My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN
QUASI-2D SYSTEMS
-0.0075
-0.008
Energy per electron
-0.0085
-0.009
-0.0095
-0.01
Ewald
MPC
-0.0105
0 200 400 600 800 1000
Number of electrons in simulation cell
Figure 6.2: The total energy per electron as a function of system size, obtained using fixed-node
DMC.
this reason, the exchange energy 2 is also plotted in figure 6.1:
∑
∫∫
[
vE (r − r ′ ) − ξ ] φ ∗ n(r)φ n (r ′ )φ n ′(r)φ ∗ n ′(r′ ) dr dr ′ (6.25)
E X = − ∑ n
n ′
cell
where φ n is the nth single-electron LDA orbital. Since the Jastrow factor cannot
alter the location of the nodes, and therefore does not affect the fixed-node DMC
energy, it was included in the DMC trial wave functions to improve efficiency.
A correction of the form (E LDA
∞
− EN
LDA ) has been applied to all the results to
account for the independent-particle finite-size effect (see section 4.1.1).
It is evident from figures 6.1 and 6.2 that the results obtained using the MPC and
Ewald interactions are in good agreement, both displaying a fairly slow convergence
with respect to the system size.
This is unexpected, and contrasts with the 3D
case, where the MPC interaction improves the rate of convergence significantly; the
2 This quantity is not the true exchange energy, which requires optimised Hartree-Fock orbitals;
rather, it is a hybrid of the LDA and Hartree-Fock theory.
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