My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 4.
ERRORS IN QMC SIMULATIONS
4.4.2 The surface energy
The energy per electron of a finite slab system is
ɛ slab = ɛ bulk + 2Aσ
N
(4.27)
where A is the area, σ is the surface energy, N is the number of electrons and ɛ bulk
is the energy per electron in the bulk material. The factor of 2 appears because
there are two surfaces in a slab. In fact, this formula strictly only holds in the limit
of an infinitely wide slab; for slabs of finite width, the energy per electron may also
display oscillations [72]. In order to calculate the surface energy, one must therefore
consider a slab which is sufficiently wide to render these oscillations unimportant;
unfortunately, the wider the slab, the closer ɛ slab becomes to ɛ bulk , and the harder is
the calculation of the surface energy:
σ = N 2A (ɛ slab − ɛ bulk ). (4.28)
This is one of the major problems in surface energy calculations: extremely high
accuracy is often required, because the difference of two very similar numbers must
be taken.
The situation is helped if the systematic errors in ɛ bulk and ɛ slab are the same,
and cancel; however, the fundamental differences between slab and bulk calculations
mean that this cannot generally be relied on.
The difficulties of surface energy calculations are exemplified by the simplest of
surface systems — a slab of electron gas — which is described in the next chapter.
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