Uniform Electron Gases

Uniform electron gases

Peter M. W. Gill∗ and Pierre-François Loos† Research School of Chemistry, Australian National University, Canberra, ACT 0200, Australia (Dated: February 7, 2012) We show that the traditional concept of the uniform electron gas (UEG) — a homogeneous system of finite density, consisting of an infinite number of electrons in an infinite volume — is inadequate to model the UEGs that arise in finite systems. We argue that, in general, a UEG is characterized by at least two parameters, viz. the usual one-electron density parameter ρ and a new two-electron parameter η. We outline a systematic strategy to determine a new density functional E(ρ,η) across the spectrum of possible ρ and η values.

PACS numbers: 31.15.E-, 71.15.Mb, 71.10.Ca, 73.20.-r Keywords: Uniform electron gas; Homogeneous electron gas; jellium; density functional theory

I. INTRODUCTION For example, significant improvements in accuracy can be achieved by using both the density ρ(r) and its gradient ∇ρ(r), The year 2012 is notable for both the journal and one of us an approach called gradient-corrected DFT [30]. Even better for, in the months ahead, both TCA and PMWG will achieve results can be achieved by including a fraction of Hartree- their half-centuries. It is inevitable and desirable that such Fock exchange (yielding hybrid methods [31, 32]) or higher occasions lead to retrospection, for it is often by looking back- derivatives of ρ(r) (leading to meta-GGAs [33]). wards that we can perceive most clearly the way ahead. Thus, However, notwithstanding the impressive progress since the as we ruminate on the things that we ought not to have done, 1970s, modern DFT approximations still exhibit fundamen- we also dream of the things that we ought now to do. tal deficiencies in large systems [34], conjugated molecules The final decade of the 20th century witnessed a major rev- [35], charge-transfer excited states [36], dispersion-stabilized olution in quantum chemistry, as the subject progressed from systems [37], systems with fractional spin or charge [38], an esoteric instrument of an erudite cognoscenti to a common- isodesmic reactions [39] and elsewhere. Because DFT is in place tool of the chemical proletariat. The fuel for this revo- principle an exact theory, many of these problems can be traced lution was the advent of density functional theory (DFT) [1] ultimately to the use of jellium as a reference system and the models and software that were sufficiently accurate and user- ad hoc corrections that its use subsequently necessitates. It is friendly to save the experimental chemist some time. These not a good idea to build one’s house on sand! days, DFT so dominates the popular perception of molecular In an attempt to avoid some of the weaknesses of jellium- orbital calculations that many non-specialists now regard the based DFT, we have invented and explored an alternative two as synonymous. paradigm called Intracule Functional Theory (IFT) [40–42]. In In principle, DFT is founded in the Hohenberg-Kohn theo- this approach, the one-electron density ρ(r) is abandoned in rem [2] but, in practice, much of its success can be traced to favour of two-electron variables (such as the interelectronic the similarity between the electron density in a molecule and distance r12) and we have discovered that the latter offer an the electron density in a hypothetical substance known as the efficient and accurate route to the calculation of molecular uniform electron gas (UEG) or jellium [3–18]. The idea — the energies [43–48]. Nonetheless, IFT is not perfect and has Local Density Approximation (LDA) — is attractively simple: shortcomings that are complementary to those of DFT. As a if we know the properties of jellium, we can understand the result, one should seek to combine the best features of each, electron cloud in a molecule by dividing it into tiny chunks of to obtain an approach superior to both. That is the goal of density and treating each as a piece of jellium. the present work and we will use atomic units throughout this The good news is that the properties of jellium are known article. from near-exact Quantum Monte Carlo calculations [19–29]. Such calculations are possible because jellium is characterized by just a single parameter ρ, the electron density. II. ELECTRONS ON SPHERES The bad news is that jellium has an infinite number of elec- arXiv:1202.1092v1 [physics.chem-ph] 6 Feb 2012 trons in an infinite volume and this unboundedness renders it, In recent research, we were led to consider the behavior of in some respects, a poor model for the electrons in molecules. electrons that are confined to the surface of a ball. This work Indeed, the simple LDA described above predicts bond ener- yielded a number of unexpected discoveries [49–57] but the gies that are much too large and this led many chemists in the one of relevance here is that such systems provide a beautiful 1970s to dismiss DFT as a quantitatively worthless theory. new family of uniform electron gases (see also [58]). Most of the progress since those dark days has resulted from concocting ingenious corrections for jellium’s deficiencies. A. Spherium atoms

∗ Corresponding author; [email protected] The surface of a three-dimensional ball is called a 2-sphere † [email protected] (for it is two-dimensional) and its free-particle orbitals (Table 2

