This study highlights the potential benefits of using terdentate over bidentate ligands in the construction of organometallic complexes as organic light-emitting diode (OLED) emitters offering better color purity, and explores in detail the molecular origins of the differences between the two. A pair of closely related platinum(II) complexes has been selected, incorporating a bidentate and a terdentate cyclometallating ligand, respectively, namely, Pt(4,6-dFppy)(acac) (1) {4,6-dFppy = 2-(4,6-difluorophenyl)pyridine metalated at C-2 of the phenyl ring} and Pt(4,6-dFdpyb)Cl (2) {4,6-dFdpyb = 4,6-difluoro-1,3-di(2-pyridyl)benzene, metalated at C-2 of the phenyl ring}. The emission properties over the range of temperatures from 1.2 to 300 K have been investigated, including optical high-resolution studies. The results reveal a detailed insight into the electronic and vibronic structures of the two compounds. In particular, the Huang-Rhys parameter S that serves to quantify the degree of molecular distortion in the excited state with respect to the ground state, though small in both cases, is smaller by a factor of 2 for the terdentate than the bidentate complex (S approximate to 0.1 and respectively). The smaller value for the former reflects the greater degree of rigidity induced by the terdentate ligand, leading to a lesser contribution of intraligand Franck-Condon vibrational modes in the green spectral range of the emission spectra. Consequently, an enhanced color purity with respect to blue light emission results. The high rigidity and the short Pt-C bond in Pt(4,6-dFdpyb)Cl also serve to disfavor nonradiative decay pathways, including those involving higher-lying dd* states. These effects account for the greatly superior luminescence quantum yield of the terdentate complex in fluid solution, amounting to phi(PL) = 80% versus only 2% found for the bidentate complex.