12.3 Diffusion Monte Carlo 397
(=0)+(=0) −(=0)−()()+()(a)(b)(c)(d)Figure 12.2. Evolution of the distributions in the transient energy estimator
method. The wave functionφ(τ= 0 )is shown in (a); it can be written as the
difference of theφ+andφ−. These two functions evolve separately and tend
therefore to the same boson ground state solution, as shown in (c). Subtracting the
two wave functions in (c) gives the small difference in (d), and this will be soon
buried in the noise in the solutions in (c).boson energy. Therefore we use the ‘transient estimator’:
ETE(τ )=∫
dXφ(τ)Hφ(τ= 0 )
∫
dXφ(τ)φ(τ= 0 )=∫
dXφ−(τ )Hφ(τ= 0 )
∫
dX[φ+(τ )−φ−(τ )]φ(τ= 0 )−
∫
dXφ+(τ )Hφ(τ= 0 )
∫
dX[φ+(τ )−φ−(τ )]φ(τ= 0 ).
(12.66)
As the wave functionφ(τ)converges to the exact fermion ground state, this estim-
ator will indeed relax to the exact fermion energy. As mentioned already, the
problem resides inφ(τ)to be extracted as the small difference between two large
distributions.
The estimator(12.66)is evaluated as follows. At timeτ, the walkers occupy
points in configuration space which are distributed according toφ±(τ ). For a walker
at the pointXin theφ+-simulation we evaluateHφ(X,τ= 0 )(for the numerator)
andφ(X,τ= 0 )(for the denominator), and sum over walkers. We do the same
with theφ−simulation, but now give the contributions a minus sign. The quantity
Hφ(X,τ= 0 )can be evaluated becauseφ(X,τ= 0 )is a trial function, given in
analytic form. The sum is divided by the sum ofφ(X,τ= 0 )over all the walkers.
There exist several extensions to and refinements of the transient estimator
method, which are beyond the scope of this book. A common characteristic of
these methods is that they are subject to instability in the errors for largeτ.