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International Journal of Quantum Chemistry

Simulated annealing‐based optimal control over tunneling process through SDWP and E ckart barrier: A momentum basis representation

Srijeeta Talukder, Pinaki Chaudhury, Subhasree Ghosh

Published Apr 25, 2017• 6 citations
Publisher

Abstract

Abstract For a reaction to proceed via tunneling mechanism, it is essential that the reactants will cross the potential barrier (EP), where its initial energy (E0) is below the potential barrier EP. Tunneling probability τ is defined as the probability of having momentum higher than km, where . In the momentum basis representation, τ can be directly calculated by integrating from the limit km to infinity, where is the wave function in the momentum space. Instead of the continuous basis, if we chose momentum grid space, τ can be expressed as . Our target here is to increase this τ by applying a polychromatic field, so that the reaction rate can be enhanced. By applying Simulated Annealing technique we have designed some polychromatic electric fields, spatially symmetric and asymmetric type, which enhances the tunneling rate in symmetric double well system and Eckart barrier confined in an infinite well.

DOI
10.1002/qua.25388

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Abstract For a reaction to proceed via tunneling mechanism, it is essential that the reactants will cross the potential barrier (EP), where its initial energy (E0) is below the potential barrier EP. Tunneling probability τ is defined as the probability of having momentum higher than km, where . In the momentum basis representation, τ can be directly calculated by integrating from the limit km to infinity, where is the wave function in the momentum space. Instead of the continuous basis, if we chose momentum grid space, τ can be expressed as . Our target here is to increase this τ by applying a polychromatic field, so that the reaction rate can be enhanced. By applying Simulated Annealing technique we have designed some polychromatic electric fields, spatially symmetric and asymmetric type, which enhances the tunneling rate in symmetric double well system and Eckart barrier confined in an infinite well.
Control of tunneling in a symmetric double well coupled to a bath via an external field is demonstrated. Optimal control theory is employed to design a laser field which couples to the reaction coordinate and drives a localized wave packet on the reactant side to the product side at a specified target time τ. Both a very quantumlike system (high barrier) and a low barrier double well are examined; excellent results are obtained for a range of reduced bath viscosities. Analysis of the control fields and corresponding localization dynamics shows that the frequency components of the control field are more or less in resonance with eigenstates of the double well and that the laser field enhances the natural dynamics of the individual wells. Future extension to more complicated models where the field couples to the bath is discussed.
Tunnelling in a one-dimensional double-well system with five turning points can be considered as one of the simplest models for a unimolecular chemical reaction. It is assumed that a potential barrier separating reactants from products is sufficiently high, so that the reaction rate constant decreases exponentially. The height of the second potential barrier is related to the widths of the energy levels in a product well imitating relaxation processes of reaction products. An analytical expression is derived for the width of an initial energy level in a reactant well. It depends exponentially on the value of 2W 1 (W 1 is a quasiclassical phase integral under the potential barrier separating reactants from products). The preexponential factor is expressed in terms of matrix elements of the propagator connecting solutions of the Schrödinger equation in asymptotic regions. Quantum and quasiclassical calculations of the propagator were performed. Under resonance conditions, when reactant and product energy levels coincide, the width of a reactant level is shown to be 4 exp 2W 2 times as large as its value in the absence of the resonance (W 2 is a quasiclassical phase integral under the second potential barrier).
For two-body s-wave collisions subject to tunnelling though an Eckart potential barrier, cross-sections and rate coefficients in the low-energy regime can be evaluated in analytic form. These provide criteria for approach to the Wigner limit and a generic plot applicable to ultracold collisions (<1 mK). The Eckart barrier shape is found to fit fairly well accurate potential curves available for F(2P) + F(2P) and He(1S) + He*(1S). Also discussed are tunnelling-dominated ultracold chemical reactions, exemplified by F + H2. The very slow approach of the reactants allows averaging over rovibrational motions, so the effective adiabatic (‘dressed’) potential surface differs substantially from the Born–Oppenheimer (‘bare’) surface obtained from electronic structure calculations. The dressed barrier is more than two-fold lower and also thinner. For it the Eckart model gives an ultracold rate coefficient for forming HF in its v′ = 2, j′ = 0 rovibrational state (6.5 × 10−14 cm3/s) that is about 200-fold higher than for the bare barrier and of the same magnitude found from a full-scale 3-D quantum scattering calculation.
Abstract We suggest a recipe for calculating the time taken by a particle to tunnel out from an initially localized state in one of the wells of symmetrical double‐well potential into the other well. We calculate average velocity 〈v̄〉 of the tunneling particle by solving the Schrödinger equation numerically for Ψ(x, t), then estimating 〈v(t)〉 = (d/dt)〈x(t)〉 and time‐averaging it. The time taken to tunnel is measured by calculating τav = l0/〈v̄〉 where l0 is an idealized estimate of the barrier width. We suggest an approximate partitioning of τav into an intrinsic decay time (τd) and a barrier crossing time (τb), τd being obtained from energy spread of the packet. τav − τd is shown to be close to the Wentzel‐Brillouin‐Kramers estimate of barrier crossing time τsc. The response of τav to an external driving field and nonzero temperatures are tested. We also apply the method to a purely barrier penetration problem as well as to the problem of tunneling through a fluctuating barrier. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
In this paper, we develop the formalism of entangled trajectory molecular dynamics (ETMD), introduced by Donoso and Martens [Phys. Rev. Lett. 2001, 87, 223202] in a form applicable to the treatment of general (i.e., nonpolynomial) potentials. We formulate our approach directly in terms of the integrodifferential equation obeyed by the Wigner function, without assuming a Taylor series expansion of the potential in powers of the coordinate. This alternative formalism has distinct advantages for propagating distribution functions represented by finite trajectory ensembles and for nonpolynomial potentials. We use a numerical implementation of the new approach to calculate the reaction probabilities for three model systems: the cubic polynomial potential, symmetric Eckart barrier and asymmetric Eckart barrier. Our results are in excellent agreement with the results of exact quantum calculation.
In this research, the optimization of the tunneling probability through a potential barrier by superimposing ultra high frequency (UHF) acoustic wave over the source of the incoming particle wave function was examined theoretically and was shown to result in acoustically augmented phonons (AAP). The graph of the tunneling probability against the kinetic energy fraction [( E/V 0 )= x ] of the particle shows a line of inflection at a non-dimensionalized critical height y c ≈ 3.12879, where y c is universal tunneling constant (UTC). As the barrier height ( y ) is increased further ( y > y c ), the reflection increases, and the tunneling probability sharply declines, in general. The lowest possible value of y is guided by the inherent particle kinetic energy, the superimposed wave number n and the material parameter β . The’ gradient of the increase in probability’ rises with a drop in the wave number n and is larger at higher values of y . A higher ratio ( E/V 0 ) coupled with a permissible-smaller wave number ( n ) of the applied UHF acoustic wave, leads to a higher tunneling probability. For increasing values of the UHF wave numbers and decreasing x -values, the potential barrier becomes increasingly opaque to tunneling. The higher the value of y is, the higher the tunneling opacity of the potential barrier becomes. The tunneling probability is highest (=0.98673) at y = 4, x = 0.9, when the orders of β and k are comparable.