  We demonstrate the ability to calculate electromagnetic sum rules with the
\textit{ab initio} symmetry-adapted no-core shell model. By implementing the
Lanczos algorithm, we compute non-energy weighted, energy weighted, and inverse
energy weighted sum rules for electric monopole, dipole, and quadrupole
transitions in $^4$He using realistic interactions. We benchmark the results
with the hyperspherical harmonics method and show agreement within $2\sigma$,
where the uncertainties are estimated from the use of the many-body technique.
We investigate the dependence of the results on three different interactions,
including chiral potentials, and we report on the $^4$He electric dipole
polarizability calculated in the SA-NCSM that reproduces the experimental data
and earlier theoretical outcomes. We also detail a novel use of the Lawson
procedure to remove the spurious center-of-mass contribution to the sum rules
that arises from using laboratory-frame coordinates. We further show that this
same technique can be applied in the Lorentz integral transform method, with a
view toward studies of electromagnetic reactions for light through medium-mass
nuclei.
