## 2 Materials and Methods (Theoretical Bases and Methods)
At low temperatures, when \(\mathrm{T}\ll\mathrm{T}_{\sigma}\) and \(\Lambda\ll\mathrm{X}\), according to \(\frac{1}{1-\frac{\Lambda}{\mathrm{X}}}\approx 1+\frac{\Lambda}{\mathrm{X}}\), we have \(D_{quant,tum}^{q-classic;(0)}(\mathrm{T})\approx\left(1+\frac{\Lambda}{ \mathrm{X}}\right)\exp(-\Lambda)\). Near the temperature of absolute zero, the kinetics of migratory polarization in HBC are only determined by the parameters of the potential pattern and are practically independent of temperature \(D_{quant,tum}^{q-classic;(0)}(0)=\exp(-\Lambda)\). With \(\mathrm{T}=\mathrm{T}_{\sigma}\), taking \(\mathrm{X}\to\Lambda\), we have \(D_{quant,tum}^{q-classic;(0)}(\mathrm{T}_{\sigma})=\Lambda\exp(-\Lambda)\). The point of intersection of function graphs \(D_{quant,tum}^{q-classic;(0)}(\mathrm{T})\), \(D_{therm}^{classic;(0)}(\mathrm{T})=\exp\left(-\frac{\mathrm{U}_{0}}{\mathrm{K} \mathrm{T}}\right)\), calculated from the equation \(\left(2+\frac{\Lambda}{\mathrm{X}}\right)\exp(-\mathrm{X})=\exp(-\Lambda)\), indicates temperature \(\mathrm{T}_{c}>\mathrm{T}_{\sigma}\). The maximum point of the functions \(D_{quant,tum}^{q-classic;(0)}(\mathrm{T})\), calculated from the following equation  
\[\mathrm{X}^{2}\left(\frac{\Lambda}{\mathrm{X}}\left(1+\frac{1}{\mathrm{X}} \right)-1\right)\exp(-\mathrm{X})=\Lambda\exp(-\Lambda),\]  
indicates temperature \(\mathrm{T}_{\mathrm{max}}>\mathrm{T}_{c}\). Here, \(\Lambda=-\mathrm{ln}\left(D_{quant,tum}^{q-classic;(0)}(0)\right)\). Thus, the conditions are met:  
\[\mathrm{T}_{\sigma}<\mathrm{T}_{c}<\mathrm{T}_{\mathrm{max}};\ D_{quant,tum}^ {q-classic;(0)}(\mathrm{T}_{\sigma})<D_{quant,tum}^{q-classic;(0)}(\mathrm{T} _{c})<D_{quant,tum}^{q-classic;(0)}(\mathrm{T}_{\mathrm{max}}).\]  
Obviously, at \(\mathrm{T}\to\infty\), \(D_{quant,tum}^{q-classic;(0)}(\mathrm{T})\to 0\). The behavior of the function \(D_{therm}^{classic;(0)}(\mathrm{T})\) corresponds to the classical statistical theory. The quasi-classical probability rate of proton transfer through the unperturbed potential barrier is determined considering both thermal activation and tunneling [1,7]  
\[W^{q-classic;(0)}(\mathrm{T})=\frac{\mathrm{v}_{0}}{2}\left(\exp(-\mathrm{X}) +D_{quant,tum}^{q-classic;(0)}(\mathrm{T})\right)=\frac{\mathrm{v}_{0}}{2}. \frac{\exp(-\Lambda)-\frac{\Lambda}{\mathrm{X}}\exp(-\mathrm{X})}{1-\frac{ \Lambda}{\mathrm{X}}} \tag{1}\]  
Here, \(\mathrm{v}_{0}\) denotes the natural frequency of oscillations in the proton in a potential well. In the temperature area \(\mathrm{T}\ll\mathrm{T}_{\sigma}\), Equation (1) is converted as follows:  
\[W^{q,-classic;(0)}(\mathrm{T})\approx\ W_{quant,tum}^{q,-classic;(0)}=\frac{ \mathrm{v}_{0}}{2}\times\frac{\exp(-\Lambda)-\exp(-\mathrm{X})}{1-\frac{ \Lambda}{\mathrm{X}}}. \tag{2}\]  
In the temperature area \(\mathrm{T}\gg\mathrm{T}_{\sigma}\), Equation (1) is converted as follows:  
\[W^{q,-classic;(0)}(\mathrm{T})\approx W_{therm}^{classic;(0)}=\frac{\mathrm{v}_{0 }}{2}\times\exp(-\mathrm{X}). \tag{3}\]Within the model of parabolic potential, between parameters \(\delta_{0}\) and U\({}_{0}\), there is an obvious communication U\({}_{0}=\frac{\mathrm{m}\omega_{0}^{2}\xi_{0}^{2}}{8}\). Taking the energy spectrum of the proton \(E_{\mathrm{n}}^{(0)}=\hbar\omega_{0}\Big{(}\mathrm{n}+\frac{1}{2}\Big{)}\) as quasi-continuous \(\Big{|}E_{\mathrm{n}\pm 1}^{(0)}-E_{\mathrm{n}}^{(0)}\Big{|}=\hbar\omega_{0} \gg\mathrm{k}_{\mathrm{B}}\mathrm{T}\), write down the condition for the quasi-classicity of proton motion in HBC \(\frac{\mathrm{U}_{0}}{E_{0}^{(0)}}=\frac{\mathrm{m}\omega_{0}\xi_{0}^{2}}{4 \hbar}\gg 1\). Here, \(\omega_{0}=2\pi\mathrm{v}_{0}\). The analysis of Equations (1)-(3) shows that, for any type of relaxation, the value of the critical temperature calculated for a rectangular barrier is 78.5% of the critical temperature in the parabolic barrier model.