Thesis for the degree of Doctor of Philosophy in
Natural Science, specialising in Chemistry
Formation of small molecules in
space through the process of
radiative association
Daria Jones (nee Burdakova)
Department of Chemistry and Molecular Biology
Gothenburg, Sweden, 2023
i
Abstract
This thesis is dedicated to calculating the reaction rate constants for
the formation of molecules through radiative association (RA) under
interstellar conditions. The molecules that have been studied are
CH, AlCl, Na+ − H +2 and Al − H2. The CH molecule was chosen
for this study because of its occurrence in several chemical reactions
in the interstellar medium, the sun and comets. We decided to
study AlCl because it plays a crucial role in the formation of other
important interstellar compounds. The choice of studying formation
of Na+ − H2 through RA is motivated by the fact that this reaction
is the first and limiting step in the process of formation of NaH in
the interstellar gas and is therefore of interest in astrochemistry. RA
of Al+ − H2 is connected to formation of AlH in a similar manner.
In order to obtain reaction rate constants, reaction cross sections
had to be computed. For the diatomic molecules, the reaction cross
sections were obtained using a semiclassical method and a classical
method, and then compared with the cross sections calculated using
quantum mechanical perturbation theory. The triatomic systems
were treated using theory developed for photodissociation together
with the driven equations method in the form of a driven Schrödinger
equation.
ISBN 978-91-8069-493-3 (TRYCK)
ISBN 978-91-8069-494-0 (PDF)
NMÄ
Part I available at http://hdl.handle.net/2077/78539 NEANENMÄRV RKKEVA E
Thesis © Daria Jones, Gothenburg, Sweden, 2023
Printed by Stema Trycksak3T0r4y1c k0s2a3k43041 0234
ii iii
SS
TT
Abstract
This thesis is dedicated to calculating the reaction rate constants for
the formation of molecules through radiative association (RA) under
interstellar conditions. The molecules that have been studied are
CH, AlCl, Na+ − H2 and Al+ − H2. The CH molecule was chosen
for this study because of its occurrence in several chemical reactions
in the interstellar medium, the sun and comets. We decided to
study AlCl because it plays a crucial role in the formation of other
important interstellar compounds. The choice of studying formation
of Na+ − H2 through RA is motivated by the fact that this reaction
is the first and limiting step in the process of formation of NaH in
the interstellar gas and is therefore of interest in astrochemistry. RA
of Al+ − H2 is connected to formation of AlH in a similar manner.
In order to obtain reaction rate constants, reaction cross sections
had to be computed. For the diatomic molecules, the reaction cross
sections were obtained using a semiclassical method and a classical
method, and then compared with the cross sections calculated using
quantum mechanical perturbation theory. The triatomic systems
were treated using theory developed for photodissociation together
with the driven equations method in the form of a driven Schrödinger
equation.
ISBN 978-91-8069-493-3 (TRYCK)
ISBN 978-91-8069-494-0 (PDF)
Part I available at http://hdl.handle.net/2077/78539
Thesis © Daria Jones, Gothenburg, Sweden, 2023
Printed by Stema
ii iii
Summary in Swedish
Det här arbetet fokuserar p̊a str̊alningsassociation som är en bety-
delsefull reaktion inom astrokemi. Astrokemi handlar om att studera
förekomsten av olika molekyler och ämnen i rymden samt hur de
kan skapas och förstöras.
När en ny molekyl ska skapas fr̊an tv̊a mindre molekyler eller
atomer måste de först kollidera. Efter kollisionen ska överskotts-
energin avges för att den nya molekylen ska kunna skapas. I gasfas
kan detta ske p̊a tv̊a olika sätt. Om en tredje kropp är närvarande
vid kollisionen kan den ta hand om överskottsenergin. P̊a grund av
den extremt l̊aga tätheten i rymden är chansen att tre molekyler
eller atomer ska kollidera mycket liten. Det andra sättet att bli
av med energin är att utstr̊ala en foton i en process som kallas
str̊alningsassociation.
Jag har utfört numeriska beräkningar för att studera hur CH,
AlCl, Al+ −H2 och Na+ −H2 molekylerna skapas genom str̊alnings-
association. Anledningen till att de molekylerna valdes är att de
är viktiga inom astrokemin. CH och AlCl deltar i flera kemiska
reaktioner b̊ade i det interstellära mediet, solen och kometer. Anled-
ningen till att det är intressant att studera hur Na+ −H2 molekylen
skapas är att den är avgörande för skapandet av NaH i interstellär
gas. Al+ − H2 valdes av liknande skäl. Målet med beräkningarna
var att f̊a fram reaktionshastighetskonstanterna som kan användas i
astrokemiska modeller.
v
Summary in Swedish
Det här arbetet fokuserar p̊a str̊alningsassociation som är en bety-
delsefull reaktion inom astrokemi. Astrokemi handlar om att studera
förekomsten av olika molekyler och ämnen i rymden samt hur de
kan skapas och förstöras.
När en ny molekyl ska skapas fr̊an tv̊a mindre molekyler eller
atomer måste de först kollidera. Efter kollisionen ska överskotts-
energin avges för att den nya molekylen ska kunna skapas. I gasfas
kan detta ske p̊a tv̊a olika sätt. Om en tredje kropp är närvarande
vid kollisionen kan den ta hand om överskottsenergin. P̊a grund av
den extremt l̊aga tätheten i rymden är chansen att tre molekyler
eller atomer ska kollidera mycket liten. Det andra sättet att bli
av med energin är att utstr̊ala en foton i en process som kallas
str̊alningsassociation.
Jag har utfört numeriska beräkningar för att studera hur CH,
AlCl, Al+ −H2 och Na+ −H2 molekylerna skapas genom str̊alnings-
association. Anledningen till att de molekylerna valdes är att de
är viktiga inom astrokemin. CH och AlCl deltar i flera kemiska
reaktioner b̊ade i det interstellära mediet, solen och kometer. Anled-
ningen till att det är intressant att studera hur Na+ −H2 molekylen
skapas är att den är avgörande för skapandet av NaH i interstellär
gas. Al+ − H2 valdes av liknande skäl. Målet med beräkningarna
var att f̊a fram reaktionshastighetskonstanterna som kan användas i
astrokemiska modeller.
v
Acknowledgements
I would like to thank my supervisor Gunnar Nyman for always
being there to answer my questions and giving me all the help and
guidance that I needed.
I am grateful to my co-supervisor Magnus Gustafsson for
helping me to understand the program we used for the diatomic
calculations and always being available to answer my questions.
I am also acknowledging the funding I received from MOLIM
(Molecules in Motion) that made it possible for me to visit Thierry
Stoecklin at the University of Bordeaux and be able to use the
program that he wrote for treating the triatomic molecules.
A big thanks to my office mate Karl-Mikael Svensson who has
always been there, from my first day as a young and clueless PhD
student, to answer all possible questions all the time. No matter
how busy he was with his own work, he always found time to help
me with mine.
I am very grateful to Martina, Betul and Panthea who helped
me with the printing and the nailing of the thesis!
Thanks to Javier for allowing me to use his beautiful night sky
photos.
I would like to thank the Candy Office for the constant and
reliable supply of candy!
And last but not least a big thanks to my family and friends for
supporting me! A special thanks to my wonderful husband Steve
who always supported me and believed in me even when I didn’t
believe in myself.
vii
Acknowledgements
I would like to thank my supervisor Gunnar Nyman for always
being there to answer my questions and giving me all the help and
guidance that I needed.
I am grateful to my co-supervisor Magnus Gustafsson for
helping me to understand the program we used for the diatomic
calculations and always being available to answer my questions.
I am also acknowledging the funding I received from MOLIM
(Molecules in Motion) that made it possible for me to visit Thierry
Stoecklin at the University of Bordeaux and be able to use the
program that he wrote for treating the triatomic molecules.
A big thanks to my office mate Karl-Mikael Svensson who has
always been there, from my first day as a young and clueless PhD
student, to answer all possible questions all the time. No matter
how busy he was with his own work, he always found time to help
me with mine.
I am very grateful to Martina, Betul and Panthea who helped
me with the printing and the nailing of the thesis!
Thanks to Javier for allowing me to use his beautiful night sky
photos.
I would like to thank the Candy Office for the constant and
reliable supply of candy!
And last but not least a big thanks to my family and friends for
supporting me! A special thanks to my wonderful husband Steve
who always supported me and believed in me even when I didn’t
believe in myself.
vii
List of Papers /
Contribution report
The included papers are reproduced with permission from the journal.
Paper I Formation of the CH/CD molecules through radiative
association of C with H/D
Daria Burdakova, Magnus Gustafsson and Gunnar Nyman
MNRAS, 517,4:4892–4901 (2022)
Paper II Formation of Na containing complex molecules in the
gas phase in dense molecular clouds: Quantum study
of the Na+ + H +2 and Na + D2 radiative association
step
Daria Burdakova, Gunnar Nyman and Thierry Stoecklin
MNRAS, 485, 4:5874–5879 (2019)
Paper III Formation of Al containing molecular complexes in
the gas phase in dense molecular clouds: quantum
study of the radiative association of Al++H and Al+2 +
D2
Daria Burdakova, Gunnar Nyman and Thierry Stoecklin
MNRAS, 503, 2:3089-3094 (2021)
Paper IV Formation of the AlCl molecule through radiative as-
sociation of Al with Cl
Daria Burdakova and Gunnar Nyman
Manuscript (2023)
ix
List of Papers /
Contribution report
The included papers are reproduced with permission from the journal.
Paper I Formation of the CH/CD molecules through radiative
association of C with H/D
Daria Burdakova, Magnus Gustafsson and Gunnar Nyman
MNRAS, 517,4:4892–4901 (2022)
Paper II Formation of Na containing complex molecules in the
gas phase in dense molecular clouds: Quantum study
of the Na+ + H +2 and Na + D2 radiative association
step
Daria Burdakova, Gunnar Nyman and Thierry Stoecklin
MNRAS, 485, 4:5874–5879 (2019)
Paper III Formation of Al containing molecular complexes in
the gas phase in dense molecular clouds: quantum
study of the radiative association of Al++H and Al+2 +
D2
Daria Burdakova, Gunnar Nyman and Thierry Stoecklin
MNRAS, 503, 2:3089-3094 (2021)
Paper IV Formation of the AlCl molecule through radiative as-
sociation of Al with Cl
Daria Burdakova and Gunnar Nyman
Manuscript (2023)
ix
Contributions from the Author
Paper I I performed all of the calculations and wrote the first draft of
the manuscript. I was involved in further improvements of the
manuscript.
Paper II I performed a part of the calculations (radiative association) Contents
and I wrote the first draft of the manuscript. I was involved
in further improvements of the manuscript.
