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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Presents theoretical, computational, and practical aspects of collision-induced phenomena with emphasis on the treatment of physical and chemical kinetics using quantum molecular dynamics
Quantum Molecular Dynamics provides a state-of-the-art overview of molecular collisions for energy-transfer and reactivity phenomena in gases. Grounded in the quantal theory of scattering and its semiclassical limits, this comprehensive volume covers key concepts and theory, computational approaches, and various applications for specific physical systems.
Detailed chapters describe elastic, inelastic and reactive collisions, that lead to energy transfer, electronic transitions, chemical reactions, and more. Starting from the electronic structure and atomic conformation of molecules, the text proceeds from introductory material to advanced modern treatments relevant to applications to new materials, the environment, biological phenomena, and energy and fuels production.
- Provides a thorough introduction to collision dynamics with realistic intermolecular forces
- Covers thermal rates and cross sections of molecular collisions phenomena
- Examines electronic excitation and relaxation phenomena mediated by molecular collisions
- Discusses many-atom scattering theory as an introduction to more advanced descriptions
- Presents the computational aspects required to calculate and compare cross-sections with experimental data
- Includes worked examples and applications to different physical systems
Quantum Molecular Dynamics is an important resource for researchers and advanced undergraduate and graduate students in physical, theoretical, and computational chemistry, chemical physics, materials science, as well as chemists, engineers, and biologists working in the energy and pharmaceutical industries and the environment.
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Veröffentlichungsjahr: 2025
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Preface
1 Collisional Phenomena, Cross Sections, and Rates
1.1 Electronic and Nuclear Motions in Collisional Phenomena
1.2 Collisional Cross Sections
1.3 Quantal Description of Collisions
1.4 Examples of Physical Systems and Phenomena
1.5 Transport, Energy Relaxation, and Reaction Rates in Gases
1.6 Concepts and Methods in the Quantal Modelling
References
2 Elastic Collisions
2.1 Elastic Collision Cross Sections
2.2 Integral Equation and Approximations
2.3 Partial-Wave Analysis
2.4 Numerical Methods for Scattering
2.5 Examples
References
3 Inelastic Collisions: Dynamics
3.1 Inelastic Collision Cross Sections
3.2 Coupled-Channel Equations
3.3 Matrix Form of Partial-Wave Equations
3.4 Collisions Involving Two Coupled Channels
3.5 Distorted-Waves Treatment
3.6 Optical Potential Models
References
4 Inelastic Collisions: Adiabatic Energy Transfer
4.1 Adiabatic Energy Transfer Cross Sections
4.2 Numerical Methods
4.3 Electronically Adiabatic Rotational Transitions
4.4 T–R Transfer Calculations and Comparisons with Experimental Values
4.5 Translational–Rotational–Vibrational (T–R–V) Transfer
References
5 Electronically Diabatic Collisions
5.1 Expansion in an Electronic Basis Set
5.2 Electronic Representations
5.3 Collisional Coupling of Molecular Electronic States
5.4 Semiclassical Description
5.5 Electronic Rearrangement for Several Interatomic Variables
References
6 Reactive Collisions
6.1 Arrangement Channels and Coordinate Transformations
6.2 Classical Reaction Dynamics
6.3 Quantal Theory of Adiabatic Reactions
6.4 Calculation of Adiabatic Reaction Cross Sections
6.5 Electronically Diabatic Reactions
6.6 First-Principles (Ab initio) Treatments of Reactive Collisions
6.7 Reduced Dimensionality, Optical-potential, and Machine-learning Treatments
References
7 The Quantum Scattering Operator and the Statistical Density Operator
7.1 Scattering Operators and Transition Rates
7.2 Partitioning the Space of State Wavefunctions
7.3 Many-Atom Scattering Operators
7.4 Density Operator Treatments
7.5 Density Operator Treatments for Reactive Collisions
References
Index
End User License Agreement
List of Tables
Chapter 6
Table 6.1 Scattering channel, arrangement, internal and coupling po...
Table 6.2 Conditions favored with large reaction probabilities in...
List of Illustrations
Chapter 1
Figure 1.1 Relationships mediated by quantum chemistry, molecular ...
Figure 1.2 Two stationary beams with velocities and colliding ...
Figure 1.3 Relations between LAB and CM velocities illustrated in ...
Figure 1.4 Angular distribution
I
(Θ) sin(Θ)
cross ...
Figure 1.5 Laboratory elastic differential cross sections vs LAB a...
Figure 1.6 Measured rotationally inelastic differential cross sect...
Figure 1.7 (A) Measured (dashed lines) and calculated (full lines)...
Figure 1.8 Calculated cross section for Li
+
+ CO
2
at collision...
Figure 1.9 Cross sections for Rg + LiF(001), Rg = Ne, Ar, Kr, Xe, ...
Figure 1.10 Measured and calculated cross sections for electron t...
Figure 1.11 Intensities of scattered H
+
beams vs energy loss ...
Figure 1.12 Excitation of into the excited state
a
3
Σ
induc...
Figure 1.13 Fragmentation ratio
Γ = σd(Br + Cl*)/[σ
...
Chapter 2
Figure 2.1 Effective radial potential
UL(R) = VL(R)/ε
...
Figure 2.2 Bundle of trajectories showing the deflection angle
χ(b
...
Figure 2.3 Reduced differential cross section times vs scatterin...
Figure 2.4 Experimentally measured differential cross sections
I(Θ
...
Figure 2.5 Differential cross sections for pairs of identical rare...
Figure 2.6 Zeroes of the radial wavefunctions and for no poten...
Figure 2.7 Potential energy function for Na–Hg vs radial distances...
Figure 2.8 Sketches of the phase shift
δ
l
vs orbital quantum ...
Figure 2.9 Sketches of phase shifts
δl, l = 0, 1, 2,
...
Chapter 3
Figure 3.1 (a) Collisional coordinates
(
R
,
r
,
θ
)
for atoms A,...
Chapter 4
Figure 4.1 An atom–atom collision trajectory starting at a time
t
1
Figure 4.2 (a) The
j = 0 → j′ = 2
...
Figure 4.3 Measured (dots) and calculated (lines) rotationally ine...
Figure 4.4 Leading potential energy expansion coefficients vs
R
...
Figure 4.5 Calculated and measured state-to-state differential cro...
Figure 4.6 Vibrational relaxation integral cross sections for He +...
Figure 4.7 Comparison of state-to-state ICSs for Ar + H
2
O from the...
Chapter 5
Figure 5.1 Orbital energy correlation diagram from united atoms to...
Figure 5.2 Calculated electronic energies for vs internuclear di...
