Comment on"Diffusion Monte Carlo study of jellium surfaces: Electronic densities and pair correlation functions"

In a fixed-node diffusion Monte Carlo calculation of the total energy of jellium slabs, Acioli and Ceperley [Phys. Rev. B {\bf 54}, 17199 (1996)] reported jellium surface energies that at low electron densities were significantly higher than those predicted in the local-density approximation (LDA) of density-functional theory. Assuming that the fixed-node error in the slab and the bulk calculations cancel out, we show that their data yield surface energies that are considerably closer to the LDA and in reasonable agreement with those obtained in the random-phase approximation.

Acioli and Ceperley 1 presented the results of fixednode difussion Monte Carlo (DMC) calculations of the total energy of jellium slabs at five different electron densities. Assuming that the released-node correction is very small at the electron densities of interest, these authors extracted surface energies by substracting from the fixed-node slab energy the corresponding released-node bulk energies of Ceperley and Alder, 2 as parametrized by Perdew and Zunger. 3 They concluded that at low electron densities jellium surface energies are significantly higher than those predicted in the local-density approximation (LDA) of density-functional theory (DFT).
In this Comment, we show that combining fixed-node slab and release-node bulk energies results in substantial imprecision. Instead, we expect a large degree of cancellation between the fixed-node error in the slab and the bulk calculations, and conclude that by substracting from the fixed-node slab energies the corresponding fixednode bulk energies one obtains jellium surface energies that are considerably closer to the LDA 4 and in reasonable agreement with those obtained in the random-phase approximation (RPA). 5 Acioli and Ceperley 1 considered jellium slabs of density n 0 = 3/4π (r s a 0 ) 3 (a 0 is the Bohr radius), the thickness of the positive background in each slab being L = 7.21 (r s a 0 ). Surface energies were then obtained from where ε slab and ε bulk represent slab and bulk energies per particle, respectively. Fixed-node energies per particle in bulk jellium were reported by Ceperley 6 to be higher than their release-node counterparts by 9 × 10 −4 Ry for r s = 2 and 2 × 10 −4 Ry for r s = 5. Hence, combining fixed-node slab and release-node bulk energies Eq. (1) yields for r s = 2 and 5 surface energies that are too large by ∼ 150 and 5 erg/cm 2 , respectively. These fixed-node errors represent ∼ 40% and ∼ 20% of the LDA correlation energy for r s = 2 and 5, respectively. Furthermore,  Table I. Also shown in this Table are the surface energies reported in Ref. 1 (σ AC ), together with the LDA and RPA surface energies that we have obtained for jellium slabs with L = 7.21 (r s a 0 ). Small differences of no more than 10 erg/cm 2 between these LDA and RPA surface energies I: Surface energies of jellium slabs with L = 7.21 (rs a0), as obtained from Eq. (1) by combining the Acioli-Ceperley fixed-node slab energies with the fixed-node bulk energies of either Eq. (2) (σ1) or Eq. (3) (σ2), and by combining the Acioli-Ceperley fixed-node slab energies with releasenode bulk energies (σAC). At rs = 2.07, combining the Acioli-Ceperley fixed-node slab energy with the corresponding fixed-node bulk energy ε bulk = −0.01482 Ry of Ceperley and Alder 16 yields σ = −558. σLDA and σRPA represent LDA and RPA surface energies. σ0 = σs + σes + σx represents the combined kinetic (σs), electrostatic (σes), and exchange (σx) contributions to the total surface energy σ. The correlation surface energy is simply σc = σ − σ0. Units are erg/cm 2 (1 erg/cm 2 = 6.2415 × 10 −5 eV/Å 2 ). and those reported before for the semi-infinite jellium 4,5 are due to the finite size of our jellium slabs.
An inspection of Table I shows that using in Eq. (1) the fixed-node bulk energies of either Eq. (2) or Eq. (3) brings the DMC surface energies closer to the LDA and to reasonable agreement with the RPA. This is consistent with recent work, where it was shown that upon surface formation there is a persistent cancellation of short-range correlation effects beyond the RPA. 11 Other approaches have also led to the conclusion that the actual jellium surface energies should be only slightly higher than those obtained in the LDA. 12,13, 14 Li et al. 15 calculated the fixed-node DMC surface energy of a jellium slab with r s = 2.07, the thickness of the positive background being L = 8.52 (r s a 0 ). These authors, 15 unlike Acioli and Ceperley, 1 extracted from their fixed-node slab energy the corresponding fixednode bulk energy (ε bulk = −0.01482 Ry), 16 and found σ = −465 erg/cm 2 ; 17 they also performed LDA calculations and found an LDA surface energy of −567 erg/cm 2 . We have calculated the LDA and RPA surface energies of a jellium slab with r s = 2.07 and L = 8.52 (r s a 0 ), and have found σ RPA = −485 erg/cm 2 and σ LDA = −589 erg/cm 2 , 18 which are both only ∼ 20 erg/cm 2 smaller than the corresponding DMC surface energy reported in Ref. 15. Fig. 1 exhibits the surface energies of Table I, as a function of r s . At the highest densities the agreement between DMC and RPA surface energies is good. At the smallest densities (r s ≥ 3.25) the corrected DMC surface energies are still larger than their RPA counterparts. Nevertheless, we note that DMC surface energies are very sensitive to small uncertainties in the parametrization of the bulk energy. Therefore, if one is to quantitatively account for jellium surface energies, both DMC slab and bulk energies entering Eq. (1) should be computed on the same footing. Small differences between the DMC and RPA calculations come from the correlation contribution to the surface energy. We have carried out calculations of exact kinetic (σ s ), electrostatic (σ es ), and exchange (σ x ) surface energies for jellium slabs with L = 7.21 (r s a 0 ), and we have defined the correlation contribution to the DMC surface energy as The results we have obtained from our analysis of the Acioli-Ceperley DMC slab energies are shown in Fig. 2 (see also and yield LDA surface energies that are close to their non-local counterparts.
In conclusion, assuming that the fixed-node error in the slab and the bulk calculations cancel out, the DMC data reported in Ref. 1 yields surface energies that are considerably closer to the LDA and in reasonable agreement with those obtained in the RPA. Nevertheless, at the smallest densities the corrected DMC surface energies are still larger than their RPA counterparts; at these densities, they are also larger than the jellium surface energies extracted from DMC calculations for jellium spheres, 20 which are found to be close to the LDA. 14 Surface energies are extremely sensitive to little uncertainties in both slab and bulk energies; hence, in order to quantitatively account for the impact of nonlocal xc effects on the surface energy one needs to pursue DMC evaluations of both slab and bulk energies on the same footing.
Partial support by the University of the Basque Country, the Basque Unibertsitate eta Ikerketa Saila, and the Spanish MCyT is acknowledged. I thank J. P. Perdew for stimulating discussions and D. M. Ceperley for correspondence related to the unpublished uniform-gas fixednode data for r s = 2.07.