a probability distribution for which direct sampling is difficult.
1
1
The Metropolis–Hastings algorithm works by generating a sequence of sample values in such a
way that, as more and more sample values are produced, the distribution of values more closely
approximates the desired distribution,
P
(
x
)
. These sample values are produced iteratively, with
the distribution of the next sample being dependent only on the current one. Specifically, at each
iteration, the algorithm picks a candidate for the next sample value based on the current one. Then,
with some probability, the candidate is either accepted (in which case the candidate value is used in
3.1
Structured Bodies
85
The different versions of the quantum Monte Carlo method all share the common
use of the technique to build accurate estimates to the multidimensional integrals
that arise in the different formulations of the manybody problem. The quantum
Monte Carlo methods allow for a direct treatment and description of complex many
body effects encoded in the wavefunction and offer numerically accurate solutions
of the manybody problem. In principle, any physical system can be described by
the manybody Schrödinger equation provided that the constituent particles are not
moving at a speed comparable to that of light and that, therefore, relativistic effects
can be neglected.
There are, indeed, two versions of the Monte Carlo technique that have been
applied to electronic structure problems: the variational (VMC) and the diffusion
(DMC) one.
In the QMC method we start from the consideration that the expectation value of
themanyelectron(say
K
)wavefunctionofthegroundstatereads(fortheequivalence
between function and vector notations see Appendix A1):
E
0
=
0

H

0
0

0
=
∗
0
(
r
)
H
0
(
r
)
d
r
∗
0
(
r
)
0
(
r
)
d
r
(3.1)
in which
r
is the 3
K
dimensional vector of electronic positions.
The energy associated with the tentative function
T
is
E
T
=
∗
T
(
r
)
H
T
(
r
)
d
r
∗
T
(
r
)
T
(
r
)
d
r
.
According to the variational principle,
E
T
is an upper limit to the (true) ground
state energy
E
0
. By reformulating the integral as follows:
E
T
=

T
(
r
)

2
H
T
(
r
)
T
(
r
)
d
r

T
(
r
)

2
d
r
.
(3.2)
the VMC Monte Carlo method is then applied using the Metropolis–Hastings algo
rithm and a set of
r
values are generated in configuration space and at each of these
points the energy (where
H
T
(
r
)/
T
(
r
)
is the “local energy”) is generated. For a
sufficiently large sample of points, the average value is given by
E
V MC
=
1
K
K
i
=
1
H
T
(
r
i
)
T
(
r
i
)
.
(3.3)
the next iteration) or rejected (in which case the candidate value is discarded, and current value is
reused in the next iteration). The probability of acceptance is determined by comparing the values of
the function
f
(
x
)
of the current and candidate sample values with respect to the desired distribution
P
(
x
)
.
86
3
Ab initio Electronic Structure for FewBody Systems
In the VMC method, therefore, it is crucial the choice of the tentative
T
(
r
)
that
determines the value of the observable computed by the simulation.
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 Fall '19
 dr. ahmed