Figure 5.3 Total (electronic plus nuclear repulsion) energies for
Figure 5.4 Energy level correlation diagram for a heteronuclear di...
Figure 5.5 Calculated electronic energies vs internuclear distance...
Figure 5.6 (a) Adiabatic (full line), diabatic (dashed line), and ...
Figure 5.7 NaH potential energies showing multiple pairs of avoide...
Figure 5.8 Total cross section for electron capture
σcap,1s = ∑nlσ
...
Figure 5.9 Low-energy electron exchange in H
+
+ H(1s)
→
...
Figure 5.10 Potential energy functions B
2
Σ and C
2
Σ and their co...
Figure 5.11 Oscillation of cross sections with orbital quantum nu...
Figure 5.12 Plot of
Θ sin(Θ) dσ21/dΩ
...
Figure 5.13 (a) Reduced elastic differential cross section (in de...
Figure 5.14 Illustration of three cases of conical intersections ...
Figure 5.15 (a) Adiabatic potential energy surfaces for collinear...
Chapter 6
Figure 6.1 (a) Center-of-mass, relative, and internal position vec...
Figure 6.2 Isocontours vs reaction coordinates for channels 1 and ...
Figure 6.3 PES isocontours for the three arrangement channels show...
Figure 6.4 Potential energy surfaces for A + BC → AB + C with vary...
Figure 6.5 Integral reaction cross section vs collision energy for...
Figure 6.6 Integral reaction cross section vs final rotational qua...
Figure 6.7 State-to-state differential reaction cross section
σ11
...
Figure 6.8 Comparisons of experimental (solid points) and theoreti...
Figure 6.9 (a) Sketch of changing triatomic energy levels going fr...
Figure 6.10 Simple neural network with two nodes
G
1
,
G
2
in an inp...
Chapter 7
Figure 7.1 Diabatic potential energy matrix elements
H11(R), H22(R
...
Figure 7.2 Ionic and neutral state populations vs dissociation tim...
Figure 7.3 Final distribution of phase space trajectories showing ...
Guide
Cover
Table of Contents
Title Page
Copyright
Preface
Begin Reading
Index
End User License Agreement
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Quantum Molecular Dynamics
Energy Transfer and Reactivity
David A. Micha
University of Florida, Gainesville, FL, USA
Copyright © 2026 by John Wiley & Sons, Inc. All rights reserved, including rights for text and data mining and training of artificial intelligence technologies or similar technologies.
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Library of Congress Cataloging-in-Publication Data
Names: A. Micha, David author | John Wiley & Sons publisher
Title: Quantum molecular dynamics : energy transfer and reactivity / David A. Micha, University of Florida, Gainesville, FL, USA.
Description: Hoboken, New Jersey : Wiley, [2026] | Includes index.
Identifiers: LCCN 2025022988 (print) | LCCN 2025022989 (ebook) | ISBN 9780470383735 hardback | ISBN 9781119319245 adobe pdf | ISBN 9781119319269 epub
Subjects: LCSH: Molecular dynamics | Quantum chemistry | Energy transfer
Classification: LCC QD461 .M5845 2026 (print) | LCC QD461 (ebook)
LC record available at https://lccn.loc.gov/2025022988
LC ebook record available at https://lccn.loc.gov/2025022989
Cover Design: WileyCover Image: © Photobank.kiev.ua/Shutterstock
Preface
The interaction of colliding molecules leads to energy transfer and to reactivity. A full treatment of these phenomena, as they relate to experimental measurements or to the kinetics of molecular gases, requires quantum mechanics and statistical mechanics. Those phenomena are particularly complex when they involve electronic excitation or electron transfer, or when they occur in large many-atom systems. The understanding extracted from the interplay of theory and experiment impacts many areas relevant to engineering applications and to the environment. Related concepts and methods provide results for thermodynamic properties including chemical equilibrium constants, and kinetic properties including transport coefficients, energy relaxation times, and reaction rates.
The material covered in this book originated from a sequence of courses on chemical physics and quantum theory taught at the University of Florida for graduate students and undergraduate researchers in the Chemistry, Physics, and Engineering departments. The subjects were introduced assuming only basic knowledge of molecular interactions and dynamics, familiarity with quantum and statistical mechanics, and the mathematics of partial differential equations, special functions, and linear algebra in functional spaces. Starting from fundamental concepts and methods, the book proceeds to develop the needed quantum theory and related computational methods to allow for the calculation of physical properties. These are mainly rates and cross sections of energy transfer and reaction phenomena, as needed to interpret experimental measurements or to calculate kinetic properties.
Given all the available literature on molecular collisional phenomena and applications, it is relevant to point out why there is a need for yet another presentation of these subjects. It helps to introduce subjects in some detail, starting from fundamental concepts and proceeding to advanced methods, as is done in every chapter of this book. Also, many new developments have occurred in recent years on new computational methods prompted by the derivation of new software algorithms and by the availability of more powerful computer hardware. This has allowed for the calculation of new results, involving complex phenomena, and for larger molecular systems. Recent developments in information sciences and artificial intelligence provide large amounts of organized relevant data and ways to retrieve it, and their basic concepts are included in this book to provide tools for the treatment of desired data. Some of these new methods and results are integrated in what follows, after covering the traditional subjects on molecular collisions and gaseous kinetics. There is an enormous literature on the subjects of this book, including several thousand papers and reviews. References in the chapters are a small selection of all the relevant publications, chosen to help with the chapter's materials while avoiding overwhelming complexities. This book's author apologizes in advance to the contributors of the many relevant publications that could not be referenced.
The presentation in each chapter begins at a level accessible to graduate and advanced undergraduate students, and proceeds to present recent theoretical and computational developments. Treatments of elastic, inelastic, and reaction phenomena are first developed in simple cases, containing physical models and calculated properties comparable to that of measured values. The theory and mathematics are developed as needed to obtain numerical results. In cases involving a many-atom system or electronic excitation, the computational work may become very demanding and treatments must be carefully chosen to achieve results. Practical treatments of the molecular dynamics combine quantal and classical mechanics, and one is developed in several chapters with new and more details than presently available in the literature.
Chapter 1 provides a qualitative introduction to collisional phenomena, cross sections, and rates, with examples for energy transfer and reactivity in physical systems. Chapter 2 gives a detailed treatment of classical and quantum dynamics of elastic collisions, including differential and integral equation formulations, partial-wave analysis, and numerical methods for classical, quantal, and semiclassical dynamics, with examples for atom–atom and electron–atom scattering. Chapter 3 on inelastic (energy transfer) collision dynamics displays coupled-channel equations, matrix forms of partial-wave expansions, examples of two coupled states and resonance phenomena, distorted-wave, and optical model treatments. Chapter 4 describes electronically adiabatic energy transfer, with special attention to numerical methods, wavefunction expansions in basis sets of quantum states for atom–polyatomic and molecule–molecule collisions, and translational–rotational–vibrational energy transfer models and cross-section results.