TABLE I. The lowest free-particle orbitals on a 2-sphere TABLE III. The lowest free-particle orbitals on a glome (i.e. a 3- √ sphere) Name l m 4π Ylm(θ,φ) Name l m n π Ylmn(χ,θ,φ) s 0 0 1 1s 0 0 0 2−1/2 1/2 p0 1 0 3 cosθ 1/2 2s 1 0 0 21/2 cos χ p+1 1 +1 (3/2) sinθ exp(+iφ) 1/2 2p 1 1 0 21/2 sin χ cosθ p−1 1 –1 (3/2) sinθ exp(−iφ) 0 2p+1 1 1 +1 sin χ sinθ exp(+iφ) 1/2 2 d0 2 0 (5/4) (3cos θ − 1) 2p−1 1 1 –1 sin χ sinθ exp(−iφ) 1/2 d+1 2 +1 (15/2) sinθ cosθ exp(+iφ) 1/2 3s 2 0 0 2−1/2(4cos2 χ − 1) d−1 2 –1 (15/2) sinθ cosθ exp(−iφ) 1/2 2 3p 2 1 0 121/2 sin χ cos χ cosθ d+2 2 +2 (15/8) sin θ exp(+2iφ) 0 1/2 2 3p 2 1 +1 61/2 sin χ cos χ sinθ exp(+iφ) d−2 2 –2 (15/8) sin θ exp(−2iφ) +1 1/2 3p−1 2 1 –1 6 sin χ cos χ sinθ exp(−iφ) 2 2 3d0 2 2 0 sin χ (3cos θ − 1) TABLE II. Number of electrons in L-spherium and L-glomium atoms 1/2 2 3d+1 2 2 +1 6 sin χ sinθ cosθ exp(+iφ) 1/2 2 L 0 1 2 3 4 5 6 7 3d−1 2 2 –1 6 sin χ sinθ cosθ exp(−iφ) 1/2 2 2 3d+2 2 2 +2 (3/2) sin χ sin θ exp(+2iφ) L-spherium 2 8 18 32 50 72 98 128 1/2 2 2 3d−2 2 2 –2 (3/2) sin χ sin θ exp(−2iφ) L-glomium 2 10 28 60 110 182 280 408

C. Exactly solvable systems I) are the spherical harmonics Ylm(θ,φ). It is known that l 2l + 1 One of the most exciting features of the two-electron atoms |Y (θ,φ)|2 = (1) ∑ lm 4π 0-spherium and 0-glomium is that, for certain values of the m=−l radius R, their Schrodinger¨ equations are exactly solvable [51]. and doubly occupying all the orbitals with 0 ≤ l ≤ L thus yields The basic theory is as follows. a uniform electron gas. We call this system L-spherium and The Hamiltonian for two electrons on a sphere is will compare it to two-dimensional jellium [18]. ∇2 ∇2 1 The number of electrons (Table II) in L-spherium is H = − 1 − 2 + (7) 2 2 u n = 2(L + 1)2 (2) where u is the interelectronic distance r ≡ |r − r |. If we 2 12 1 2 the volume of a 2-sphere is V = 4πR and, therefore, assume that the Hamiltonian possesses eigenfunctions that (L + 1)2 depend only on u, it becomes ρ = 2 (3) 2πR u2 d2 (2D − 1)u D − 1 d 1 H = − 1 + − + (8) 4R2 du2 4R2 u du u

B. Glomium atoms where D is the dimensionality of the sphere. Three years ago, we discovered that H has polynomial eigenfunctions, but only The surface of a four-dimensional ball is a 3-sphere (or for particular values of R. (This is analogous to the discovery “glome” [59]) and its free-particle orbitals (Table III) are the that hookium[? ] has closed-form wavefunctions, but only for hyperspherical harmonics Y (χ,θ,φ). It is known [60] that particular harmonic force constants [62, 63].) We showed that lmn there exist b(n+1)/2c nth-degree polynomials of this type and l m 2 2 (l + 1) that the associated energies and radii satisfy |Ylmn(χ,θ,φ)| = (4) ∑ ∑ 2π2 2 m=0 n=−m 4Rn,mEn,m = n(n + 2D − 2) (9) and doubly occupying all the orbitals with 0 ≤ l ≤ L thus yields where the index m = 1,...,b(n + 1)/2c. a uniform electron gas. We call this system L-glomium and For 0-spherium (i.e. D = 2), by introducing x = u/(2R) and will compare it to three-dimensional jellium [18]. using Eq. (9), we obtain the Sturm-Liouville equation The number of electrons (Table II) in L-glomium is d x(1 − x2) dΨ n(n + 2)x n = (L + 1)(L + 2)(2L + 3)/3 (5) + Ψ = RΨ (10) dx 2 dx 2 the volume of a 3-sphere is V = 2π2R3 and, therefore, The eigenradii R can then be found by diagonalization in a (L + 1)(L + 2)(2L + 3) ρ = (6) polynomial basis which is orthogonal on [0,1]. The shifted 6π2R3 Legendre polynomials are ideal for this [64]. 3

