My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
slab width s = 17.64248 and density parameter r s = 2.07. 10
At this electron density
(which corresponds to aluminium), the plasma frequency is ω p = 0.581574 and
the plasmon reciprocal-space cut-off is k c = 0.695048. Using 600 electrons means
that the length of side of the cell is L = 35.5464; in this relatively large cell, the
long-range plasmon correlations should be important, whereas in a small cell with
L ∼ kc
−1 , the short-range behaviour would be expected to dominate. In addition, it
means that L c = 5.5, so that L c ≫ k −1
c
and the short-range function u cusp is allowed
to decay naturally. As a further check, simulations were also carried out for an even
larger system containing 1600 electrons.
The trial wave functions used in the test were of the standard two-determinant
Slater-Jastrow form described in equation (3.77), with determinantal orbitals obtained
from LDA calculations. In addition to the VMC energies, it is instructive to
compare the electron density profiles generated by the different Jastrow factors:
∫
∑ N
n(z) = |Ψ(r 1 , . . . , r N )| 2 δ(z − z i ) d 3 r 1 · · · d 3 r N . (8.122)
i=1
During the VMC simulation, the z-positions of the electrons are sampled periodically.
These coordinates are taken from the distribution with probability density
function n(z)/N. To see this, consider P (a < z 1 < b) (the probability that the first
electron lies in a given z-range):
( ∫ )
|Ψ(r 1 , . . . , r N )| 2 dx 1 dy 1 d 3 r 2 · · · d 3 r N
P (a < z 1 < b) =
=
∫ b
z 1 =a
∫ b
z=a
dz 1
n(z)
N dz. (8.123)
The problem of reconstructing the probability density function n(z)/N from the set
of sampled points {z i } is addressed in appendix D.
8.4.1 Unbounded slab
Figure 8.14 shows the density profile of the original two-determinant wave function.
Because the orbitals used in the determinants were taken from an LDA calculation,
10 For the rest of this chapter, all measurements will once again be quoted in atomic units.
155