Chapter 5 deals with the challenging subject of diabatic collisions, where atomic motions and electronic transitions are dynamically coupled, and treatments require choices of electronic representations, multiple electronic potential energy surfaces, and their couplings. This chapter covers conical intersections, seams, and geometric phase effects. Semiclassical descriptions are developed with new details, based on a general eikonal formulation of wavefunctions. Reactive collisions are discussed in Chapter 6, with a modern formulation based on coupled arrangement channels for both adiabatic and nonadiabatic reactions, including time-dependent formulations and some recent developments related to machine-learning procedures. It contains many examples of calculated cross sections. Finally, Chapter 7 is dedicated to general scattering operators and density operators. Partitioning of state-space is developed to construct effective Hamiltonians, multiple scattering is treated in a many-atom system, and coupled arrangement-channel operator equations for reactions are given here. Density operator developments are covered to include statistical averages from models and from experimental setups, and quantum-classical equations and related results are presented from eikonal and Wigner-transform treatments.
This book follows the author's previous work, Molecular Interactions, and the author reiterates the acknowledgment made in that book's preface. The author thanks his quantum theory science teacher at the Institute of Physics Balseiro, Bariloche (Argentina), Dr. Jose Balseiro, and Dr. Per-Olov Lowdin for his mentoring in quantum chemistry at the Quantum Chemistry Group, Uppsala University (Sweden). Much was learned from colleagues and researchers at the University of Wisconsin (Madison), Drs. Richard Bernstein and Joseph Hirschfelder, and at the University of California, San Diego, Dr. Keith A. Brueckner. He is grateful for the collaborations and hospitality at institutions visited during sabbaticals: Gothenburg University in Sweden, Harvard University, the Max Planck Institutes for Stroemungsforschung and for Astrophysics in Germany, Imperial College in London (England), the Institute of Theoretical Physics at the University of California Santa Barbara, JILA at the University of Colorado in Boulder, the Weizmann Institute in Rehovot (Israel), Florida State University, the École Normale Supérieure in Paris (France), and the Institute for Mathematics and its Applications at the University of Minnesota in Minneapolis. The author is especially grateful to the many graduate and undergraduate students who conducted research under his direction, as well as to postdoctoral associates and visiting scientists at the University of Florida. Due to their numbers, they are not listed here; their names can be found at the website https://people.clas.ufl.edu/Micha/.
The author is greatly appreciative of the organizations that provided essential financial support for his research and collaborations: the Swedish International Development Agency; the Alfred P. Sloan Foundation; the National Science Foundation of the USA, for many years of Principal Investigator research support, and funding of US-Latin American workshops; NASA; the Alexander von Humboldt Foundation Senior Scientists Awards twice; and the Dreyfus Foundation.
Thanks are also due to reviewers at the early stages when this book and the previous one were being planned, particularly Bernard Kirtman, George Schatz, and Victor Batista. Several of the author's research colleagues have been kind enough to read and comment on some of the book's chapters, in particular John Stanton, Adrian Roitberg, and Dmitri Kilin. Apologies are extended to the many colleagues and research visitors not mentioned due to text space limits!
The author appreciates the patience and support of the editors at John Wiley & Sons, Inc., over the years it took to have this book completed. Special thanks go to project senior editor Michael Leventhal for his guidance during production and publication. A special thanks is also due to the author's wife, Rebecca A. Micha, for her help in keeping the English grammar right in many paragraphs, and for her unwavering support.
October 2024
David A. Micha
Gainesville, Florida
1Collisional Phenomena, Cross Sections, and Rates
CONTENTS
1.1 Electronic and Nuclear Motions in Collisional Phenomena
1.2 Collisional Cross Sections
1.2.1 Definition of a Cross Section
1.2.2 Conservation Laws
1.2.3 Collisions in the Center of Mass Frame
1.2.4 Classification of Collision Processes
1.3 Quantal Description of Collisions
1.3.1 Time-Dependent Quantal States
1.3.2 Time-Independent Steady States
1.4 Examples of Physical Systems and Phenomena
1.4.1 Overview of Phenomena
1.4.2 Electronically Adiabatic Heavy-Particle Collisions
1.4.3 Electronically Diabatic Heavy-Particle Collisions
1.4.4 Electron and Photon Scattering by Molecules
1.5 Transport, Energy Relaxation, and Reaction Rates in Gases
1.6 Concepts and Methods in the Quantal Modelling
References
1.1 Electronic and Nuclear Motions in Collisional Phenomena
Matter consisting of particles such as atoms or molecules in a gas, where interparticle average distances are large compared to the size of the constituent particles, can be described in terms of pair interactions during collisions, insofar the probability of finding a third particle nearby is small and its effect on the interaction pair is negligible.
Collisions in a gas can be described in terms of the properties of constituent particles, their positions and velocities, and their interaction potential energies [1–4]. This can sometimes be done with classical mechanics or more generally within quantum mechanics. For colliding particles A and B, it is convenient to describe their relative motion in terms of their interparticle distance and their relative velocities. The rate of encounters (or number of collisions per unit time) is proportional to the relative flux (or number of collisions per unit time and unit area traversed by the relative-motion trajectories), with the proportionality factor equal to a collisional cross section, a measure of the diameters of the particles dependent on their relative velocity.
If the gas has been in contact with a medium of a given temperature, it will eventually reach thermal equilibrium through energy exchange during collisions, and rate processes can be assumed to occur near thermal equilibrium. This remains stable insofar individual pair interactions occur in a diluted system with low particle densities.
There is extensive literature on concepts and applications of molecular collision phenomena and their relation to experimental methods such as molecular beam scattering and spectroscopy. At the introductory level, some (among many others) relevant books are in references [5–10]. At an intermediate level, a classic text is [11] among others [12–17], dealing with quantitative treatments. Advanced treatments with extensive use of the quantum formalism of scattering theory are found in [18–24]. Applications involving models, and calculations of cross sections of interest in the interpretation of experimental results obtained with molecular beam and spectroscopic techniques, are available in [25–35], and the use of cross sections as inputs in molecular reaction rates, gaseous transport, and kinetics, can be found in [36–45]. Mathematical and computational methods developed for the calculation of cross sections can be found in [46–51]. Applications to surface phenomena are also covered in several books, among them [52–54].