KS For 0-glomium (i.e. D = 3), proceeding similarly yields Many formulae have been proposed for EX and EC but the most famous are those explicitly designed to be exact for d x(1 − x2) dΨ n(n + 4)x -jellium. In the case of exchange, one finds w(x) + w(x)Ψ = Rw(x)Ψ (11) D dx 2 dx 2 Z 1/D √ EX = XD ρ dr (20) where the weight function w(x) = x 1 − x2. The eigenradii are found by diagonalization in a basis which is orthogonal where Dirac [5] determined the coefficient X3 in 1930 and with respect to w(x) on [0,1]. Glasser [68] found the general formula for XD in 1983. An exact energy can be partitioned into its kinetic part The correlation functional is not known exactly but accurate Quantum Monte Carlo calculations on jellium in 2D [20– 2 2 ET = (−1/4)hΨ|∇1 + ∇2|Ψi (12) 23, 25, 26, 28, 29] and 3D [19, 24, 27] have been fitted [26, 69] to functions of the form and its two-electron part jell Z −1 EC = CD (ρ)dr (21) Eee = (1/2)hΨ|u |Ψi (13) and the resulting reduced energies (i.e. the energy per electron) By construction, Eq. (21) yields the correct energy when ap- of the ground states of 0-spherium and 0-glomium, for the first plied to the uniform electron gas in jellium. But what happens two eigenradii, are shown in the left half of Table IV. when we apply it to a uniform electron gas on a sphere?

III. SINGLE-DETERMINANT METHODS IV. THE NON-UNIQUENESS PROBLEM

A. Hartree-Fock theory [65, 66] The deeply disturbing aspect of jellium-based DFT models — and the launching-pad for the remainder of this paper — is the countercultural claim, that In the Hartree-Fock (HF) partition, the reduced energy[? ] of an n-electron system is The uniform electron gas with density ρ is not unique.

E = TS + EV + EJ + EK + EC (14) Though it may seem heretical to someone who has worked with jellium for many years, or to someone who suspects that where the five contributions are the non-interacting-kinetic, ex- the claim violates the Hohenberg-Kohn theorem, we claim ternal, Hartree, exchange and correlation energies, respectively. that two D-dimensional uniform electron gases with the same The first four of these are defined by density parameter ρ may have different energies. To illustrate this, we now show that density functionals [5, 26, 69] which n Z 1 ∗ 2 are exact for jellium are wrong for 0-spherium and 0-glomium. TS = − ∑ ψi (r)∇ ψi(r)dr (15) 2n i 1 Z E = + ρ(r)v(r)dr (16) A. Illustrations from exactly solvable systems V n ZZ 1 −1 EJ = + ρ(r1)r12 ρ(r2)dr1 dr2 (17) The energy contributions for 0-spherium and 0-glomium are 2n easy to find. There is no external potential, so E = 0. The n ZZ V 1 ∗ −1 ∗ density ρ(r) is constant, so the Kohn-Sham orbital ψ(r) = EK = − ∑ ψi (r1)ψ j(r1)r12 ψ j (r2)ψi(r2)dr1 dr2 p 2n i, j ρ(r) is constant, and TS = 0. The Hartree energy is the (18) self-repulsion of a uniform spherical shell of charge of radius R and one finds [50] where ψi(r) is an occupied orbital, ρ(r) is the electron density, Γ(D − 1) Γ(D/2 + 1/2) 1 and v(r) is the external potential. The correlation energy EC is EJ = (22) defined so that Eq. (14) is exact. Γ(D − 1/2) Γ(D/2) R The exchange energy is predicted [57] by Eq. (20) to be

B. Kohn-Sham density functional theory [67] / 2D D!1 D EX = − (23) (D2 − 1)πR 2 In the Kohn-Sham (KS) partition, the energy is and the correlation energy predicted by Eq. (21) is simply KS EKS = TS + EV + EJ + EX + EC (19) jell E = CD (2/V) (24) where the last two terms, which are sometimes combined, C are the Kohn-Sham exchange and correlation energies. The Applying these formulae to the exactly solvable states of 0- KS correlation energy EC is deﬁned so that Eq. (19) is exact. spherium and 0-glomium considered in Section II C yields 4

TABLE IV. Exact and Kohn-Sham reduced energies of the ground states of 0-spherium and 0-glomium for various eigenradii R

Exact Jellium-based Kohn-Sham DFT Error

jell 2RET Eee ETS EV EJ −EX −EC EKS EKS − E √ 0-spherium 3 0.051982 0.448018 1/2 0 0 1.154701 0.490070 0.1028 0.562 0.062 √ 28 0.018594 0.124263 1/7 0 0 0.377964 0.160413 0.0593 0.158 0.015 √ 0-glomium 10 0.014213 0.235787 1/4 0 0 0.536845 0.217762 0.0437 0.275 0.025 √ 66 0.007772 0.083137 1/11 0 0 0.208967 0.084764 0.0270 0.097 0.006 the results in the right half of Table IV. In all cases, the KS- DPHuL DFT energies are too high by 10 – 20%, indicating that the correlation functional that is exact for the uniform electron gas 0.06 in jellium grossly underestimates the correlation energy of the uniform electron gases in 0-spherium and 0-glomium. 0.04