Paper III I performed most of the calculations and I wrote the first draft I 1
of the manuscript. I was involved in further improvements of
the manuscript. 1 Introduction 3
Paper IV I performed all of the calculations and wrote the first draft of
2 Background 7
the manuscript. I was involved in further improvements of the
2.1 Astrochemistry . . . . . . . . . . . . . . . . . . . . . 7
manuscript.
2.2 Radiative association . . . . . . . . . . . . . . . . . . 12
3 Computational methods for radiative association 15
3.1 Reaction rate coefficient . . . . . . . . . . . . . . . . 15
3.2 Perturbation theory . . . . . . . . . . . . . . . . . . . 17
3.3 Semi-Classical method . . . . . . . . . . . . . . . . . 18
3.4 Classical method . . . . . . . . . . . . . . . . . . . . 20
3.5 Breit-Wigner method . . . . . . . . . . . . . . . . . . 21
3.6 Driven equations method for RA . . . . . . . . . . . 22
4 Numerical methods 25
4.1 DVR . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Numerov . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 LEVEL . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 Results 31
5.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . 47
Bibliography 51
x xi
Contributions from the Author
Paper I I performed all of the calculations and wrote the first draft of
the manuscript. I was involved in further improvements of the
manuscript.
Paper II I performed a part of the calculations (radiative association) Contents
and I wrote the first draft of the manuscript. I was involved
in further improvements of the manuscript.
Paper III I performed most of the calculations and I wrote the first draft I 1
of the manuscript. I was involved in further improvements of
the manuscript. 1 Introduction 3
Paper IV I performed all of the calculations and wrote the first draft of
2 Background 7
the manuscript. I was involved in further improvements of the
2.1 Astrochemistry . . . . . . . . . . . . . . . . . . . . . 7
manuscript.
2.2 Radiative association . . . . . . . . . . . . . . . . . . 12
3 Computational methods for radiative association 15
3.1 Reaction rate coefficient . . . . . . . . . . . . . . . . 15
3.2 Perturbation theory . . . . . . . . . . . . . . . . . . . 17
3.3 Semi-Classical method . . . . . . . . . . . . . . . . . 18
3.4 Classical method . . . . . . . . . . . . . . . . . . . . 20
3.5 Breit-Wigner method . . . . . . . . . . . . . . . . . . 21
3.6 Driven equations method for RA . . . . . . . . . . . 22
4 Numerical methods 25
4.1 DVR . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Numerov . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 LEVEL . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 Results 31
5.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . 47
Bibliography 51
x xi
Contents
II Papers 57
Part I
xii 1
Contents
II Papers 57
Part I
xii 1
Chapter 1
Introduction
A starry sky has been a source of wonder for people throughout time,
giving rise to various religious and scientific models as we strive to
understand the origin of these glowing orbs. Stars have not only
captivated our curiosity, but they have also served practical purposes,
such as navigation and timekeeping. Sailors, for instance, relied on
different constellations to find their way across the seas, while the
Mayans incorporated them into their calendar.
Our understanding of the night sky has changed tremendously,
from believing that all the shiny spheres in the night sky are stars
to recognizing that some of them are actually planets. Different
models to describe the movements of the stars were proposed and,
initially, it was believed that the Earth was the centre of the universe
and all the stars and planets were revolving around it. This theory
was challenged when Galileo invented the first telescope in 1609,
allowing him to study the stars and planets in more detail. Using
his telescope Galileo could study the landscape of the Moon, see
the different phases of Venus and observe the four largest Jovian
satellites. He was the one who realized that celestial objects are not
perfect spheres and that the Earth is in fact orbiting the Sun.
The telescope invented by Galileo was, what we call today, a
refractive telescope, using multiple lenses to magnify distant objects.
However, this type of telescope had notable limitations. Observing
objects at greater distances required increasing the length of the
telescope, something that created many practical challenges. At-
tempting to overcome this limitation by making thinner lenses led
3
Chapter 1
Introduction
A starry sky has been a source of wonder for people throughout time,
giving rise to various religious and scientific models as we strive to
understand the origin of these glowing orbs. Stars have not only
captivated our curiosity, but they have also served practical purposes,
such as navigation and timekeeping. Sailors, for instance, relied on
different constellations to find their way across the seas, while the
Mayans incorporated them into their calendar.
Our understanding of the night sky has changed tremendously,
from believing that all the shiny spheres in the night sky are stars
to recognizing that some of them are actually planets. Different
models to describe the movements of the stars were proposed and,
initially, it was believed that the Earth was the centre of the universe
and all the stars and planets were revolving around it. This theory
was challenged when Galileo invented the first telescope in 1609,
allowing him to study the stars and planets in more detail. Using
his telescope Galileo could study the landscape of the Moon, see
the different phases of Venus and observe the four largest Jovian
satellites. He was the one who realized that celestial objects are not
perfect spheres and that the Earth is in fact orbiting the Sun.
The telescope invented by Galileo was, what we call today, a
refractive telescope, using multiple lenses to magnify distant objects.
However, this type of telescope had notable limitations. Observing
objects at greater distances required increasing the length of the
telescope, something that created many practical challenges. At-
tempting to overcome this limitation by making thinner lenses led
3
1. Introduction
images of distant galaxies, stars and nebulae.
While optical telescopes allowed us to observe the universe on a
larger scale, the development of radio telescopes made it possible to
detect much smaller things, namely molecules. Radio telescopes are
particularly well-suited for detecting molecular emissions because
many molecules emit light within the radio frequency domain when
they transition between different rotational and vibrational energy
levels. By studying the observed set of frequencies, or spectrum,
scientists can determine which molecule emitted them.
Figure 1.1: An image of the Orion Nebula taken with a telescopic The detection of molecules in space was a groundbreaking de-
lens of 200 mm. Image credit Javier Almansa Sobrino via AstroBin, velopment, challenging the previous belief among astronomers that
username javieralso. interstellar gas solely consisted of atoms. Additionally, when or-
ganic molecules like carbon monoxide and water were detected, the
possibility of finding life beyond our planet became less questionable.
After the discovery of molecules in space, a new scientific branch
called astrochemistry was born, and from this point, the story un-
to another issue. The lenses, despite being improved, still exhibited
folds!
some level of prism-like behavior, causing different colors of light
to refract unequally, resulting in blurry images. This is the reason
why Newton’s discovery of reflecting telescopes held immense signifi-
cance for astronomy. Unlike refracting telescopes that rely on lenses,
reflecting telescopes employ curved mirrors, eliminating any prism
effects and ensuring clear images. Additionally, increasing the size
of the mirror allows to capture fainter signals, which in turn makes
it possible to gaze deeper into space.
When light from a star or another celestial object travels through
the Earth’s atmosphere, some of it will be absorbed, scattered or
refracted, ultimately affecting the observations made by the telescope.
It is therefore important to minimize the atmospheric path and many
telescopes are strategically placed on mountains.
Light pollution is another crucial factor that must be taken into
account when choosing a telescope location. Bright cities in close
proximity can significantly hinder the telescope’s ability to detect
weak signals from space, making deserts an ideal choice for telescope
placement. However, the optimal solution is to circumvent the
Earth’s atmosphere altogether and astronomy had a big breakthrough
when the first telescope, Hubble Space Telescope, was launched into
space in 1990. It allowed to capture incredibly sharp and clear
4 5
1. Introduction
images of distant galaxies, stars and nebulae.
While optical telescopes allowed us to observe the universe on a
larger scale, the development of radio telescopes made it possible to
detect much smaller things, namely molecules. Radio telescopes are
particularly well-suited for detecting molecular emissions because
many molecules emit light within the radio frequency domain when
they transition between different rotational and vibrational energy
levels. By studying the observed set of frequencies, or spectrum,
scientists can determine which molecule emitted them.
Figure 1.1: An image of the Orion Nebula taken with a telescopic The detection of molecules in space was a groundbreaking de-
lens of 200 mm. Image credit Javier Almansa Sobrino via AstroBin, velopment, challenging the previous belief among astronomers that
username javieralso. interstellar gas solely consisted of atoms. Additionally, when or-
ganic molecules like carbon monoxide and water were detected, the
possibility of finding life beyond our planet became less questionable.
After the discovery of molecules in space, a new scientific branch
called astrochemistry was born, and from this point, the story un-
to another issue. The lenses, despite being improved, still exhibited
folds!
some level of prism-like behavior, causing different colors of light
to refract unequally, resulting in blurry images. This is the reason
why Newton’s discovery of reflecting telescopes held immense signifi-
cance for astronomy. Unlike refracting telescopes that rely on lenses,
reflecting telescopes employ curved mirrors, eliminating any prism
effects and ensuring clear images. Additionally, increasing the size
of the mirror allows to capture fainter signals, which in turn makes
it possible to gaze deeper into space.
When light from a star or another celestial object travels through
the Earth’s atmosphere, some of it will be absorbed, scattered or
refracted, ultimately affecting the observations made by the telescope.
It is therefore important to minimize the atmospheric path and many
telescopes are strategically placed on mountains.
Light pollution is another crucial factor that must be taken into
account when choosing a telescope location. Bright cities in close
proximity can significantly hinder the telescope’s ability to detect
weak signals from space, making deserts an ideal choice for telescope
placement. However, the optimal solution is to circumvent the
Earth’s atmosphere altogether and astronomy had a big breakthrough
when the first telescope, Hubble Space Telescope, was launched into
space in 1990. It allowed to capture incredibly sharp and clear
4 5
Chapter 2
Background
In this chapter, you will find a background to the topics addressed in
this thesis. It begins with a general overview of astrochemistry and
the subsequent section offers a description of the radiative association
process, which is the main focus of the thesis.
2.1 Astrochemistry
Astrochemistry, as the name states, is a discipline in the borderland
of chemistry and astronomy, focusing on the abundance of elements
and molecules in space and the way they react and interact with
radiation.
Detection of molecules and their abundances
During the last couple of decades, a large variety of molecules have
been detected in space and databases containing lists of the detected
molecules can be found in The Cologne Database for Molecular
Spectroscopy, The Astrochymist and the work of M. Araki ∗.
Around 80 percent of those molecules were detected through radio
astronomy techniques (cm to far infra-red) using wavelengths in the
centimeter, millimeter and submillimeter regions (McGuire [2018]).
In the radio region, molecules can be observed via the light emitted
when a molecule goes from a higher to a lower rotational state. The
∗http://molecules-in.space/
7
Chapter 2
Background
In this chapter, you will find a background to the topics addressed in
this thesis. It begins with a general overview of astrochemistry and
the subsequent section offers a description of the radiative association
process, which is the main focus of the thesis.
2.1 Astrochemistry
Astrochemistry, as the name states, is a discipline in the borderland
of chemistry and astronomy, focusing on the abundance of elements
and molecules in space and the way they react and interact with
radiation.
