Weiner's theory for exactly solvable Schrödinger equation with symmetric double well potential

Abstract

The Weiner's theory (WT) is developed on the basis of the exactly solvable Schrödinger equation with trigonometric double-well potential (TDWP). The symmetric case of TDWP is considered. This modified version of WT (mWT) enables one to eliminate some severe approximations of the original Weiner's approach and to obtain more accurate results. An analytic formula is derived which provides the calculation of the proton transfer rate with the help of elements implemented in {\sl {Mathematica}}. We exemplify the application of mWT by calculating the proton transfer rate constant in the hydrogen bond of the proton-bound ammonia dimer cation ${\rm{N_2H_7^{+}}}$ (${\rm{H_3N\cdot\cdot\cdot H^{+} \cdot\cdot\cdot NH_3}}$). The parameters of the model for this object are extracted from available literature data on IR spectroscopy and quantum chemical calculations. The approach yields the transition from the Arrhenius-like exponential temperature dependence characteristic of thermal activation to that of quantum tunneling. Besides it is well suited for describing the phenomenon of vibrationally enhanced tunnelling.

Figures (5)

• Figure 1: The trigonometric double-well potential (\ref{['eq2']}) at the values of the parameters $m=2$; $p=7.82971$. The parameters are chosen to describe the hydrogen bond in the proton-bound ammonia dimer cation ${\rm{N_2H_7^{+}}}$ for the case $R_{NN}=3.15\ \AA$ (they are extracted from the data of quantum chemistry Gar08). At the right of Fig.1 the energy levels (given by (\ref{['eq4']})) are presented.
• Figure 2: The dependence of the proton transfer rate constant on the inverse temperature in the proton-bound ammonia dimer cation ${\rm{N_2H_7^{+}}}$ for the various cases of the parameter $p$ at $m=2$.
• Figure 3: The dependence of the proton transfer rate constant on the inverse temperature in the proton-bound ammonia dimer cation ${\rm{N_2H_7^{+}}}$ for the case $p=7.82971$ at $m=2$ corresponding to $R_{NN}=3.15\ \AA$.
• Figure 4: The dependence of the probability flux and the quantum transmission coefficient product for the third level ($q=2$, i.e., $\bar{n}=1$) on the frequency of the external oscillator in the Zundel ion ${\rm{H_5O_2^{+}}}$ (oxonium hydrate) with $R_{OO}=3.0\ A$ (the values of the parameters for TDWP are $m=57$; $p=76$). The value of the coupling constant between the proton coordinate and that of the oscillator is $\alpha=5$ and the ratio of the mass for the proton to that of the oscillator is $\delta=1$ (see Sit25 for details).
• Figure 5: The dependence of the proton transfer rate constant on the inverse temperature in the proton-bound ammonia dimer cation ${\rm{N_2H_7^{+}}}$ for the model case $p=12$ at $m=2$ calculated within the framework of the Weiners' theory (WT) and the modified Weiners' theory (mWT). The parameters of WT in (\ref{['eq66']}) are $f_0=0.000021$; $f_2=0.1$ and $f_4=0.095$.