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.
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