The Transfer of Molecular Energies by Collision: Recent Quantum Treatments

336 Pages·1979·7.369 MB·English

by  Franco Antonio Gianturco (auth.)

Editors
Prof. Dr. Gaston Berthier Prof. Dr. Hans H. Jaffe
Universite de Paris Department of Chemistry
Institut de Biologie University of Cincinnati
Physico-Chimique Cincinnati, Ohio 452211USA
Fondation Edmond de Rothschild
13, rue Pierre et Marie Curie Prof. Joshua Jortner
F-75005 Paris Institute of Chemistry
Tel-Aviv University
61390 Ramat-Aviv
Tel-Aviv/Israel
Prof. Dr. Michael J. S. Dewar
Department of Chemistry Prof. Dr. Werner Kutzelnigg
The University of Texas Lehrstuhl fOr Theoretische Chemie
Austin, Texas 78712/USA der Universitat Bochum
Postfach 102148
0-4630 Bochum 1
Prof. Dr. Hanns Fischer
Prof. Dr. Klaus Ruedenberg
Physikalisch-Chemisches Institut
Department of Chemistry
der Universitat ZOrich
Iowa State University
Ramistr.76
Ames, Iowa 50010/USA
CH-8001 ZOrich
Prof. Dr. Eolo Scrocco
Universita di Pisa
Prof. Kenichi Fukui
Istituto di Chi mica Fisica
Kyoto University
Via Risorgimento, 35
Dept. of Hydrocarbon Chemistry
1-56100 Pisa
Kyoto/Japan
Prof. Dr. Werner Zeil
Direktor des Instituts
Prof. Dr. Hermann Hartmann fOr Physikalische und
Akademie der Wissenschaften Theoretische Chemie
und der Literatur zu Mainz der Universitat TObingen
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0-6500 Mainz 0-7406 Mossingen bei TObingen
Lecture Notes in
Chemistry
Edited by G. Berthier, M. J. S. Dewar, H. Fischer
K. Fukui, H. Hartmann, H. H. Jaffe, J. Jortner
w. Kutzelnigg, K. Ruedenberg, E. Scrocco, W. Zeil
11
Franco A Gianturco
The Transfer of Molecular Energies
by Call ision:
Recent Quantum Treatments
Springer-Verlag
Berlin Heidelberg New York 1979
Author
Franco Antonio Gianturco
Institute of Physical Chemistry
The University of Bari
Via Amendola 173, Bari/ltaly
and
Quantum Chemistry Laboratory
(L.C.O.E.M.), C.NR, Pisa/ltaly
ISBN-13 :978-3-540-09701-3 e-ISBN-13 :978-3-642-93122-2
DOl: 10.1007/978-3-642-93122-2
This work is subject to copyright. All rights are reserved, whether the whole
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agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1979
Softcover reprint of the hardcover 1s t edition 1979
"The abnihilisation of the etym by the grisning of the grosning of
the grinder of the grunder of the first Lord of Hurtreford explodo
tonates through parsuralia with an ivanmorinthorrumble fragoromboa~
sity amidwhiches general uttermosts confussion are perceivable mole
tons skaping with mulicules"
J. Joyce, Finnegan's wake
ABSTRACT
Volume 11: F.A. GIANTURCO
THE TRANSFER Of MOLECULAR ENERGIES BY COLLISIOM: RECENT Q.UANTWI TREATMENTS
17 figures, 327 pages, 1979
The aim of this study is to provide a summary of the currently used theoretical
and computational techniques and a revie~ of the most recent results that have de
alt .mainly with the simplest type of energy transfer in a molecular collision
process, namely the conversion of the translational energy of a structureless atom
in its ground electronic state into the internal rotational and vibrational energy
of a diatomic molecule, also in its ground electronic state. This is probably one
of the most common events in chemical processes and while the experimental data
from molecular beam measurements have become increasingly more meaningful and
reliable, only recent years have seen any thorough, ab initio approach to the
various computational aspects of this problem. A brief resume of potential scatte
ring results is used as a preparation of the working formalism. The ab initio po
tential surface calculations are then examined and the various angular momentum
coupling representations, those that yield the multichannel scattering formalism
of vibrorotational inelastic cross sections in atom-molecule and molecule-molecule
encounters, are discussed. Some of the numerical techniques that have lent themsel
ves to the most recent applications are also reviewed together with the various
decoupling schemes that are necessary when dealing with more complex cases.
