My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 3.
QUANTUM MONTE CARLO METHODS
This ideal form unfortunately cannot be achieved because it requires the evaluation
of the difficult integral ∫ |g(r ′ )| dr ′ . However, choosing a good importance function,
close to the optimum, reduces the statistical error in the estimate of I significantly.
3.2 Variational Monte Carlo
The various forms of quantum Monte Carlo implement the techniques outlined above
in different ways. Variational Monte Carlo is very direct: Monte Carlo methods are
used to evaluate expressions like
E T =
∫
Ψ
∗
T (R)ĤΨ T (R) dR
∫
Ψ
∗
T
(R)Ψ T (R) dR . (3.12)
E T is an energy expectation value for a system with Hamiltonian operator Ĥ; Ψ T (R)
is a trial wave function depending on all the spatial coordinates, 2 which are represented
here by the vector R. The expectation value is variational: E T ≥ E 0 , where
E 0 is the exact ground-state energy. As the quality of the trial wave function improves,
E T becomes closer to E 0 . The two become equal when Ψ T ∝ Ψ 0 , where Ψ 0
is the ground-state wave function.
Consider a trial wave function close to the exact ground-state eigenfunction:
Ψ T = Ψ 0 + ∑ ɛ i Ψ i (3.13)
i>0
so that the coefficients {ɛ i } are small. The energy expectation value is then
E T = E 0 + ∑ i>0
|ɛ i | 2 (E i − E 0 ) + O[ɛ 4 i ]. (3.14)
The error in the energy is of order ɛ 2 i ; this demonstrates that the energy expectation
value for a given trial wave function is more accurate than the wave function itself.
The quality of the trial wave function is clearly very important. The subject of
the form and optimisation of the trial wave function will be discussed in more detail
later in this thesis.
2 In many cases, it is convenient to think of the electron spins as fixed; the spin-dependence of
the wave function will be suppressed from now on.
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