We have combined the detailed He I recombination model of Smits with the co
llisional transitions of Sawey & Berrington in order to produce new accurat
e helium emissivities that include the effects of collisional excitation fr
om both the 2(3)S and 2(1)S levels. We present a grid of emissivities for a
range of temperature and densities along with analytical fits and error es
timates. These grids eliminate the necessity of making corrections for coll
isional enhancements as in the work of Clegg or Kingdon & Ferland for lines
with upper levels below n = 5. For densities greater than n(e) approximate
to 10(6) cm(-3), inclusion of collisional excitation from the 21S level is
also necessary if accuracies of greater than a few percent are required. A
tomic data for a model atom with 29 levels (n(max) = 5) are presented that
match the recombination model of Smits to within 5% over the temperature ra
nge T = 5000-20,000 K. Collisional effects are calculated self-consistently
using the algorithm of Almog & Netzer. This model atom will be useful in m
odels of radiative transfer. A notable feature of this technique is an algo
rithm that calculates the "indirect" recombination rates, the recombination
to individual levels that go through n > n(max) first. Fits accurate to wi
thin 1% are given for the emissivities of the brightest lines over a restri
cted range for estimates of primordial helium abundance. We characterize th
e analysis uncertainties associated with uncertainties in temperature, dens
ity, fitting functions, and input atomic data. We estimate that atomic data
uncertainties alone may limit abundance estimates to an accuracy of simila
r to 1.5%; systematic errors may be greater than this. This analysis uncert
ainty must be incorporated when attempting to make high-accuracy estimates
of the helium abundance. For example, in recent determinations of the primo
rdial helium abundance, uncertainties in the input atomic data have been ne
glected. Finally, we compare our theoretical calculations to the measured s
trengths of a few dozen helium emission lines in three nebulae, Orion (NGC
1976) and the planetary nebulae NGC 6572 and IC 4997. Incorporation of coll
isional effects yields noticeable improvements for some lines, but some not
able discrepancies remain.