B. Limitations of the one-electron density parameter ρ 0.02

u The results in Table IV demonstrate conclusively that not all 0.5 1.0 1.5 uniform electron gases with the density ρ are equivalent. The simplest example of this√ is the ground state of two electrons on -0.02 a 2-sphere with R = 3/2. The exact wavefunction, reduced energy and density of this system are √FIG. 1. Specific Coulomb holes for 0-spherium (dotted) with R = Ψ = 1 + u (25) 3/2 and 2D jellium (solid). Both are uniform gases with ρ = 2/(3π). E = 1/2 (26) ρ(r) = 2/(3π) (27) key insight is that the probability of finding two electrons in but, when fed this uniform density, the exchange-correlation that volume is different. functional that is exact for two-dimensional jellium grossly This is illustrated in Figs 1 and 2, which compare the proba- overestimates the energy, yielding EKS = 0.562. bility distributions of the interelectronic distance [52, 70, 71] This discovery has worrying chemical implications. Con- in various two-dimensional uniform electron gases. These trary to the widespread belief that the LDA (e.g. the S-VWN reveal that, although similar for u ≈ 0 (because of the Kato functional) is accurate when applied to regions of a molecule cusp condition [72]), the specific Coulomb holes (i.e. the holes where ρ(r) is almost uniform (such as near bond midpoints), per unit volume [40]) in two gases with the same one-electron our results reveal that it actually performs rather poorly. density ρ can be strikingly different. In each case, the jellium The discovery also has counterintuitive implications at a hole is both deeper and wider than the corresponding spherium theoretical level. It implies that the years of effort that have hole, indicating that the jellium electrons exclude one another been expended in calculating the properties of jellium do not more strongly, and one is much less likely to find two electrons provide us with a complete picture of homogeneous electron in a given small volume of jellium than in the same volume of gases. On the contrary, although they inform us in detail about spherium. the infinite uniform electron gas, they tell us very little about We conclude from these comparisons that (at least) two the properties of finite electron gases. parameters are required to characterize a uniform electron gas. In a nutshell, the results in Table IV reveal that a UEG is not Although the parameter choice is not unique, we believe that completely characterized by its one-electron density parameter the first should be a one-electron quantity, such as the density ρ. Evidently, something else is required. . . ρ (or, equivalently, the Seitz radius rs) and the second should be a two-electron quantity such as η = h(r,r), where h is the pair correlation function defined by C. Virtues of two-electron density parameters 1 ρ (r ,r ) = ρ(r )ρ(r )[1 + h(r ,r )] (28) We know that it is possible for two uniform electron gases 2 1 2 2 1 2 1 2 to have the same density ρ but different reduced energies E. But how can this be, given that the probability of finding an and ρ2 is the diagonal part of the second-order spinless density electron in a given volume is identical in the two systems? The matrix [1]. 5

DPHuL Accordingly, we propose to embark on a comprehensive study of L-spherium and L-glomium atoms, in order eventually 0.005 to liberate the LDA from jellium’s yoke through a process 0.004 of radical generalization. The results of these spherium and glomium calculations will generalize the known properties 0.003 of jellium, because we have shown that the energies of L- 0.002 spherium and L-glomium approach those of 2D jellium and 3D jellium, as L becomes large. 0.001 In the remaining sections, we will conﬁne our attention to u the (two-dimensional) L-spherium atoms. However, exactly 1 2 3 4 5 the same approach can and will be used to address the (three- - 0.001 dimensional) L-glomium atoms in the future. -0.002 √ FIG. 2. Speciﬁc Coulomb holes for 0-spherium (dotted) with R = 7 B. Basis sets and integrals and 2D jellium (solid). Both are uniform gases with ρ = 1/(14π). The Hamiltonian for L-spherium is V. LESSONS FROM SPHERIUM AND GLOMIUM 1 n n 1 H = − ∇2 + (32) ∑ i ∑ r A. A modest proposal 2 i=1 i< j i j

