  Approximate full configuration interaction (FCI) calculations have recently
become tractable for systems of unforeseen size thanks to stochastic and
adaptive approximations to the exponentially scaling FCI problem. The result of
an FCI calculation is a weighted set of electronic configurations, which can
also be expressed in terms of excitations from a reference configuration. The
excitation amplitudes contain information on the complexity of the electronic
wave function, but this information is contaminated by contributions from
disconnected excitations, i.e. those excitations that are just products of
independent lower-level excitations. The unwanted contributions can be removed
via a cluster decomposition procedure, making it possible to examine the
importance of connected excitations in complicated multireference molecules
which are outside the reach of conventional algorithms. We present an
implementation of the cluster decomposition analysis and apply it to both true
FCI wave functions, as well as wave functions generated from the adaptive
sampling CI (ASCI) algorithm. The cluster decomposition is useful for
interpreting calculations in chemical studies, as a diagnostic for the
convergence of various excitation manifolds, as well as as a guidepost for
polynomially scaling electronic structure models. Applications are presented
for (i) the double dissociation of water, (ii) the carbon dimer, (iii) the
{\pi} space of polyacenes, as well as (iv) the chromium dimer. While the
cluster amplitudes exhibit rapid decay with increasing rank for the first three
systems, even connected octuple excitations still appear important in Cr$_2$,
suggesting that spin-restricted single-reference coupled-cluster approaches may
not be tractable for some problems in transition metal chemistry.
