlated systems
System 1 (2e harmonic well) . We first consider a pairof interacting electrons in a strongly confining harmonicpotential well ( ω = a.u.) where correlation is veryweak . We calculate the exact many-body electron den-sity using iDEA, and compare it against the densities ob-tained from applying the LDAs self-consistently. Thereis a progression from the 1 e –2 e –3 e –HEG LDA and so wechoose to plot the 1 e and HEG LDA densities (i.e., the 2 e and 3 e LDA densities lie between these) against the exact[Fig. 6(a)]. Both LDAs match the exact density well, andso we plot their absolute errors ( δn = n LDA − n exact ) tomore clearly identify their differences [Fig. 6(b)]. The 1 e LDA has a slightly smaller net absolute error ( R | δn | dx ).While the HEG LDA gives a slightly better electron den-sity in the central region (dip in the density), the 1 e LDAbetter matches the decay of the density towards the edgesof the system, and perhaps more interestingly, the twopeaks in the density where the self-interaction correction . . . . n (a.u.) . . . . " c ( a . u . ) e LDA e LDAHEG LDA
FIG. 4. We calculate the exact ε c for the 2 e (solid green line)and 3 e (dotted blue line) slab systems through Hartree-Fockcalculations. We plot these against the ε c curve in the HEGLDA (dotted-dashed black line). The ε c in the HEG LDA ismuch larger ( ∼ e LDA and ∼ e LDA). While not a perfect comparison due to the refinementprocess used in the construction of the finite LDAs, it gives auseful indication of the size of ε c in their ε xc curves. . . . . n (a.u.) . . . . . " x ( a . u . ) HEG e LDA
FIG. 5. The ε x curve in the 1 e LDA ( ε x = ε xc ) is used as areference here. Plotted is its difference ( δε x = ε x − ε e x ) withthe ε x curve in the HEG LDA ( ε x = ε xc − ε c ). It can be seenthat the 1 e LDA yields a larger exchange energy density thanthe HEG LDA at all densities. Note: This is not true in thevery low-density region ( n < . is largest.Due to the importance of energies in DFT calcula-tions, we also compare the exact E xc and total energy E total , with those obtained from applying the LDAs self-consistently (Table IV). While all the LDAs give goodapproximations to both quantities, there are some signif-icant differences due to this system being dominated byregions of high density, and the ε xc curves separating in x (a.u.) . . . . n , V e x t ( a . u . ) V ext Exact e LDAHEG LDA x (a.u.) . . . . . . . n ( a . u . ) e LDAHEG LDA
FIG. 6. System 1 (two electrons in a harmonic potential well).(a) The external potential (dotted-dashed blue line), togetherwith the exact electron density (solid red line), and the den-sities obtained from applying the 1 e (dashed green line) andHEG (dotted black line) LDAs. Both LDAs are in very goodagreement with the exact result. (b) The absolute error inthe density ( δn = n LDA − n exact ) in the 1 e (dashed green line)and HEG (dotted black line) LDAs, allowing their differencesto be more clearly identified. this limit (see Fig. 3). As with the approximations to theelectron density, there is a progression from the 1 e –2 e –3 e –HEG LDA, with the 1 e LDA reducing the absoluteerrors ( δE xc E xc , δEE ) in the HEG LDA by a factor of 5 − System 2 (3e harmonic well) . Next, we consider a har-monic potential well with three electrons, but slightly lessconfining ( ω = ), in order to avoid an unphysically highelectron density ( n > . e harmonicwell system, we find a progression from the 1 e –2 e –3 e –HEG LDA, with all LDAs giving good electron densities(see Fig. 7(a) for the 1 e and HEG LDA densities plottedagainst the exact). Again, the 1 e LDA has the smallestnet absolute error, and outperforms the rest of the LDAsin the regions where the density peaks [Fig. 7(b)].We also compare the exact E xc and E total against theLDAs (Table IV). All LDAs give good energies, withsome noticeable differences between them due to this sys-tem being dominated by regions of high density, like inthe 2 e harmonic well system. However, the magnitude of x (a.u.) . . . . n , V e x t ( a . u . ) V ext Exact e LDAHEG LDA x (a.u.) . . . . . . . n ( a . u . ) e LDAHEG LDA
FIG. 7. System 2 (three electrons in a harmonic potentialwell). (a) The external potential (dotted-dashed blue line),together with the exact electron density (solid red line), andthe densities obtained from applying the 1 e (dashed greenline) and HEG (dotted black line) LDAs. Much like the 2 e harmonic well system, both LDAs match the exact densitywell. (b) The absolute error in the density in the 1 e (dashedgreen line) and HEG (dotted black line) LDAs. Again, the 1 e LDA outperforms the HEG LDA in the density peaks, whichis dominated by the self-interaction correction. E xc in the 1 e LDA is greater than the exact (i.e., it over-estimates the amount of exchange + correlation), andsubsequently it gives a total energy lower than the exact.While the absolute error in E xc for each LDA is similarto that in E total , this overestimation of exchange + cor-relation in the 1 e LDA results in the 2 e LDA giving thebest total energy.