The interaction of molecules with photons leading to photoinduced phenomena such as dissociation and ionization and related theory can be found in [15, 55–57], and scattering of electrons by atoms and molecules are found in [11, 58, 59]. In addition, many recent relevant chapters about molecular collisions in gases and at surfaces and about molecular interactions with photons are found in volumes of several edited series on advances, such as Advances in Chemical Physics; Advances in Quantum Chemistry; Advances in Atomic, Molecular and Optical Physics; and Annual Review of Physical Chemistry.
Given all this literature on molecular collisional phenomena and applications, it is relevant to point out why there is a need for yet another presentation of these subjects. It helps to introduce subjects in some detail, starting from fundamental concepts and proceeding to advanced methods, as done in what follows here in every chapter. Also, many new developments have occurred in recent years in new computational methods prompted by the derivation of new algorithms and by the availability of more powerful computer hardware and software. Many new results have been generated involving complex phenomena and for larger molecular systems. Recent developments in information sciences and artificial intelligence provide large amounts of organized relevant data and ways to retrieve it. The language of concepts and methods in what follows provides tools to access desired data. They provide new insights and data useful in many applications. Some of these new methods and results are integrated into what follows after covering the traditional subjects on molecular collisions and gaseous kinetics. The presentation in each chapter begins at a level accessible to undergraduate students with knowledge of differential equations and special functions, who are familiar with introductions to quantum mechanics, and proceeds to present recent developments.
Insights on the quantum dynamics of molecular interactions involving a many-atom system can best be derived from the interplay of accurate experimental measurements, such as those obtained from crossed molecular beams or from time-resolved spectroscopy, and detailed theoretical treatments from quantum mechanics and statistics. Theory provides an interpretation of measurements and resulting data on molecular interactions and dynamics, while experimental measurements provide checks on the accuracy of models and calculations.
Figure 1.1 shows relationships mediated by quantum chemistry, molecular dynamics, and statistical mechanics, leading from electrons, nuclei, and photons present in molecules and electromagnetic fields, to physical properties. Collisional cross sections, transport coefficients, and reaction rates follow from potential energy functions by means of molecular dynamics and statistical mechanics. Obtained from reference [4].
Figure 1.1 Relationships mediated by quantum chemistry, molecular dynamics, and statistical mechanics, leading from electrons, nuclei, and photons present in molecules and electromagnetic fields, to physical properties. Collisional cross sections, transport coefficients, and reaction rates follow from potential energy functions by means of molecular dynamics and statistical mechanics.
Obtained from reference [4] / John Wiley & Sons.
1.2 Collisional Cross Sections
1.2.1 Definition of a Cross Section
Let us consider a collision process
where the collision species A, B, C, and D may be electrons, atoms, molecules, ions, photons, or even a surface. The index p is a collection of quantum numbers specifying the internal (electronic and rovibrational) state of A, and similarly for the others. The pair (p, q) and the initial relative motion momentum define the reactant-channel state , and similarly β is used for products C and D and final momentum .
For two stationary beams with velocities and colliding in a laboratory (LAB) frame, as shown in Figure 1.2, with an incoming flux of A relative to B, equal to the number of A particles moving toward B per unit area and unit time, with nA (or nB) the number of particles A (or B) per unit volume, the increment of the α → β reaction rate (pairs/unit time) may be expressed as
Here = is the relative velocity of the particles, τ is the reaction volume, and dΩC is an increment of solid angle subtended by the detector of emerging product species C.
Figure 1.2 Two stationary beams with velocities and colliding in a laboratory (LAB) frame, leading to the formation of species C and D. Here τ is the reaction volume and dΩC is an increment of solid angle subtended by the detector of particle C.
Hence nB τ is the number of particles B in the reaction volume and nA is the flux of particles A relative to each B. The function is a differential cross section in the laboratory frame, with units of area. The integral cross section is defined by
1.2.2 Conservation Laws
As long as each pair collision may be considered an isolated event, we must have conservation of the total mass, momentum, angular momentum, and energy of the system. Indicating the final values with primed symbols, we have conservation of
(a) Mass,
M
tot
=
M
A
+
M
B
=
M
C
+
M
D
=
M
′
tot
(b) Momentum,
(c) Angular Momentum
(d) Energy
where MA is the total mass of A, is the momentum of the center of mass (CM) of A, is the orbital angular momentum of the CM of A located at position , and is the internal angular momentum of species A, which can be nuclear-rotational, electronic orbital or spin, or a combination of them.
Also, is the kinetic energy of A, is the internal excitation energy of the state A(p), is the binding energy (−IA for ionization or −DA for dissociation) of A, of the “elementary particles” making up A, B, C, and D. An example is provided by the reaction F + H2 → H + FH, where the “elementary particles” are F, H, and H, and the binding energies are the dissociation energies of H2 and FH with negative signs.
1.2.3 Collisions in the Center of Mass Frame
The CM frame is defined by introducing CM and relative positions and as
and fixing the orientation of the frame axes with respect to the LAB frame. The positions of the CM of A and B are and . Letting the frame origin move with the CM velocity that satisfies (differentiating the above equations with respect to time)
it follows that
where, using the relative velocity and reduced mass M = MAMB/Mtot, one finds , Erel = M/2, , and , which is the relative angular momentum of A with respect to B in terms of the relative momentum .
Hence in the CM frame with its origin at one finds that , and conservation laws apply with the modifications:
Relations between LAB and CM velocities are illustrated in the so-called Newton velocity diagram of Figure 1.2, where O and G are the origins of the LAB and CM frames, and
where the end points of velocities for particles A and B are labelled by A and B in Figure 1.3, and particle velocities in the CM frame are and , and similarly for products C and D, written with primed symbols . Detection of species C follows from its flux through the element of surface in the CM frame or alternatively through in the LAB frame.
Figure 1.3 Relations between LAB and CM velocities illustrated in the so-called Newton velocity diagram. Here G is the location of the center-of-mass of the (A, B) and (C, D) pairs and detection of species C follows from its flux through the element of surface in the LAB frame or alternatively through .
Reaction cross sections and rates for α → β collisions of two molecules A and B can be described in both LAB or CM frames, in terms of their incoming molecular-pair flux, or J0[60]. In the LAB frame, the rate of pair-transitions per unit time is
This can be converted to a CM frame using that and . It leads to with the relation between fluxes given by
In an experimental set up where the distribution of initial velocities is thermal at a temperature T, given by the Maxwell–Boltzmann distribution
the incoming volume densities are nI = NIfI, and J0 contains only . Integration over VCM provides a cross-sectional increment for events where the product species C emerges within solid angles in dΩC.
