CASINO manual - Theory of Condensed Matter

energy missing at the SJ-VMC level, while BF-DMC achieves about 38% with respect to SJ-DMC.
The case of AE Carbon is even more pronounced, the figures being 45% for VMC and a mere 17% for
DMC.
The reason for this is clear: the backflow parameters are optimized within VMC, where the nodal
error is far less important than the bulk error—bulk meaning ‘the wave-function away from the nodes’
in this context—simply because of the difference in the size of the two regions in configuration space.
Theoretically, it is even possible to worsen the DMC results by using a wave-function that improves
the VMC energies, but this case has not been found in practice. It is unknown whether a different
optimization scheme may be used to systematically improve the nodal surface of a wave-function
regardless of its bulk.
Even so, there are two solid points that encourage the use of backflow on many problems:
• It reduces the fixed-node error by a statistically significant amount in VMC and DMC at little
additional cost in many cases.
• BF-VMC energies are often found to be very close to the SJ-DMC ones. The VMC method is
thus turned into a powerful, yet simple tool delivering highly reliable results.
24 Statistical analysis of data
24.1 The reblocking method
Each configuration generated by the QMC algorithms is related to a configuration from the previous
iteration, so the raw QMC data in the vmc.hist and dmc.hist files are serially correlated. A naïve
calculation of the variance of the energy (and hence the standard error in the mean) is an underestimate,
because successive local energies are more similar on average than they would be if the
configurations were independent.
The effects of serial correlation can be eliminated by gathering successive data points into blocks and
averaging over the data in each block [48]. The variance of the set of block averages can then be
calculated. If the block length is greater than the correlation length between data points then the
block averages are uncorrelated and an unbiased estimate of their variance is obtained. Thus, if the
estimated standard error in the mean energy is plotted against block length then it should, for large
enough blocks, be distributed about a constant value, which is the true standard error in the mean.
If it does not reach such a plateau then there is insufficient data to estimate the standard error in the
energy estimate.
Each iteration is equally weighted in a VMC calculation; however, for DMC, each iteration is weighted
by the total weight of the configurations multiplied by the Π-weight. In either case, let the iteration
weights be w i , the total number of data points be M and the energy from iteration i be e i .
Consider the bth reblocking transformation, in which the block length is B b = 2 b−1 . The data range
may be divided into M b blocks, the last of which is usually incomplete.
For each block j, the block weight is
W bj = ∑ i∈j
w i , (180)
and the corresponding block energy is
E bj =
∑
i∈j e iw i
W bj
. (181)
The ‘reblocked’ energy is
E b =
=
∑
j E bjW bj
∑
j W bj
∑
∑ i e iw i
i w i
≡ E, (182)
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