B. A system dominated by the self-interactioncorrection
The self-interaction correction (SIC) is absent in xcfunctionals constructed from the HEG. However, the xcenergy of the 1 e slab systems (which were used to con-struct the 1 e LDA) consists entirely of SIC. In the firsttwo model systems, we found that the 1 e LDA (and in-deed the other finite LDAs) better describes the electrondensity in regions where the SIC is strongest, than the
TABLE IV. Total energies and xc energies for the set of weakly correlated systems (1–3), from exact calculations and fromapplying the four LDAs self-consistently ( δE LDA = E LDA − E exact ). Estimated errors are ± E total (a.u.) E xc (a.u.)Exact δE e total δE e total δE e total δE HEGtotal
Exact δE e xc δE e xc δE e xc δE HEGxc e harmonic well 1.6932 0.0037 0.0126 0.0153 0.0211 − . e harmonic well 3.1875 − . − . − . e double well − . − . HEG LDA. We now investigate this further.
System 3 (2e double well) . We choose a system withtwo electrons confined to a double-well potential. Thewells are separated, such that the electrons are highly lo-calized and can be considered as two separate subsystems[Fig. 8(a)]. This results in the Hartree potential beingsmall outside of the wells, and being dominated by theelectron self-interaction within the wells. Consequently,a large proportion of the xc potential is self-interactioncorrection. Applying the LDAs, we find the usual pro-gression 1 e –2 e –3 e –HEG. Focusing on the peaks in theelectron density, the 1 e LDA substantially reduces theerror present in the HEG LDA [Fig. 8(b)]. To under-stand this, we analyze the xc potential [Fig. 8(c)]. The1 e LDA better reproduces the large dips in V xc , corre-sponding to the peaks in the electron density. Hence, theSIC is more effectively captured.While the LDA errors in E xc are larger than in the firsttwo systems, they are still small (4.8–6.8%) (Table IV).The absolute errors in E total are similar. C. Systems where correlation is stronger
System 4 (2e atom) . We now consider a systemwhere the relative size of electron correlation increasessignificantly : two electrons confined to a softened atomiclike potential, V ext = − ( | ax | + 1) − , where a = .Although we find the same progression (1 e –2 e –3 e –HEG)as seen in the first three model systems, in which corre-lation was weak, all LDAs give inadequate electron den-sities. This can be seen by plotting the 1 e and HEGLDA densities against the exact [Fig. 9(a)]. The LDAsgive densities that are not even qualitatively correct, e.g.,predicting a single peak in the center of the system, whichis absent in the exact density. The net absolute errors aremuch larger than in the weakly correlated systems, how-ever, the 1 e LDA once again gives the smallest [Fig. 9(b)].We find that although the LDA densities are poor, thexc energies are surprisingly good (Table V). This can beattributed somewhat (see Sec. III D for investigation offurther causes) to errors in the density being partiallycanceled by errors inherent in the approximate xc en-ergy functional . We infer this by noting the progres-sion (HEG–3 e –2 e –1 e ) when we apply the LDAs to the exact density, in contrast to the self-consistent solutions in Table V. As in the weakly correlated systems, the ab-solute errors in E total are smaller than in E xc , due toa partial cancellation of errors from the Hartree energycomponent. It is much more apparent in this system dueto the LDAs incorrectly predicting a central peak in theelectron density [Fig. 9(a)]. System 5 (3e atom) . Finally, we consider three