Unbiasing the initiator approximation in Full Configuration Interaction Quantum Monte Carlo

We identify and rectify a crucial source of bias in the initiator FCIQMC algorithm. Non-initiator determinants (i.e. determinants whose population is below the initiator threshold) are subject to a systematic {\em undersampling} bias, which in large systems leads to a bias in the energy when an insufficient number of walkers is used. We show that the acceptance probability ($p_{acc}$), that a non-initiator determinant has its spawns accepted, can be used to unbias the initiator bias, in a simple and accurate manner, by reducing the applied shift to the non-initiator proportionately to $p_{acc}$. This modification preserves the property that in the large walker limit, when $p_{acc}\rightarrow1$, the unbiasing procedure disappears, and the initiator approximation becomes exact. We demonstrate that this algorithm shows rapid convergence to the FCI limit with respect to walker number, and furthermore largely removes the dependence of the algorithm on the initiator threshold, enabling highly accurate results to be obtained even with large values of the threshold. This is exemplified in the case of butadiene/ANO-L-pVDZ and benzene/cc-pVDZ, correlating 22 and 30 electrons in 82 and 108 orbitals respectively. In butadiene $5\times 10^7$ and in benzene $10^8$ walkers suffice to obtain an energy to within a milli-Hartree of the CCSDT(Q) result, in Hilbert spaces of $10^{26}$ and $10^{35}$ respectively. Essentially converged results require $\sim 10^8$ walkers for butadiene and $\sim 10^9$ walkers for benzene, and lie slightly lower than CCSDT(Q). Owing to large-scale parallelisability, these calculations can be executed in a matter of hours on a few hundred processors. The present method largely solves the initiator-bias problems that the initiator method suffered from when applied to medium-sized molecules.