Corresponding to dΩC, the solid angle in CM is dΩ, and from invariance of the cross sections under → transformations, it follows that for a single collision event,
which defines the CM differential cross section Iβα(Ω, ). A similar relation for a thermal distribution gives in CM .
The relation between solid angle increments,
obtained from the geometry of collision kinematics in velocity space, gives the relation between differential cross sections.
1.2.4 Classification of Collision Processes
For the generic reaction A(p) + B(q) → C(r) + D(s) and fixed reactant conditions (i.e. for fixed Etot), a product channel is called open if E′rel > 0 and closed if E′rel < 0. For open channels, we may have
(a) Elastic collisions
(b) Inelastic collisions
(c) Reactive collisions
with
the reaction energy.
Elastic collisions are coherent if the states of A and B do not change and incoherent otherwise. Non-elastic collisions are exoergic if > Erel and endoergic otherwise. Choosing and to indicate initial and final directions, respectively, collisions are forward if and backward if .
An additional classification item can be introduced when an initial collision partner is a photon φ absorbed by a compound AB and leading to its dissociation, as in
(d) Photodissociation
Also, one of the initial collisional partners can be a solid surface, with the particle–surface collision being elastic, inelastic, reactive, or dissociative.
1.3 Quantal Description of Collisions
1.3.1 Time-Dependent Quantal States
We consider again a general case A(p) + B(q) → C(r) + D(s). A quantal description is based on state wavefunctions Ψ(t) that are solutions, evolving with time, of the Schrodinger equation for given initial and boundary conditions. The equation contains a Hamiltonian operator that can be constructed by adding the kinetic energies of all the atomic ion-cores and electrons, and their interaction energies, which can be Coulombic and other long- and short-range interparticle potential energies. The Hamiltonian may also contain spin–orbit coupling energies and time-dependent energies resulting from externally applied electromagnetic fields.
Starting from the time-dependent Schrodinger equation [18] and its boundary conditions (or normalization) when the collision partners A and B are initially at time tin far apart,
Here ℏ = h/(2π) is the reduced Planck constant, and an initial quantal state Ψ(in) = Ψ(tin) is given by the product of a pair of non-overlapping wavepackets Ψ(w), = A, B, with velocities and and initial many-atom states for each reactant species. The bracket notation indicates integration over all the wavefunction variables. The total Hamiltonian operator can be written as a sum of terms for relative and internal motions plus their coupling energy operator and also plus an external time-dependent energy from applied fields or due to an environment,
where is the operator in the CM frame for the relative motion kinetic energy, + is the Hamiltonian for the internal motions of isolated A and B in a set of internal states {φn}, and is an interaction potential energy between relative and internal motions. Similar expressions apply for C + D. The coupling goes to zero when reactants or products are far apart. The wavefunction evolves in time as a wavepacket Ψ(t) with asymptotic boundary conditions after collision at a final time tfn given by
where Ψ(0) is the evolved initial wavepacket and Ψ(sc) is a scattered component. After a collision, the probability at times t ≥ tfn for transition into a final state Ψ(fn) = Ψ(C)Ψ(D) is obtained from P(fn)(t) = |〈Ψ(fn) | Ψ(t)〉|2 and the transition rate is dP(fn)/dt.
A usual experimental setup for measurement of cross sections involves scattering of a stationary beam of molecules A and B in a pair-state with given translational velocities and , respectively, and relative velocity = ending in a state β for the C + D pair. In this case, it is relevant to introduce an initial flux , the number of incoming colliding pairs per unit time and unit traversed area given as a product of velocity times a particle density ρα per unit volume, and to calculate a steady-state flux for transitions between initial and final states [12, 15, 20, 61]. This can be done for the time-independent Hamiltonian of an isolated collision event, by first calculating steady states, with as solutions of the equation , for total energies , and relative momentum , with scattering boundary conditions, and by expressing the wavepacket Ψα(t) as a superposition
shown as a symbolic addition and integration over states ν, with Aα(ν) selected to form the incoming scattering wavepacket. The connection between time-dependent and time-independent treatments of collisions is described in detail in several books [12, 20] and reviews, and in applications with emphasis on molecular systems [15, 61].
1.3.2 Time-Independent Steady States
The total Hamiltonian operator for a collision in an isolated space is
For an isolated collision in a collision region without any external forces, the Hamiltonian is time-independent and the time variable can be separated in the equation of motion of the wavefunction, to work within a time-independent description, where
with the state index for internal states n = (p, q) for A and B (or pair I), and R is the relative distance between the CMs of the final species C and D. The total energy is conserved during a collision and given for reactants by for the products. The stationary wave is an eigenfunction of the Hamiltonian as shown, and the previous normalization boundary condition for wavepackets must now be replaced by a scattering asymptotic condition at large distances R from the collision region. In regions where we can write the total wavefunction in the CM frame as a product of the eigenfunctions of and of , which we assume are known. This allows us to separate the free motion and internal motions states in the total wavefunction at large distances.
At the large distances where the reaction products are detected, the wavefunction is given by a sum of the incoming unperturbed wave plus a scattering wave generated by interactions in the collision region, leading to products C and D in internal states β. Consistent with this, the flux of particle C with respect to D detected in state is .
The flux operator along the direction can be obtained from the momentum operator , where ∇R is a gradient along the pair relative position , and from the density operator as [18]
with an expectation value Writing the density operator as in terms of the Dirac delta function for a position density and with a ket-bra notation for internal states, it is possible to identify the state-to-state scattered flux . The differential cross sections Iβα(Ω, ) = dσβα/dΩ for C, emerging through a detection area dA = R2dΩ within solid angle dΩ in the direction given by a unit vector , can be obtained from the incoming and outgoing flux terms as
With the denominator given by the incoming unperturbed flux. This cross section can be obtained from the asymptotic wavefunction components in Ψα(sc).
1.4 Examples of Physical Systems and Phenomena
1.4.1 Overview of Phenomena
An overview of phenomena for frequently encountered physical systems can be organized as
(A) Electronically adiabatic heavy-particle collisions.
(B) Electronically diabatic heavy-particle collisions.
(C) Electron and photon scattering by molecules.
Their description can be done in terms of stationary eigenfunctions ΨE for fixed energy, if the Hamiltonian of the system is time-independent, or using time-dependent wavefunctions Ψ(t) if there are external time-dependent fields or if it is convenient to construct initial localized wavepackets that evolve in time. These wavefunctions depend on all the electronic variables (positions and spins) X and all the nuclear (or ion core) positions .
