My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 9.
ENERGY
A NEW CALCULATION OF THE JELLIUM SURFACE
determinants; the form of the Jastrow factor was determined after consideration of
the results of chapter 8. The short-ranged two-body term
u cusp (x i , x j ) =
α
2(1 + δ σi σ j
) e−r ij/α−r 2 ij /L2 c
(9.7)
has the advantage that the corresponding one-body function required to restore the
original density is derived analytically: the one- and two-body terms may therefore
be optimised by varying the single parameter α. 2
This speeds up the otherwise
painfully slow manual optimisation process; the failure of automatic optimisation
based on variance minimisation is documented in chapter 5.
The evaluation of
the Jastrow factor within a simulation is also inexpensive compared with the full
plasmonic form.
9.3.1 Results
The first stage of the QMC simulations is the optimisation of the trial wave function.
Short VMC calculations were carried out for different values of α; an example of the
dependence of the VMC energy on α is displayed in figure 9.5. The optimal value
of α, obtained by a quadratic fit to these curves, was found to increase slightly with
the cell size L.
Having obtained optimised wave functions, the next step is to carry out longer
VMC runs, the results of which are plotted in figure 9.6. Systems of size ranging
from 300 up to 3000 electrons were simulated; the number of time steps required
ranged from tens of thousands for the larger systems to around a million for the
smaller ones. The MPC interaction was used for these calculations: testing at one
slab width revealed that the quasi-2D Ewald and MPC interactions returned values
which were in agreement with each other, confirming the results of chapter 6.
The two quasi-2D interactions are compared with the 3D Ewald sum in figure
9.7. The 3D Ewald sum gave slightly lower energies than the other interactions;
2 Note that L c is fixed by the requirement that u cusp (x i , x j ) be effectively zero when |x i −x j | ∼ L,
and is not an optimiseable parameter.
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