My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 9.
ENERGY
A NEW CALCULATION OF THE JELLIUM SURFACE
Type of potential
VMC energy (mHa)
No image tail 32.54 ± 0.08
Image plane at z 0 = 0.72 32.54 ± 0.08
Image plane at z 0 = 1.49 32.70 ± 0.08
Table 9.1: Comparison of energies calculated with different versions of the exchange-correlation
potential. The DFT energies are identical for all potentials. The trial wave function for the VMC
calculations contained no Jastrow factor; the simulations used a simulation cell containing 600
electrons.
The simulation cell is square in the xy-plane, extending from −∞ to ∞ in the
z-direction. The boundary conditions therefore imply that
k x = 2πm x
L
where m x = 0, 1, 2, . . . (9.3)
with a corresponding relation for k y .
However, this is not the only possible choice. The aim is to model an infinite
slab, in which k ‖ takes on a continuous range of values. Recent studies [74, 55]
have used ideas from band-structure theory [6, 59] in QMC simulations to improve
the extrapolation to the thermodynamic limit. The idea is to introduce some fixed
phase shift across the simulation cell, so that
φ(r + L x ) = e iθx φ(r) (9.4)
where L x is a vector of length L in the x-direction. Conventional periodic boundary
conditions correspond to choosing θ x = 0. In the work of Lin et al. [55], an average
over several different phase shifts is taken. Rajagopal and coworkers [74] found that
using θ x = π (‘antiperiodic’ boundary conditions) gave good results. Under these
boundary conditions, equation (9.3) becomes
k x = π L (1 + 2m x). (9.5)
This demonstrates the advantage of using antiperiodic or periodic boundary conditions:
it is always possible to construct real orbitals by taking pairs of k ‖ and −k ‖ .
This is not true for an arbitrary phase shift.
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