  Selected configuration interaction (SCI) methods are currently enjoying a
resurgence due to several recent developments which improve either the overall
computational efficiency or the compactness of the resulting SCI vector. These
recent advances have made it possible to get full CI (FCI) quality results for
much larger orbital active spaces, compared to conventional approaches.
However, due to the starting assumption that the FCI vector has only a small
number of significant Slater determinants, SCI becomes intractable for systems
with strong correlation. This paper introduces a method for developing SCI
algorithms in a way which exploits local molecular structure to significantly
reduce the number of SCI variables. The proposed method is defined by first
grouping the orbitals into clusters over which we can define many particle
cluster states. We then directly perform the SCI algorithm in a basis of tensor
products of cluster states instead of Slater determinants. While the approach
is general for arbitrarily defined cluster states, we find significantly
improved performance by defining cluster states through a Tucker decomposition
of the global (and sparse) SCI vector. To demonstrate the potential of this
method, called tensor product selected configuration interaction (TPSCI), we
present numerical results for a diverse set of examples: 1) modified Hubbard
model with different inter- and intra-cluster hopping terms, 2) less obviously
clusterable cases of bond breaking in N2 and F2, and 3) ground state energies
of large planar {\pi}-conjugated systems with active spaces of up to 42
electrons in 42 orbitals. These numerical results show that TPSCI can be used
to significantly reduce the number of SCI variables in the variational space,
and thus paving a path for extending these deterministic and variational SCI
approaches to a wider range of physical systems.
