  We present new collisional-radiative models (CRMs) for helium in the
quiescent solar corona, and predict the emissivities of the He and He$^+$ lines
to be observed by DKIST, Solar Orbiter, and Proba-3. We discuss in detail the
rates we selected for these models, highlighting several shortcomings we have
found in previous work. As no previous complete and self-consistent coronal CRM
for helium existed, we have benchmarked our largest model at a density of
10$^{6}$ cm$^{-3}$ and temperature of 20,000 K against recent CRMs developed
for photoionised nebulae. We then present results for the outer solar corona,
using new dielectronic recombination rates we have calculated, which increase
the abundance of neutral helium by about a factor of two. We also find that all
the optical triplet He I lines, and in particular the well known He I 10830 and
5876 A lines are strongly affected by both photo-excitation and
photo-ionisation from the disk radiation, and that extensive CRM models are
required to obtain correct estimates. Close to the Sun, at an electron density
of 10$^{8}$ cm$^{-3}$ and temperature of 1 MK, we predict the emissivity of the
He I 10830 A to be comparable to that of the strong Fe XIII coronal line at
10798 A. However, we expect the He I emissivity to sharply fall in the outer
corona, with respect to Fe XIII. We confirm that the He$^+$ Lyman $\alpha$ at
304 A is also significantly affected by photo-excitation and is expected to be
detectable as a strong coronal line up to several solar radii.
