  The exploration of the root structure of coupled cluster equations holds both
foundational and practical significance for computational quantum chemistry.
This study provides insight into the intricate root structures of these
non-linear equations at both the CCD and CCSD level of theory. We utilize
computational techniques from algebraic geometry, specifically the monodromy
and parametric homotopy continuation methods, to calculate the full solution
set. We compare the computed CC roots against various established theoretical
upper bounds, shedding light on the accuracy and efficiency of these bounds. We
hereby focus on the dissociation processes of four-electron systems such as
(H$_2$)$_2$ in both D$_{2{\rm h}}$ and D$_{\infty {\rm h}}$ configurations,
H$_4$ symmetrically distorted on a circle, and lithium hydride. We moreover
investigate the ability of single-reference coupled cluster solutions to
approximate excited state energies. We find that multiple CC roots describe
energies of excited states with high accuracy. Remarkably, our investigations
reveal that for systems like lithium hydride, CC not only provides
high-accuracy approximations to several excited state energies but also to the
states themselves.
