  This work explores the ability of classical electronic structure methods to
efficiently represent (compress) the information content of full configuration
interaction (FCI) wave functions. We introduce a benchmark set of four hydrogen
model systems of different dimensionality and distinctive electronic
structures: a 1D chain, a 1D ring, a 2D triangular lattice, and a 3D
close-packed pyramid. To assess the ability of a computational method to
produce accurate and compact wave functions, we introduce the accuracy volume,
a metric that measures the number of variational parameters necessary to
achieve a target energy error. Using this metric and the hydrogen models, we
examine the performance of three classical deterministic methods: i) selected
configuration interaction (sCI) realized both via an a posteriori and
variational selection of the most important determinants, ii) rank-reduced FCI,
obtained by an a posteriori singular value decomposition of the FCI tensor
(SVD-FCI), and iii) the matrix product state representation obtained via the
density matrix renormalization group (DMRG). We find that DMRG generally gives
the most efficient wave function representation for all systems, particularly
in the 1D chain with a localized basis. For the 2D and 3D systems, all methods
perform best with a delocalized basis, and the efficiency of sCI is closer to
that of DMRG, with the former having and accuracy volume approximately twice as
large in the strong correlation regime. Compared to sCI, the SVD-FCI scheme is
generally found to require a slightly larger number of parameters to achieve
the same energy accuracy.
