Transition state theory: Variational formulation, dynamical corrections, and error estimates

Eric Vanden Eijnden, Fabio A. Tal

Research output: Contribution to journalArticle

Abstract

Transition state theory (TST) is revisited, as well as evolutions upon TST such as variational TST in which the TST dividing surface is optimized so as to minimize the rate of recrossing through this surface and methods which aim at computing dynamical corrections to the TST transition rate constant. The theory is discussed from an original viewpoint. It is shown how to compute exactly the mean frequency of transition between two predefined sets which either partition phase space (as in TST) or are taken to be well-separated metastable sets corresponding to long-lived conformation states (as necessary to obtain the actual transition rate constants between these states). Exact and approximate criterions for the optimal TST dividing surface with minimum recrossing rate are derived. Some issues about the definition and meaning of the free energy in the context of TST are also discussed. Finally precise error estimates for the numerical procedure to evaluate the transmission coefficient κS of the TST dividing surface are given, and it is shown that the relative error on κS scales as 1 κS when κS is small. This implies that dynamical corrections to the TST rate constant can be computed efficiently if and only if the TST dividing surface has a transmission coefficient κS which is not too small. In particular, the TST dividing surface must be optimized upon (for otherwise κS is generally very small), but this may not be sufficient to make the procedure numerically efficient (because the optimal dividing surface has maximum κS, but this coefficient may still be very small).

Original languageEnglish (US)
Article number184103
JournalJournal of Chemical Physics
Volume123
Issue number18
DOIs
StatePublished - 2005

Fingerprint

formulations
estimates
Rate constants
Free energy
Conformations
coefficients
partitions
free energy

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics

Cite this

Transition state theory : Variational formulation, dynamical corrections, and error estimates. / Vanden Eijnden, Eric; Tal, Fabio A.

In: Journal of Chemical Physics, Vol. 123, No. 18, 184103, 2005.

Research output: Contribution to journalArticle

@article{684c3685c64c4c38b98de13260a4b791,
title = "Transition state theory: Variational formulation, dynamical corrections, and error estimates",
abstract = "Transition state theory (TST) is revisited, as well as evolutions upon TST such as variational TST in which the TST dividing surface is optimized so as to minimize the rate of recrossing through this surface and methods which aim at computing dynamical corrections to the TST transition rate constant. The theory is discussed from an original viewpoint. It is shown how to compute exactly the mean frequency of transition between two predefined sets which either partition phase space (as in TST) or are taken to be well-separated metastable sets corresponding to long-lived conformation states (as necessary to obtain the actual transition rate constants between these states). Exact and approximate criterions for the optimal TST dividing surface with minimum recrossing rate are derived. Some issues about the definition and meaning of the free energy in the context of TST are also discussed. Finally precise error estimates for the numerical procedure to evaluate the transmission coefficient κS of the TST dividing surface are given, and it is shown that the relative error on κS scales as 1 κS when κS is small. This implies that dynamical corrections to the TST rate constant can be computed efficiently if and only if the TST dividing surface has a transmission coefficient κS which is not too small. In particular, the TST dividing surface must be optimized upon (for otherwise κS is generally very small), but this may not be sufficient to make the procedure numerically efficient (because the optimal dividing surface has maximum κS, but this coefficient may still be very small).",
author = "{Vanden Eijnden}, Eric and Tal, {Fabio A.}",
year = "2005",
doi = "10.1063/1.2102898",
language = "English (US)",
volume = "123",
journal = "Journal of Chemical Physics",
issn = "0021-9606",
publisher = "American Institute of Physics Publising LLC",
number = "18",

}

TY - JOUR

T1 - Transition state theory

T2 - Variational formulation, dynamical corrections, and error estimates

AU - Vanden Eijnden, Eric

AU - Tal, Fabio A.

PY - 2005

Y1 - 2005

N2 - Transition state theory (TST) is revisited, as well as evolutions upon TST such as variational TST in which the TST dividing surface is optimized so as to minimize the rate of recrossing through this surface and methods which aim at computing dynamical corrections to the TST transition rate constant. The theory is discussed from an original viewpoint. It is shown how to compute exactly the mean frequency of transition between two predefined sets which either partition phase space (as in TST) or are taken to be well-separated metastable sets corresponding to long-lived conformation states (as necessary to obtain the actual transition rate constants between these states). Exact and approximate criterions for the optimal TST dividing surface with minimum recrossing rate are derived. Some issues about the definition and meaning of the free energy in the context of TST are also discussed. Finally precise error estimates for the numerical procedure to evaluate the transmission coefficient κS of the TST dividing surface are given, and it is shown that the relative error on κS scales as 1 κS when κS is small. This implies that dynamical corrections to the TST rate constant can be computed efficiently if and only if the TST dividing surface has a transmission coefficient κS which is not too small. In particular, the TST dividing surface must be optimized upon (for otherwise κS is generally very small), but this may not be sufficient to make the procedure numerically efficient (because the optimal dividing surface has maximum κS, but this coefficient may still be very small).

AB - Transition state theory (TST) is revisited, as well as evolutions upon TST such as variational TST in which the TST dividing surface is optimized so as to minimize the rate of recrossing through this surface and methods which aim at computing dynamical corrections to the TST transition rate constant. The theory is discussed from an original viewpoint. It is shown how to compute exactly the mean frequency of transition between two predefined sets which either partition phase space (as in TST) or are taken to be well-separated metastable sets corresponding to long-lived conformation states (as necessary to obtain the actual transition rate constants between these states). Exact and approximate criterions for the optimal TST dividing surface with minimum recrossing rate are derived. Some issues about the definition and meaning of the free energy in the context of TST are also discussed. Finally precise error estimates for the numerical procedure to evaluate the transmission coefficient κS of the TST dividing surface are given, and it is shown that the relative error on κS scales as 1 κS when κS is small. This implies that dynamical corrections to the TST rate constant can be computed efficiently if and only if the TST dividing surface has a transmission coefficient κS which is not too small. In particular, the TST dividing surface must be optimized upon (for otherwise κS is generally very small), but this may not be sufficient to make the procedure numerically efficient (because the optimal dividing surface has maximum κS, but this coefficient may still be very small).

UR - http://www.scopus.com/inward/record.url?scp=27644558553&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=27644558553&partnerID=8YFLogxK

U2 - 10.1063/1.2102898

DO - 10.1063/1.2102898

M3 - Article

AN - SCOPUS:27644558553

VL - 123

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 18

M1 - 184103

ER -