My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 3.
QUANTUM MONTE CARLO METHODS
3.1 Random sampling
To calculate expectation values of observables in quantum mechanics, we need to
evaluate expressions of the form:
∫
Ψ
∗ 〈Ô〉 = (X)ÔΨ(X) dX
∫
Ψ∗ (X)Ψ(X) dX , (3.1)
where
Ô is a Hermitian operator and X is a vector representing all the electron
coordinates.
For large systems, the dimension of the integrals is correspondingly large. Computation
of such integrals by conventional quadrature methods becomes impractical,
because the error in these calculations scales poorly in high dimensions. For example,
the error in a Simpson’s Rule calculation scales as M −4/d , where M is the
number of grid points and d is the dimensionality of the integral. As d increases,
improving the accuracy of the calculation becomes prohibitively expensive.
In contrast, the Monte Carlo method of integration generates a statistical error
in the value of the integral which scales as M −1/2 , independent of the dimension.
While this is worse than grid-based techniques when d is small, it is far superior for
large d.
One way of evaluating the integral
∫
I = g(r) dr (3.2)
Ω
is to sample a set of M random vectors {r i } from a distribution which is uniform over
the region of integration Ω. Each vector is assigned a score g(r i ) and the integral is
estimated as
I ≈ Ω M
M∑
g(r i ). (3.3)
i=1
This is the Monte Carlo method of integration, and in this form, it is not very
efficient. It can be much improved by using importance sampling. The integral is
first rewritten as
∫
I =
f(r)P (r) dr, (3.4)
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