Energy levels governed by golden section in OH+...O moiety under proton sharing coupled with spin-orbit interactions

Resonance Raman spectra of protonated meso-tetraphenylporphine (TPP) dimers measured earlier in aqueous solutions have been interpreted in terms of polaronic exciton theory. Consideration of the incident photon interaction with H3O+ producing polaronic exciton shows up spin-orbit interaction depending (βq = 3.741657387) and almost non-depending (β = 1.19100654) on quaternary molecule coordination in the water clusters. Both β parameters that take into consideration proton and electron spin orientations have been involved in major relations derived under the development of polaronic exciton theory. Theoretically estimated energy gap between the levels with different proton spin orientations, i.e. e↓↑p↑e and e↓↓p↑e in the OHδ+...O moiety is found to be 40.766 cm−1. Experimental energy gap 25.5 ± 1.3 cm−1 averaged for several doublets in the Raman spectrum with λex  = 441.6 nm is consistent with the theoretical 25.19 cm−1 obtained using golden section as a denominator. It appears that golden section governs the fine structure under the proton sharing in the OH+...O moiety of the water cluster embedded between TPP units of the dimers.