We examine the ordering of the Lower Consolute Solution Temperatures (LCSTs) for a set of dimethylpyridines. Density functional theory (DFT) is used. The equilibrium geometries and binding energies of dimers, each comprised of a pair of dimethylpyridines in a sandwich conformation and one H2O molecule at a pivotal site between the nitrogens (the 2:1 dimer), are calculated. It was shown previously that dimer formation in the water-rich zone of the phase diagram has a crucial role in dimethylpyridine demixing. In the resulting dimer diffusion, large hydrophobic clusters of mostly organic content, which expel water and promote phase separation, are assembled. In this description, phase separation requires the formation of 2:1 dimers, but it is the cleavage of hydrogen bonds of the neighboring H2O molecules, which stimulates the diffusion and the subsequent separation dynamics at the LCST. In the present study, we investigate this model and calculate the interaction strength of the external hydrogen bonds. This is obtained as the difference in electronic energy between the 2:1 dimer and the dimer augmented by one or two H2O molecules. The results are compared to the known LCST hierarchy in five dimethylpyridines (DMP): 2,6-DMP > 2,4-DMP > 2,5-DMP > 3,4-DMP > 3,5-DMP. The complexes are derived using high level Kohn-Sham DFT including dispersion terms. The hydrophobic-hydrophilic properties are accounted for by the solvation model, employed for the mixed medium of 60%-water and 40%-organic content. This is simulated by combination of model descriptors of water and DMP in the parameterization scheme of the polarizable continuum model. The calculation results agree with the experimental evidence.