Structure of a proton wire in the harmonic model with allowance for the interproton interaction for the first and second neighbors

Differential equations based on the one-component harmonic model of a water chain are proposed for description of the proton wire structure with allowance for the interproton interaction of near and far neighbors. The solution to the Sturm-Liouville problem is considered for a fourth-order differential equation that takes into account interproton interactions with the first and second adjacent links of a one-dimensional chain. The function of proton displacements in such a wire is shown to describe a quasi-periodical structure, depending on the ratio of constants D ₁ and D ₂ for the interproton interaction of the first and second neighbors. According to calculations using the parameters characteristic of the water chain, the curve of the proton displacement is a plot of function y = cosk ₁ x + sink ₂ x and is similar to the curve of the hydrogen bond length distribution obtained earlier in the quantum-chemical calculations of the proton channel model.