Lingering Time of the Proton in the Wells of the Double-Minimum Potential of Hydrogen Bonds

Abstract

The movement of a proton which moves in a one‐dimensional effective double‐minimum potential V(x) and can pass by means of the tunnel effect from one well into the other was investigated. The lingering times of the proton in the potential wells (a) and (b) were calculated for more than 150 potential models as functions of the height V max of the potential barrier separating the two wells, the energy difference ΔV , and separation d of the minima. In the determination of the lingering times, it was assumed that the proton at time t = 0 is to be found with certainty in one of the wells and can be represented by means of a statistical operator related to the eigenstates of a one‐well Hamiltonian H (a) or H (b) , respectively. The times determined on this basis were compared with those which result with a quasiclassical method from the classical frequency of oscillation of a particle in a well and from the penetration coefficient of the potential barrier. The deviation between the values determined with both methods is largest if an eigenvalue of H (a) becomes approximately equal to an eigenvalue of H (b) . For the potential model V(x) a polynomial of fourth degree in the position coordinate x was chosen. The eigenfunctions of H (a) and H (b) as well as those of the effective Hamiltonian H corresponding to V(x) were represented as linear combinations of eigenfunctions of the harmonic oscillator with angular frequency ω . ω was chosen so that the mean square deviations of the approximated eigenvalues of H became as small as possible for the states of lowest energy.