An electronically adiabatic collision involves a single electronic state and its related potential energy dependent on chosen nuclear positions; electronic motions are separated from nuclear motions by the Born–Oppenheimer decoupling approximation [4, 18], which fixes nuclear positions, and the collision involves only atomic CM degrees of freedom and motion on a single potential energy surface (or PES) function of atomic positions. Electronic rearrangement due to excitation or electron transfer instead involves several electronic states and the related PESs as well as the couplings of electronic states mediated by nuclear motions, in diabatic (or non-adiabatic) phenomena. The following examples show involved degrees of freedom and Hamiltonians needed for quantal descriptions.
The scattering of electrons or of photons by a single molecule requires introduction of a Hamiltonian that includes the scattering particle (electron or photon), molecule, and their interaction energies to begin with. Electronic excitations however may lead to dissociation into two or more interacting fragments, which can be treated as in the final scattering stage in a collision of heavy particles. These subjects are treated in detail in several published books on electron scattering [11, 58, 59] and photon scattering [55, 56, 62–64] and in review chapters [65, 66]. They are briefly described in what follows but are not otherwise developed in this book.
The following examples are a selection of results briefly illustrating calculated and measured cross sections. The subjects are expanded in the following chapters.
1.4.2 Electronically Adiabatic Heavy-Particle Collisions
(a) Two structureless atoms (e.g. Na + Hg, He + He)
Indicating with the relative position vector between their CMs, the Hamiltonian contains
written in terms of the gradient operator ∇R, where is a functional form of the interaction potential energy, with a long-range attraction, short-range strong repulsion and a well in between, and ε for the energy of the well depth at position Rm[4]. The wavefunction is for relative momentum and can be conveniently expanded in spherical harmonic functions of the relative position vector angles.
Differential and integral cross sections can be calculated from accurate potential energy functions obtained from electronic structure calculations or by parametrization. Elastic scattering differential cross sections in the CM frame with relative collision energy E = M/2 and into solid angles Ω = (ϑ, ϕ) are given by dσ/dΩ = I(ϑ, ) due to axial symmetry, as shown in Figure 1.4 for Na + Hg. They display broad oscillations for varying angles called rainbow fixtures and fast oscillations resulting from quantal wave superposition.
Cross sectional features are affected by the size and masses of colliding partners, as shown in Figure 1.5, from LAB measurements for pairs of noble gas atoms from He to Xe. In addition to rainbow and quantal oscillations, this figure shows large oscillations due to quantal identical-particle exchange symmetry that are more prominent in the lighter atom pairs and disappear for the heavier pairs.
Figure 1.4 Angular distribution I(Θ) sin(Θ) cross section vs scattering angle Θ for Na + Hg for changing CM collisional energies E. The broad oscillations are rainbow fixtures, and the fast oscillations result from quantal wave superposition.
Adapted from [67].
(b) Nonreactive atom–diatomic pair (e.g. He + N
2
)
Let be the position vector of the atom relative to the CM of the diatomic, and be the relative position vector of the atoms in the diatomic with a reduced mass m of internal motion and an internal potential energy . Then
Figure 1.5 Laboratory elastic differential cross sections vs LAB angle Θ for identical pairs of noble gas atoms. In addition to rainbow and quantal oscillations, this figure shows large oscillations due to quantal identical-particle exchange symmetry that are more prominent in the lighter atom pairs and disappear for the heavier ones.
[68] / AIP Publishing.
where (r) can be an expansion in the displacement r − re from the equilibrium diatomic distance, or a Morse potential, and the coupling potential can be expanded in terms of Legendre polynomials Pl(cosγ) as [4]
where . The wavefunction is of form and can be expanded in spherical harmonic functions of the angles of and .
Cross sections can be obtained vs scattering angles and collisional energies, for vibrational–rotational transitions (, j) → (, j′) between diatomic states of quantum vibrational and rotational numbers (, j) and energy levels . In a CM reference frame, with relative collision energy E = M/2 and relative momentum change into solid angles Ω = (ϑ, ϕ), cross sections are
Figure 1.6 Measured rotationally inelastic differential cross section vs scattering angle for He–N2 collisions in the CM frame, for the collision-induced rotational transitions j = 0 → 2 and j = 1 → 3 in N2.
Source: Adapted from [69].
Comparisons of measured and calculated cross sections vs scattering angles for selected collision energies provide information on the shapes of intermolecular potentials. Results for He + N2 are shown in Figure 1.6.
(c) Reactive atom–diatomic pair (e.g. H + H
2
)
We can use the same coordinates as in (b), but there are three equivalent sets for the three possible rearrangements A + BC, B + CA, and C + AB, defining the arrangements j = 1, 2, and 3. The total potential energy E is conveniently expressed as a function of the distances rAB, rBC, and rCA of each of the three atom–atom pairs, or alternatively in terms of Jacobi vectors . Isocontours of the potential energy (such as for H + H2) in a linear conformation show valleys for reactants and products and an activation barrier between them [4]. The total Hamiltonian can be written in three equivalent ways for the three arrangements, as
The coupling in arrangement j is a function of the Jacobi vectors . The wavefunction is a superposition of coupled reaction-channel states that are functions of the channel variables, so that . Calculations can be done expanding the arrangement components in basis sets of diatomic vibrational–rotational functions [12, 13, 15], or as a sequence of coupled atom–diatomic scattering amplitudes using transition operators [70].
Cross sections can be calculated and measured for transitions between specific arrangement channels 1 and j, for given collisional energies E in CM, and as functions of the scattering solid angle for product species detected in arrangement j. Indicating with symbols α and ν the collection of state quantum numbers for arrangements 1 and j, differential reaction cross sections are of form dσjν,1α/dΩj for products emerging in arrangement j. Quantal treatments are generally needed to identify reaction mechanisms and to clarify whether long-lived states (or scattering resonance states) are created during reactions. These aspects can be clarified generating partial state-to-state cross sections for each total angular momentum J of the triatomic system [71, 72].
Results for reactive D + H2 (, j) → HD (, j′) + H at the CM collision energy E = 0.78 eV are shown in panels (a), (b), and (c) of Figure 1.7 part (A) vs CM scattering angles, for a sequence of transitions ( = 0, j = 0) → ( = 0, j′ = 0 − 11) and ( = 0, j = 0) → (′ = 1, j′ = 0 − 6). Full lines are calculated values and dashed lines are fits to measured values. Theoretical values obtained from accurate potential energy surfaces are compared with experimental values and show good agreements in demanding tests of the PES accuracy and quantal dynamics treatment.
