My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 9.
ENERGY
A NEW CALCULATION OF THE JELLIUM SURFACE
9.3 Surface energy calculations
On reviewing the earlier sections of this thesis, it is apparent that even with the
many improvements introduced here, finite-size errors cannot be eliminated. In
particular, the Coulomb finite-size error conjectured to be caused by ‘squashing’ of
the exchange-correlation hole must still be dealt with. In order to minimise the effect
of these errors on the calculation of the surface energy, great care must be taken
with the extrapolation to infinite system size; in addition, it is unwise to compare
results from slab and bulk systems.
This latter constraint means that simulations must be carried out for a range
of slab widths; the bulk and surface energies may then be extracted by a fitting
procedure, using equation (5.6), repeated here for convenience:
( ) 8πr
3
ɛ slab = ɛ bulk + s σ 1
3 s . (9.6)
The motivation for this approach is that the errors arising in bulk and slab calculations
may differ significantly; using only slab results should ensure a better
cancellation of errors.
As was noted in chapter 5, the surface energy calculated using equation (9.6)
displays oscillations of decreasing amplitude as the slab width is increased; the true
surface energy is the limiting value of σ as s → ∞. To minimise the errors introduced
by these oscillations, it is possible to choose slab widths for which the LDA value
of σ matches the extrapolated value: three such special slab widths are 11.7783,
15.1317 and 18.4851.
For each slab width, calculations must be performed for a range of cell sizes; the
results can then be extrapolated to the limit of infinite cell size. By selecting sizes
for which the LDA slab energy per electron corresponds to the infinite-cell value for
the same slab width, the need to apply the independent-particle finite-size correction
described in section 4.1.1 is obviated.
Each system to be simulated requires an optimised trial wave function. As usual,
LDA orbitals (with the image-tail correction of section 9.1) make up the Slater
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