and the natural basis functions for HF and correlated calcula- The discovery that uniform electron gases are characterized tions on this are the spherical harmonics Y (θ,φ) introduced by two parameters (ρ and η) has many ramifications but one lm in Section II. These functions are orthonormal [64] of the most obvious is that the foundations of the venerable Local Density Approximation need to be rebuilt. Y Y 0 0 = δ 0 δ 0 (33) The traditional LDA writes the correlation energy of a molec- lm l m l,l m,m ular system as and are eigenfunctions of the Laplacian, so that Z EKS ≈ C(ρ)dr (29) 2 C Ylm ∇ Yl0m0 = −l(l + 1) δl,l0 δm,m0 (34) thereby assuming that the contribution from each point r de- The required two-electron repulsion integrals can be found pends only on the one-electron density ρ(r) at that point. How- using the standard methods of two-electron integral theory ever, now that we know that the energy of a uniform electron [74]. For example, the spherical harmonic resolution of the gas depends on ρ and η, it is natural to replace Eq. (29) by the Coulomb operator [75–79] generalized expression −1 −1 4π ∗ Z r = R Y (r1)Ylm(r2) (35) KS 12 ∑ 2l + 1 lm EC ≈ C(ρ,η)dr (30) lm yields the general formula In a sense, this two-parameter LDA represents a convergence in the evolution of Density Functional Theory (which stresses the −1 one-electron density) and Intracule Functional Theory (which Yl1m1Ya1b1 r12 Yl2m2Ya2b2 focuses on the two-electron density). 4π = R−1 Y Y Y Y Y Y (36) How can we find the new density functional C(ρ,η)? One ∑ l1m1 l2m2 lm a1b1 a2b2 lm lm 2l + 1 could take an empirical approach but that is an overused option within the DFT community [73] and we feel that it is more where the one-electron integrals over three spherical harmonics satisfactory to derive it from the uniform electron gas. As we involve Wigner 3 j symbols [64] and the sum over l and m is show in Section V C, it turns out that it is easy to compute limited by the Clebsch-Gordan selection rules. the exact values of TS, EV, EJ and EX for any L-spherium or L-glomium atom and, therefore, if one knew the exact wavefunctions and energies of L-spherium and L-glomium for a wide range of L and R, one could extract the exact Kohn-Sham C. Hartree-Fock calculations correlation energies Unlike our calculations for electrons in a cube [80], the HF KS EC = E − TS − EV − EJ − EX (31) calculations for our present systems are trivial. Because the shells in L-spherium are filled, the restricted[? ] HF and KS and determine the exact dependence of these on ρ and η. orbitals are identical and are simply the spherical harmonics. 6

TABLE V. Reduced energies, densities and η values of L-spherium with the four smallest eigenradii† R

Wavefunction-based energies Kohn-Sham energies Ingredients of the new model

KS LREHF −EC ETS EJ −EX −EC ρ η

0 R1 0.577350 0.077350 0.500000 0.000000 1.154701 0.490070 0.164630 0.212207 -0.896037 R2 0.188982 0.046125 0.142857 0.000000 0.377964 0.160413 0.074694 0.022736 -0.991159 R3 0.092061 0.028497 0.063564 0.000000 0.184122 0.078144 0.042414 0.005396 -0.999496 R4 0.054224 0.018941 0.035282 0.000000 0.108447 0.046026 0.027138 0.001872 -0.999976

1 R1 4.579572 1.000000 4.618802 0.980140 0.848826 R2 1.278833 0.107143 1.511858 0.320826 0.090946 R3 0.596204 0.025426 0.736488 0.156288 0.021582 R4 0.345007 0.008821 0.433789 0.092053 0.007487

2 R1 11.543198 2.666667 10.392305 1.470210 1.909859 R2 3.191241 0.285714 3.401680 0.481239 0.204628 R3 1.483203 0.067802 1.657098 0.234431 0.048560 R4 0.857188 0.023522 0.976025 0.138079 0.016846

3 R1 21.477457 5.000000 18.475209 1.960281 3.395305 R2 5.929228 0.535714 6.047432 0.641652 0.363783 R3 2.754531 0.127128 2.945952 0.312575 0.086328 R4 1.591634 0.044103 1.735156 0.184106 0.029949 √ √ √ √ † 1 1 1 p 1 p R1 = 2 3; R2 = 2 28; R3 = 2 63 + 12 21; R4 = 2 198 + 6 561