FinallY,correlations of the state-to-state deexcitation cross sections with bulk
measurements of relaxation times are also briefly presented. (498 References).
Contents: A resume of quantum mechanical potential scattering. - Potential energy
hypersurface calculations for simple systems. - Rotational and vibrational inela
sticity in molecular encounters. - Dimensionality reduction methods for rotovibr~
tional cross sections calculations.-Numerical methods for the coupled equations:
a survey.-Rotovibrational relaxation models in simple gases.
I
I
FOREWORD
These Lecture Notes are intended as an introduction to the theoretical
formulation and computational aspects of the molecular energy transfer processes
which take place in an increasingly sophisticated range of molecular scattering
experiments. They are directed to chemistry graduate students and emphasize the
quantum mechanical approach, with little or no attention to classical and semi
classical treatments or to formal presentations.
Several Sections of the first Chapters are based on lectures given at
the Graduate School of Physics of the University of Genoa a few years ago and I
thank the students for their sense of duty in following to the end all those no
tation-filled blackboards and transparencies.
The kind patience of my wife Carolyn in reading the whole manuscript
and improving its form is gratefully acknowledged.
Franco A. Gianturco
Bari, September 1978
CON TEN T S
FOREWORD
I NTRODUCTI ON Page
1. A RESUME OF QUANTUM MECHANICAL POTENTIAL SCATTERING
1.1. General formulation of the problem Page 5
1.2. Solutions of the radial equation " 10
1.3. The method of partial waves 13
1.4. Some properties of 61. The Born appro~imation 18
1.5. Properties of the S-matrix: bound states and resonances 23
1.6. Classical and semiclassical scattering,a set of defini-
tions 34
References 44
2. POTENTIAL ENERGY HYPERSURFACE CALCULATIONS FOR SIMPLE SYSTEMS
2.1. Kinematic considerations 45
2.2. General development of a priori method 52
2.3. Some approximate treatments 68
2.4. The electron gas model 73
2.5. A survey of recent applications 93
References 99
3. ROTATIONAL AND VIBRATIONAL INELASTICITY IN MOLECULAR ENCOUNTERS
3.1. Introduction 104
3.2. Quantum treatments of inelastic collisions 105
3.3. The rotational behaviour of molecules 112
3.4. Rotational excitation in atom-molecule collisions: the
Sff refenence· frame 117
3.5. Rotational excitation in atom-molecule collision: the
helicity representation 131
3.6. The vibro-rotational extension 141
3.7. Molecu1e- molecule inelastic encounters 147
3.8. Applications 161
References 174
4. DIMENSIONALITY REDUCTION METHODS FOR ROTOVIBRATIONAL CROSS SECTION
CALCULATIONS
4.1. Introduction 177
4.2. The CS approach 180
4.3. The sudden approximation methods 205
4.4. The effective potential treatment 226
4.5. The BSA treatments of purely vibrational inelasticity 234
4.6. The LD simplifications 243
4.7. The distorted wave approximations 248
4.8. General conclusions 257
VIII
References 260
5. NUMERICAL METHODS FOR THE COUPLED EQUATIONS: A SURVEY
5.1. Introduction 265
5.2. The De Voge1aere's method 270
5.3. The Numerov methods 276
5.4. The methods of piecewise analytic solutions 278
5.5. The solutions via integral equations 285
5.6. The coupled channel R-matrix methods 289
5.7. The variable phase methods 295
References 299
6. ROTOVIBRATIONAL RELAXATION MODELS IN SH1PLE GASES
6.1. Introductions 301
6.2. An outline of experiments 303
6.3. The rate equations 305
6.4. The H2 - He relaxations and other examples 317
References 326
I NTRODUCTI ON
The study of chemistry at a microscopic level, both from a theoretical and
an 'experimental' viewpoint, is mainly the study of the numerous processes that
before, during and after a chemical reaction (or in the absence of a chemical
reaction) control either the transformation of one type of molecular energy into
another type or its transfer into the 'envi ronment'. Withi n the domai n of \~hat
is commonly known as physical chemistry these studies have a very long history
indeed and the many theoretical models suggested through the years for each spe
cific case outline the progress we have made in understanding the workings of
chemi stry at a physi ca 1 1eve 1.
