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Physical Review A

Thermal rate of transmission through a barrier: Exact expansion of up to and including terms of order ℏ 4

Eli Pollak, Jianshu Cao

Published Feb 6, 2023• 11 citations
Publisher

Abstract

Nine decades after Wigner's formulation of quantum rate theory, his celebrated result was recently generalized to the asymmetric barrier using an exact first-order expansion of the transmission probability in terms of ${\ensuremath{\hbar}}^{2}$. This paper extends the first-order quantum correction to second-order correction of order ${\ensuremath{\hbar}}^{4}$ for the thermally averaged transmission probability through an arbitrary barrier. The derivation employs a systematic expansion of the projection operator onto products and the thermal distribution which involves a Taylor expansion of the potential about the barrier up to eighth order. The resulting exact analytical expression is calibrated with numerical calculations of several model potentials and shows excellent agreement when the ${\ensuremath{\hbar}}^{4}$ term is included. In comparison, the semiclassical transition state theory cannot reproduce the correct ${\ensuremath{\hbar}}^{4}$ terms when the anharmonicity is treated on the level of VPT-4 (vibrational perturbation theory---fourth order) and will potentially need a VPT-6 expansion. Further analysis of the quartic barrier reveals suppressed transmission due to the dominant role of quantum reflection above the barrier. These results not only provide a conceptual framework but can also be applied to heavy atom tunneling and machine learning.

DOI
10.1103/physreva.107.022203

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Nine decades after Wigner's formulation of quantum rate theory, his celebrated result was recently generalized to the asymmetric barrier using an exact first-order expansion of the transmission probability in terms of \ensuremathℏ2{\ensuremath{\hbar}}^{2}\ensuremathℏ2. This paper extends the first-order quantum correction to second-order correction of order \ensuremathℏ4{\ensuremath{\hbar}}^{4}\ensuremathℏ4 for the thermally averaged transmission probability through an arbitrary barrier. The derivation employs a systematic expansion of the projection operator onto products and the thermal distribution which involves a Taylor expansion of the potential about the barrier up to eighth order. The resulting exact analytical expression is calibrated with numerical calculations of several model potentials and shows excellent agreement when the \ensuremathℏ4{\ensuremath{\hbar}}^{4}\ensuremathℏ4 term is included. In comparison, the semiclassical transition state theory cannot reproduce the correct \ensuremathℏ4{\ensuremath{\hbar}}^{4}\ensuremathℏ4 terms when the anharmonicity is treated on the level of VPT-4 (vibrational perturbation theory---fourth order) and will potentially need a VPT-6 expansion. Further analysis of the quartic barrier reveals suppressed transmission due to the dominant role of quantum reflection above the barrier. These results not only provide a conceptual framework but can also be applied to heavy atom tunneling and machine learning.
We derive the quantum-mechanical tunneling transmission coefficient for energies near the peak of a general potential barrier, to fourth order in the Wentzel-Kramers-Brillouin (WKB) approximation, using a method based on an eighth-order Taylor expansion of the potential near the peak. The result agrees with contour-integral formulations of WKB tunneling. For the P\"oschl-Teller potential the WKB series converges rapidly to the known exact transmission coefficient. Higher-order WKB corrections may be important in attempts to determine the internuclear potential by inversion using low-energy fusion cross-section data.
The dissipationless quantum-thermal escape rate from a metastable well is often written as a Boltzmann average of the Hill-Wheeler flux over a continuum of energies. This continuum treatment diverges exponentially at low temperature, because it pairs the quantum partition function of the well with a classical flux integral. Retaining the discrete nature of the quasibound spectrum removes the divergence exactly, and the resulting rate is obtained in closed form on both sides of the crossover temperature T0T_0T0​. Evaluating a uniform Kemble transmission probability on Bohr-Sommerfeld levels further removes the parabolic-barrier error and recovers the WKB decay rate of the lowest resonance at zero temperature. Benchmarked against the resonances of cubic and quintic potentials, obtained by complex scaling, this uniform semiclassical rate stays within 9%9\%9% of the exact one over eleven orders of magnitude. Below T0T_0T0​, the rate rises above its zero-temperature plateau with the level spacing as activation energy, and the plateau ends at a temperature T1T_1T1​ noticeably below T0T_0T0​. In an isolated well, whose levels are not replenished, the thermal rate is only the initial slope of a decay which ceases to be single exponential at T1T_1T1​, while at T0T_0T0​ the initial escape moves from tunneling inside the well to the barrier top. We discuss implications for Josephson junctions, single-electron pumps, and nuclear burning in dense stellar matter.
Tunneling through the one-dimensional Eckart barrier is treated by applying a higher order semiclassical approximation which adds corrections proportional to powers of ℏ to the Herman–Kluk (HK) initial value approximation. Although the usual, zero-order HK treatment is very poor in this case, the first- and second-order corrections substantially improve the accuracy of the computed tunneling probabilities. To investigate how this works, the HK expression is shown to be equivalent, for the present purposes, to a formula involving a single integral over the initial momentum, with an integrand that has a simple analytical form. Similarly, the most important part of the first-order correction term is shown to be expressible in a very simple form. For a particular range of energies, the integral can be analyzed in terms of a steepest descent treatment along a path through a caustic. In this way, it is verified that the zero-order approximation does not approach the correct classical limit as ℏ → 0, but the first-order term, which does not vanish in this limit, improves the accuracy of the result. More generally, the corrections of each order contain terms that are of all orders in ℏ 1/3 , including those that are O (ℏ 0 ) and which survive in the classical limit. The infinite sum over such terms is performed analytically and shown to yield the correct classical limit for the tunneling amplitude.
Ninety years ago, Wigner derived the leading order expansion term in ℏ2 for the tunneling rate through a symmetric barrier. His derivation included two contributions: one came from the parabolic barrier, but a second term involved the fourth-order derivative of the potential at the barrier top. He left us with a challenge, which is answered in this paper, to derive the same but for an asymmetric barrier. A crucial element of the derivation is obtaining the ℏ2 expansion term for the projection operator, which appears in the flux-side expression for the rate. It is also reassuring that an analytical calculation of semiclassical transition state theory (TST) reproduces the anharmonic corrections to the leading order of ℏ2. The efficacy of the resulting expression is demonstrated for an Eckart barrier, leading to the conclusion that especially when considering heavy atom tunneling, one should use the expansion derived in this paper, rather than the parabolic barrier approximation. The rate expression derived here reveals how the classical TST limit is approached as a function of ℏ and, thus, provides critical insights to understand the validity of popular approximate theories, such as the classical Wigner, centroid molecular dynamics, and ring polymer molecular dynamics methods.
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.