elec-trons in an external potential of the same form as the 2 e atom, but less confining, with a = . Along with theusual progression (1 e –2 e –3 e –HEG), we find a similar re-sult to the 2 e atom, with the LDAs giving poor electrondensities [Fig. 10(a)]. Although the densities are qual-itatively correct, unlike in the 2 e atom, the LDAs sig-nificantly underestimate the peaks in the electron den-sity. Subsequently, the absolute errors are very large[Fig. 10(b)]. The 1 e LDA, along with giving the low-est net absolute error, most accurately reproduces thepeaks in the density, where the SIC is largest.While the absolute errors in E xc are larger than in the2 e atom, they are still small (Table V). Again, this par-tially arises from applying approximate xc energy func-tionals to incorrect densities. As in the 2 e atom, the ab-solute errors in E total are much lower than those in E xc ,due to a partial cancellation of errors from the Hartreeenergy component. D. Cancellation of errors between exchange andcorrelation
HEG-based LDAs have been known to typically under-estimate the magnitude of the exchange energy E x , whileoverestimating the magnitude of the correlation energy E c . Consequently, while the total E xc is underestimatedin magnitude, the approximation proves to be better thanwas originally expected due to a partial cancellation oferrors.We investigate how well our HEG LDA approximates E x and E c in the model systems, and how this con-tributes to accurate values for E xc . To do this we performHartree-Fock calculations for each of the model systems,and together with the exact solutions obtained throughiDEA, are able to divide the exact E xc into its exchangeand correlation components. We then apply the HEGLDA, which is split into separate E x and E c function-als, for comparison (Table VI). In all systems, the HEG TABLE V. Total energies and xc energies for the set of strongly correlated systems (4-5), from exact calculations and fromapplying the four LDAs self-consistently ( δE LDA = E LDA − E exact ). Estimated errors are ± E total (a.u.) E xc (a.u.)Exact δE e total δE e total δE e total δE HEGtotal
Exact δE e xc δE e xc δE e xc δE HEGxc e atom -1.5099 0.0053 0.0044 0.0032 0.0022 -0.3728 0.0084 0.0101 0.0099 0.01113 e atom -2.3282(5) 0.0121(5) 0.0085(5) 0.0057(5) 0.0029(5) -0.493(4) 0.029(4) 0.029(4) 0.027(4) 0.028(4)TABLE VI. Exchange energies and correlation energies for allsystems (1-5), from exact calculations and from applying theHEG LDA self-consistently ( δE LDA = E LDA − E exact ). Esti-mated errors are ± System E x (a.u.)Exact δE HEGx e harmonic well − . . e harmonic well − . . e double well − . . e atom − . . e atom − . . E c (a.u.)Exact δE HEGc e harmonic well − . − . e harmonic well − . − . e double well − . − . e atom − . − . e atom − . − . LDA underestimates the magnitude of E x , while it over-estimates the magnitude of E c . However, due to the ex-change energy being the dominant component of E xc ,even in strongly correlated systems, this only leads to apartial cancellation of errors.The 1 e LDA yields a larger magnitude for ε x than theHEG LDA across the entire density range studied (up to0.6 a.u.) (Fig. 5), which arises from a better descriptionof the SIC (Sec. III B). In the 1 e LDA correlation is ab-sent. Consequently, the 1 e xc energies that follow fromTables IV and V can be considered as approximationsto E x . We note that the 1 e LDA substantially reducesthe error in E x that arises in the HEG LDA . We inferthat this error reduction will also extend to the 2 e and3 e LDAs.
