Chemical Reactions Using a Non-Equilibrium Wigner Function Approach

Abstract

Abstract:A three-dimensional model of binary chemical reactions is studied. We consider an ab initio quantum two-particle system subjected to an attractive interaction potential and to a heat bath at thermal equilibrium at absolute temperature T > 0. Under the sole action of the attraction potential, the two particles can either be bound or unbound to each other. While at T = 0, there is no transition between both states, such a transition is possible when T > 0 (due to the heat bath) and plays a key role as k B T approaches the magnitude of the attractive potential. We focus on a quantum regime, typical of chemical reactions, such that: (a) the thermal wavelength is shorter than the range of the attractive potential (lower limit on T) and (b) (3/2)k B T does not exceed the magnitude of the attractive potential (upper limit on T). In this regime, we extend several methods previously applied to analyze the time duration of DNA thermal denaturation. The two-particle system is then described by a non-equilibrium Wigner function. Under Assumptions (a) and (b), and for sufficiently long times, defined by a characteristic time scale D that is subsequently estimated, the general dissipationless non-equilibrium equation for the Wigner function is approximated by a Smoluchowski-like equation displaying dissipation and quantum effects. A comparison with the standard chemical kinetic equations is made. The time τ required for the two particles to transition from the bound state to unbound configurations is studied by means of the mean first passage time formalism. An approximate formula for τ, in terms of D and exhibiting the Arrhenius exponential factor, is obtained. Recombination processes are also briefly studied within our framework and compared with previous well-known methods.

Non-Equilibrium Liouville and Wigner Equations: Classical Statistical Mechanics and Chemical Reactions for Long Times

“…Assumption. As | q |→ +∞, W(x, q; t) → 0 quickly, for fixed x and t, so that +∞ −∞ dqW(x, q; t)q n converges, for any integer n ≥ 0 [32,34].…”

“…M n,n =0 = 0 for any n, except for n = 1 (with n = 0). In [32,34], we have obtained the first five equations in that quantum hierarchy. For any V = 0, the equation for ∂W n /∂t for any n contains a contribution from W n+1 (always with the same structure, with n-independent coefficients) given in Equation (45), but does not on higher order moments W n , with n > n + 1.…”

“…No approximation has been carried out, so far, in this section for brevity, as they will be performed in the following sections. See [32,34] for details and further analysis and approximations in the one-dimensional case (in particular, for the long term).…”

“…The present author has carried out several detailed analyses of the classical Liouville and quantum Wigner equation, through infinite hierarchies for suitable non-equilibrium moments, in order to analyze approaches to thermal equilibrium in long-term approximations [25][26][27][28][29][30][31][32]. Motivated by previous work on non-equilibrium DNA thermal denaturation (the dynamics of two interacting classical three-dimensional macromolecular chains) [33], binary chemical reactions have been studied quantum-mechanically, based on the non-equilibrium Wigner function [34].…”

“…The intention of this work is to review and extend the work mentioned above in [25][26][27][28][29][30][31][32] and in [34]. To outline, we shall focus on the dynamics of non-equilibrium non-relativistic systems implied by the classical Liouville equation and the quantum Wigner one [2].…”