My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 5.
THE JELLIUM SLAB
The DMC results clearly lie above those obtained in DFT. More evidence is provided
by Almeida et al. [3], who studied the jellium sphere system, and were able to match
their DFT surface energies with Sottile and Ballone’s finite-system DMC results over
a range of densities (although the precise value r s = 2.07 was not included in their
calculations). With the correction suggested by Pitarke, the calculation of Acioli
and Ceperley is brought closer to the DFT results, but this correction cannot be
applied to the calculations of Li et al.
In addition, both groups (Li et al. and Acioli and Ceperley) applied only the
independent-particle finite-size correction 4 to the slab energies, and not the Coulomb
correction. The Coulomb finite-size correction would increase the slab energies;
because the corresponding bulk energies were fully finite-size corrected, making the
correction would raise the DMC surface energy still further, increasing the difference
between the DMC and DFT results. Almeida believes the combined DFT-RPA
calculations to be currently the most accurate, putting the surface energy between
−550 and −590 erg cm −2 .
At this density, the separate contributions to the surface energy are individually
sizeable, but cancel as a whole; Pitarke and Eguiluz [72] give the following
breakdown:
• σ s = −4643 erg cm −2 (kinetic);
• σ es = 1072 erg cm −2 (electrostatic);
• σ xc = 3007 erg cm −2 (exchange-correlation).
This is the opposite of the ideal situation, and is partly why jellium surface energy
calculations are particularly difficult.
A relative error of 10% in the surface energy σ corresponds to around 50 erg
cm −2 or 0.03 mHa bohr −2 . Using equation (4.28) to calculate σ, and assuming that
there is no error in the bulk energy gives
∆σ = N 2A ∆ɛ slab =
3s ∆ɛ
8πrs
3 slab . (5.4)
4 These finite-size errors are discussed in chapter 4.
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