CASINO manual - Theory of Condensed Matter
CASINO manual - Theory of Condensed Matter
CASINO manual - Theory of Condensed Matter
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To each walker r j , we assign a linked list L j (or an order set) <strong>of</strong> N FW elements, where N FW is the total<br />
number <strong>of</strong> FW time steps. Each element in the list is a scalar or vector. At each time step, the local<br />
quantities (forces, energies, etc.) evaluated at r j are written into one element, which is then added to<br />
the linked list on one side. Simultaneously, one element is deleted on the other side. Following this<br />
updating method, the nth element in the linked list always contains quantities that were evaluated n<br />
time steps ago so that the last element in the linked list was calculated N FW time steps ago. When<br />
the walker drifts, diffuses and branches in a standard DMC simulation, the linked list is copied when<br />
the walker is copied, and the linked list is deleted when the walker is deleted. The result <strong>of</strong> this<br />
procedure is crucial: all walkers that have a common ancestor from N FW time steps ago also have the<br />
same N FW th element in the linked list. Therefore, the numerator <strong>of</strong> the pure estimator in Eq. (434)<br />
at each time step can be written as<br />
∑<br />
A(r j )ω(r j ) = ∑ L j (N FW ), (443)<br />
j<br />
j<br />
where L j (N FW ) is the N FW th element <strong>of</strong> the linked list associated with walker r j . Hence, the sum<br />
<strong>of</strong> the product A j ω j can be calculated as the sum over the N FW th elements <strong>of</strong> all linked lists for a<br />
given time step. Similarly, the sum over all weights in the denominator <strong>of</strong> Eq. (434) reduces to the<br />
population number N pop at a given time step,<br />
∑<br />
ω(r j ) = N pop . (444)<br />
j<br />
So far, we assumed a DMC simulation without reweighting. Since casino uses by default the reweighting<br />
scheme (ibran=T), the additional weights p j from the branching factor need to be included in the<br />
pure estimate,<br />
∑<br />
j p ∑<br />
jω j A(r j )<br />
∑<br />
j p =<br />
j<br />
j p jL j (N FW )<br />
∑<br />
j p . (445)<br />
j<br />
The difference between the original algorithm by CB and the one implemented in casino and presented<br />
above is that the former averages over all N FW elements in one linked list and only stores the averages<br />
for each linked list. In particular, after N FW initial time steps, all elements <strong>of</strong> the linked lists are<br />
averaged, and additional N FW time steps are required before these averaged values are used to calculate<br />
the pure estimate. The advantage <strong>of</strong> this CB algorithm is that it does not require the storage <strong>of</strong> the<br />
linked lists. The disadvantage is that the contribution to the pure estimate is only evaluated as an<br />
average over one block, which makes it impossible to reblock the data and to properly determine the<br />
statistical error bar. The FW implementation chosen in casino, in contrast, keeps and writes out the<br />
contributions to the pure estimate for each time step, which can be used to properly decorrelate the<br />
statistical data. The additional required storage <strong>of</strong> the linked lists is negligible for the systems tested<br />
so far.<br />
35.3 Some practical advice<br />
In principle, the FW estimator <strong>of</strong> Eq. (434) is only exact when the FW time is infinite. In practice,<br />
however, this is not possible: if the FW time is too long, it is very likely that most asymptotic walkers<br />
are descendents from only a few (possibly just one) walkers, since the total population is kept around<br />
a target population. Therefore, most wave-function ratios will be zero and the FW estimate is only<br />
calculated from a few (possibly just one) independent samples, and the FW estimate is inaccurate. In<br />
contrast, if the FW time is too short, higher-states in Eq. (442) may not have decayed away. Therefore,<br />
an optimal FW time must be chosen. For parallel computing, a larger target population is possible<br />
than on a single computer, resulting in a larger optimal FW time.<br />
For calculations with a target population <strong>of</strong> around 10,000, a FW time <strong>of</strong> 10 a.u. was found to be<br />
sufficient in calculations for some small molecules [101, 102, 110] and no significant changes in the<br />
pure FW estimates were found for longer FW times. Therefore, the FW time is currently hardwired<br />
to 10 a.u. in casino.<br />
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