The molecular composition of the planet-forming regions of protoplanetary disks across the luminosity regime

(Abridged) Near- to mid-IR observations of protoplanetary disks show that the inner regions (<10AU) are rich in small organic volatiles (e.g., C2H2 and HCN). Trends in the data suggest that disks around cooler stars (~3000K) are potentially more carbon- and molecule-rich than their hotter counterparts. Our aims are to explore the composition of the planet-forming region of disks around stars from M dwarf to Herbig Ae and compare with the observed trends. Models of the disk physical structure are coupled with a gas-grain chemical network to map the abundances in the planet-forming zone. N2 self shielding, X-ray-induced chemistry, and initial abundances, are investigated. The composition in the 'observable' atmosphere is compared with that in the midplane where the planet-building reservoir resides. M dwarf disk atmospheres are relatively more molecule rich than those for T Tauri or Herbig Ae disks. The weak far-UV flux helps retain this complexity which is enhanced by X-ray-induced ion-molecule chemistry. N2 self shielding has only a small effect and does not explain the higher C2H2/HCN ratios observed towards cooler stars. The models underproduce the OH/H2O column density ratios constrained in Herbig Ae disks, despite reproducing the absolute value for OH: H2O self shielding only increases this discrepency. The disk midplane content is sensitive to the initial main elemental reservoirs. The gas in the inner disk is generally more carbon rich than the midplane ices and is most significant for disks around cooler stars. The atmospheric C/O ratio appears larger than it actually is when calculated using observable tracers only because gas-phase O2 is predicted to be a significant oxygen reservoir. The models suggest that the gas in the inner regions of disks around cooler stars is more carbon rich; however, calculations of the molecular emission are necessary to confirm the observed trends.