My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 9.
ENERGY
A NEW CALCULATION OF THE JELLIUM SURFACE
-0.5
-1
-1.5
ε slab (mHa)
-2
-2.5
-3
-3.5
MPC
Q2D Ewald
3D Ewald
-4
0.005 0.01 0.015 0.02 0.025 0.03 0.035
s/N
Figure 9.7: Comparing the MPC and Ewald interactions when s = 18.4851.
± 0.042 respectively. From equation (9.6), these points should lie on a straight line
when plotted against 1/s; figure 9.8 shows that this is in fact the case. The gradient
of the line passing through these points is -28.5 ± 2.3, which gives the surface energy
of jellium as -0.384 ± 0.031 mHa bohr −2 or -600 ± 50 erg cm −2 . This figure brings
the QMC result into line with the other calculations referred to in chapter 5.
In contrast, the values of σ quoted in table 9.5 were calculated using the method
of Li et al. [53]: no Coulomb finite-size correction was applied, and the slab energy
was simply combined with Ceperley and Alder’s fixed-node value 3 for ɛ bulk in equation
(9.6). This demonstrates that, although the Coulomb finite-size error acts to
decrease the energy (see the three calculations with s = 11.7783 in table 9.5), the
greatest error is positive and comes from comparing the results of bulk and slab
simulations. The quality of the nodal surface of the trial wave function is better in
the homogeneous electron gas than in the jellium slab.
3 The fixed-node bulk calculation of Ceperley and Alder referred to by Li is unpublished; the
quoted energy per electron is ɛ bulk = −0.2017 eV per electron.
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