CASINO manual - Theory of Condensed Matter

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