Detection of molecules and their abundances
During the last couple of decades, a large variety of molecules have
been detected in space and databases containing lists of the detected
molecules can be found in The Cologne Database for Molecular
Spectroscopy, The Astrochymist and the work of M. Araki ∗.
Around 80 percent of those molecules were detected through radio
astronomy techniques (cm to far infra-red) using wavelengths in the
centimeter, millimeter and submillimeter regions (McGuire [2018]).
In the radio region, molecules can be observed via the light emitted
when a molecule goes from a higher to a lower rotational state. The
∗http://molecules-in.space/
7
2. Background 2.1. Astrochemistry
spacing between the rotational energy levels is proportional to the Reaction Rate coefficient
rotational constant B = h2 where I is the moment of inertia. For Neutral-neutral A + B → C + D 4 · 10−11 cm3s−1 8π cI
diatomic molecules the moment of inertia is simply equal to µr2 Ion-molecule A+ + B → C+ + D 2 · 10−9 cm3s−1 †
where µ is the reduced mass and r is the bond length. Charge transfer A+ + B → B+ + A 10−9 cm3s−1 †
Some molecules can be detected directly, while for others tracers Radiative association A + B → AB + hν *
−9 −1
are used. A molecular tracer is a species whose density is correlated Photodissociation AB + hν → A + B 10 s 
Dissociative recombination A+ + e− → C + D 10−7 cm3s−1 †
to the density of the molecule of interest. Therefore, if the molecule
Collisional association A + B + C → AB + C 10−32 cm6s−1 †
of interest is hard to detect, its density can be deduced from the
Associative detachment A− + B → AB + e− 10−9 cm3s−1 †
tracer.
A wide array of molecular species, ranging in size from two to 13 Table 2.1: A list of common reactions and their typical reaction rate
atoms and comprising both neutrals and ions, has been discovered in constants (Tielens [2005]). * Strongly depends on reaction † Assuming
exothermic reaction  Assuming exothermic reaction and no barrier  Assuming
space so far. Larger molecules like C4H7N, C60 and C70 have also been unshielded radiation field
detected (Millar [2015]). These molecules often contain hydrogen,
which is the most abundant element in the interstellar medium (ISM).
The abundances of all of the elements are represented as a fraction
In gas phase, various types of reactions can take place, such as
of the abundance of H. The second most common element is He with
neutral-neutral reactions, ion-neutral reactions, ion-electron disso-
an abundance of a tenth of the abundance of H while other heavier
ciative reactions and radiative recombination or association (Nuth
element, like C, N, O, F, Na, Mg, Si, P S and Cl, have relative
−4 −8 et al. [2006]). These reactions involve the creation, breaking, orabundances ranging from 10 to 10 .
rearrangement of chemical bonds. The driving forces behind these
reactions are usually photons and cosmic rays. These reactions and
Reactions in space their typical reaction rate constants can be seen in Table 2.1.
It can be seen from the table that dissociative recombination
The species present in space can collide and react with each other. is the fastest reaction followed by ion-molecule, charge transfer,
Due to low temperatures and density in space, these reactions are photodissociation and associative detachment reactions. The slowest
considerably slower than those occurring on Earth, but taking into reactions are the neutral-neutral and collisional association reactions.
account the long lifetime of interstellar clouds it is still possible for The reaction rate constant for RA varies significantly from one
molecules to be created and detected (Millar [2015]). reaction to another, with values typically ranging from 10−20 cm3s−1
Reactions can occur either in gas phase or on grain surfaces. to 10−16 cm3s−1 (Julienne and Krauss [1973]).
Grains play an important role in astrochemistry providing a surface
where the species can accrete, react and more importantly give
away excess energy, making it possible to create a molecule. The
reactions on grains go through 4 steps: accretion, diffusion, reaction Astrochemical networks
and ejection. The first two steps (accretion and diffusion) set the
timescale for the overall reaction since they decide how fast the Knowing which reactions occur in space and their reaction rates
species will meet on the surface. After the reaction, the new molecule makes it possible to deduce the abundances of different molecules,
can detach from the surface and start interacting with molecules in which can be connected by a set of differential equations (Tielens
the gas phase. [2005]). The abundance of the species i can be obtained from:
8 9
2. Background 2.1. Astrochemistry
spacing between the rotational energy levels is proportional to the Reaction Rate coefficient
rotational constant B = h2 where I is the moment of inertia. For Neutral-neutral A + B → C + D 4 · 10−11 cm3s−1 8π cI
diatomic molecules the moment of inertia is simply equal to µr2 Ion-molecule A+ + B → C+ + D 2 · 10−9 cm3s−1 †
where µ is the reduced mass and r is the bond length. Charge transfer A+ + B → B+ + A 10−9 cm3s−1 †
Some molecules can be detected directly, while for others tracers Radiative association A + B → AB + hν *
−9 −1
are used. A molecular tracer is a species whose density is correlated Photodissociation AB + hν → A + B 10 s 
Dissociative recombination A+ + e− → C + D 10−7 cm3s−1 †
to the density of the molecule of interest. Therefore, if the molecule
Collisional association A + B + C → AB + C 10−32 cm6s−1 †
of interest is hard to detect, its density can be deduced from the
Associative detachment A− + B → AB + e− 10−9 cm3s−1 †
tracer.
A wide array of molecular species, ranging in size from two to 13 Table 2.1: A list of common reactions and their typical reaction rate
atoms and comprising both neutrals and ions, has been discovered in constants (Tielens [2005]). * Strongly depends on reaction † Assuming
exothermic reaction  Assuming exothermic reaction and no barrier  Assuming
space so far. Larger molecules like C4H7N, C60 and C70 have also been unshielded radiation field
detected (Millar [2015]). These molecules often contain hydrogen,
which is the most abundant element in the interstellar medium (ISM).
The abundances of all of the elements are represented as a fraction
In gas phase, various types of reactions can take place, such as
of the abundance of H. The second most common element is He with
neutral-neutral reactions, ion-neutral reactions, ion-electron disso-
an abundance of a tenth of the abundance of H while other heavier
ciative reactions and radiative recombination or association (Nuth
element, like C, N, O, F, Na, Mg, Si, P S and Cl, have relative
−4 −8 et al. [2006]). These reactions involve the creation, breaking, orabundances ranging from 10 to 10 .
rearrangement of chemical bonds. The driving forces behind these
reactions are usually photons and cosmic rays. These reactions and
Reactions in space their typical reaction rate constants can be seen in Table 2.1.
It can be seen from the table that dissociative recombination
The species present in space can collide and react with each other. is the fastest reaction followed by ion-molecule, charge transfer,
Due to low temperatures and density in space, these reactions are photodissociation and associative detachment reactions. The slowest
considerably slower than those occurring on Earth, but taking into reactions are the neutral-neutral and collisional association reactions.
account the long lifetime of interstellar clouds it is still possible for The reaction rate constant for RA varies significantly from one
molecules to be created and detected (Millar [2015]). reaction to another, with values typically ranging from 10−20 cm3s−1
Reactions can occur either in gas phase or on grain surfaces. to 10−16 cm3s−1 (Julienne and Krauss [1973]).
Grains play an important role in astrochemistry providing a surface
where the species can accrete, react and more importantly give
away excess energy, making it possible to create a molecule. The
reactions on grains go through 4 steps: accretion, diffusion, reaction Astrochemical networks
and ejection. The first two steps (accretion and diffusion) set the
timescale for the overall reaction since they decide how fast the Knowing which reactions occur in space and their reaction rates
species will meet on the surface. After the reaction, the new molecule makes it possible to deduce the abundances of different molecules,
can detach from the surface and start interacting with molecules in which can be connected by a set of differential equations (Tielens
the gas phase. [2005]). The abundance of the species i can be obtained from:
8 9
2. Background 2.1. Astrochemistry
Component T(K) n(cm−3) Fractional ionization
∑ ∑ ∑ ∑ Molecular 10-20 10
2-106 < 10−6
dn(i)
= −n(i) n(j)k + n(j)n(k)k + n(j)k −n(i) k Cold atomic 50-100 20-50 ∼ 10
−4
ij jk j j
dt Warm atomic 6000-10 000 0.2-0.5 ∼ 0.1
j jk j j
Warm ionized ∼ 8000 0.2-0.5 1
(2.1)
Hot ionized ∼ 106 ∼ 0.0065 1
where n() denote the densities and k are the reaction rate constants. Table 2.2: The properties of the different components of the ISM.
The first sum includes all bimolecular reactions where the species The values are taken from the works of Ferrière [2001], Caselli et al.
i is a reactant and the second sum consists of all the bimolecular [1998], Jenkins [2013], Wolfire et al. [1995].
reactions that form species i. The last two sums correspond to
unimolecular reactions that create or destroy species i.
The set of differential equations can be written in matrix form so-called components, with different physical properties that are
(Halász et al. [2013]) and solved assuming steady state, meaning summarized in Table 2.2.
the derivatives of all included species are equal to zero. To obtain Approximately half of the interstellar matter can be found in
a time-dependent solution, the equations can be integrated with a clouds and the rest of it is spread out between them (Ferrière [2001]).
specific starting point that aligns with the physical conditions of the The clouds are usually divided into three categories: dark clouds,
location of interest. These physical conditions encompass aspects diffuse clouds and translucent clouds.
like temperature, density, strength of ionization, and intensity of Dark clouds (also known as cold dense clouds) consist mostly of
UV radiation (van Dishoeck and Black [1986]). As the molecular molecular gas which stops the light from travelling deeper into the
abundances evolve, the physical conditions must be updated accord- cloud. This leads to negligible photodissociation and photoionization,
ingly. Models that simulate these networks of reactions can vary which in turn results in low fractional ionization. Having a weak
in complexity, ranging from simpler versions with fewer than 10 photon field also makes it possible for hydrogen to stay in molecular
reactions to more intricate ones that involve over 1000 reactions in form.
the calculations (Herbst and Klemperer [1972], Prasad and Huntress Diffuse clouds have typically low temperature and density and
[1980], Tielens and Hagen [1982], Hasegawa et al. [1992] and Garrod are made up by atomic gas. The radiation can penetrate deep
and Herbst [2006]). into the cloud and ionize atoms (mostly C and S). Together with
A model is deemed to be good if it can recreate observations photoelectric emission from the dust grains, electrons produced from
such as the abundances of molecular and atomic hydrogen and some ionization of atoms help heating the gas to a temperature of 50-
small molecules like CH, CO, OH, CH+ and CN. Other important 100 K (Millar [2015]). Cooling, on the other hand, occurs through
observables that the model should be able to reproduce are the rotational transitions of H2 and fine structure transitions within C
+,
distribution among the rotational levels of the ground state of H2, S+ and O. Diffuse clouds tend to have a high fractional ionization
the H/D ratio and the amount of ionized atoms. due to C+ being easily produced by the photon field.
