Tunneling rates in bound systems using smooth exterior complex scaling within the framework of the finite basis set approximation

Abstract

A basis-set time-independent method to calculate tunneling rates in bound systems through a potential barrier is presented. The tunneling decay rates are associated with the imaginary parts of the complex eigenvalues of the Schrödinger equation where the reaction coordinate r′ is complex scaled such that, dr = dr′[1/cos θ(r′)]exp (iθ(r′)), where tan θ(r′) = tan θ∞g(r′). The function g(r′) fulfills 0 ≤ g(r′) ≤ 1 and shows a smooth transition from 0 to 1 near r′ = r0 which is the location of the top of the barrier. The value of θ∞ should be larger than a critical value for which a sharp transition from a real eigenvalue spectrum to a complex one is obtained. Illustrative numerical applications to two isomerization reaction models are given.

Hybrid approach to molecular continuum processes combining Gaussian basis functions and the discrete variable representation

“…The inclusion of increasingly diffuse Gaussians or expansively 1 3 2 5 4 # 6 0 6 8 72 @ 9 A B D C 7E @ 9 A B 2 @ F H G P I R Q A B S C 7E 5 9 T E @ U W V S F B # X G P I R Q A B S C 7E 5 large basis sets as a remedy to their locality eventually fails due to the development of linear dependencies within the Gaussian basis. Furthermore, in connection with exterior complex scaling, Gaussian functions, like other analytic basis functions, require a cumbersome "smoothed" exterior scaling treatment that can produce unwanted physical consequences [15][16][17].…”

Calculation of the Quantum-Mechanical Tunneling in Bound Potentials

“…Coordinates 1 and 2 are the positions of the minima in the reactant and product regions, respectively, and also define the barrier region. and are inflection points of the potential given by (20). This is shown in Figure 2 for a piecewise quadratic continuous potential.…”

“…Even though the approximate treatments of tunneling can be surprisingly accurate, the search for more conceptually rigorous expressions led to the development of less traditional approaches, such as the stationary state decomposition for quadratic potentials [17] via the quantum Hamilton-Jacobi equation adapted for energy eigenstates [18,19], the exterior complex scaling method [20], and, more recently, the quantum instanton theory [21], ring polymer dynamics [22], and perturbative "beyond instantons" correction to the WKB approximation [23]. The quantum tunneling is known to be very sensitive to the barrier shape and energy, as shown, for example, in the context of enzyme catalysis by Hay and coworkers [9].…”

Formulae for partial widths derived from the Lindblad equation

“…It may, e.g., arise by introducing an artificial imaginary potential which is zero in some interior region and increasing towards some boundary (a complex absorbing potential -CAP) [17][18][19]. Other frequently used techniques are (uniform) complex scaling [20][21][22], exterior complex scaling (ECS) [23,24] and smooth exterior complex scaling [25,26].…”