Quantum instanton approximation for thermal rate constants of chemical reactions

Abstract

A quantum mechanical theory for chemical reaction rates is presented which is modeled after the ͓semiclassical ͑SC͔͒ instanton approximation. It incorporates the desirable aspects of the instanton picture, which involves only properties of the ͑SC approximation to the͒ Boltzmann operator, but corrects its quantitative deficiencies by replacing the SC approximation for the Boltzmann operator by the quantum Boltzmann operator, exp(Ϫ␤Ĥ ). Since a calculation of the quantum Boltzmann operator is feasible for quite complex molecular systems ͑by Monte Carlo path integral methods͒, having an accurate rate theory that involves only the Boltzmann operator could be quite useful. The application of this quantum instanton approximation to several one-and two-dimensional model problems illustrates its potential; e.g., it is able to describe thermal rate constants accurately (ϳ10-20% error͒ from high to low temperatures deep in the tunneling regime, and applies equally well to asymmetric and symmetric potentials.

Using the thermal Gaussian approximation for the Boltzmann operator in semiclassical initial value time correlation functions

“…Therefore, since matrix elements of the Boltzmann operator are given within the TGA by is typically bimodal; i.e., the TGA is intrinsically incapable of capturing the two saddle points (along the off-diagonal direction Äx ) of the coordinate Boltzmann matrix element for a barrier potential, which is characteristic of the low temperature regime, as shown in Fig. 1(a) in ref 46. This bimodal character, however, is described semiquantitively by the more rigorous imaginary time Van Vleck and the Herman-Kluk propagators in imaginary times 42 .…”

Efficient estimators for quantum instanton evaluation of the kinetic isotope effects: Application to the intramolecular hydrogen transfer in pentadiene

“…9 This approximation is similar to the semiclassical instanton theory, 10 but has an advantage that the Boltzmann operator is treated fully quantum mechanically rather than within the SC approximation. The quantum instanton theory thus incorporates all the tunneling, corner-cutting, and quantum-fluctuation effects correctly and is expected to overcome the quantitative deficiency of the SC instanton model.…”

“…Indeed, several test applications have shown the QI theory to give accurate quantum rates over a wide temperature range, from the "deep" tunneling regimes at low temperatures to the regime of overbarrier dynamics at high temperature. 9,[11][12][13][14] In Ref. 8 the QI was successfully applied to compute KIE's in a simple hydrogen exchange reaction.…”

Skirting the transition state, a new paradigm in reaction rate theory

“…Today, computer codes (8) based on this theory and important refinements to it, for example, to approximately include quantum mechanical tunneling, are in widespread use. Attempts to develop fully quantum versions of TST have also been pursued (9). The fundamental ideas of TST were incorporated into statistical theories of reaction rates 55 years ago by Marcus (10) into what is now termed RRKM theory (for Rice, Ramsperger, Kassel, and Marcus).…”