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arXiv:1312.4409 · Condensed Matter Physics

Time evolution of a small reactive system

J. P. Badiali

Published Jan 1, 2013• 1 citation
arXivHTMLPDF

Abstract

We investigate the irreversible evolution of a small system in which a chemical reaction takes place.We have two main goals: the first requires to find an equation to produce a time-irreversible behavior,the second consists in introducing a simple exactly solvable model in order to understand basic facts in chemical kinetics.Our basic tool is the transition function counting the number of paths joining two points in the reactive coordinates system.An exact quantum Smoluchowski equation is derived for the reactive system in vacuum, in the presence of a solvent in equilibrium at any time with the reactive system a new Smoluchowski equation is obtained.The transition from a quantum regime to a classical one is discussed.The case of a reactive system not in equilibrium with its neighborhood is investigated in terms of path integral and via a partial differential function.Memory effects and closure assumptions are discussed.Using a simple potential model, the chemical rate constant is exactly calculated and questions such as the meaning of the activation energy or the physical content of the so-called prefactor are investigated.

DOI
10.5488/cmp.16.43003

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The applicability of the time-dependent Schrödinger equation to molecular reaction dynamics is discussed from several perspectives. First, with reference to Schrödinger's original paper, we point out that he himself stated that what is now called the time-dependent Schrödinger equation is insufficient to solve the problems of non-conservative systems like chemical reactions. Second, the detonation velocity of some explosives has been demonstrated as a new parameter that directly illustrates the timescales of elementary chemical reactions. The calculated values of ~30 fs are in good agreement with the timescale of 10–100 fs for chemical reactions, as experimentally demonstrated by the Nobel laureate Zewail. Third, we consider the state of an electron in a box when the length of the box suddenly increases from L to 2L, which is often found in textbooks as the simplest example of the application of the time-dependent Schrödinger equation to chemical reactions, and highlight the inconsistency between this model and an actual chemical reaction. Finally, we re-iterate that only quantum steady states can be accurately described by the Schrödinger equation and that it is the most natural to connect changes between adjacent quantum steady states by the variational principle.