Translucent clouds consist of both molecular and atomic gas. The
Components of the ISM visual extinction of the translucent clouds is therefore somewhere in
between the ones for dark and diffuse clouds.
The ISM consists of a mix of gas in atomic, molecular and ionic Since hydrogen is the most abundant element, the form it is
form together with dust particles. The gas exists in different forms, present in will have a great effect on the chemical reactions. The
10 11
2. Background 2.1. Astrochemistry
Component T(K) n(cm−3) Fractional ionization
∑ ∑ ∑ ∑ Molecular 10-20 10
2-106 < 10−6
dn(i)
= −n(i) n(j)k + n(j)n(k)k + n(j)k −n(i) k Cold atomic 50-100 20-50 ∼ 10
−4
ij jk j j
dt Warm atomic 6000-10 000 0.2-0.5 ∼ 0.1
j jk j j
Warm ionized ∼ 8000 0.2-0.5 1
(2.1)
Hot ionized ∼ 106 ∼ 0.0065 1
where n() denote the densities and k are the reaction rate constants. Table 2.2: The properties of the different components of the ISM.
The first sum includes all bimolecular reactions where the species The values are taken from the works of Ferrière [2001], Caselli et al.
i is a reactant and the second sum consists of all the bimolecular [1998], Jenkins [2013], Wolfire et al. [1995].
reactions that form species i. The last two sums correspond to
unimolecular reactions that create or destroy species i.
The set of differential equations can be written in matrix form so-called components, with different physical properties that are
(Halász et al. [2013]) and solved assuming steady state, meaning summarized in Table 2.2.
the derivatives of all included species are equal to zero. To obtain Approximately half of the interstellar matter can be found in
a time-dependent solution, the equations can be integrated with a clouds and the rest of it is spread out between them (Ferrière [2001]).
specific starting point that aligns with the physical conditions of the The clouds are usually divided into three categories: dark clouds,
location of interest. These physical conditions encompass aspects diffuse clouds and translucent clouds.
like temperature, density, strength of ionization, and intensity of Dark clouds (also known as cold dense clouds) consist mostly of
UV radiation (van Dishoeck and Black [1986]). As the molecular molecular gas which stops the light from travelling deeper into the
abundances evolve, the physical conditions must be updated accord- cloud. This leads to negligible photodissociation and photoionization,
ingly. Models that simulate these networks of reactions can vary which in turn results in low fractional ionization. Having a weak
in complexity, ranging from simpler versions with fewer than 10 photon field also makes it possible for hydrogen to stay in molecular
reactions to more intricate ones that involve over 1000 reactions in form.
the calculations (Herbst and Klemperer [1972], Prasad and Huntress Diffuse clouds have typically low temperature and density and
[1980], Tielens and Hagen [1982], Hasegawa et al. [1992] and Garrod are made up by atomic gas. The radiation can penetrate deep
and Herbst [2006]). into the cloud and ionize atoms (mostly C and S). Together with
A model is deemed to be good if it can recreate observations photoelectric emission from the dust grains, electrons produced from
such as the abundances of molecular and atomic hydrogen and some ionization of atoms help heating the gas to a temperature of 50-
small molecules like CH, CO, OH, CH+ and CN. Other important 100 K (Millar [2015]). Cooling, on the other hand, occurs through
observables that the model should be able to reproduce are the rotational transitions of H2 and fine structure transitions within C
+,
distribution among the rotational levels of the ground state of H2, S+ and O. Diffuse clouds tend to have a high fractional ionization
the H/D ratio and the amount of ionized atoms. due to C+ being easily produced by the photon field.
Translucent clouds consist of both molecular and atomic gas. The
Components of the ISM visual extinction of the translucent clouds is therefore somewhere in
between the ones for dark and diffuse clouds.
The ISM consists of a mix of gas in atomic, molecular and ionic Since hydrogen is the most abundant element, the form it is
form together with dust particles. The gas exists in different forms, present in will have a great effect on the chemical reactions. The
10 11
2. Background 2.2. Radiative association
hydrogen can be either neutral (HI region) or ionized (HII region). (Zámečńıková et al. [2020]), CF+ (Öström et al. [2016]), HF (Gustafs-
son et al. [2014]), AlCl (Andreazza et al. [2018b]), NaCl (Šimsová
et al. [2022]), O2 (Babb and Dalgarno [1995]), C2 (Krauss and Juli-
2.2 Radiative association enne [1973]) and the triatomic systems H+2 − He ([Mrugala et al.,
2003]), H− − H2 ([Ayouz et al., 2011]), H− − N2 ([Stoecklin et al.,
In order to form a molecule from two atoms or smaller molecules,
2013b]), H − CO ([Stoecklin et al., 2018b]), H− − CO (Stoecklin
the species need to collide. After the collision, the new system will
et al. [2018a]). All the above studies were performed using dynamics,
have an excess of energy that will force it to dissociate unless this
i.e. where the atomic motion is computed either with quantum
energy is released in some manner.
or classical mechanics. Formation of a number of larger molecules
The energy can be carried away by a third body that is present
through RA has also been studied, using statistical methods, for
in the collision but not participating in the reaction. However, a
example see the works of Herbst and Dunbar [1991], Tennis et al.
three-body collision is highly unlikely in space due to low density
[2021].
(Bates [1951], Babb et al. [1998]) and another way to get rid of excess
In this thesis RA to form CH, AlCl, Na+ −H2 and Al+ −H2 has
energy is through emission of a photon:
been studied using quantum and, to some extent, classical dynamics.
The astrochemical relevance of these molecules is described in the
A+ B → AB + hν (2.2) following sections.
This process is called radiative association (RA) and, since the CH
photon has to be emitted during a very short time period when the
species are close to each other, this process is rather improbable and CH is the first molecule that has been identified in space (Swings
yields very low reaction rate constants. Keeping in mind the large and Rosenfeld [1937], Herbig [1968]). Most of the CH present in the
timescales considered in astrochemistry, RA can nevertheless explain ISM has been created in diffuse cold neutral gas and the RA process
formation of several molecules in space. is one of the possible pathways to form this molecule (Solomon and
The low reaction rate constants make RA difficult to study Klemperer [1972]):
experimentally (Gerlich and Horning [1992]) and, although it is
+ +
possible through setups like ion traps and ion cyclotron apparatus, C + H → CH + hν (2.3)
these approaches are limited to studying reactions involving ions
and neutral particles. followed by:
RA is also hard to compute numerically due to the large contri-
CH+ + e− → CH+ hν (2.4)
bution from tunneling and the fact that all scattering and bound
states need to be considered in the calculations. There are several
An alternative route has been suggested by Black and Dalgarno
numerical methods to treat RA but so far they are only applicable
[1973] assuming high abundance of molecular hydrogen:
to formation of smaller molecules.
The RA process has previously been studied computationally C+ +H2 → CH+2 + hν (2.5)
for forming several diatomic systems, including OH (Svensson et al.
[2015]), CH+ (Barinovs and van Hemert [2006]), NH (Szabo and followed by:
Gustafsson [2019]), HD (Stancil and Dalgarno [1997]), CN (An-
tipov et al. [2009]), CO (Gustafsson and Nyman [2015]), CO+ CH+ −2 + e → CH+ H (2.6)
12 13
2. Background 2.2. Radiative association
hydrogen can be either neutral (HI region) or ionized (HII region). (Zámečńıková et al. [2020]), CF+ (Öström et al. [2016]), HF (Gustafs-
son et al. [2014]), AlCl (Andreazza et al. [2018b]), NaCl (Šimsová
et al. [2022]), O2 (Babb and Dalgarno [1995]), C2 (Krauss and Juli-
2.2 Radiative association enne [1973]) and the triatomic systems H+2 − He ([Mrugala et al.,
2003]), H− − H2 ([Ayouz et al., 2011]), H− − N2 ([Stoecklin et al.,
In order to form a molecule from two atoms or smaller molecules,
2013b]), H − CO ([Stoecklin et al., 2018b]), H− − CO (Stoecklin
the species need to collide. After the collision, the new system will
et al. [2018a]). All the above studies were performed using dynamics,
have an excess of energy that will force it to dissociate unless this
i.e. where the atomic motion is computed either with quantum
energy is released in some manner.
or classical mechanics. Formation of a number of larger molecules
The energy can be carried away by a third body that is present
through RA has also been studied, using statistical methods, for
in the collision but not participating in the reaction. However, a
example see the works of Herbst and Dunbar [1991], Tennis et al.
three-body collision is highly unlikely in space due to low density
[2021].
(Bates [1951], Babb et al. [1998]) and another way to get rid of excess
In this thesis RA to form CH, AlCl, Na+ −H2 and Al+ −H2 has
energy is through emission of a photon:
been studied using quantum and, to some extent, classical dynamics.
The astrochemical relevance of these molecules is described in the
A+ B → AB + hν (2.2) following sections.
This process is called radiative association (RA) and, since the CH
photon has to be emitted during a very short time period when the
species are close to each other, this process is rather improbable and CH is the first molecule that has been identified in space (Swings
yields very low reaction rate constants. Keeping in mind the large and Rosenfeld [1937], Herbig [1968]). Most of the CH present in the
timescales considered in astrochemistry, RA can nevertheless explain ISM has been created in diffuse cold neutral gas and the RA process
formation of several molecules in space. is one of the possible pathways to form this molecule (Solomon and
The low reaction rate constants make RA difficult to study Klemperer [1972]):
experimentally (Gerlich and Horning [1992]) and, although it is
+ +
possible through setups like ion traps and ion cyclotron apparatus, C + H → CH + hν (2.3)
these approaches are limited to studying reactions involving ions
and neutral particles. followed by:
RA is also hard to compute numerically due to the large contri-
CH+ + e− → CH+ hν (2.4)
bution from tunneling and the fact that all scattering and bound
states need to be considered in the calculations. There are several
An alternative route has been suggested by Black and Dalgarno
numerical methods to treat RA but so far they are only applicable
[1973] assuming high abundance of molecular hydrogen:
to formation of smaller molecules.
The RA process has previously been studied computationally C+ +H2 → CH+2 + hν (2.5)
for forming several diatomic systems, including OH (Svensson et al.
[2015]), CH+ (Barinovs and van Hemert [2006]), NH (Szabo and followed by:
Gustafsson [2019]), HD (Stancil and Dalgarno [1997]), CN (An-
tipov et al. [2009]), CO (Gustafsson and Nyman [2015]), CO+ CH+ −2 + e → CH+ H (2.6)
12 13
2. Background
Since the first reaction in the second pathway (equation 2.5) is the
limiting one, the amount of CH created will be restricted by the
amount of H2. CH can therefore be used as a proxy for molecular hy-
drogen (Wiesemeyer et al. [2018]) assuming that CH is predominantly
formed by the process suggested by Black and Dalgarno.