IV. CONCLUSIONS
We have constructed an LDA based on the homo-geneous electron gas (HEG) through suitable quantumMonte Carlo techniques and find that it is remarkablysimilar in many regards to a set of three LDAs con-structed from finite systems. Applying them to test sys-tems to explore the differences between them, we findthat the finite LDAs give better densities and energiesin highly confined systems in which correlation is weak.Most interestingly, the LDA constructed from systemsof just one electron most accurately describes the self-interaction correction. All LDAs give poor densities insystems where correlation is stronger, but give reason-ably good energies, with the HEG LDA giving the besttotal energies. Across all test systems, the HEG LDA un-derestimates the magnitude of the exchange energy andoverestimates the magnitude of the correlation energy,leading to a partial cancellation of errors. As a conse-quence of the finite LDAs giving a better description ofthe self-interaction correction, we infer that they wouldreduce the error in the exchange energy. Furthermore,we expect that finite LDA functionals will also provide abetter treatment of the SIC for spinful electrons. Theirderivation and usage could lead to an improved descrip-tion of the electronic structure in a variety of situations,such as at the onset of Wigner oscillations.Data created during this research is available by re-quest from the York Research Database . ACKNOWLEDGMENTS
M.C. acknowledges the GENCI allocation for computerresources under Project No. 0906493. We thank LeopoldTalirz for recent developments in the iDEA code, andMatt Hodgson and Jack Wetherell for helpful discussions. P. Hohenberg and W. Kohn, Phys. Rev. , B864 (1964). R. Dreizler and E. Gross,
Density Functional Theory: AnApproach to the Quantum Many-Body Problem (SpringerBerlin Heidelberg, 2012). R. Parr and W. Yang,
Density-Functional Theory of Atomsand Molecules , International Series of Monographs onChemistry (Oxford University Press, USA, 1994). C. Fiolhais, F. Nogueira, and M. Marques,
A Primerin Density Functional Theory , Lecture Notes in Physics(Springer Berlin Heidelberg, 2003). K. Burke, The Journal of Chemical Physics , 150901(2012), https://doi.org/10.1063/1.4704546. R. M. Martin,
Electronic Structure: Basic Theory andPractical Methods (Cambridge University Press, 2004). x (a.u.) . . . . n , V e x t ( a . u . ) V ext Exact e LDAHEG LDA x (a.u.) . . . . . . n ( a . u . ) e LDAHEG LDA x (a.u.) . . . . . . V x c ( a . u . ) Exact e LDAHEG LDA
FIG. 8. System 3 (two electrons in a double-well potential).(a) The external potential (dotted-dashed blue line), togetherwith the exact electron density (solid red line), and the den-sities obtained from applying the 1 e (dashed green line) andHEG (dotted black line) LDAs. The wells are separated, suchthat the electrons are highly localized. (b) The absolute errorin the density in the 1 e (dashed green line) and HEG (dottedblack line) LDAs. The 1 e LDA is far superior in the regionswhere the density peaks, and hence where the Hartree poten-tial is large and dominated by the electron self-interaction.(c) The exact xc potential (solid red line), and the xc poten-tials given by the 1 e (dashed green line) and HEG (dottedblack line) LDAs. The dips in V xc are more closely matchedby the 1 e LDA due to it better capturing the self-interactioncorrection, present in the exact V xc . x (a.u.) . . . . . n ( a . u . ) Exact e LDAHEG LDA . . . . V e x t ( a . u . ) V ext x (a.u.) . . . . . n ( a . u . ) e LDAHEG LDA
FIG. 9. System 4 (two electrons in a softened atomiclikepotential). (a) The external potential (dotted-dashed blueline), together with the exact electron density (solid red line),and the densities obtained from applying the 1 e (dashed greenline) and HEG (dotted black line) LDAs. Unlike in the weaklycorrelated systems, the LDAs give poor electron densities. (b)The absolute error in the density in the 1 e (dashed green line)and HEG (dotted black line) LDAs. While the net absoluteerrors are much larger than in the weakly correlated systems,the 1 e LDA still performs the best. E. Engel and R. Dreizler,
Density Functional Theory: AnAdvanced Course , Theoretical and Mathematical Physics(Springer Berlin Heidelberg, 2011). W. Kohn and L. J. Sham, Phys. Rev. , A1133 (1965). D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. , 566(1980). Throughout this paper, we take the exchange energy tobe the exchange energy of a self-consistent Hartree-Fockcalculation. Throughout this paper, we take the correlation energy tobe the difference between the exact energy of the many-electron system and the energy of a self-consistent Hartree-Fock calculation. G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. ,601 (2002). M. Lein, E. K. U. Gross, and J. P. Perdew, Phys. Rev. B , 13431 (2000). O. Gritsenko, S. Van Gisbergen, A. G¨orling, E. Baerends, et al. , J. Chem. Phys , 8478 (2000). N. T. Maitra, K. Burke, and C. Woodward, Phys. Rev.Lett. , 023002 (2002). K. Burke, J. Werschnik, and E. Gross, J. Chem. Phys ,062206 (2005).
