### Abstract

The use of stochastic reduction approximations in studies of vibrational energy transfer in collinear atom plus diatomic molecule (harmonic oscillator) collisions is investigated. We begin by deriving exact coupled Liouville equations for the time evolution of vibrational and translational degrees of freedom in these collisions. Neglect of terms involving T-V correlation leads directly to the well known self-consistent approximations (SCA), and with further simplification to impulsive approximations. A self-consistent stochastic (SCS) method is then introduced, and it is found to provide an intermediate level of approximation between SCA and impulsive. For the simple hamiltonian considered, the SCS approximation leads to closed form expressions for the translational equations of motion. A special partitioning of translation and vibration first developed by Rosen is then introduced via two canonical transformations and is shown to lead to an improved description of translation-vibration correlations. Application of the SCS approximation within the framework of this partitioning leads to an energy conserving closed form trajectory which reduces to the refined impulse approximation of Matahan in the appropriate limit. Consideration of collision ensembles corresponding to both forward and reverse trajectories leads to a velocity symmetrized method analogous to that used in the ITFITS method. Numerical applications of the refined SCS method to He + H_{2} and (N_{2}) + N_{2} indicates that this method is more accurate than ITFITS (with only a small increase in complexity) in predicting low order moments of the energy transfer, at least at low collision energies (E_{T}/ñω <3-4). Transition probabilities for single quantum jump transitions are also accurately predicted at these energies (in comparing with exact quantum results) but probabilities of multiple quantum jump transitions, and of single quantum jump transitions at high collision energies are often less quantitative, primarily due to inaccuracies in the treatment of vibrational motion by master equation methods.

Original language | English |
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Pages (from-to) | 263-286 |

Number of pages | 24 |

Journal | Chemical Physics |

Volume | 24 |

Issue number | 2 |

DOIs | |

Publication status | Published - Sep 1 1977 |

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### ASJC Scopus subject areas

- Physical and Theoretical Chemistry
- Spectroscopy
- Atomic and Molecular Physics, and Optics

### Cite this

**On self-consistent and stochastic treatments of V-T energy transfer in collinear atom-diatom collisions.** / Schatz, George C.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - On self-consistent and stochastic treatments of V-T energy transfer in collinear atom-diatom collisions

AU - Schatz, George C

PY - 1977/9/1

Y1 - 1977/9/1

N2 - The use of stochastic reduction approximations in studies of vibrational energy transfer in collinear atom plus diatomic molecule (harmonic oscillator) collisions is investigated. We begin by deriving exact coupled Liouville equations for the time evolution of vibrational and translational degrees of freedom in these collisions. Neglect of terms involving T-V correlation leads directly to the well known self-consistent approximations (SCA), and with further simplification to impulsive approximations. A self-consistent stochastic (SCS) method is then introduced, and it is found to provide an intermediate level of approximation between SCA and impulsive. For the simple hamiltonian considered, the SCS approximation leads to closed form expressions for the translational equations of motion. A special partitioning of translation and vibration first developed by Rosen is then introduced via two canonical transformations and is shown to lead to an improved description of translation-vibration correlations. Application of the SCS approximation within the framework of this partitioning leads to an energy conserving closed form trajectory which reduces to the refined impulse approximation of Matahan in the appropriate limit. Consideration of collision ensembles corresponding to both forward and reverse trajectories leads to a velocity symmetrized method analogous to that used in the ITFITS method. Numerical applications of the refined SCS method to He + H2 and (N2) + N2 indicates that this method is more accurate than ITFITS (with only a small increase in complexity) in predicting low order moments of the energy transfer, at least at low collision energies (ET/ñω <3-4). Transition probabilities for single quantum jump transitions are also accurately predicted at these energies (in comparing with exact quantum results) but probabilities of multiple quantum jump transitions, and of single quantum jump transitions at high collision energies are often less quantitative, primarily due to inaccuracies in the treatment of vibrational motion by master equation methods.

AB - The use of stochastic reduction approximations in studies of vibrational energy transfer in collinear atom plus diatomic molecule (harmonic oscillator) collisions is investigated. We begin by deriving exact coupled Liouville equations for the time evolution of vibrational and translational degrees of freedom in these collisions. Neglect of terms involving T-V correlation leads directly to the well known self-consistent approximations (SCA), and with further simplification to impulsive approximations. A self-consistent stochastic (SCS) method is then introduced, and it is found to provide an intermediate level of approximation between SCA and impulsive. For the simple hamiltonian considered, the SCS approximation leads to closed form expressions for the translational equations of motion. A special partitioning of translation and vibration first developed by Rosen is then introduced via two canonical transformations and is shown to lead to an improved description of translation-vibration correlations. Application of the SCS approximation within the framework of this partitioning leads to an energy conserving closed form trajectory which reduces to the refined impulse approximation of Matahan in the appropriate limit. Consideration of collision ensembles corresponding to both forward and reverse trajectories leads to a velocity symmetrized method analogous to that used in the ITFITS method. Numerical applications of the refined SCS method to He + H2 and (N2) + N2 indicates that this method is more accurate than ITFITS (with only a small increase in complexity) in predicting low order moments of the energy transfer, at least at low collision energies (ET/ñω <3-4). Transition probabilities for single quantum jump transitions are also accurately predicted at these energies (in comparing with exact quantum results) but probabilities of multiple quantum jump transitions, and of single quantum jump transitions at high collision energies are often less quantitative, primarily due to inaccuracies in the treatment of vibrational motion by master equation methods.

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U2 - 10.1016/0301-0104(77)85233-6

DO - 10.1016/0301-0104(77)85233-6

M3 - Article

AN - SCOPUS:0038762171

VL - 24

SP - 263

EP - 286

JO - Chemical Physics

JF - Chemical Physics

SN - 0301-0104

IS - 2

ER -