My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 9.
ENERGY
A NEW CALCULATION OF THE JELLIUM SURFACE
9.1 Improved orbitals
The single-electron orbitals which make up the Slater determinants in the QMC
trial wave function must be obtained from some prior calculation. Typically, this is
an LDA calculation; the earlier QMC simulations of the jellium slab relied on LDA
wave functions.
As described in chapter 2, a key component of any density-functional theory
calculation is the exchange-correlation potential V XC . It was shown by Lang and
Kohn [48] that outside a metal surface, the correct asymptotic form of the potential
is image-like:
1
V XC (z) = −
4(z − z 0 ) . (9.1)
Here z is the coordinate normal to the surface and z 0 is the position of the image
plane. However, the LDA gives a potential which decays exponentially. Recently,
Eguiluz and coworkers investigated the form of V XC at a metal surface from first
principles [19, 20]; their potential reproduced the asymptotic form given in equation
(9.1), and matched the conventional LDA value inside the metal.
Having the correct image tail in the potential is important for studying several
processes relevant to experiment: Eguiluz cites binding energies and lifetimes of
image-potential-bound surface states, tunnelling currents in the scanning-tunnelling
microscope, and resonant-tunnelling rates for ion-surface collisions as examples. It is
not clear whether it will prove equally important when calculating the ground-state
energy in QMC.
To investigate this, density-functional theory calculations were carried out using
a version of V XC containing the image potential, with two different positions 1 for
the image plane. The resulting wave functions were tested in VMC simulations, and
compared with the traditional LDA wave functions.
Figure 9.1 shows the original and modified forms of the potential. In the vacuum
1 The two image-plane positions (relative to the slab edge) were z 0 = 0.72 and z 0 = 1.49; these
values were obtained by Eguiluz, the first by fitting to the image tail, the second from the linear
response.
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