Figure 1.7 part (B) shows an alternative presentation of experimental measurements of the reaction F + p-H2 → HF () + H at a LAB collision energy of 1.84 kcal/mol, with product HF ( = 1, 2, 3) flux intensities vs molecular velocity in isocontour plots. For an incoming F atom beam from right to left, the figures show that product intensities peak primarily in the back direction and that secondly some = 3 intensity appears in the forward direction. Reproduction of these experimental plots with quantal dynamics calculations requires very accurate potential energy surfaces, as shown in [72].
(d) Atom–polyatomic and molecule–polyatomic pairs
Collisions involving polyatomic molecules can be described starting from the total potential energy function of all the atomic positions. For scattering of molecule A with atoms at positions and molecule B with positions , the total potential energy is of form E(R) = ∑a,bab + ∑a,b,b′abb′ + ⋯, a sum over pairs of atoms, triplets, and so on. Introducing the molecular CMs and and relative atomic positions such as , the interaction potential energy is , with and [4].
A general treatment of the quantal dynamics of scattering can be done introducing dynamical atom–atom correlation functions for the whole system and by relating cross sections to the correlation functions [75]. Measured cross sections give the intensities of projectile energy distributions for initial collision energy E vs scattering angle Ω following target energy transfer ε. For impulsive collisions at hyperthermal energies of a projectile atom A with a target B containing atoms b, the double differential inelastic cross section is
Figure 1.7 (A) Measured (dashed lines) and calculated (full lines) differential cross sections for reactive D + H2 (, j) → HD (′, j′) + H at the CM collision energy E = 0.78 eV. (B) Detailed measured cross sections for F + p-H2 → HF () + H at a LAB collision energy of 1.84 kcal/mol, shown as velocity flux isocontours for emerging HF molecules in vibrational states = 1–3.
Source: (A) Adapted from [73]; (B) [74] / AIP Publishing.
where dσb/dΩ is an atomic cross-sectional component of the target and dPb/dε is a rate of energy transfer obtained from each atom correlation function in the target [76].
An example is Li+ + CO2, which can be described in terms of the position of the atom relative to the center of mass of the triatomic, angles ΩM giving the orientation of the triatomic in the CM frame, and three internal vibrational coordinates qV for its two bond distances and one interbond angle. For impulsive collisions at energy E = 2.81 eV and a vibrational temperature TV = 300 K leading to fast vibrations and slow rotations of the target, energy loss measurements provide cross sections for vibrational normal mode transitions with changing quantum numbers nj = nj, fn − nj,in for symmetric stretch, asymmetric stretch, and bending modes j = 1, 2, 3, vs CM scattering angles, as shown in Figure 1.8.
(e) Atom–surface collisions
The mechanisms of atom–surface collisions are similar to those in atom–polyatomic collisions, except that surfaces can display atomic periodicities leading to collisional intensity variations related to quantal interference of scattered waves [4, 53].
Figure 1.8 Calculated cross section for Li+ + CO2 at collisional energy E = 2.81 eV and a vibrational temperature TV = 300 K with fast vibrations and slow rotations of the target, leading to vibrational normal mode transitions with changing quantum numbers nj, j = 1, 2, 3. The crosses are experimental results.
Source: [76] / with permission of Elsevier.
An example such as Rg + LiF(001) is described by the position of the rare-gas atom over the (001) crystal surface of solid LiF and by the positions QS of all the atoms on the surface and below it, which interact with He, in a reference frame attached to the surface. The interaction potential energy function V(X, Y, Z) for this system displays the translational symmetry of the (001) surface containing Li+ and F− ions. The Z-coordinate is perpendicular to the LiF(001) fcc surface, and the X-coordinate runs through F− ions at the surface.
Cross sections for Rg + LiF(001), Rg = Ne, Ar, Kr, Xe, vs final scattering angle θF for fixed initial angle θS = 40° from the surface axis are shown in Figure 1.9[77]. Here, the temperatures are TS = 300 K for the surface and TG = 573 K for the Rg gas. The scattering of lighter atoms display a double maximum corresponding to rainbow scattering angles.
1.4.3 Electronically Diabatic Heavy-Particle Collisions
(a) Electronically diabatic atom–atom collisions
Electronic rearrangement during collisions may involve excitation and charge transfer or ionization of product species when excited reactants are present. These phenomena may be simultaneously present. Examples are:
H
+
+ H(1s) → H
+
+ H(
nlm
) (e
−
excitation)
H
+
+ H(1s) → H(
nlm
) + H
+
(e
−
transfer)
He* (1s2s
1
S) + Ar(3p
6 1
S) → He(1s
2
1
S) + Ar
+
(3p
5 3
P) + e
−
(ionization)
where (nlm) are the quantum numbers of a hydrogen atomic state.
Figure 1.9 Cross sections for Rg + LiF(001), Rg = Ne, Ar, Kr, Xe, vs final scattering angle θF for fixed initial angle θS = 40° from the surface axis.
Source: [77] / AIP Publishing.
The Hamiltonian contains now both nuclear (Q) and electronic (X) variables and must be written as a sum over energies of interacting nuclei and electrons or a sum over many-electron atoms A and B Hamiltonians and their interaction energy,
In a CM reference frame with relative interatomic position , the interaction potential is . An eigenfunction Ψα( X) can be constructed as sums of molecular AB many-electron functions in a configuration interaction with its terms multiplying scattering functions of relative nuclear positions, or as products of atomic functions of electron variables for A and B, times relative motion functions, in a valence-bond description. Many-electron functions must be made invariant under electron exchange permutations between the collision partners [4]. Their expansion in a basis set of electronic states Γ introduce a matrix of diagonal and non-diagonal potential energy functions, and scattering leads to momentum-induced transitions Γ → Γ′ among the electronic states.
The treatment simplifies when only one electron is active in the collision. Figure 1.10 shows a collection of theoretical and experimental total cross sections σ(tr)(EL) for electron transfer in H+ + H(1s) (or D(1s)) vs LAB collision energy, with theory results shown as full and dashed lines and experimental measurements as dots and crosses [44].
Figure 1.10 Measured and calculated cross sections for electron transfer H+ + H(1s) (or D(1s)) → H(1s) + H+ (or D+) vs LAB collision energy EL, with theory results shown as full and dashed lines and experimental measurements as dots and crosses.
Source: [44] / Oxford University Press.
(b) Electronically diabatic atom–diatom collisions
An example is given by the following set of coupled arrangements,
H
+
+ H
2
→ H(1s) + H
2
+
(e
−
transfer)
H
+
+ H
2
→ H
2
+ H
+
(H displacement),
H
+
+ H
2
→ H
2
+
+ H(1s) (H − transfer)
which illustrates the complexity resulting from coupling of several electronic arrangements. In a treatment where the three arrangement channels are allowed to couple, the total wavefunction for the three atoms is of form , and all terms must be made invariant under electron exchange between collision partners [4]. Here again transitions between electronic states are mediated by nuclear momentum couplings.
