CASINO manual - Theory of Condensed Matter
CASINO manual - Theory of Condensed Matter
CASINO manual - Theory of Condensed Matter
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energy missing at the SJ-VMC level, while BF-DMC achieves about 38% with respect to SJ-DMC.<br />
The case <strong>of</strong> AE Carbon is even more pronounced, the figures being 45% for VMC and a mere 17% for<br />
DMC.<br />
The reason for this is clear: the backflow parameters are optimized within VMC, where the nodal<br />
error is far less important than the bulk error—bulk meaning ‘the wave-function away from the nodes’<br />
in this context—simply because <strong>of</strong> the difference in the size <strong>of</strong> the two regions in configuration space.<br />
Theoretically, it is even possible to worsen the DMC results by using a wave-function that improves<br />
the VMC energies, but this case has not been found in practice. It is unknown whether a different<br />
optimization scheme may be used to systematically improve the nodal surface <strong>of</strong> a wave-function<br />
regardless <strong>of</strong> its bulk.<br />
Even so, there are two solid points that encourage the use <strong>of</strong> backflow on many problems:<br />
• It reduces the fixed-node error by a statistically significant amount in VMC and DMC at little<br />
additional cost in many cases.<br />
• BF-VMC energies are <strong>of</strong>ten found to be very close to the SJ-DMC ones. The VMC method is<br />
thus turned into a powerful, yet simple tool delivering highly reliable results.<br />
24 Statistical analysis <strong>of</strong> data<br />
24.1 The reblocking method<br />
Each configuration generated by the QMC algorithms is related to a configuration from the previous<br />
iteration, so the raw QMC data in the vmc.hist and dmc.hist files are serially correlated. A naïve<br />
calculation <strong>of</strong> the variance <strong>of</strong> the energy (and hence the standard error in the mean) is an underestimate,<br />
because successive local energies are more similar on average than they would be if the<br />
configurations were independent.<br />
The effects <strong>of</strong> serial correlation can be eliminated by gathering successive data points into blocks and<br />
averaging over the data in each block [48]. The variance <strong>of</strong> the set <strong>of</strong> block averages can then be<br />
calculated. If the block length is greater than the correlation length between data points then the<br />
block averages are uncorrelated and an unbiased estimate <strong>of</strong> their variance is obtained. Thus, if the<br />
estimated standard error in the mean energy is plotted against block length then it should, for large<br />
enough blocks, be distributed about a constant value, which is the true standard error in the mean.<br />
If it does not reach such a plateau then there is insufficient data to estimate the standard error in the<br />
energy estimate.<br />
Each iteration is equally weighted in a VMC calculation; however, for DMC, each iteration is weighted<br />
by the total weight <strong>of</strong> the configurations multiplied by the Π-weight. In either case, let the iteration<br />
weights be w i , the total number <strong>of</strong> data points be M and the energy from iteration i be e i .<br />
Consider the bth reblocking transformation, in which the block length is B b = 2 b−1 . The data range<br />
may be divided into M b blocks, the last <strong>of</strong> which is usually incomplete.<br />
For each block j, the block weight is<br />
W bj = ∑ i∈j<br />
w i , (180)<br />
and the corresponding block energy is<br />
E bj =<br />
∑<br />
i∈j e iw i<br />
W bj<br />
. (181)<br />
The ‘reblocked’ energy is<br />
E b =<br />
=<br />
∑<br />
j E bjW bj<br />
∑<br />
j W bj<br />
∑<br />
∑ i e iw i<br />
i w i<br />
≡ E, (182)<br />
152