Another possible pathway to create CH is through the endother- Chapter 3
mic reaction:
C+ +H2 + 0.396eV → CH+ +H (2.7)
Computational methods for
CH+ is then converted to CH as shown in equation 2.4. Since energy
is required for the endothermic reaction (equation 2.7), CH can also radiative association
be used as a tracer for shocks (Xu and Li [2016]).
Metal ions
This chapter begins by defining reaction rate constants and reaction
The importance of gas phase metals has been questioned until several cross sections. Later the methods used to calculate the reaction rate
molecules containing the metals Na, Mg, Al and K were detected coefficient and reaction cross sections are described.
in the star IRC+10216: NaCl, AlF, AlCl, KCl, Na(CN), MgCN,
and MgNC (Cernicharo and Guelin [1987], Kawaguchi et al. [1993],
Turner et al. [1994], Ziurys et al. [1994], Ziurys et al. [1995]), prov- 3.1 Reaction rate coefficient
ing that metals can form molecules in gas phase under interstellar
conditions. Consider a reaction involving a transition between two different
Formation of molecules containing metal ions can have a big im- states A and B, which could be, for example, different electronic or
pact on the amount of electrons in the gas as radiative recombination rovibrational levels. In this framework, the reaction rate coefficient
with an electron is more efficient if the metal is in molecular form is given by (Greene and Kuppermann [1968]):
compared to it being in atomic form. In other words the reaction ∫ ∞
k(T )A→B = v(E)f(E, T )σ(E)A→BdE (3.1)
MX+ + e− → M+X+ hν (2.8) 0
is significantly faster than the reaction where v(E) is the velocity of the colliding species relative to each
other and E is their energy. f(E, T ) is the Boltzmann distribution:
M+ + e− → M+ hν (2.9) √
E −E/kBT
where M+ is an arbitrary metal ion. This effect is important for f(E, T ) = 2 e (3.2)π(kBT )3
astrochemical models where having a large amount of electrons in
the gas inhibits the chemical evolution of H, C, N, O, and S via σ(E)A→B is the reaction cross section for the transition from the
ion/molecule reactions (Mitchell [1978], Graedel et al. [1982]). scattering electronic state characterised by A to the electronic state
characterized by B. A reaction cross section is a combination of
14 15
2. Background
Since the first reaction in the second pathway (equation 2.5) is the
limiting one, the amount of CH created will be restricted by the
amount of H2. CH can therefore be used as a proxy for molecular hy-
drogen (Wiesemeyer et al. [2018]) assuming that CH is predominantly
formed by the process suggested by Black and Dalgarno.
Another possible pathway to create CH is through the endother- Chapter 3
mic reaction:
C+ +H2 + 0.396eV → CH+ +H (2.7)
Computational methods for
CH+ is then converted to CH as shown in equation 2.4. Since energy
is required for the endothermic reaction (equation 2.7), CH can also radiative association
be used as a tracer for shocks (Xu and Li [2016]).
Metal ions
This chapter begins by defining reaction rate constants and reaction
The importance of gas phase metals has been questioned until several cross sections. Later the methods used to calculate the reaction rate
molecules containing the metals Na, Mg, Al and K were detected coefficient and reaction cross sections are described.
in the star IRC+10216: NaCl, AlF, AlCl, KCl, Na(CN), MgCN,
and MgNC (Cernicharo and Guelin [1987], Kawaguchi et al. [1993],
Turner et al. [1994], Ziurys et al. [1994], Ziurys et al. [1995]), prov- 3.1 Reaction rate coefficient
ing that metals can form molecules in gas phase under interstellar
conditions. Consider a reaction involving a transition between two different
Formation of molecules containing metal ions can have a big im- states A and B, which could be, for example, different electronic or
pact on the amount of electrons in the gas as radiative recombination rovibrational levels. In this framework, the reaction rate coefficient
with an electron is more efficient if the metal is in molecular form is given by (Greene and Kuppermann [1968]):
compared to it being in atomic form. In other words the reaction ∫ ∞
k(T )A→B = v(E)f(E, T )σ(E)A→BdE (3.1)
MX+ + e− → M+X+ hν (2.8) 0
is significantly faster than the reaction where v(E) is the velocity of the colliding species relative to each
other and E is their energy. f(E, T ) is the Boltzmann distribution:
M+ + e− → M+ hν (2.9) √
E −E/kBT
where M+ is an arbitrary metal ion. This effect is important for f(E, T ) = 2 e (3.2)π(kBT )3
astrochemical models where having a large amount of electrons in
the gas inhibits the chemical evolution of H, C, N, O, and S via σ(E)A→B is the reaction cross section for the transition from the
ion/molecule reactions (Mitchell [1978], Graedel et al. [1982]). scattering electronic state characterised by A to the electronic state
characterized by B. A reaction cross section is a combination of
14 15
3. Computational methods for radiative association 3.2. Perturbation theory
from tunneling through a barrier, which is not allowed classically,
the semiclassical and classical methods can be complemented by
Breit–Wigner theory in order to treat the resonance contribution.
For the triatomic system we used a method developed by Stoecklin
(Stoecklin et al. [2013b]) based on the photodissociation theory by
Band et al. (Band et al. [1981]) and Balint-Kurti et al. (Balint-Kurti
and Shapiro [1981]) together with the driven equations method by
Heather and Light [1983]. Since photodissociation is the reverse
process of RA some changes were made by Stoecklin et al. [2013a].
A more detailed descriptions of these methods will be given below.
3.2 Perturbation theory
The quantum mechanical method based on perturbation theory is a
rigorous but computationally expensive way to obtain the reaction
Figure 3.1: A schematic illustration of the potential energy curve cross sections compared to the semiclassical and classical methods.
showing how quasibound states give rise to resonances. The first step in obtaining the reaction cross sections is to calcu-
late the wave functions of the scattering and bound states, which
can be done by solving the diatomic Schrödinger equation (SE):
[ 2 2 ]
collision cross section and reaction probability. For the RA process, − d + Veff (r, J) Ψ = EΨ (3.3)
the reaction cross sections will include both direct contribution 2µ dr2
from passing over the potential barrier and resonance contribution
associated with quasibound states that are illustrated in Figure 3.1. where µ is the reduced mass and Veff (r, J) is the effective potential
The quasibound states arise due to a barrier in the potential energy as a function of the inter-atomic separation r. The effective potential
2
curve prolonging the lifetime of the state. The longer the lifetime of consists of the potential energy V (r) and the centrifugal term  J(J+
2
the quasibound state the higher is the probability of transitioning 1)/2µr :
into a bound state. This increase in probability appears as peaks in 2J(J + 1)
the cross sections as a function of energy and are called resonances. Veff (r, J) = V (r) + (3.4)2
The positions of the peaks match the energies of the quasibound 2µr
states. When calculating the wave functions the problem is divided into
In this study, several different methods were used to obtain the two cases that correspond to E > 0 and E < 0. E = 0 is set to be
reaction cross sections. For the diatomic molecules the semiclassical equal to the energy of the reactants. The E > 0 case corresponds to
and classical methods were used and compared to the results ob- scattering states and the wave functions are obtained by numerically
tained from a method based on perturbation theory. Using quantum integrating the SE using Numerov’s method, see next chapter. When
mechanical perturbation theory one can compute both the direct and E < 0, the states are bound and their corresponding wave functions
the indirect contributions. Since the resonance contribution comes are found by using the DVR method, as explained in the next chapter.
16 17
3. Computational methods for radiative association 3.2. Perturbation theory
from tunneling through a barrier, which is not allowed classically,
the semiclassical and classical methods can be complemented by
Breit–Wigner theory in order to treat the resonance contribution.
For the triatomic system we used a method developed by Stoecklin
(Stoecklin et al. [2013b]) based on the photodissociation theory by
Band et al. (Band et al. [1981]) and Balint-Kurti et al. (Balint-Kurti
and Shapiro [1981]) together with the driven equations method by
Heather and Light [1983]. Since photodissociation is the reverse
process of RA some changes were made by Stoecklin et al. [2013a].
A more detailed descriptions of these methods will be given below.
3.2 Perturbation theory
The quantum mechanical method based on perturbation theory is a
rigorous but computationally expensive way to obtain the reaction
Figure 3.1: A schematic illustration of the potential energy curve cross sections compared to the semiclassical and classical methods.
showing how quasibound states give rise to resonances. The first step in obtaining the reaction cross sections is to calcu-
late the wave functions of the scattering and bound states, which
can be done by solving the diatomic Schrödinger equation (SE):
[ 2 2 ]
collision cross section and reaction probability. For the RA process, − d + Veff (r, J) Ψ = EΨ (3.3)
the reaction cross sections will include both direct contribution 2µ dr2
from passing over the potential barrier and resonance contribution
associated with quasibound states that are illustrated in Figure 3.1. where µ is the reduced mass and Veff (r, J) is the effective potential
The quasibound states arise due to a barrier in the potential energy as a function of the inter-atomic separation r. The effective potential
2
curve prolonging the lifetime of the state. The longer the lifetime of consists of the potential energy V (r) and the centrifugal term  J(J+
2
the quasibound state the higher is the probability of transitioning 1)/2µr :
into a bound state. This increase in probability appears as peaks in 2J(J + 1)
the cross sections as a function of energy and are called resonances. Veff (r, J) = V (r) + (3.4)2
The positions of the peaks match the energies of the quasibound 2µr
states. When calculating the wave functions the problem is divided into
In this study, several different methods were used to obtain the two cases that correspond to E > 0 and E < 0. E = 0 is set to be
reaction cross sections. For the diatomic molecules the semiclassical equal to the energy of the reactants. The E > 0 case corresponds to
and classical methods were used and compared to the results ob- scattering states and the wave functions are obtained by numerically
tained from a method based on perturbation theory. Using quantum integrating the SE using Numerov’s method, see next chapter. When
mechanical perturbation theory one can compute both the direct and E < 0, the states are bound and their corresponding wave functions
the indirect contributions. Since the resonance contribution comes are found by using the DVR method, as explained in the next chapter.
