This series of papers comprises a systematic exploration of the hypothesis that the far-ultraviolet radiation from star clusters and elliptical galaxies originates from extremely hot horizontal-branch (HB) stars and their post-H B progeny. This first paper presents an extensive grid of calculations of stellar models from the zero-age horizontal branch (ZAHB) through to a point late in post-HB evolution or a point on the white dwarf cooling track. The grid will be used to produce synthesized UV fluxes for the interpretation of existing and future short-wavelength (900-3000 Å) observations. Our sequences have been computed for a range of masses which concentrates on models that begin their HB evolution very close to the hot end of the ZAHB. We have calculated tracks for three metal-poor compositions ([Fe/H] = -2.26, -1.48, -0.47 with [O/Fe] > 0), for use with globular cluster observations. We have also chosen three metal rich compositions (Z = 0.017 = Z_sun, Z = 0.04, 0.06) for use in the study of elliptical galaxy populations. For each of the two super-metal-rich compositions, for which the helium abundance is unconstrained by observation, we have computed two sets of sequences: one assuming no additional helium, and a second with a large enhancement (Y_HB = 0.29 and 0.36 for Z = 0.04), and (Y_HB = 0.29 and 0.46 for Z = 0.06). For each set of sequences, our lowest ZAHB envelope masses (M⁰_ENV) are in the range 0.002 < M⁰_ENV < 0.006 M_sun.

We use the term extreme horizontal branch (EHB) to refer to HB sequences of constant mass that do not reach the thermally pulsing stage on the AGB. These models evolve after core helium exhaustion into post- early asymptotic giant branch (AGB) stars, which leave the AGB before thermal pulsing, and AGB-manque' stars, which never reach the AGB. We describe various features of the evolution of post-HB stars, discussing the correspondence between slow phases of evolution at high temperature and the early-AGB evolution. We note that the relationship between core mass and luminosity for stars on the upper AGB is not straightforward, because stars arrive on the ZAHB with a range of masses and subsequently burn different amounts of fuel. We determine from our models an upper bound to the masses of EHB stars, finding that it varies little for [Fe/H] <0, but that it is sensitive to the helium abundance. We show that for each composition there is a range of M⁰_ENV (at least a few hundredths M_sun) in which the models have a slow phase of evolution at high temperature. The duration of this phase is found to increase with the metallicity, but its luminosity is lower, so that total UV energy output is not significantly different from metal-poor sequences. The properties of very metal rich stars are, however, made uncertain by our lack of knowledge of the helium abundance for [Fe/ H] > 0; the range of stellar masses in which high temperatures are attained for significant periods of time increases with Y. There is no intrinsic composition dependence of the peak UV output from evolved stars; the output from a stellar population depends most directly on the mass distribution of stars arriving on the ZAHB. This is determined mainly by the mass loss that occurs on the red giant branch.

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