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The Journal of Chemical Physics

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

Jürgen Brickmann, Herbert W. Zimmermann

Published Feb 15, 1969• 90 citations
Publisher

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 Vmax 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)orH(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)andH(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.

DOI
10.1063/1.1671247

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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 Vmax 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)orH(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)andH(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.
Proton transfer plays a key role in the applications of advanced energy materials as well as in the functionalities of biological systems. In this work, based on the transfer matrix method, we study the quantum effects of proton transfer in a series of one-dimensional (1D) model potentials and numerically calculate the quantum probability of transferring across single and double barriers (wells). In the case of single barriers, when the incident energies of protons are above the barrier height, the quantum oscillations in the transmission coefficients depend on the geometric shape of the barriers. It is found that atomic resonant tunneling (ART) not only presents in the rectangular single well and rectangular double barriers as expected, but also exists in the other types of potential wells and double barriers. For hetero-structured double barriers, there is no resonant tunneling in the classical forbidden zone, i.e., in the case when the incident energy (E i) is lower than the barrier height (E b). Furthermore, we have provided generalized analysis on the characteristics of transmission coefficients of hetero-structured rectangular double barriers.
The time evolution of quantum mechanical states in a square well of infinite depth with a Dirac δ function at its center has been examined for cases where the initial state was localized in one of the wells. Let well A denote the well in which the initial state is localized, and let P(t) denote the integrated probability density Ψ*Ψ in well A. For opaque barriers the time-dependent system is adequately described by a two-state model in which only the pair of stationary states of even and odd parity are considered whose wave functions in well A, apart from an arbitrary phase factor, are largely identical with the wave function of the initial state. For P(t) a harmonic oscillation is observed whose frequency νt is well approximated by the well-known formula for the tunneling frequency νt≂ΔE/h, where ΔE represents the energy separation among the pair of states in the model. For the present model of a symmetric double well it has been shown that for highly transparent barriers a three-state model can describe the time evolution adequately. The three stationary states involved in this model are a state of odd parity whose wave function in well A is largely the same as the wave function of the initial state and a pair of stationary states of even parity which on the energy scale are immediately above and below the first state. In this three-state model the function P(t) is a superposition of two sinusoidal functions with nearly identical amplitudes and frequencies plus a constant. As a consequence, the amplitude of P(t) changes harmonically. In the present model a δ function has been used as a barrier in order to minimize the mathematical detail involved in the time-dependent treatment. It is to be expected that the beating in P(t) can be observed also in the time evolution of a state localized in one of the wells of a symmetric double-minimum potential with a more realistic ‘‘low’’ barrier if the density of energy levels near the energies of the levels to be considered in the corresponding three-state model varies slowly with energy.
A system consisting of two protons in two double minimum potentials coupled by a bilinear interaction was studied. Explicit calculations were performed for local potentials described by a combination of harmonic oscillator and Gaussian terms. To determine eigenfunctions and -values variational procedures were employed which made use of numerically determined 1-particle solutions. Eigenvalues and (graphical representations of) 2d eigenfunctions are presented, classified according to their symmetry, and used to determine the time evolution of the system for given initial conditions.
Abstract A recently published methodology based on the semiclassical path integral theory was applied in a double well structure and gave the analytic form of the system's Green's function. This type of potential can describe the ammonia molecule as far as the motion of the nitrogen atom perpendicular to the hydrogen plane is discussed. Because of the fact that a double well describes a bound system and correspondingly stationary states (constructed by the symmetric and antisymmetric superposition of the eigenstates of the two unperturbed wells), it was expected that the energy spectrum would be real, in a form of doublets due to the splitting effect that takes place. However, the result was a pair of complex poles, which had a clearly positive imaginary part for each member. The present work explains the role of the imaginary parts of the complex poles as the decay rate of quantities defined as the energy indeterminacies, which are directly related to the fact that energy is not well determined in a classically forbidden region of motion. These quantities come as a function of (dκ)/dE, which is the derivative of the classical action inside the potential barrier, with respect to energy. The major contribution comes from the turning points, and then the imaginary parts are responsible, not only for the conservation of energy, but for the correct sign of time as well. In this way, a different approach for the tunneling process is adopted, in which the entry or exit of the particle from the potential barrier takes place inside a neighborhood of the turning point, as though the latter was broadened and fluctuating. The magnitude of the previously mentioned decay rate is equal to ω/π, where ω is the frequency of the classical oscillations inside one well. In contrast, the inversion frequency is generated by the part of the complex pole that is unrelated to (dκ)/dE and is much smaller in magnitude than the classical frequency, since it is given as ω/π exp(−κ). In this way, the period of the energy fluctuations is much smaller than the internal period of the system produced by the oscillating communication of the two classically allowed regions of motion. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007
The tunneling between double wells of atom in crossed electromagnetic fields is investigated by a one-dimensional Hamiltonian model. The crossed fields induced outer well is apart from the nuclear origin and it is very difficult to access by means of spectroscopy but it will be possible if there exists the tunneling of the electron between the outer well and the Coulomb potential predominated well at the nuclear origin. A one-dimensional quantum calculation with B-spline basis has been performed for hydrogen atom in crossed fields accessible in our laboratory, at B = 0.8 T and F = −220 V·cm −1 . The calculation shows that the wavefunctions of some excited states close to the Stark saddle point in the outer well extend over to the Coulomb potential well, making it possible to penetrate the quantum information of the outer well. However, the tunneling rate is very small and the spectral measurement of the transitions from the ground state should be of a high resolution and high sensitivity.