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Evolutionary models have been calculated for Population II stars of 0.5-1.0 M☉ from the pre-main sequence to the lower part of the giant branch. Rosseland opacities and radiative accelerations were calculated taking into account the concentration variations of 28 chemical species, including all species contributing to Rosseland opacities in the OPAL tables. The effects of radiative accelerations, thermal diffusion, and gravitational settling are included. While models were calculated for both Z = 0.00017 and 0.0017, we concentrate on models with Z = 0.00017 in this paper. These are the first Population II models calculated taking radiative acceleration into account. It is shown that, at least in a 0.8 M☉ star, it is a better approximation not to let Fe diffuse than to calculate its gravitational settling without including the effects of grad(Fe). In the absence of any turbulence outside of convection zones, the effects of atomic diffusion are large mainly for stars more massive than 0.7 M☉. Overabundances are expected in some stars with Teff ≥ 6000 K. Most chemical species heavier than CNO are affected. At 12 Gyr, overabundance factors may reach 10 in some cases (e.g., for Al or Ni), while others are limited to 3 (e.g., for Fe). The calculated surface abundances are compared to recent observations of abundances in globular clusters as well as to observations of Li in halo stars. It is shown that, as in the case of Population I stars, additional turbulence appears to be present. Series of models with different assumptions about the strength of turbulence were then calculated. One series minimizes the spread on the Li plateau, while another was chosen with turbulence similar to that present in AmFm stars of Population I. Even when turbulence is adjusted to minimize the reduction of Li abundance, there remains a reduction by a factor of at least 1.6 from the original Li abundance. Independent of the degree of turbulence in the outer regions, gravitational settling of He in the central region reduces the lifetime of Population II stars by 4%-7% depending on the criterion used. The effect on the age of the oldest clusters is discussed in a forthcoming paper (Paper II). Just as in Population I stars where only a fraction of stars, such as AmFm stars, have abundance anomalies, one should look for the possibility of abundance anomalies of metals in some Population II turnoff stars but not necessarily in all. Expected abundance anomalies are calculated for 28 species and compared to observations of M92 as well as to Li observations in halo field stars.