My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 9.
ENERGY
A NEW CALCULATION OF THE JELLIUM SURFACE
-0.5
-1
-1.5
-2
ε slab (mHa)
-2.5
-3
-3.5
-4
s = 11.7783
-4.5
s = 15.1317
s = 18.4851
-5
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
s/N
Figure 9.6: The slab energy per electron, plotted against s/N, where s is the slab width and N is
the number of electrons. The abscissa is proportional to 1/L 2 , and was chosen to demonstrate the
form of the finite-size errors.
imately as 1/L 2 . The reason that the Jastrow factor performs less well in small
systems is almost certainly the parameter L c . This parameter, which keeps the
wave function free of unwanted cusps, depends on the cell size, and is not allowed to
be optimised. It modifies the long-range behaviour of the two-body term; it does not
appear to affect the electron density adversely (by rendering the derived one-body
term less efficient).
Most importantly, this finite-size error is independent of the slab width: the
evidence for this is the fact that the different curves in figure 9.6 remain equallyspaced.
This allows an accurate calculation of the surface energy to be made.
First, the curves s = 11.7783 and s = 15.1317 are interpolated on the abscissae
of the third curve, s = 18.4851. Working with the largest possible system size
(and thus effectively eliminating the Coulomb finite-size error), the slab energies
per electron at this point are (in mHa) -3.713 ± 0.063, -4.060 ± 0.051 and -4.593
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