These orbitals yield the reduced energy contributions orbital-based correlation methods may be particularly effective. We now consider some of these. L(L + 2) a. Configuration interaction [81] This was the original TS = + (37) 4R2 scheme for proceeding beyond the HF approximation. It has EV = +0 (38) fallen out of favor with many quantum chemists, because its (L + 1)2 size-inconsistency and size-inextensivity seriously hamper its EJ = + (39) efficacy for computing the energetics of chemical reactions. R " # Nevertheless, for calculations of the energy of L-spherium, L + 1 −L, −1/2, 1/2, L + 2 its variational character, systematic improvability and lack of EK = − 4F3 ; 1 (40) 2R −L − 1/2, 2, L + 3/2 convergence issues make it an attractive option. 4(L + 1) b. Møller-Plesset perturbation theory [82] In an earlier EX = − (41) paper [49], we showed that the MP2, MP3, MP4 and MP5 3πR energies of 0-spherium can be found in closed form, for any where 4F3 is a balanced hypergeometric function [64] of unit value of R. However, although we observed that the MP series argument[? ]. It is encouraging to discover that Eq. (40) seems to converge rapidly for small R, its convergence was approaches Eq. (41) in the large-L limit [57]. much less satisfactory for R & 1, where ρ . 0.15. Unfortu- We have used Eqs (37) – (40) to compute the exact HF nately, this is useless for our present purposes, because many energies of L-spherium for a number of the eigenradii discussed of the L-spherium systems in Table V have much larger radii in Section II C. These, together with the TS, EJ and EX values, and lower densities than these values. and ρ values from Eq. (3), are shown in Table V. c. Coupled-cluster theory [83] The coupled-cluster hier- archy (viz. CCSD, CCSDT, CCSDTQ, . . . ) probably converges much better than the MPn series, and this should certainly be D. Orbital-based correlation methods explored in the future, but we suspect that it will nonetheless perform poorly in the large-R, small-ρ systems where static Although it is easy to find the HF energy of L-spherium, the correlation dominates dynamic correlation [49, 84, 85] and the calculation of its exact energy is not a trivial matter. How can single-configuration HF wavefunction is an inadequate starting this best be achieved? The fact that the occupied and virtual point. orbitals are simple functions (spherical harmonics), so the AO d. Explicitly-correlated methods [86] The CI, MP and → MO integral transformation is unnecessary, suggests that CC approaches expand the exact wavefunction as a linear com- 7 bination of determinants and it has been known since the early F. Results and holes days of quantum mechanics that this ansatz struggles to de- scribe the interelectronic cusps [72]. The R12 methods over- The discovery [51] that the Schrodinger¨ equation for 0- come this deficiency by explicitly including terms that are spherium (and 0-glomium) is exactly solvable for each of its linear in ri j in the wavefunction, and converge much more eigenradii is extremely helpful, for it allows us to generate the rapidly as the basis set is enlarged but, unfortunately, they also exact energies E and resulting Kohn-Sham correlation energies require a number of non-standard integrals. If these integrals KS EC for the first four atoms in Table V, without needing to are not computed exactly, they are approximated by resolu- perform any of the correlated calculations described above. tions in an auxiliary basis set. In either case, the computer However, these are the easiest cases, the “low-hanging fruit” implementation is complicated. so to speak, and there remain large gaps in the Table. We could have filled these gaps with rough estimates of the exact energies but we prefer to leave them empty, to emphasize that there is much to do in this field and to challenge the correlation ex- perts in the wavefunction and density functional communities E. Other correlation methods to address these beautifully simple systems. Once this is done, the data in the final three columns of Table V will provide the ingredients for the construction of a new It is possible that the unusually high symmetry of L- correlation density functional EKS(ρ,η) that will be exact for spherium makes it well suited to correlation methods that are C all L-spherium atoms and for 2D jellium. not based on the HF orbitals. We now consider two of these. Finally, of course, an analogous strategy will yield a new e. Iterative Complement Interaction (ICI) method [87, 88] functional that is exact for all L-glomium atoms and for 3D In this approach, the Schrodinger¨ equation itself is used to jellium. We will recommend that these functionals replace the generate a large set of n-electron functions that are then linearly LDA correlation functionals that are now being used. combined to approximate the true wavefunction. It has been spectacularly successful in systems with a small number of electrons, but has not yet been applied to systems as large as 2- VI. CONCLUDING REMARK or 3-spherium (with 18 and 32 electrons, respectively). f. Quantum Monte Carlo methods [89, 90] Of the var- All uniform electron gases are equal, but some are more ious methods in this family, Diffusion Monte Carlo (DMC) equal than others. usually yields the greatest accuracy. In this approach, the Schrodinger¨ equation is transformed into a diffusion equation in imaginary time τ and, in the limit as τ → ∞, the ground state ACKNOWLEDGMENTS energy is approached. Unfortunately, to be practically feasible, the method normally requires that the wavefunction’s nodes be We would like to thank the Australian Research Council for known and, despite some recent progress [91], the node prob- funding (Grants DP0771978, DP0984806 and DP1094170) and lem remains unsolved. The quality of the trial wavefunction the National Computational Infrastructure (NCI) for generous and finite-size errors are other potential restrictions [92, 93]. supercomputer grants.

