My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 3.
QUANTUM MONTE CARLO METHODS
• the weights are liable to vary wildly, since V is not a well-behaved function;
sampling is dominated by one or two walkers with much larger weight than
the others.
Fortunately, these problems can be addressed.
3.3.4 Importance-sampled diffusion Monte Carlo
The simple DMC algorithm can be dramatically improved by the use of importance
sampling, which was introduced in section 3.1 and applied to VMC in section 3.2.1.
The first step is to multiply equation (3.28) on the left with a trial wave function
Ψ T (R):
∂Ψ(R, τ)
Ψ T (R)
∂τ
= −Ψ T (R)(Ĥ − E T )Ψ(R, τ). (3.36)
An energy shift E T has also been introduced; this will be used later to control the
walker weights.
Without the energy shift, equation (3.35) shows that the wave
function ultimately decays (or grows) exponentially as e −τE 0
; introducing E T
setting E T ≈ E 0 allows this to be avoided.
The new analogue of the particle density is the product of the trial wave function
with the solution of the Schrödinger equation:
and
f(R, τ) = Ψ T (R)Ψ(R, τ). (3.37)
Expressing equation (3.36) in terms of f is a matter of algebra, which eventually
gives
∂f
∂τ = 1 2 ∇2 f − ∇ · (vf) + (E T − E L )f (3.38)
where a new quantity has been introduced: the drift velocity
The local energy
v(R) = 1 2 ∇ ln ( |Ψ T (R)| 2) . (3.39)
E L (R) = ĤΨ T (R)
Ψ T (R)
(3.40)
was introduced in section 3.2.1, and represents the energy of a configuration R
calculated with respect to the trial wave function. Because the trial wave function
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