My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 3.
QUANTUM MONTE CARLO METHODS
Ψ T Φ FN
0 , where Φ FN
0 is known as the fixed-node ground state. It may be shown
[75, 60] that the fixed-node energy is variational: that is, E FN
0 ≥ E 0 , where E 0 is
the energy of the true fermionic ground state. Typically, for the trial wave functions
used in QMC, all the nodal pockets are equivalent [23], so that the calculated energy
does not depend on which pockets are populated. The errors associated with the
fixed-node approximation will be mentioned in chapter 4, along with some of the
techniques which aim to go beyond it.
The nodes of the wave function cause other problems, because both the drift
velocity and the local energy diverge here. The approximation for the drift-diffusion
Green’s function G D uses the fact that the potential energy of the system does not
change much during the course of a move. However, near the nodal surface, the
move size can become large (because v diverges) and the energy can change rapidly;
the result is that the approximation is no longer a good one. A better approximation
can be obtained by limiting both the drift velocity and the local energy [79].
3.3.6 Estimators
In the preceding sections, the fixed-node diffusion Monte Carlo method has been
described; the result of applying this technique is a set of walkers with weights
distributed according to Ψ T Φ FN
0 , where the fixed-node ground state Φ FN
0 is usually
a good approximation to the true ground state Φ 0 . For the method to be useful,
these walkers and weights must provide a way of estimating operator expectation
values; this is the link between simulation and measurable reality.
Two estimators of the ground-state energy have in fact been described already:
E T , the trial energy, and 〈 E L
〉
, the average local energy. The expectation value of
the local energy in the limit of large imaginary time is
〈 ∑ (
lim E L Ri (τ) ) 〉 ∫
w i (τ) = lim f(R, τ)E L (R) dR. (3.62)
τ→∞
τ→∞
i
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