[1] R.G. Parr, W. Yang, Density-functional theory of atoms and [18] G.F. Giuliani, G. Vignale, Quantum Theory of Electron Liquid molecules (Oxford, Clarendon Press, 1989) (Cambridge University Press, Cambridge, England, 2005) [2] P. Hohenberg, W. Kohn, Phys. Rev. 136, B864 (1964) [19] D.M. Ceperley, B.J. Alder, Phys. Rev. Lett. 45, 566 (1980) [3] E. Fermi, Z. Phys. 36, 902 (1926) [20] B. Tanatar, D.M. Ceperley, Phys. Rev. B 39, 5005 (1989) [4] L.H. Thomas, Proc. Cam. Phil. Soc. 23, 542 (1927) [21] Y. Kwon, D.M. Ceperley, R.M. Martin, Phys. Rev. B 48, 12037 [5] P.A.M. Dirac, Proc. Cam. Phil. Soc. 26, 376 (1930) (1993) [6] E. Wigner, Phys. Rev. 46, 1002 (1934) [22] G. Ortiz, P. Ballone, Phys. Rev. B 50, 1391 (1994) [7] W. Macke, Z. Naturforsch. A 5a, 192 (1950) [23] F. Rapisarda, G. Senatore, Aust. J. Phys. 49, 161 (1996) [8] M. Gell-Mann, K.A. Brueckner, Phys. Rev. 106, 364 (1957) [24] Y. Kwon, D.M. Ceperley, R.M. Martin, Phys. Rev. B 58, 6800 [9] L. Onsager, L. Mittag, M.J. Stephen, Ann. Phys. 18, 71 (1966) (1998) [10] F. Stern, Phys. Rev. Lett. 30, 278 (1973) [25] G. Ortiz, M. Harris, P. Ballone, Phys. Rev. Lett. 82, 5317 (1999) [11] A.K. Rajagopal, J.C. Kimball, Phys. Rev. B 15, 2819 (1977) [26] C. Attaccalite, S. Moroni, P. Gori-Giorgi, G.B. Bachelet, Phys. [12] A. Isihara, L. Ioriatti, Phys. Rev. B 22, 214 (1980) Rev. Lett. 88, 256601 (2002) [13] G.G. Hoffman, Phys. Rev. B 45, 8730 (1992) [27] F.H. Zong, C. Lin, D.M. Ceperley, Phys. Rev. E 66, 036703 [14] M. Seidl, Phys. Rev. B 70, 073101 (2004) (2002) [15] J. Sun, J.P. Perdew, M. Seidl, Phys. Rev. B 81, 085123 (2010) [28] N.D. Drummond, R.J. Needs, Phys. Rev. Lett. 102, 126402 [16] P.F. Loos, P.M.W. Gill, Phys. Rev. B 83, 233102 (2011) (2009) [17] P.F. Loos, P.M.W. Gill, Phys. Rev. B (in press) [29] N.D. Drummond, R.J. Needs, Phys. Rev. B 79, 085414 (2009) 8

