Testing and Using the Lewin-Lieb Bound in Density Functional Theory

Lewin and Lieb have recently proven several new bounds on the exchange-correlation energy that complement the Lieb-Oxford bound. We test these bounds for atoms, for slowly-varying gases, and for Hooke's atom, finding them usually less strict than the Lieb-Oxford bound. However, we also show that, if a generalized gradient approximation (GGA) is to guarantee satisfaction of the new bounds for all densities, new restrictions on the the exchange-correlation enhancement factor are implied.


I. INTRODUCTION
The Lieb-Oxford (LO) bound [1] is a cornerstone of exact conditions in modern density functional theory. [2] Rigorously proven for non-relativistic quantum systems, the LO bound provides a strict upper bound on the magnitude of the exchange-correlation energy, E XC , of any system relative to a simple integral over its density. The constant in the LO bound was built into the construction of the PBE generalized gradient approximation [3], one of the most popular approximations in use in density functional theory (DFT) today. [4] The use of the bound to construct approximations remains somewhat controversial, as most systems' E XC does not come close to reaching this bound. [5] Recently, Lewin and Lieb [6] have proven several alternative forms for the bound that are distinct from the original LO bound. In each, some fraction of the density integral is traded for an integral over a density gradient. Thus, the new bounds are tighter for uniform and slowly varying gases, and could be hoped to be tighter for real systems. If so, they would be more useful than the LO bound in construction and testing of approximate density functionals. We show below that they are not tighter for atoms or for Hooke's atom (two electrons in a parabolic well). However, they do lead to new restrictions on the enhancement factor in GGAs that are constructed to guarantee satisfaction of the bounds for all possible densities.
To begin, the LO bound can be written as where C LO is a constant that Lieb and Oxford [1] showed is no larger than 1.68 (Chan and Handy showed it to be no larger than 1.6358). [7] For simplicity we define the following density integrals: Two new families of bounds are derived by Lewin and Lieb [6]: where C LL = 3(9π/2) 1/3 /5 ≈ 1.4508, c 1 = 1.206 × 10 −3 , c 2 = 0.2097, k = 5 − p, and α is any positive number. Here U XC is the potential energy contribution to the exchange-correlation energy. We can convert these to a family of conditions on E XC with several simple steps. We utilize the adiabatic connection formula [8,9] in terms of the scaled density: where n γ (r) = γ 3 n(γr) is the density scaled uniformly [10,11] by positive constant γ. Examining each of the integrals in the LL bounds, we find so that applying the bounds to every value of λ between 0 and 1 yields a bound on the DFT exchange-correlation energy directly: There is a Faustian dilemma when it comes to the value of α: A very small value makes the first term smaller in magnitude than that of the LO bound, but increases the of the gradient additions. A very large value will make those additions negligible, but also make the first term larger than that of LO bound. Choosing α in each case to maximize the right-hand-side, Lewin and Lieb find This yieldsc 1 = 4(3c 1 ) 1/4 /3 ≈ 0.3207 andc 2 = 3(2c 2 ) 1/3 /2 ≈ 1.1227. Lewin  bound by combining the p = 2 case with the Schwarz inequality: 2 ) ≈ 0.7650. We refer to these as the optimized LL bounds with LL1 and LL2 given by (9) with p = 1, 2 respectively. LL3 is given by (11).

II. LEWIN-LIEB BOUNDS FOR SPHERICALLY SYMMETRIC ATOMS
To test each of these bounds, we performed calculations using the non-relativistic atomic OEP code of Engel. [12] We used the PBE functional [3] to find selfconsistent atomic densities, and evaluated all I p . We did this for a simple subset of atoms for which highly accurate correlation energies are available. [13] The results are fully converged with respect to the radial grid. We use accurate E XC from Ref. 14 In Table I, we list the results. We see immediately that, unfortunately, the new bounds are less restrictive than the current LO bound. Atoms have gradients that are sufficiently large as to make the corrections larger than the density-integral term. To be sure that no atom behaves differently, we examine the large-Z limit, where Thomas-Fermi theory applies. [15,16] It has recently been shown [14] that for atoms, where C X = 0.2201, A = 0.020.., and B XC ≈ 0.039. The dominant term, which is an exchange contribution, was proven by Schwinger [17], and can be easily calculated by inserting the TF density [16] into the local approximation for E X . Since where A X = 3 4 3 π 1/3 ≈ 0.738, this easily satisfies all bounds, including any LL bound with the gradient terms ignored. In Fig. 1, we plot I 1 as a function of Z −1/3 , to show that it approaches its large Z limit, which can be extracted from the TF density: where we find d 1 = 3.58749. In Fig. 2, we plot the ratio I 2 /I 1 , showing that, although I 2 diverges in TF theory (as noted by Lewin and Lieb), it appears to vanish relative to I 1 in this limit. Thus all the additions in the LL bounds become relatively small in this limit, and no change in behavior occurs. As Z → ∞, the LL1 bound eventually becomes more restrictive than the LO bound, but only at unrealistically [18] large values of Z.