16 17
3. Computational methods for radiative association 3.3. Semi-Classical method
The calculated wave functions are used to obtain the transition dipole follows the Franck-Condon principle (Atkins and Paula [2006]) and
moment matrix elements: has a significantly higher probability compared to the case without
∫ ∞ conservation of position and momenta. The optical potential can
M ΛEΛJ,v′Λ′J ′ = FEJ(r)DΛΛ′(r)ΨΛ′v′J ′(r)dr (3.5) then be written as:
0
i
where DΛΛ′ is either the transition dipole moment or the permanent Vopt(R) = − AΛ→Λ′(R) (3.8)2
dipole moment, FΛEJ(r) is the initial wave function and ΨΛ′v′J ′ is the
final wave function. After obtaining the transition dipole moment where AΛ→Λ′(R) is the probability to transition from an electronic
matrix elements, it is possible to calculate the reaction cross sections: state denoted by Λ to another electronic state denoted by Λ′:
( ) 32π3k ( ) 2e 2− δ0,Λ+Λ′ DΛΛ′(R)3
32 π42 1 ω A ′(R) = (3.9)EΛ′v′J ′ Λ→Λ 3
σ 2ΛJ,Λ′v′J ′ = PΛ SΛJ→Λ′J ′ |MEΛJ,v′Λ′J ′ | (3.6) 3 2− δ0,Λ λΛΛ′(R)
12  c30 µE 2π
where DΛΛ′(r) is the transition dipole moment, λΛΛ′(r) is the wave-
where c is the velocity of light, ω is the the frequency of the emitted length corresponding to the transition between two electronic states
photon, SΛJ→Λ′J ′ are the the Hönl-London factors (Hönl and London and ke is the Coulomb constant. Since bound states are the ones of
[1925]) and PΛ is the statistical weight factor: interest, AΛ→Λ′(R) needs to fulfill requirements necessary to enable
(2S + 1)(2− δ ) transition to a bound state otherwise it is set to zero. The two0,Λ
PΛ = (3.7) requirements are that the relative translational energy E should be
(2LA + 1)(2SA + 1)(2LB + 1)(2SB + 1)
smaller than the energy of the emitted photon Eph and also that for
where S is the spin of the diatom, δ0,Λ is the Kronecker delta. L , L a certain J there should be at least one R for which the potentialA B
and S , S are the electronic angular momentum and spin quantum energy is smaller than the continuum potential energy:A B
numbers corresponding to the A and B atoms.
2J(J + 1)
V (R) + < V (R → ∞) (3.10)
2µR2
3.3 Semi-Classical method
Considering the two conditions above, the expression for the transi-
The semiclassical method is computationally cheaper than the quan- tion probability can be written as:
tum mechanical method mentioned in the section above and is based {
on the optical potential method. The optical potential is a negative AΛ→Λ′(R), It is possible to form a bound state
′
imaginary term which is added to the Hamiltonian to absorb a part AΛ→Λ (R) = (3.11)0, Otherwise
of the scattering wave function, thus introducing a loss of flux due
to spontaneous emission. Including the optical potential in the SE gives:
Zygelman and Dalgarno [1988] have developed an optical po-
tential when studying radiative quenching. By imposing that the [ ]2 d2 2(J(J + 1)− Λ2) i ′
energy of the emitted photon has to be exactly equal to the energy − +VΛ(R)+ − AEJΛ→Λ′−E ΨΛ→ΛEJ = 02 2
difference between the two states involved in the transition, the 2µ dR 2µR 2
wave vectors and positions are conserved. This type of transition (3.12)
18 19
3. Computational methods for radiative association 3.3. Semi-Classical method
The calculated wave functions are used to obtain the transition dipole follows the Franck-Condon principle (Atkins and Paula [2006]) and
moment matrix elements: has a significantly higher probability compared to the case without
∫ ∞ conservation of position and momenta. The optical potential can
M ΛEΛJ,v′Λ′J ′ = FEJ(r)DΛΛ′(r)ΨΛ′v′J ′(r)dr (3.5) then be written as:
0
i
where DΛΛ′ is either the transition dipole moment or the permanent Vopt(R) = − AΛ→Λ′(R) (3.8)2
dipole moment, FΛEJ(r) is the initial wave function and ΨΛ′v′J ′ is the
final wave function. After obtaining the transition dipole moment where AΛ→Λ′(R) is the probability to transition from an electronic
matrix elements, it is possible to calculate the reaction cross sections: state denoted by Λ to another electronic state denoted by Λ′:
( ) 32π3k ( ) 2e 2− δ0,Λ+Λ′ DΛΛ′(R)3
32 π42 1 ω A ′(R) = (3.9)EΛ′v′J ′ Λ→Λ 3
σ 2ΛJ,Λ′v′J ′ = PΛ SΛJ→Λ′J ′ |MEΛJ,v′Λ′J ′ | (3.6) 3 2− δ0,Λ λΛΛ′(R)
12  c30 µE 2π
where DΛΛ′(r) is the transition dipole moment, λΛΛ′(r) is the wave-
where c is the velocity of light, ω is the the frequency of the emitted length corresponding to the transition between two electronic states
photon, SΛJ→Λ′J ′ are the the Hönl-London factors (Hönl and London and ke is the Coulomb constant. Since bound states are the ones of
[1925]) and PΛ is the statistical weight factor: interest, AΛ→Λ′(R) needs to fulfill requirements necessary to enable
(2S + 1)(2− δ ) transition to a bound state otherwise it is set to zero. The two0,Λ
PΛ = (3.7) requirements are that the relative translational energy E should be
(2LA + 1)(2SA + 1)(2LB + 1)(2SB + 1)
smaller than the energy of the emitted photon Eph and also that for
where S is the spin of the diatom, δ0,Λ is the Kronecker delta. L , L a certain J there should be at least one R for which the potentialA B
and S , S are the electronic angular momentum and spin quantum energy is smaller than the continuum potential energy:A B
numbers corresponding to the A and B atoms.
2J(J + 1)
V (R) + < V (R → ∞) (3.10)
2µR2
3.3 Semi-Classical method
Considering the two conditions above, the expression for the transi-
The semiclassical method is computationally cheaper than the quan- tion probability can be written as:
tum mechanical method mentioned in the section above and is based {
on the optical potential method. The optical potential is a negative AΛ→Λ′(R), It is possible to form a bound state
′
imaginary term which is added to the Hamiltonian to absorb a part AΛ→Λ (R) = (3.11)0, Otherwise
of the scattering wave function, thus introducing a loss of flux due
to spontaneous emission. Including the optical potential in the SE gives:
Zygelman and Dalgarno [1988] have developed an optical po-
tential when studying radiative quenching. By imposing that the [ ]2 d2 2(J(J + 1)− Λ2) i ′
energy of the emitted photon has to be exactly equal to the energy − +VΛ(R)+ − AEJΛ→Λ′−E ΨΛ→ΛEJ = 02 2
difference between the two states involved in the transition, the 2µ dR 2µR 2
wave vectors and positions are conserved. This type of transition (3.12)
18 19
3. Computational methods for radiative association 3.5. Breit-Wigner method
Solving the equation above results in the following expression for where ω is the frequency of the emitted photon, v is the relative
the cross section: velocity and N(b, V, T ) is the number of collisions. Ĩ(b, v, ω) is
2 ∑∞ ( ) the emission intensity and can be obtained by taking the Fourier2π Λ→Λ′
σΛ→Λ′(E) = PΛ (2J + 1) 1− E−4Im(δEJ ) (3.13) transform of the Larmor power formula:
µE
J=0 ∣ ∣
4 ∣ ∫ ∞ ∣2
The expression for the semiclassical cross section can be obtained 2ωĨ(b, v, ω) = P ∣∣ eiωtΛ D(b, v, t)dt∣∣ (3.18)Λ→Λ′ 3c3under the assumption that δ is small, and by summing over π(4π0) −∞EJ
many J :
∫ where D(b, v, t) is the molecular dipole moment vector and PΛ en-
2π2 ∞ Λ→Λ sures that only collisions on the chosen potential energy surfaceσΛ→Λ′(E) = PΛ (2J + 1)Im(δ
µE EJ
)dJ (3.14)
0 are included. The rate constant can be obtained by dividing the
emission intensity by the photon energy and then integrating over
In the next step the J ’s can be excluded from the equation by taking
2 2 2 all photon frequencies with specific boundary conditions that onlythe classical limit J(J + 1) = (µνb) = 2µEb where b is the
allow transitions to bound states. The boundaries are ω
impact parameter: min
= E/
∫ and ωmax = (E − Vmin)/. The lower boundary is making sure that∞
Λ→Λ′ at least the collision energy E is carried away by the photon. OnσΛ→Λ′(E) = 8πPΛ bIm(δEb )db (3.15)
0 the other hand, the photon cannot carry away more energy than the
difference between the collision energy E and the lowest part of the
Using the WKB (Bransden and Joachain [2000]) approximation for
potential energy. Keeping the above mentioned conditions in mind,
the phase shift in the classically allowed region one can obtain:
the rate constant can be written as:
( ) 1 ∫ ∞ ∫ ( ) 1( ) 3 ∫∞ 8 2 1 2 ∞µ 2 AEb
σ (E) = 4π P b √ Λ→Λ′(R)dRdb k(T ) = EΛ→Λ′ Λ
2E − µπ k T0 Rc 1 VΛ(R)/E − b2/R2 ∫ ∞ ∫
b 0 (3.19)
ωmax Ĩ(b, E, ω)
(3.16) 2πb e−E/kbTdωdbdE
0 E/ ω
where Rc is the classical turning point.