[30] J.P. Perdew, Y. Wang, Phys. Rev. B 33, 8800 (1986) [62] S. Kais, D.R. Herschbach, R.D. Levine, J. Chem. Phys. 91, 7791 [31] P.M.W. Gill, B.G. Johnson, J.A. Pople, M.J. Frisch, Int. J. Quan- (1989) tum Chem., Symp. 26, 319 (1992) [63] M. Taut, Phys. Rev. A 48, 3561 (1993) [32] A.D. Becke, J. Chem. Phys. 98, 5648 (1993) [64] F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark (eds.), [33] J. Tau, J.P. Perdew, V.N. Staroverov, G.E. Scuseria, Phys. Rev. NIST handbook of mathematical functions (Cambridge Univer- Lett. 91, 146401 (2003) sity Press, New York, 2010) [34] L.A. Curtiss, K. Raghavachari, P.C. Redfern, J.A. Pople, J. Chem. [65] D.R. Hartree, Proc. Cam. Phil. Soc. 24, 89 (1928) Phys. 112, 7374 (2000) [66] V. Fock, Z. Phys. 61, 126 (1930) [35] H.L. Woodcock III, H.F. Schaefer III, P.R. Schreiner, J. Phys. [67] W. Kohn, L.J. Sham, Phys. Rev. 140, A1133 (1965) Chem. A 106, 11923 (2002) [68] L. Glasser, J. Boersma, SIAM J. Appl. Math. 43, 535 (1983) [36] A. Dreuw, M. Head-Gordon, J. Am. Chem. Soc. 126, 4007 [69] J.P. Perdew, Y. Wang, Phys. Rev. B 45, 13244 (1992) (2004) [70] C.A. Coulson, A.H. Neilson, Proc. Phys. Soc. (London) 78, 831 [37] M.D. Wodrich, C. Corminboeuf, P. Schleyer, Org. Lett. 8, 3631 (1961) (2006) [71] P. Gori-Giorgi, S. Moroni, G.B. Bachelet, Phys. Rev. B 70, [38] A.J. Cohen, P. Mori-Sanchez, W. Yang, Science 321, 792 (2008) 115102 (2004) [39] D.R.B. Brittain, C.Y. Lin, A.T.B. Gilbert, E.I. Izgorodina, P.M.W. [72] T. Kato, Comm. Pure. Appl. Math 10, 151 (1957) Gill, M.L. Coote, Phys. Chem. Chem. Phys. 11, 1138 (2009) [73] P.M.W. Gill, Aust. J. Chem. 54, 661 (2001) [40] P.M.W. Gill, D.P. O’Neill, N.A. Besley, Theor. Chem. Acc. 109, [74] P.M.W. Gill, Adv. Quantum Chem. 25, 141 (1994) 241 (2003) [75] S.A. Varganov, A.T.B. Gilbert, E. Deplazes, P.M.W. Gill, J. [41] P.M.W. Gill, D.L. Crittenden, D.P. O’Neill, N.A. Besley, Phys. Chem. Phys. 128, 201104 (2008) Chem. Chem. Phys. 8, 15 (2006) [76] P.M.W. Gill, A.T.B. Gilbert, Chem. Phys. 356, 86 (2009) [42] P.M.W. Gill, Annu. Rep. Prog. Chem., Sect. C 107, 229 (2011) [77] T. Limpanuparb, P.M.W. Gill, Phys. Chem. Chem. Phys. 11, [43] E.E. Dumont, D.L. Crittenden, P.M.W. Gill, Phys. Chem. Chem. 9176 (2009) Phys. 9, 5340 (2007) [78] T. Limpanuparb, A.T.B. Gilbert, P.M.W. Gill, J. Chem. Theory [44] D.L. Crittenden, E.E. Dumont, P.M.W. Gill, J. Chem. Phys. 127, Comput. 7, 830 (2011) 141103 (2007) [79] T. Limpanuparb, P.M.W. Gill, J. Chem. Theory Comput. (in [45] D.L. Crittenden, P.M.W. Gill, J. Chem. Phys. 127, 014101 press) (2007) [80] S. Ghosh, P.M.W. Gill, J. Chem. Phys. 122, 154108 (2005) [46] Y.A. Bernard, D.L. Crittenden, P.M.W. Gill, Phys. Chem. Chem. [81] J.C. Slater, Phys. Rev. 34, 1293 (1929) Phys. 10, 3447 (2008) [82] C. Møller, M.S. Plesset, Phys. Rev. 46, 681 (1934) [47] J.K. Pearson, D.L. Crittenden, P.M.W. Gill, J. Chem. Phys. 130, [83] G.D. Purvis, R.J. Bartlett, J. Chem. Phys. 76, 1910 (1982) 164110 (2009) [84] J.W. Hollett, P.M.W. Gill, J. Chem. Phys. 134, 114111 (2011) [48] J.W. Hollett, P.M.W. Gill, Phys. Chem. Chem. Phys. 13, 2972 [85] J.W. Hollett, L.K. McKemmish, P.M.W. Gill, J. Chem. Phys. (2011) 134, 224103 (2011) [49] P.F. Loos, P.M.W. Gill, Phys. Rev. A 79, 062517 (2009) [86] W. Kutzelnigg, W. Klopper, J. Chem. Phys. 94, 1985 (1991) [50] P.F. Loos, P.M.W. Gill, J. Chem. Phys. 131, 241101 (2009) [87] H. Nakatsuji, Phys. Rev. Lett. 93, 030403 (2004) [51] P.F. Loos, P.M.W. Gill, Phys. Rev. Lett. 103, 123008 (2009) [88] H. Nakatsuji, H. Nakashima, Y. Kurokawa, A. Ishikawa, Phys. [52] P.F. Loos, P.M.W. Gill, Phys. Rev. A 81, 052510 (2010) Rev. Lett. 99, 240402 (2007) [53] P.F. Loos, Phys. Rev. A 81, 032510 (2010) [89] A. Luchow,¨ J.B. Anderson, Annu. Rev. Phys. Chem. 51, 501 [54] P.F. Loos, P.M.W. Gill, Phys. Rev. Lett. 105, 113001 (2010) (2000) [55] P.F. Loos, P.M.W. Gill, Mol. Phys. 108, 2527 (2010) [90] W.M.C. Foulkes, L. Mitas, R.J. Needs, G. Rajagopal, Rev. Mod. [56] P.F. Loos, P.M.W. Gill, Chem. Phys. Lett. 500, 1 (2010) Phys. 73, 33 (2001) [57] P.F. Loos, P.M.W. Gill, Phys. Rev. B (submitted) [91] L. Mitas, Phys. Rev. Lett. 96, 240402 (2006) [58] M. Seidl, Phys. Rev. A 75, 062506 (2007) [92] N.D. Drummond, M.D. Towler, R.J. Needs, Phys. Rev. B 70, [59] URL http://mathworld.wolfram.com/Glome.html 235119 (2004) [60] J. Avery, Hyperspherical harmonics: applications in quantum [93] N.D. Drummond, R.J. Needs, A. Sorouri, W.M.C. Foulkes, Phys. theory (Kluwer Academic, Dordrecht, 1989) Rev. B 78, 125106 (2008) [61] N.R. Kestner, O. Sinanoglu, Phys. Rev. 128, 2687 (1962)