Assessment of Multireference Approaches to Explicitly Correlated Full Configuration Interaction Quantum Monte Carlo

The Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method has proved able to provide near-exact solutions to the electronic Schr\"odinger equation within a finite orbital basis set, without relying on an expansion about a reference state. However, a drawback to the approach is that being based on an expansion of Slater determinants, the FCIQMC method suffers from a basis set incompleteness error that decays very slowly with the size of the employed single particle basis. The FCIQMC results obtained in a small basis set can be improved significantly with explicitly correlated techniques. Here, we present a study that assesses and compares two contrasting `universal' explicitly correlated approaches that fit into the FCIQMC framework; the $[2]_{R12}$ method of Valeev {\em et al.}, and the explicitly correlated canonical transcorrelation approach of Yanai {\em et al}. The former is an {\em a posteriori} internally-contracted perturbative approach, while the latter transforms the Hamiltonian prior to the FCIQMC simulation. These comparisons are made across the 55 molecules of the G1 standard set. We found that both methods consistently reduce the basis set incompleteness, for accurate atomization energies in small basis sets, reducing the error from 28~mE$_{\text h}$ to 3-4~mE$_{\text h}$. While many of the conclusions hold in general for any combination of multireference approaches with these methodologies, we also consider FCIQMC-specific advantages of each approach.