Comparing this equation to equation 3.1, one can deduce the expres-
sion for the classical reaction cross section:
3.4 Classical method ∫ ∞ ∫ ∞ ∣∣ ∫ ∞ ∣4 ∣2
σ (E) = P b ω3 iωtThe classical method treats the cases when the transition happens Λ→ ∣Λ Λ e D(b, E, t)dt∣ dωdb
3c34π ∣ ∣
within the same electronic state. Similarly to the semiclassical 0 0 0 −∞
approach, this method does not require computing the wave functions (3.20)
and is also computationally cheaper than the quantum mechanical
method. The starting point when deducing this method (Gustafsson 3.5 Breit-Wigner method
[2013]) is the classically radiated intensity of accelerating molecules:
∫ ∞ ∫ ∞ Since neither the classical nor the semiclassical method can account
I(T, ω) = N(b, V, T )Ĩ(b, v, ω)dbdv (3.17) for resonances, Breit-Wigner theory (Breit and Wigner [1936]) is used
0 0
20 21
3. Computational methods for radiative association 3.5. Breit-Wigner method
Solving the equation above results in the following expression for where ω is the frequency of the emitted photon, v is the relative
the cross section: velocity and N(b, V, T ) is the number of collisions. Ĩ(b, v, ω) is
2 ∑∞ ( ) the emission intensity and can be obtained by taking the Fourier2π Λ→Λ′
σΛ→Λ′(E) = PΛ (2J + 1) 1− E−4Im(δEJ ) (3.13) transform of the Larmor power formula:
µE
J=0 ∣ ∣
4 ∣ ∫ ∞ ∣2
The expression for the semiclassical cross section can be obtained 2ωĨ(b, v, ω) = P ∣∣ eiωtΛ D(b, v, t)dt∣∣ (3.18)Λ→Λ′ 3c3under the assumption that δ is small, and by summing over π(4π0) −∞EJ
many J :
∫ where D(b, v, t) is the molecular dipole moment vector and PΛ en-
2π2 ∞ Λ→Λ sures that only collisions on the chosen potential energy surfaceσΛ→Λ′(E) = PΛ (2J + 1)Im(δ
µE EJ
)dJ (3.14)
0 are included. The rate constant can be obtained by dividing the
emission intensity by the photon energy and then integrating over
In the next step the J ’s can be excluded from the equation by taking
2 2 2 all photon frequencies with specific boundary conditions that onlythe classical limit J(J + 1) = (µνb) = 2µEb where b is the
allow transitions to bound states. The boundaries are ω
impact parameter: min
= E/
∫ and ωmax = (E − Vmin)/. The lower boundary is making sure that∞
Λ→Λ′ at least the collision energy E is carried away by the photon. OnσΛ→Λ′(E) = 8πPΛ bIm(δEb )db (3.15)
0 the other hand, the photon cannot carry away more energy than the
difference between the collision energy E and the lowest part of the
Using the WKB (Bransden and Joachain [2000]) approximation for
potential energy. Keeping the above mentioned conditions in mind,
the phase shift in the classically allowed region one can obtain:
the rate constant can be written as:
( ) 1 ∫ ∞ ∫ ( ) 1( ) 3 ∫∞ 8 2 1 2 ∞µ 2 AEb
σ (E) = 4π P b √ Λ→Λ′(R)dRdb k(T ) = EΛ→Λ′ Λ
2E − µπ k T0 Rc 1 VΛ(R)/E − b2/R2 ∫ ∞ ∫
b 0 (3.19)
ωmax Ĩ(b, E, ω)
(3.16) 2πb e−E/kbTdωdbdE
0 E/ ω
where Rc is the classical turning point.
Comparing this equation to equation 3.1, one can deduce the expres-
sion for the classical reaction cross section:
3.4 Classical method ∫ ∞ ∫ ∞ ∣∣ ∫ ∞ ∣4 ∣2
σ (E) = P b ω3 iωtThe classical method treats the cases when the transition happens Λ→ ∣Λ Λ e D(b, E, t)dt∣ dωdb
3c34π ∣ ∣
within the same electronic state. Similarly to the semiclassical 0 0 0 −∞
approach, this method does not require computing the wave functions (3.20)
and is also computationally cheaper than the quantum mechanical
method. The starting point when deducing this method (Gustafsson 3.5 Breit-Wigner method
[2013]) is the classically radiated intensity of accelerating molecules:
∫ ∞ ∫ ∞ Since neither the classical nor the semiclassical method can account
I(T, ω) = N(b, V, T )Ĩ(b, v, ω)dbdv (3.17) for resonances, Breit-Wigner theory (Breit and Wigner [1936]) is used
0 0
20 21
3. Computational methods for radiative association 3.6. Driven equations method for RA
to add the resonance contribution to the classical and semiclassical coupled angular basis sets are used to expand both the initial and
cross sections. the final wave functions:
Breit-Wigner theory is based on lifetimes of a state that can ∑
Y JM
m
decay either by tunneling through a barrier or via radiation. Both jl (r̂, R̂) = 〈jmjlml‖JM〉y
j(r̂)ymlj l (R̂) (3.25)
processes have a corresponding lifetime, τ rad and τ tunvJΛ→Λ′ vJΛ, and taking
mj ,ml
the inverse of them gives the resonance widths: where j, l and J are the angular momentum quantum numbers
( )
1 1 associated with the operators ĵ, l̂ and Ĵ . mj, ml and M represent
 + = Γ tun + Γ rad Total m
rad tun ν′J ′Λ′ ν′J ′Λ′→Λ′′ = Γν′J ′Λ′Λ′′ (3.21) the projections on the z space fixed axis. y
j
l (r̂) is a diatomicτvJΛ→Λ′ τvJΛ spherical harmonic function. The wave function ϕvj(r) can obtained
The total resonance width is the sum of the radiative and the by solving:
tunneling widths, and if both widths are known, the BW cross m m
hdiatψvj(r)y
j(r̂) =  ψ y j(r̂) (3.26)
sections can be computed: l vj vj(r) l
π2 ∑ (2J ′ + 1)Γ tun Γ rad where vj is the eigenvalue of the free diatomic Hamiltonian hdiat.ν′J ′Λ′ ν′J ′Λ′σ (E) = P →Λ′′Λ′→Λ′′ Λ′ (3.22) When treating a homonuclear diatomic molecule, the two spin
2µE 2′ ′ (E − Eν′J ′Λ′) + (Γ Total 2′ ′ ′ ′′/2)ν J ν J Λ Λ states (para/ortho) can be treated separately. In order to obtain the
radial parts of the total wave function after RA has occurred, the
where Eν′J ′Λ′ is the position of the resonance on the energy axis. following equation has to be solved:
[ ]
d2 l(l + 1) ′ ′ ′ ′ ′ ′
3.6 Driven equations method for RA − + k2 (E)− U v j l v j l
dR2 R2 vj vjl
(R) χvjl (R) =
In the driven equation method (Band et al. [1981], Balint-Kurti αJ ′M ′λvjlJM(R) (3.27)
and Shapiro [1981], Heather and Light [1983]) the matter-radiation
′ ′ ′
interaction operator is used to connect the initial (scattering) and where kvj are channel wave vectors and U
v j l
vjl are the matrix elements
the final (bound) wave functions. The wave functions are expanded ′of the intermolecular potential. The λαJ M
′
vjlJM(R) term in the right
using coupled angular basis sets and, here, Jacobi coordinates (Zhang hand side is called the driving term and it arises from the dipolar
[1998], Arthurs and Dalgarno [1960]) in a space fixed frame are used: coupling of the initial scattering and final bound states within the
1 ∑ dipolar approximation.
ΨJMi (R, r, θ) = χ
JM JM
Rr v,j,l
(R)ϕvj(r)Yjl (R̂, r̂) (3.23) ∫
v,j,l λαJ
′M ′ ∗
vjlJM(R) =− 2µ dR̂dr̂drϕνj(r)[Y JMjl (r̂, R̂)]∗
(3.28)
′ ′
αJ ′M ′ 1
∑ ′ ′ ′ ′ 
 α,J ,M
Ψf (R, r, θ) = ω
αJ M J M
v′,j′,l′ (R)ϕv′j′(r)Yj′l′ (R̂, r̂) (3.24)
× µ(R,
r)Ψf (R, r, θ)
Rr
v′,j′,l′
When a charged atom is reacting with a neutral diatom a reasonable
where Ψi(R, r, θ) is the initial wave function and Ψ
α
f (R, r, θ) is the approximation to make is that the dipole moment lies along the
′ ′
final wave function. χJMv,j,l(R) and ω
αJ M
v′,j′,l′ (R) are the R-dependent intermolecular axis. This assumption has been previously tested
radial parts of the initial and final wave functions while ϕvj(r) and and confirmed by Stoecklin et al. [2013a]. Another approximation
ϕv′j′(r) are the the r-dependent rovibrational parts. The same that was made was that the dependence of the dipole moment on
22 23
3. Computational methods for radiative association 3.6. Driven equations method for RA
to add the resonance contribution to the classical and semiclassical coupled angular basis sets are used to expand both the initial and
cross sections. the final wave functions:
Breit-Wigner theory is based on lifetimes of a state that can ∑
Y JM
m
decay either by tunneling through a barrier or via radiation. Both jl (r̂, R̂) = 〈jmjlml‖JM〉y
j(r̂)ymlj l (R̂) (3.25)
processes have a corresponding lifetime, τ rad and τ tunvJΛ→Λ′ vJΛ, and taking
mj ,ml
the inverse of them gives the resonance widths: where j, l and J are the angular momentum quantum numbers
( )
1 1 associated with the operators ĵ, l̂ and Ĵ . mj, ml and M represent
 + = Γ tun + Γ rad Total m
rad tun ν′J ′Λ′ ν′J ′Λ′→Λ′′ = Γν′J ′Λ′Λ′′ (3.21) the projections on the z space fixed axis. y
j
l (r̂) is a diatomicτvJΛ→Λ′ τvJΛ spherical harmonic function. The wave function ϕvj(r) can obtained
The total resonance width is the sum of the radiative and the by solving:
tunneling widths, and if both widths are known, the BW cross m m
hdiatψvj(r)y
j(r̂) =  ψ y j(r̂) (3.26)
sections can be computed: l vj vj(r) l
π2 ∑ (2J ′ + 1)Γ tun Γ rad where vj is the eigenvalue of the free diatomic Hamiltonian hdiat.ν′J ′Λ′ ν′J ′Λ′σ (E) = P →Λ′′Λ′→Λ′′ Λ′ (3.22) When treating a homonuclear diatomic molecule, the two spin
2µE 2′ ′ (E − Eν′J ′Λ′) + (Γ Total 2′ ′ ′ ′′/2)ν J ν J Λ Λ states (para/ortho) can be treated separately. In order to obtain the
radial parts of the total wave function after RA has occurred, the
where Eν′J ′Λ′ is the position of the resonance on the energy axis. following equation has to be solved:
[ ]
d2 l(l + 1) ′ ′ ′ ′ ′ ′
3.6 Driven equations method for RA − + k2 (E)− U v j l v j l
dR2 R2 vj vjl
(R) χvjl (R) =
In the driven equation method (Band et al. [1981], Balint-Kurti αJ ′M ′λvjlJM(R) (3.27)
and Shapiro [1981], Heather and Light [1983]) the matter-radiation
′ ′ ′
interaction operator is used to connect the initial (scattering) and where kvj are channel wave vectors and U
v j l
vjl are the matrix elements
the final (bound) wave functions. The wave functions are expanded ′of the intermolecular potential. The λαJ M
′
vjlJM(R) term in the right
using coupled angular basis sets and, here, Jacobi coordinates (Zhang hand side is called the driving term and it arises from the dipolar
[1998], Arthurs and Dalgarno [1960]) in a space fixed frame are used: coupling of the initial scattering and final bound states within the
1 ∑ dipolar approximation.
