  Nuclei exhibit both single-particle and collective degrees of freedom, with
the latter often subdivided into vibrational and rotational motions.
Experimentally identifying the relative roles of these collective modes is
extremely challenging, particularly in the face of possible shape coexistence.
Model-independent, invariant quantities describing the deformation of a nucleus
in the intrinsic frame have long been known but their determination potentially
requires a large quantity of experimental data to achieve convergence. Through
comparison with the nuclear shell model, the question of convergence will be
addressed. Shell-model calculations performed in the $sd$- and $pf$-shell model
spaces are used to determine electric-quadrupole matrix elements for a
multitude of low-lying states. Relative contributions to the rotationally
invariant quantities from multiple states can therefore be determined. It is
found that on average, the inclusion of four intermediate states results in the
leading-order invariant, $\left<Q^2\right>$, converging to within 10% of its
true value and the triaxiality term, $\left<\cos{\left(3\delta\right)}\right>$,
converging to its true value, though some variance remains. Higher-order
quantities relating to the softness of the nuclear shape are found to converge
more slowly. The convergence of quadrupole rotationally invariant sum rules was
quantified in the $sd$- and $pf$-shell model spaces and indicates the challenge
inherent in a full determination of nuclear shape. The present study is limited
to relatively small valence spaces. Larger spaces, such as the rare-earth
region, potentially offer faster convergence.