10 0 10 20 30 x (a.u.) . . . . . n ( a . u . ) Exact e LDAHEG LDA . . . V e x t ( a . u . ) V ext
10 0 10 20 30 x (a.u.) . . . . . . . n ( a . u . ) e LDAHEG LDA
FIG. 10. System 5 (three electrons in a softened atomiclikepotential). (a) The external potential (dotted-dashed blueline), together with the exact electron density (solid red line),and the densities obtained from applying the 1 e (dashed greenline) and HEG (dotted black line) LDAs. Like in the 2 e atom,the LDAs give poor electron densities. The 1 e LDA more ac-curately reproduces the peaks in the density, where the SIC islargest. (b) The absolute error in the density in the 1 e (dashedgreen line) and HEG (dotted black line) LDAs. Again, the netabsolute errors are large, with the 1 e LDA giving the smallest. D. Varsano, A. Marini, and A. Rubio, Phys. Rev. Lett. , 133002 (2008). R. O. Jones and O. Gunnarsson, Rev. Mod. Phys. , 689(1989). W. E. Pickett, Computer Physics Reports , 115 (1989). X. Gonze, P. Ghosez, and R. W. Godby, Phys. Rev. Lett. , 294 (1997). J. P. Perdew and A. Zunger, Phys. Rev. B , 5048 (1981). D. J. Tozer and N. C. Handy, J. Chem. Phys , 10180(1998). A. Dreuw, J. L. Weisman, and M. Head-Gordon, J. Chem.Phys , 2943 (2003). M. Levy, J. P. Perdew, and V. Sahni, Phys. Rev. A ,2745 (1984). C.-O. Almbladh and U. von Barth, Phys. Rev. B , 3231(1985). P. Mori-Sanchez and A. J. Cohen, Phys. Chem. Chem.Phys. , 14378 (2014). M. A. Mosquera and A. Wasserman, Phys. Rev. A ,052506 (2014). M. A. Mosquera and A. Wasserman,Molecular Physics , 2997 (2014), https://doi.org/10.1080/00268976.2014.968650. M. T. Entwistle, M. J. P. Hodgson, J. Wetherell,B. Longstaff, J. D. Ramsden, and R. W. Godby, Phys.Rev. B , 205134 (2016). M. Casula, C. Filippi, and S. Sorella, Phys. Rev. Lett. ,100201 (2005). M. J. P. Hodgson, J. D. Ramsden, J. B. J. Chapman,P. Lillystone, and R. W. Godby, Phys. Rev. B , 241102(2013). Spinless electrons obey the Pauli principle but are re-stricted to a single spin type. Systems of two or threespinless electrons exhibit features that would need a largernumber of spin-half electrons to become apparent. For ex-ample, two spinless electrons experience the exchange ef-fect, which is not the case for two spin-half electrons in an S = 0 state. Furthermore, spinless KS electrons occupy agreater number of KS orbitals. A. Gordon, R. Santra, and F. X. K¨artner, Phys. Rev. A , 063411 (2005). We use Hartree atomic units: m e = ~ = e = 4 πε = 1. We have significantly increased the precision of the calcula-tions for the slab systems in order to do this. The numericaldifference between the new seven-parameter fits and orig-inal four-parameter fits is less than 1% in ε xc across thedensity range used in constructing the LDAs (except inthe very low-density region n < .
06 a.u.). This has al-lowed us to resolve the differences between the four LDAsin fine detail. See Supplemental Material for the parametric form of thexc potential in the finite LDAs. See Supplemental Material for the parametric form of theexchange potential in the HEG LDA. See Supplemental Material for the parametric form of thecorrelation potential in the HEG LDA. M. Casula, S. Sorella, and G. Senatore, Phys. Rev. B ,245427 (2006). L. Shulenburger, M. Casula, G. Senatore, and R. M. Mar-tin, Journal of Physics A: Mathematical and Theoretical , 214021 (2009). See Supplemental Material for the parameters of the modelsystems, and details on our calculations to obtain con-verged results. We calculate the absolute error between the exact elec-tron density and the density obtained from a self-consistentHartree-Fock calculation ( δn = n HF − n exact ), and find thenet absolute error to be R | δn | dx ≈ . × − . The corre-lation energy is 0.13% of the exchange-correlation energy, − .
62 a.u. We calculate the absolute error between the exact elec-tron density and the density obtained from a self-consistentHartree-Fock calculation ( δn = n HF − n exact ), and find thenet absolute error to be R | δn | dx ≈ . × − . The cor-relation energy is 1.1% of the exchange-correlation energy, − .