Figure 1.11 shows scattering beam intensities of ions H+ in collisions at an initial CM energy E = 20 eV with a target of H2(i = 0), leading to (a) inelastic H+ + H2(f) and (b) charge transfer H + H2+(f), vs energy loss [78]. Peaks correspond to excitation of vibrational states of the target, with their locations slightly changed due to rotational excitation of the diatomic. The endoergicity of ΔE = 1.84 eV in charge transfer is clearly seen in the late appearance of vibrational peaks for (b). The locations of peaks are in good agreement with the vibrational excitation energies of H2(f) and H2+(f).
Figure 1.11 Intensities of scattered H+ beams vs energy loss for (a) inelastic and (b) charge transfer measured, for H+ + H2.
Source: [78] / AIP Publishing.
1.4.4 Electron and Photon Scattering by Molecules
(a) Electron–atom and electron–molecule systems
The total Hamiltonian here is a sum of terms for the electron kinetic energy, internal molecular motions, and the electron–molecule interaction, given by
with the system described by a wavefunction for a projectile electron of position and spin variables x0 and wavevector , scattered by a target with Nel electrons and nuclei (or atomic ions) at locations Q in an initial state Γ0. This wavefunction can be expanded in a basis set to electronic states Γ of the target, with expansion coefficients describing the scattering of the electron.
Generally for a molecule with Nel electrons, taking the CM frame origin at the position of the molecular CM, let be the position of the incoming electron and be those of the molecular electrons. The Hamiltonian terms involve electrons (0 or e) and nuclei (n), or atomic ion cores, interacting through Coulomb potential as
The corresponding wavefunction for electronic position and spin variables can be expanded in a basis set of molecular electronic states leading to a coupled-states (or close-coupling) treatment of the scattering, with coefficients dependent on the position of the projectile electron. Their asymptotic form provides scattering amplitudes and cross sections.
A simple example and thoroughly studied system is e + He(1s2), leading to elastic and inelastic scattering [59],
which can be described as the excitation of a two-electron system including polarization of the target. The corresponding wavefunction is of form Ψα(x0, x1, x2), for electronic position and spin variables and the position of the incoming electron, with the He nucleus at the origin of the reference frame. However, accurate calculations require a more detailed treatment as a three-electron system with full consideration of polarization and electron exchange effects.
In some cases, it is possible to describe elastic collisions with a one-electron model by representing the atom by means of a static potential energy derived from the atomic charge density nA(r0), so that and
where φA is the electric potential function obtained from the Poisson equation, given the charge density. Alternatively, one can introduce an effective atomic charge function eZA(r0) and write , where ZA(r0) reproduces the atomic selfconsistent field at radius r0 and generally satisfies ZA(0) = ZA (the nuclear charge number) and ZA(∞) = ZA − N, the net asymptotic charge number for the atom A.
An example of electron–diatom systems is e + H2, with the diatomic initially in its ground electronic and vibrational–rotational state , which may lead to elastic scattering where the diatomic stays in its initial state or to inelastic scattering with the diatomic undergoing electronic and (or) vibrational–rotational excitation. Figure 1.12 shows cross sections for excitation of into the excited state 1σg2σga3Σg, with a differential cross section vs scattering angle in (a) for collision energy of 20 eV and an integral cross section vs collision energy in (b). From [79], their Figure 25. The collision can be treated as a three-electron system undergoing state changes for nuclei at different orientations and bond-stretch values.
In the case where there is no electronic excitation, it is again possible to introduce an effective potential energy dependent only on the position of the scattered electron but with additional dependence on diatomic vibrational and orientation variables r and ω.
Figure 1.12 Excitation of into the excited state a3Σ induced by electron collisions. (a) Differential cross section for collision energy of 20 eV and (b) the integral cross section.
Source: [79] / with permission of Elsevier.
Electronic or vibrational excitation may lead to fragmentation of the molecule, with the fragments emerging from the interaction region along measurable angles, and with given translational energies.
(b) Photon–atom and photon–molecule systems
Here the photon is described as a radiation particle (a quantum of electromagnetic field ), and its interaction with a molecule is treated with Hamiltonian terms for the radiation and for its coupling to the molecule [56]. The field is a superposition of waves with wavevector and wavelength .
In the dipole approximation for the coupling, valid for photon field wavelengths large compared to the molecular size, and using creation and annihilation operators for photons of given wavevector and polarization σ, the total Hamiltonian is a sum of terms for radiation, molecular motions, and their interaction, given by
with an electric field operator
and a molecular electric dipole operator , where is a photon-annihilation operator and the dipole is given in terms of molecular constituent charges cI (of electrons and nuclei) located at positions .
The wavefunction for photons initially incoming in a state with wavevector and polarization σin and for the molecule in the ground state Γg is of the form with total energy and with the initial scattering states specified as a collection of quantum numbers for the states of the photon and molecule. The final state after scattering may be the result of an elastic or inelastic collision, where the internal molecular state and the wavelength of the photon change, or it may be a photoinduced dissociation state of the molecule.
An example is the photolysis of the BrCl( = 0, 1, 2) molecule in three parent vibrational states, induced by absorption of a photon with wavelengths in the range 330–570 nm, shown in Figure 1.13. Photofragmentation leads to Br(4p5 P3/2) + Cl(3p5 P3/2) and also to vibrationally excited Cl* in Br(4p5 P3/2) + Cl*(3p5 P1/2). Dissociation cross sections σd for the two products have been measured, and the ratio Γ = σd(Br + Cl*)/[σd(Br + Cl) + σd(Br + Cl*)] vs photon energies is shown in Figure 1.13, as well as calculated ratios from wavepacket modeling of the photolysis, showing excellent agreement. This comparison is detailed in [80].
Figure 1.13 Fragmentation ratio Γ = σd(Br + Cl*)/[σd(Br + Cl) + σd(Br + Cl*)] vs photon energies of the BrCl( = 0, 1, 2) molecule induced by absorption of a photon. Experimental points are compared to calculations from a wavepacket treatment of the photolysis.
Source: [80] / AIP Publishing.
1.5 Transport, Energy Relaxation, and Reaction Rates in Gases
Given state-to-state transition probabilities resulting from molecular interactions, it is possible to combine them with statistical mechanical distributions of the molecules in the gas phase at or near thermal equilibrium, to generate macroscopic properties [1, 81]. Among these, transport properties deal with diffusion (mass transport), viscosity
