Dye sensitizers with wideband absorption covering the near-IR region have long been of interest because they potentially harvest a wide range of solar energies essential to promote photocurrent power conversion efficiencies. In this study, we used time-dependent density functional theory with spin-orbit (SO) interactions to theoretically explore the long-wavelength absorptions and spin-forbidden triplet transitions activated by SO interactions for terpyridyl ruthenium/osmium complex dyes. These dyes feature a Ru(II) sensitizer coordinated with a phosphine ligand and are exemplified by DX1, denoted as [trans-dichloro-(phenyldimethoxyphosphine)(2,2';6',2″-terpyridyl-4,4',4″-tricarboxylic)Ru]. We found that ancillary ligands significantly affected the longest wavelength spin-allowed absorption, with NCS- ligands yielding longer wavelength S1 transitions than halides. High atomic number halide ligands caused blue shifts in the S1 transition. Os complexes consistently exhibited longer wavelength S1 transitions than Ru complexes with identical ligands. In Ru/Os complexes, ancillary ligands with higher atomic numbers have a more pronounced effect in activating spin-forbidden triplet transitions through spin-orbit coupling (SOC) than those with lower atomic numbers. The absorption wavelength of the SOC-activated transition primarily depended on the energy of lower lying triplet states. Some complexes exhibited T1 states activated by SOC, leading to longer wavelength absorption than that of SOC-activated T2 states. Our study revealed the significance of ancillary ligands and SOC interactions in Ru/Os complexes, offering insights for optimizing materials with enhanced long-wavelength absorption properties, particularly in the near-IR range, for photovoltaic and optoelectronic applications.