  In this work we present an extension of the popular selected configuration
interaction (SCI) algorithms to the Transcorrelated (TC) framework. Although we
used in this work the recently introduced one-parameter correlation factor [E.
Giner, J. Chem. Phys., 154, 084119 (2021)], the theory presented here is valid
for any correlation factor. Thanks to the formalization of the non Hermitian TC
eigenvalue problem as a search of stationary points for a specific functional
depending both on left-and right-functions, we obtain a general framework
allowing different choices for both the selection criterion in SCI and the
second order perturbative correction to the energy. After numerical
investigations on different second-row atomic and molecular systems in
increasingly large basis sets, we found that taking into account the non
Hermitian character of the TC Hamiltonian in the selection criterion is
mandatory to obtain a fast convergence of the TC energy. Also, selection
criteria based on either the first order coefficient or the second order energy
lead to significantly different convergence rates, which is typically not the
case in the usual Hermitian SCI. Regarding the convergence of the total second
order perturbation energy, we find that the quality of the left-function used
in the equations strongly affects the quality of the results. Within the
near-optimal algorithm proposed here we find that the SCI expansion in the TC
framework converges faster than the usual SCI both in terms of basis set and
number of Slater determinants.
