My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 3.
QUANTUM MONTE CARLO METHODS
where the cusp conditions require that F σi σ j
= √ (1 + δ σi σ j
)/ω p .
The introduction of the u function inevitably modifies the density (except in
a homogeneous system). However, it is often the case that the original density is
very close to the true value, especially if it is derived from density-functional theory
calculations. It is therefore desirable to restore the original density profile, and this
motivates the introduction of a one-body term,
J 1 (X) = ∑ i
χ(r i ). (3.80)
Many and various forms of the function χ are in use, but the primary aim is always
to restore the desired one-electron density. With this in mind, a useful estimate [21]
of the optimal function is
( ) 1/2 ρ0
χ(r) ∝ ln . (3.81)
ρ u
Here ρ 0 is the original density, obtained before the introduction of the Jastrow factor;
ρ u is the density obtained after the introduction of the two-body term, but before
the introduction of χ.
3.4.2 Optimisation
In practice, both one- and two-body terms include variational parameters which
are optimised to generate the best possible Jastrow factor. 7
It would seem natural
to optimise the Jastrow factor by minimising the variational energy produced by a
VMC simulation,
E V (α) =
∫
Ψ 2 (X; α)E L (X; α) dX
∫ , (3.82)
Ψ2 (X; α) dX
where α represents the set of variational parameters and E L is the local energy
defined in section 3.3.4. However, it is more common to minimise the variance of
the local energy:
σ 2 E L
(α) =
∫
Ψ 2 (X; α)(E L (X; α) − E V (α)) 2 dX
∫ . (3.83)
Ψ2 (X; α) dX
7 Three-body terms are also often included [36].
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