Chemical and Biological Physics Department, Weizmann Institute of Science, 76100 Rehovoth, Israel.
Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.
J Chem Phys. 2022 Aug 21;157(7):074109. doi: 10.1063/5.0106649.
Ninety years ago, Wigner derived the leading order expansion term in ℏ 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 ℏ 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 ℏ. 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.
90 年前，维格纳推导出了通过对称势垒的隧穿率中 ℏ 的一阶展开项。他的推导包括两个贡献：一个来自于抛物形势垒，但第二个项涉及势垒顶部的势的四阶导数。他给我们留下了一个挑战，本文回答了这个问题，即推导出对于非对称势垒的相同结果。推导的一个关键要素是获得投影算符的 ℏ 展开项，它出现在速率的通量侧表达式中。令人欣慰的是，半经典过渡态理论（TST）的解析计算再现了 ℏ 阶的非谐修正。对于 Eckart 势垒的计算结果表明，特别是在考虑重原子隧穿时，应该使用本文中推导的展开式，而不是抛物形势垒近似。这里推导出的速率表达式揭示了经典 TST 极限如何作为 ℏ 的函数逼近，从而为理解流行的近似理论（如经典 Wigner、质心分子动力学和环聚合物分子动力学方法）的有效性提供了关键见解。
