Exact quantum thermal flux–flux correlations are obtained for the inverted parabolic barrier and free particle by smearing and scaling corresponding two-surface classical correlations. The smearing width is identified as a dynamical uncertainty in x associated with imaginary time evolution by −iħβ/2. The scaling is just the ratio of the high temperature asymptotic form of this dynamical uncertainty to the dynamical uncertainty itself. It represents a sort of quantum amplification. This quantum adaptation of classical flux–flux correlation is generalized to generic one-dimensional barrier systems, and applied to the symmetric and asymmetric Eckart barriers. Good agreement between model and converged numerical results is obtained down to temperatures where tunnelling begins to predominate above-barrier transmission. The model does, however, fail further into the tunnelling regime.
Exact quantum thermal flux–flux correlations are obtained for the inverted parabolic barrier and free particle by smearing and scaling corresponding two-surface classical correlations. The smearing width is identified as a dynamical uncertainty in x associated with imaginary time evolution by −iħβ/2. The scaling is just the ratio of the high temperature asymptotic form of this dynamical uncertainty to the dynamical uncertainty itself. It represents a sort of quantum amplification. This quantum adaptation of classical flux–flux correlation is generalized to generic one-dimensional barrier systems, and applied to the symmetric and asymmetric Eckart barriers. Good agreement between model and converged numerical results is obtained down to temperatures where tunnelling begins to predominate above-barrier transmission. The model does, however, fail further into the tunnelling regime.
A coherent state phase space representation of operators, based on the Husimi distribution, is used to derive an exact expression for the symmetrized version of thermal correlation functions. In addition to the time and temperature independent phase space representation of the two operators whose correlation function is of interest, the integrand includes a non-negative distribution function where only one imaginary time and one real time propagation are needed to compute it. The methodology is exemplified for the flux side correlation function used in rate theory. The coherent state representation necessitates the use of a smeared Gaussian flux operator whose coherent state phase space representation is identical to the classical flux expression. The resulting coherent state expression for the flux side correlation function has a number of advantages as compared to previous formulations. Since only one time propagation is needed, it is much easier to converge it with a semiclassical initial value representation. There is no need for forward-backward approximations, and in principle, the computation may be implemented on the fly. It also provides a route for analytic semiclassical approximations for the thermal rate, as exemplified by a computation of the transmission factor through symmetric and asymmetric Eckart barriers using a thawed Gaussian approximation for both imaginary and real time propagations. As a by-product, this example shows that one may obtain "good" tunneling rates using only above barrier classical trajectories even in the deep tunneling regime.
The transition-path-time distribution is formalized for quantum systems and applied to a number of examples. Using a symmetrized thermal density, transition times are studied for the free particle, a $\ensuremath{\delta}$-function potential, a square-barrier potential, and symmetric-double-well dynamics at very low temperature. These studies exemplify extreme nonlocality for motion in $\ensuremath{\delta}$-function potentials, vanishing tunneling times for the square-barrier potential, and varying transit times in the symmetric-double-well potential. In all cases, there are regions where the longer the distance traversed, the shorter the mean transit time is. For the thermal density correlation functions studied here, the Hartman effect exemplifies itself through the independence of the transit time on the barrier height. However, due to the thermal distribution, the transit time does depend on the barrier width, initially decreasing with increasing width but then increasing again.
The standard approaches to tunneling times are replaced by considering time correlation functions. A class of correlation functions that is always positive is identified and used to define quantum mechanical transition time probability distributions. The formalism is used to study the quantum dynamics of a thermal position correlation function of a parabolic barrier Hamiltonian. The transition time probability distribution between two points distributed symmetrically about the barrier top shifts to shorter times as the temperature is reduced and tunneling is increased. A study of the mean transition time as a function of the distance between the center of the initial and final densities shows that when the temperature is sufficiently low and tunneling dominates the dynamics, increasing the length of the path traversed decreases the mean transition time. The introduction of friction to the dynamics does not "destroy" this phenomenon, except when the friction coefficient is very large.
Expressions for the singular flux operator eigenfunctions and eigenvalues are given in terms of the Dirac delta-function representable as a localized Gaussian wavepacket. This functional form enables computation of the cumulative reaction probability N(E) from the wavepacket time-correlation functions. The Gaussian based form of the flux eigenfunctions, which is not tied to a finite basis of a quantum-mechanical calculation, is particularly useful for approximate calculation of N(E) with the trajectory based wavepacket propagation techniques. Numerical illustration is given for the Eckart barrier using the conventional quantum-mechanical propagation and the quantum trajectory dynamics with the approximate quantum potential. N(E) converges with respect to the Gaussian width parameter, and the convergence is faster at low energy. The approximate trajectory calculation overestimates tunneling in the low energy regime, but gives a significant improvement over the parabolic estimate of the tunneling probability.
Tunneling speeds up barrier crossing; quantum reflection can slow it down. For any one-dimensional barrier, kBT[κ(T) − 1], with κ the tunneling factor, falls with temperature to A, the signed area between quantum and classical transmission. Hence κ < 1 occurs if and only if A < 0, then above a unique temperature. A is the shift of the mean crossing threshold below the barrier top. For any regular thin barrier with heights Vmin ≤ Vmax, A → (7Vmin − Vmax)/6, as a step raises the mean threshold by 1/6 of its height. Numerically, the Eckart threshold Vmax/Vmin lies between 6.8159 and 7 + 4√3. The upload contains the Letter, the Supplemental Material (proofs, ball-arithmetic certificates, numerical methods, reaction data) and the reproducibility package (Python: mpmath, SymPy, NumPy, SciPy, Matplotlib, python-flint/Arb, ELIAH, formerly NETI) with the LaTeX sources.
Abstract The calculation of the thermal rate constant as a time integral over flux‐flux correlation functions is a challenging task when the potential energy along the reaction coordinate cannot be associated with a distinctive single barrier. In the case of resonant tunneling through a double barrier potential, the calculations may become formidable due to the population of long‐lived resonance states and the corresponding long time‐decay of the flux‐flux correlation functions. The flux averaging method was introduced recently in order to circumvent this problem in cases where the long time dynamics is due to a single resonance state with the longest lifetime in the system. In this work we generalize the method for calculations of thermal resonant‐tunneling rates in systems of many resonances, where the long time‐decay is accompanied by an internal dynamics within the quasi‐bound system. This extra complication is handled by additional averaging of flux‐flux correlation funcation over the time period of the internal dynamics. The result is an exact expression for the rate constant in terms of a linear combination of time integrals over flux‐flux correlation functions, which reaches its asymptotic time limit in a short (direct scattering) time, regardless of the long time‐decay of the flux‐flux correlation functions. This is derived for an analytic model system, and demonstrated in a numerical simulation of resonant tunneling through a double barrier potential.