37 a.u. M.-C. Kim, E. Sim, and K. Burke, Phys. Rev. Lett. ,073003 (2013). This is also true in the 2 e double-well system where corre-lation is negligible, and the exchange energy is dominatedby the SIC. M. T. Entwistle, M. Casula, and R. W. Godby, (2018),doi:10.15124/65cd1dd8-c240-45a5-9a47-0ac6ff870f51. omparison of local density functionals based on electron gas and finite systems(Supplemental Material)
M. T. Entwistle, M. Casula, and R. W. Godby Department of Physics, University of York, and European TheoreticalSpectroscopy Facility, Heslington, York YO10 5DD, United Kingdom Institut de Min´eralogie, de Physique des Mat´eriaux et de Cosmochimie (IMPMC),Sorbonne Universit´e, CNRS UMR 7590, IRD UMR 206, MNHN, 4 Place Jussieu, 75252 Paris, France (Dated: June 27, 2018)
EXCHANGE-CORRELATION POTENTIALS
The xc potential is defined as the functional derivative of the xc energy which in the LDA reduces to a simple form: V LDAxc ( n ) = δE LDAxc [ n ] δn ( x ) = ε xc ( n ( x )) + n ( x ) dε xc dn (cid:12)(cid:12)(cid:12)(cid:12) n ( x ) . (1) Finite LDAs
The xc energy density in the three finite LDAs is parameterized by (we use Hartree atomic units): ε xc ( n ) = ( A + Bn + Cn + Dn + En + F n ) n G . (2)The xc potential is obtained using Eq. (1) : V LDAxc ( n ) = (cid:2) A (1 + G ) + B (2 + G ) n + C (3 + G ) n + D (4 + G ) n + E (5 + G ) n + F (6 + G ) n (cid:3) n G . (3) HEG LDA
In the HEG LDA, the xc energy density is split into separate exchange and correlation parts. Consequently, Eq. (1)is also split into separate exchange and correlation parts.The exchange energy density in the HEG LDA is parameterized as in Eq. (2), and so the exchange potentialis of the same form as Eq. (3).The correlation energy density in the HEG LDA is parameterized by: ε c ( r s ) = − A RPA r s + Er Br s + Cr + Dr ln(1 + αr s + βr ) α , (4)where 2 r s = 1 /n . The correlation potential is given by: V c ( r s ) = ε c ( r s ) − r s α (1 + Br s + Cr + Dr ) " − ( A + 2 Er s ) ln(1 + αr s + βr )+ ( Ar s + Er )( B + 2 Cr s + 3 Dr ) ln(1 + αr s + βr )1 + Br s + Cr + Dr − ( Ar s + Er )( α + 2 βr s )(1 + αr s + βr ) . (5) a r X i v : . [ c ond - m a t . o t h e r] J un SYSTEM 1 (TWO-ELECTRON HARMONIC WELL)
The external potential is: V ext ( x ) = 12 ω x , (6)where ω = a.u. For this system converged results are obtained with δx = 0 .
01 a.u. for exact and Hartree-Fockcalculations, and δx = 0 .
005 a.u. for LDA calculations.
SYSTEM 2 (THREE-ELECTRON HARMONIC WELL)
The external potential is of the same form as Eq. (6) with ω = a.u. The grid spacing is δx = 0 .
04 a.u. for exactand Hartree-Fock calculations, and δx = 0 .
004 a.u. for LDA calculations.
SYSTEM 3 (TWO-ELECTRON DOUBLE WELL)
The external potential is: V ext ( x ) = αx − βx , (7)where α = 5 × − and β = 1 . × − . The grid spacing is δx = 0 .
015 a.u. for exact and Hartree-Fock calculations,and δx = 0 . SYSTEM 4 (TWO-ELECTRON ATOM)
The external potential is: V ext ( x ) = − | ax | + 1 , (8)where a = . The grid spacing is δx = 0 .
03 a.u. for exact and Hartree-Fock calculations, and δx = 0 .
015 a.u. forLDA calculations.
SYSTEM 5 (THREE-ELECTRON ATOM)
The external potential is of the same form as Eq. (8) with a = a.u. The grid spacing is δx = 0 .
225 a.u. for exactand Hartree-Fock calculations, and δx = 0 ..