  Selected configuration interaction (SCI) methods have emerged as
state-of-the-art methodologies for achieving high accuracy and generating
benchmark reference data for ground and excited states in small molecular
systems. However, their precision relies heavily on extrapolation procedures to
produce a final estimate of the exact result. Using the structure of the exact
electronic energy landscape, we provide a rationale for the common linear
extrapolation of the variational energy as a function of the second-order
perturbative correction. In particular, we demonstrate that the energy gap and
the coupling between the so-called internal and external spaces are the key
factors determining the rate at which the linear regime is reached. Starting
from first principles, we also derive a new non-linear extrapolation formula
that improves the post-processing of data generated from SCI methods and can be
applied to both ground- and excited-state energies.
