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28.1 The FCIQMC Method

Details of the FCIQMC algorithm are best obtained from various publications on the method, which include:

G. H. Booth, A. J. W. Thom, and A. Alavi, J. Chem. Phys. 131, 054106 (2009) (Full FCIQMC),
D. M. Cleland, G. H. Booth, and A. Alavi, J. Chem. Phys. 134, 024112 (2011) ($i$-FCIQMC) and
G. H. Booth, D. M. Cleland, A. J. W. Thom, and A. Alavi, J. Chem. Phys. 135, 084104 (2011),
and which should be cited in published work resulting from the use of this module.

Briefly, the FCIQMC method involves discretizing the wavefunction amplitudes over the full Slater determinant space into a number of signed `walkers'. These walkers represent a coarse-grained snapshot of the wavefunction at any given instant. Through a population dynamics of these walkers evolving in imaginary-time, the ground-state energy and wavefunction can be resolved to arbitrary accuracy (in principle FCI quality), in a time-averaged fashion, without the need for any explicit diagonalization steps. The master equations which are stochastically realized and govern this dynamic, are given by

\begin{displaymath}
-\frac{d N_i}{d \tau} = (H_{ii} - E_{ref} - S) N_i + \sum_{j \neq i} H_{ij} N_j \; ,
\end{displaymath} (54)

where $\tau$ represents the evolution in imaginary-time, $N_i$ is the walker population on configuration $i$, and $S$ is the energy `shift', which can be used as a measure of the correlation energy once the walker population is stable. In the original formulation, this dynamic is integrated exactly over imaginary-time, and converges to the exact basis-set correlation energy of the system within systematically reducible random errors, as long as the number of walkers in the system exceeds a system-dependent number, required to overcome the `sign-problem' present in the space. This number is indicated by the appearance of a plateau in the growth profile of the walkers for a given value of $S$.

However, in a modification to the algorithm which allows for a generally smooth convergence to the FCIQMC result with increasing walker number, the sum in Eq. 56 is truncated, such that when considering a configuration $i$ whose population is zero, this sum then only runs over those configurations who are deemed `initiator' configurations. These initiator configurations are ones which have an instantaneous population of above a parameter $n_{add}$, or are the chosen `reference' configuration. This modification to the algorithm rigorously converges to the full FCIQMC result in the limit of a large number of walkers, or as $n_{add} \rightarrow 0$, and is dubbed `$i$-FCIQMC'. This is the default algorithm used in the FCIQMC module.

Note that the choice of reference configuration, or indeed orbital space, should be independent of the final energy obtained, and the method thus constitutes a multiconfigurational correlation treatment, suitable for strongly-correlated problems. However, the choice of reference configuration and orbital space may affect rate of convergence and random error decay.



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