Aiming to settle the controversial observations for halogen-bridged binuclear transition-metal (MMX) complexes, finite-temperature Hartree-Fock calculations are performed for a relevant twoband Peierls-Hubbard model. Thermal, as well as quantum, phase transitions are investigated with particular emphasis on the competition between electron itinerancy, electron-phonon interaction and electron-electron correlation. Recently observed distinct thermal behaviors of two typical MMX compounds Pt2(CH3CS2)4I and (NH4)4[Pt2(P2O5H2)4I]·2H2O are supported and further tuning of their electronic states is predicted.PACS numbers: 71.10. Hf, 71.45Lr, 75.30.Fv, 65.50.+m There is a class of quasi-one-dimensional materials containing chains of transition-metal (M) complexes bridged by halogens (X). Their representatives such as Wolffram's red (M = Pt, X = Cl) and Reihlen's green (M = Pt, X = Br) salts [1] are composed of mononuclear metal complexes and are thus referred to as MX chain compounds. These PtX chains generally exhibit a mixed-valence ground state with strong dimerization of the X sublattice. When Pt is replaced by Ni [2,3], a mono-valence regular-chain structure is instead stabilized, which is nothing but a tuning from the Peierls to Mott insulator. The electronic state of the MX chain system can such widely be tuned that the optical energy gap changes from 0.8 to 3.3 eV by varying the transition metals, the surrounding ligands, the bridging halogens and the counter ions [4]. In recent years, further efforts have been devoted to bridging polynuclear metal complexes by halogens and physical as well as chemical interest in this system has been renewed. A new group, MMX, consists of two families, [9,10], which are generically called the pop and dta complexes, respectively. The MMX chain system potentially provides not a few topics of qualitatively new aspect. The formal oxidation state of the metal ions is 3+ in MX chains, whereas 2.5+ in MMX chains. Therefore, MMX chains possess an unpaired electron per metal dimer even in their trapped-valence states, which is in contrast with the valence-trapped state consisting of M 2+ and M 4+ in MX chains. The M(d z 2 )-M(d z 2 ) direct overlap in MMX chains effectively reduces the on-site Coulomb repulsion due to its d σ * character and may therefore enhance the electrical conductivity. In the dta complexes with no counter ion, the chains weakly interact with each other through van der Waals contacts but are free from relatively strong hydrogen bonds, raising an increased possibility of the metal sublattice, as well as the halogen sublattice, being distorted. Nowadays, the chemical explorations along this line have reached a novel MMMMX chain compound HH-In response to recent stimulative observations for MMX chain compounds, we here study quantum and thermal phase competitions in one-dimensional unitassembled spin-charge-lattice coupling systems. We employ the where n j:m,s = a † j:m,s a j:m,s with a † j:m,s being the creation operator of an electron with spin s = ± (up and down...