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

Intermolecular Interactions Required for the Formation of Liquid Microcrystals Produced by the Precursors Self-Organized from Protonated TPP Dimers

“…It should be noted that proton sharing usually occurs between hydrogen-bonded water molecules in the excited state that requires an energy of 0.81422 eV in the case of vibrational mode of proton sharing [18,19]. Similar proton motion between the hydrogen-bonded molecules that is accompanied by the proton spin turnover as depicted in Scheme 1 requires considerably less energy, which occurs because of spin-orbit coupling between the electron and proton establishing the hydrogen bond [20]. In this case, the spin-orbit coupling parameter β = 1.19100654 is almost non-dependent on quaternary molecule coordination like under X-ray absorption [19].…”

“…accompanied by the change of the nucleus spin direction to the opposite resulting in the forward proton moving [20]. The difference (ћω − ћω') = 1.41153 cm -1 arisen because of spin-spin interaction between the coupled proton and electron establishing the hydrogen bond is estimated with the hydrogen's Lamb shift [21]; oxygen atoms and electrons are depicted by red and pink, respectively, blue lines show the bonds and hydrogen bonding.…”

“…However, this H/H + change should be accompanied by the change of two hydrogen bonds location in the structure. The latter occurs under proton motion between the hydrogen-bonded molecules with the change of proton spin direction to the opposite accompanied by an energy absorption producing so-called boson peak [20]. But if the change of proton spin direction occurs together with an excitation that results in the change of two hydrogen bonds location, then the absorbed energy should be less by ћω' (see Scheme 1).…”

“…Thus, for the proton moving through water in the solution with THF, the frequency of the absorbed energy can be evaluated theoretically using the frequency estimation in the case of boson peak. The following relation for a frequency (ƒHb =c/ν -1 ) absorption required for the proton motion along the direction between the hydrogen-bonded molecules was derived to explain the origination of boson peak [20].…”

“…where rex is the polaronic exciton radius that is the average distance between the opposite charges, mr * is the reduced effective mass of polaronic exciton, mr * = m h ef m e ef/(m h ef + m e ef), m h ef and m e ef are respectively the effective masses of hole and electronic polarons, β is spin-orbit coupling parameter (β = 1.19100654 [19,20]), ge and gp are respectively the electron and proton g-factors, and m is quantum number. With rex = 2.54 Å, ge = 2.002319304, gp = 5.5856947, m h ef = 9.51me, mp = 1836.152667me, and m = 1 for proton, the calculated ƒHb = 1.3294299 THz or νHb = 44.345 cm -1 .…”