When the events under study take place in the absence of external radiative
fields, and when the systems are in their gaseous states, one is obviously faced
with a simpler situation, albeit one that is still general enough to give rise a
grea t variety of phenomena and processes. It is however evi dent that, at thi s le
vel, collisions between molecules wi 11 playa very important role in bringing
about such events and that our understanding of most of the phenomena will have
necessarily to go through a careful examination of the collisional process itself.
It has only been in recent years however that such examination has reached 'criti
cality' in the chemical community and that several aspects of the energy transfer
which takes place during molecular collisions have been closely scrutinized.
The scope of the present Lecture Notes is therefore to provide newcomers to
the subject with a compact presentation of the theoretical and computational tech
niques that have been imp1err~nted to yield realistic predictions for some of the
possible outcomes of the molecular encounters. Because of the wide variety of
energy transformations that can take place, a drastic reduction of the area exam
ined was obviously necessary. Thus, only the simplest type of energy transfer in
molecular collisions has been considered, namely that involving the conversion
of the translational energy of either a structure1ess atom or a small molecule
(both in their closed-shell, ground electronic states) into the internal rota
tional and vibrational energies of a diatomic molecule (also in its ground e1ec-
2
tronic state). This is probably still one of the most common events among those
that happen during a chemical reaction, but while the experimental data from mo
lecular beam measurements have become increasingly more· meaningful and reliable,
only recently have we been able to perform any thorough, ab initio approach to
the various computational aspects of this problem.
From a general viewpoint, the analysis of energy transfer in molecular col
lisions in the gas kinetic regime can be divided into two independent stages.
This is possible because the duration of a given collision between two molecules
is usually much shorter than the mean time between successive collisions. This
means that it is always possible to choose a time interval that is short compared
to the mean time between collisions but still long enough when compared with the
duration of a single collision. This allows for carrying on the t~atment within
a binary encounter modelling of the whole process and the system consisting of
two particular colliding molecules can then be considered as isolated from all
the remaining molecules over the chosen time interval.
Returning to the first stage, then, the state of the system can be described
by quantum mechanical equations that only take into account the degrees of free
dom of the two molecules under consideration. In this approximation the influence
of all the other molecules manifests itself only through initial conditions that
specify the states of the target or of both molecules before they collide. The
first dynamic stage of this analysis is then completed when the appropriate solu
tions of the quantum mechanical problem are obtained. Such solutions in fact
yield the relevant transition probabilities or state-to-state scattering cross
sections which in turn provide the microscopic characteristics of the fundamental
collision processes.
In the second stage of the analysis one can compute some macroscopic quanti
ties directly accessible to experiment, namely, the rates of the different and
possible relaxation processes. The aim of the theory is then to provide a micro
scopic interpretation of macroscopic parameters which describe a given reaction
kinetics. At this essentially statistical stage of the analysis one relies on
various types of kinetic equations which govern the temporal evolution (relaxa
tion) of the distribution of molecules among various states which arises as a
result of many successive binary collisions.
Such an analysis of the sequence of physical events leading to molecular
energy transfer in the gas phase can thus provide a logical chapter plan for the