Context. Planets are thought to eventually form from the mostly gaseous (~99% of the mass) disks around young stars. The density structure and chemical composition of protoplanetary disks are affected by the incident radiation field at optical, FUV, and X-ray wavelengths, as well as by the dust properties. Aims. The effect of FUV and X-rays on the disk structure and the gas chemical composition are investigated. This work forms the basis of a second paper, which discusses the impact on diagnostic lines of, e.g., C+, O, H2O, and Ne+ observed with facilities such as Spitzer and Herschel. Methods. A grid of 240 models is computed in which the X-ray and FUV luminosity, minimum grain size, dust size distribution, and surface density distribution are varied in a systematic way. The hydrostatic structure and the thermo-chemical structure are calculated using ProDiMo. Results. The abundance structure of neutral oxygen is stable to changes in the X-ray and FUV luminosity, and the emission lines will thus be useful tracers of the disk mass and temperature. The C+ abundance distribution is sensitive to both X-rays and FUV. The radial column density profile shows two peaks, one at the inner rim and a second one at a radius r=5-10 AU. Ne+ and other heavy elements have a very strong response to X-rays, and the column density in the inner disk increases by two orders of magnitude from the lowest (LX = 1e29 erg/s) to the highest considered X-ray flux (LX = 1e32 erg/s). FUV confines the Ne+ ionized region to areas closer to the star at low X-ray luminosities (LX = 1e29 erg/s). H2O abundances are enhanced by X-rays due to higher temperatures in the inner disk and higher ionization fractions in the outer disk. The line fluxes and profiles are affected by the effects on these species, thus providing diagnostic value in the study of FUV and X-ray irradiated disks around T Tauri stars. (abridged)
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