My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 3.
QUANTUM MONTE CARLO METHODS
Substituting the expressions for the acceptance probabilities,
(
)
n(R ′ ) T (R ′ ← R) min 1, T (R←R′ )P (R ′ )
n(R) = T (R ′ ←R)P (R)
(
) (3.21)
T (R ← R ′ ) min 1, T (R′ ←R)P (R)
T (R←R ′ )P (R ′ )
= P (R′ )
P (R) . (3.22)
This illustrates that the equilibrium walker density is proportional to the required
probability density; the generated points are indeed sampled from this distribution.
3.2.3 Advantages and disadvantages of VMC
The degree of success of any VMC calculation is heavily dependent on the quality
of the trial wave function: a bad wave function will lead to a bad estimate of the
energy. The variational nature of the calculation makes it useful for establishing
an upper bound to the ground-state energy; the expectation values of other operators
(besides Ĥ) may also be calculated using this method, but these values are
not variational. The sampling and accuracy are better for operators Ô for which
the commutator [Ĥ, Ô] = 0, since the ground-state wave function is then also an
eigenfunction of Ô (if the ground state is not degenerate). The arguments of the
preceding sections regarding optimal sampling (leading to equation (3.17)) and the
error in the estimated energy (leading to equation (3.14)) may then be duplicated,
replacing Ĥ with Ô.
The main disadvantage of VMC, the total reliance on the trial wave function,
does not apply to diffusion Monte Carlo.
3.3 Diffusion Monte Carlo
The Diffusion Monte Carlo method provides a means of improving an estimated
ground-state wave function. It is a projector method: in theory, the ground-state
component of any trial wave function is projected out, giving the exact ground
state (subject to statistical errors). The factors which render this ideal projection
unachievable will be described in later chapters.
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