III. HOOKE'S ATOM AND THE SLOWLY-VARYING ELECTRON GAS
We also performed calculations on the model system of two electrons in a harmonic potential, the Hooke's atom.
[19] One might imagine that, for higher or lower densities, the bounds might tighten, or their order reverse, given the different external potential. We report three distinct results. For k → ∞, where k is the spring constant, the density becomes large, and E XC → E X . All energies and integrals scale as ω 1/2 , where ω = √ k. The first line of Table II shows the results, which are analogous to those of the two-electron ions (with different constants). The order of the bounds remains the same as in Table I. In the next line, we report actual energies for the largest value of k for which there exists an analytic solution, k = 1/4. Again we see the same behavior.
The most interesting case is the low-density limit, k → 0. In this limit, the kinetic energy becomes negligible, and the electrons arrange themselves to minimize the potential energy, on opposite sides of the center. This regime provides a system where correlation energy becomes comparable to exchange energy. The third line of Table II shows that none of the bounds is tight in this limit (the XC energy vanishes relative to any of them) and that the LL bounds diverge relative to the LO bound.

IV. EFFECTS ON FXC
In the rest of this paper, we show how the LL bounds can be used to derive interesting and new restrictions on the enhancement factor of generalized gradient approximations (GGA's). Begin with the definition of the enhancement factor for a GGA for spin unpolarized systems: where e unif X (n) = −A X n 4/3 is the exchange energy density of a spin-unpolarized uniform gas, r S = (3/(4πn)) 1/3 is the local Wigner-Seitz radius, and s = |∇n|/(2k F n) is the (exchange) dimensionless measure of the gradient, where k F = (3π 2 n) 1/3 is the local Fermi wavevector. Most famously, the PBE approximation was constructed to ensure it satisfies the LO bound for any density. A sufficient condition to guarantee this is Here we digress slightly, to correct a popular misconception in the literature. [20] The LO bound applies to the XC energy. There is no unique choice of XC energy density, and more than one choice was used in the derivation of PBE. [21] Thus the enhancement factor in PBE should not (and does not) correspond to any choice of energy density. No bound has ever been defined, much less proven, for a specific energy density. As others [5] and Table I have shown, real systems do not come close to saturating the LO bound. In fact, the B88 exchange functional does not satisfy Eq. (16) due to the logarithmic dependence on s. But for the present purposes, B88 gives exchange energies almost identical to PBE and very close to exact exchange energies for atoms. While one can design densities that cause B88 to violate the LO bound, they look nothing like densities of real systems. [22] Now we apply the logic of PBE to the LL bounds. We wish to find conditions on the enhancement factor that guarantee satisfaction of those bounds for all possible densities. In this context, the optimum bounds are not useful, since they contain denominators different from the local term. Dividing each term of Eq. (8) by e unif X , we find that is a sufficient condition to ensure satisfaction of the LL bounds for any density, withC LL = C LL /A X = 6(2π/3) 2/3 /5 = 1.9643..., and We now find the most restrictive value of α for each value of s, to yield Writing these out explicitly yields and F XC ≤ 1.9643 + 3.06212 s 2/3 , (LL2) (22) in contrast to the LO bound In Fig 3, we plot all three bounds and see that for small values of s, the LL2 bound is tighter than the LO bound. We therefore define the new Lieb-Oxford-Lewin (LOL) bound to be the LL2 bound for small s and the old LO bound otherwise. This new bound is more restrictive than the LO bound when s is less than 0.032. The existence of a tighter bound on the enhancement factor for all s has been empirically suggested. [23] We also plot the PBE enhancement factor, showing that it satisfies the LOL bound, but is much closer for small s than the old LO bound. Spin-polarization is handled different for exchange than for correlation. For exchange-correlation together, it does not raise F XC beyond its maximum for unpolarized systems, as that is achieved in the low-density limit, which is independent of spin. But in the opposite, high-density limit, exchange dominates, and its spindependence is determined by the exact spin-scaling relation for exchange:

V. CONCLUSION
In summary, we have tested the optimum LL bounds (which have already been applied to DFT by other groups [24]) for a variety of simple systems, finding they are less restrictive for those systems than the LO bound. However, the LL bounds are clearly more restrictive for a uniform gas, and the family of bounds that the LL bounds come from can be used to place limits on the enhancement factor of GGA's. With this in mind, we constructed the combined LOL bound and we recommend that the LOL bound be used whenever relevant for all future functional development and testing of GGAs.