ΨJMi (R, r, θ) = χ
JM JM
Rr v,j,l
(R)ϕvj(r)Yjl (R̂, r̂) (3.23) ∫
v,j,l λαJ
′M ′ ∗
vjlJM(R) =− 2µ dR̂dr̂drϕνj(r)[Y JMjl (r̂, R̂)]∗
(3.28)
′ ′
αJ ′M ′ 1
∑ ′ ′ ′ ′ 
 α,J ,M
Ψf (R, r, θ) = ω
αJ M J M
v′,j′,l′ (R)ϕv′j′(r)Yj′l′ (R̂, r̂) (3.24)
× µ(R,
r)Ψf (R, r, θ)
Rr
v′,j′,l′
When a charged atom is reacting with a neutral diatom a reasonable
where Ψi(R, r, θ) is the initial wave function and Ψ
α
f (R, r, θ) is the approximation to make is that the dipole moment lies along the
′ ′
final wave function. χJMv,j,l(R) and ω
αJ M
v′,j′,l′ (R) are the R-dependent intermolecular axis. This assumption has been previously tested
radial parts of the initial and final wave functions while ϕvj(r) and and confirmed by Stoecklin et al. [2013a]. Another approximation
ϕv′j′(r) are the the r-dependent rovibrational parts. The same that was made was that the dependence of the dipole moment on
22 23
3. Computational methods for radiative association
the vibration of the diatom could be neglected and thus the dipole
moment could be taken as proportional to R. After making these
approximations, equation 3.28 can be simplified:
2 ∑
λαJ
′M ′ −2µ νj′l′J ′ α,J ′ ′
vjlJM(R) = δ
,M
ν,ν′ ΓνjlJ Rωνj′l′ (R) (3.29)m
j′,l′ Chapter 4
′ ′ ′
where Γ νj l JνjlJ are the Clebsch-Gordon coefficients (Zare [1988]).
A Magnus propagator (Guillon and Stoecklin [2009]) was used
to propagate the scattering wave function starting at the classically Numerical methods
forbidden region into the asymptotic region, see previous work by
Stoecklin et al. [2013a] for more details. Finally the reaction cross
section can be expressed as:
[ ] Since wave functions for all scattering and bound states are needed2
8π2 ∑ ′ to obtain the cross sections using the PT method, solving the time-
σv,j(E) = ω
3 Mα,Jα v,j,J(E) (3.30)3k2 c3 independent SE to obtain them is a big part of the computationalvj J,J ′,α problem. Below, two methods commonly used to numerically solve
the SE will be described - the DVR (Lill et al. [1986], Light et al.
where ωα is the frequency of the emitted photon. The driving
α,J ′ [1985]) and the Numerov method (Johnson [1977]). The DVRterms Mv,j,J(E) are integrated over R and summed over all angular method is used to obtain the bound wave functions because it is
momenta and bound states.
best suited for spatially localized wave functions. When treating
more delocalized wave functions the Numerov method is used instead
which was done to obtain the scattering states of the diatomic system.
The quasibound rovibrational energy levels we obtained using the
LEVEL program by Le Roy [2017].
For the triatomic system the scattering states were obtained by
solving the driven equation described in the RA methods chapter.
4.1 DVR
When using the DVR method, the time independent Schrödinger
equation (TISE) is solved on a grid of N points by transforming an
N -dimensional functional space to a point space with the same di-
mension. The first step is to define a variational basis representation
(VBR), where a variable AV BRi,j is represented as:
∫ ∞
AV BR ∗ij = ψi (x)Âψ
∗
j (x)dx (4.1)
−∞
24 25
3. Computational methods for radiative association
the vibration of the diatom could be neglected and thus the dipole
moment could be taken as proportional to R. After making these
approximations, equation 3.28 can be simplified:
2 ∑
λαJ
′M ′ −2µ νj′l′J ′ α,J ′ ′
vjlJM(R) = δ
,M
ν,ν′ ΓνjlJ Rωνj′l′ (R) (3.29)m
j′,l′ Chapter 4
′ ′ ′
where Γ νj l JνjlJ are the Clebsch-Gordon coefficients (Zare [1988]).
A Magnus propagator (Guillon and Stoecklin [2009]) was used
to propagate the scattering wave function starting at the classically Numerical methods
forbidden region into the asymptotic region, see previous work by
Stoecklin et al. [2013a] for more details. Finally the reaction cross
section can be expressed as:
[ ] Since wave functions for all scattering and bound states are needed2
8π2 ∑ ′ to obtain the cross sections using the PT method, solving the time-
σv,j(E) = ω
3 Mα,Jα v,j,J(E) (3.30)3k2 c3 independent SE to obtain them is a big part of the computationalvj J,J ′,α problem. Below, two methods commonly used to numerically solve
the SE will be described - the DVR (Lill et al. [1986], Light et al.
where ωα is the frequency of the emitted photon. The driving
α,J ′ [1985]) and the Numerov method (Johnson [1977]). The DVRterms Mv,j,J(E) are integrated over R and summed over all angular method is used to obtain the bound wave functions because it is
momenta and bound states.
best suited for spatially localized wave functions. When treating
more delocalized wave functions the Numerov method is used instead
which was done to obtain the scattering states of the diatomic system.
The quasibound rovibrational energy levels we obtained using the
LEVEL program by Le Roy [2017].
For the triatomic system the scattering states were obtained by
solving the driven equation described in the RA methods chapter.
4.1 DVR
When using the DVR method, the time independent Schrödinger
equation (TISE) is solved on a grid of N points by transforming an
N -dimensional functional space to a point space with the same di-
mension. The first step is to define a variational basis representation
(VBR), where a variable AV BRi,j is represented as:
∫ ∞
AV BR ∗ij = ψi (x)Âψ
∗
j (x)dx (4.1)
−∞
24 25
4. Numerical methods 4.2. Numerov
where ψ∗i (x) and ψ
∗
j (x) are basis functions. It is then possible to different trajectories are tested until one that fulfills those boundary
use a set of N points {xk} to approximate the integral above as a conditions is found.
quadrature: Starting by rewriting the TISE:
∑ ( ) [N d2 ]
V BR ≈  +Q(R) ψ(R) = 0 (4.7)Aij wkψi (xk) Âψj (xk) (4.2) dR2
k=1
The second term in the equation above, Q(R), is given by:
where wk is the quadrature weight. The following step is to use a 2µ[ ]
unitary transformation that transforms the VBR functional space Q(R) = E − Vl(R) (4.8)2
into a DVR point space:
where µ is the reduced mass and Vl(R) is the effective potential
ADV R = UAV BRU † (4.3) energy. One can solve the TISE by integrating using three point
recursion:
where the transformation matrix U is: [ ] [ ] [ ]
1−Tk+1 ψ(Rk+1)− 2+10Tk ψ(Rk)+ 1−Tk−1 ψ(Rk−1) = 0 (4.9)
1
U = (Y †Y )− 2Y † (4.4)
where Rk are grid points on a uniform grid ranging from R0 up to
and Y †ij = ψ
∗
i (xj). RK and Tk is:
It is then possible to rewrite the potential and kinetic energy [∆R]2
operators in DVR. One of the advantages of this approach is that the Tk = − Q(Rk) (4.10)
12
matrix corresponding to the potential energy operator is diagonal:
A more numerically stable method is the renormalized Numerov
V DV Rij = V (xi)δij (4.5) method where the transformation:
(1− T )ψ
The kinetic energy operator can be expressed as: k+1 k+1Rk = (4.11)
(1− Tk)ψk
2 N∑−1 d2
TDV R
ψn(xj)
ij = − ∆x ψn(xi) (4.6)
is made to obtain a two point recursion instead of the three point
2m dx2 recursion:
n=1
−1
where ∆x is the grid spacing. It is most common to use a uniform Rk = Uk −Rk−1 (4.12)
grid, but another type of grid can be chosen if it better suits the where Uk = (2 + 10Tk)/(1− Tk). Equation 4.12 can be used to find
problem at hand. the solution to the TISE (equation 4.7) at the outer boundary RK
by starting the iteration at R0.
4.2 Numerov When searching for the solution to the TISE at the inner bound-
ary R0 an expression similar to equation 4.12 is used:
The Numerov method belongs to the group called shooting methods, R̂k = U − R̂−1k (4.13)
where integration of differential equations is done iteratively. It is k+1
called a shooting method because in order to solve the differential If the right boundary conditions are applied, the Numerov method
equation with certain boundary conditions, i.e. for bound states, can be used to obtain both the bound and scattering wave functions.
26 27
4. Numerical methods 4.2. Numerov
where ψ∗i (x) and ψ
∗
j (x) are basis functions. It is then possible to different trajectories are tested until one that fulfills those boundary
use a set of N points {xk} to approximate the integral above as a conditions is found.
quadrature: Starting by rewriting the TISE:
∑ ( ) [N d2 ]
V BR ≈  +Q(R) ψ(R) = 0 (4.7)Aij wkψi (xk) Âψj (xk) (4.2) dR2
k=1
The second term in the equation above, Q(R), is given by:
where wk is the quadrature weight. The following step is to use a 2µ[ ]
unitary transformation that transforms the VBR functional space Q(R) = E − Vl(R) (4.8)2
into a DVR point space:
where µ is the reduced mass and Vl(R) is the effective potential
ADV R = UAV BRU † (4.3) energy. One can solve the TISE by integrating using three point
recursion:
where the transformation matrix U is: [ ] [ ] [ ]
1−Tk+1 ψ(Rk+1)− 2+10Tk ψ(Rk)+ 1−Tk−1 ψ(Rk−1) = 0 (4.9)
1
U = (Y †Y )− 2Y † (4.4)
where Rk are grid points on a uniform grid ranging from R0 up to
and Y †ij = ψ
∗
i (xj). RK and Tk is:
It is then possible to rewrite the potential and kinetic energy [∆R]2
operators in DVR. One of the advantages of this approach is that the Tk = − Q(Rk) (4.10)
12
matrix corresponding to the potential energy operator is diagonal:
A more numerically stable method is the renormalized Numerov
V DV Rij = V (xi)δij (4.5) method where the transformation:
(1− T )ψ
The kinetic energy operator can be expressed as: k+1 k+1Rk = (4.11)
(1− Tk)ψk
2 N∑−1 d2
TDV R
ψn(xj)
ij = − ∆x ψn(xi) (4.6)
is made to obtain a two point recursion instead of the three point
2m dx2 recursion:
n=1
−1
where ∆x is the grid spacing. It is most common to use a uniform Rk = Uk −Rk−1 (4.12)
grid, but another type of grid can be chosen if it better suits the where Uk = (2 + 10Tk)/(1− Tk). Equation 4.12 can be used to find
problem at hand. the solution to the TISE (equation 4.7) at the outer boundary RK
by starting the iteration at R0.
4.2 Numerov When searching for the solution to the TISE at the inner bound-
ary R0 an expression similar to equation 4.12 is used:
The Numerov method belongs to the group called shooting methods, R̂k = U − R̂−1k (4.13)
where integration of differential equations is done iteratively. It is k+1
called a shooting method because in order to solve the differential If the right boundary conditions are applied, the Numerov method
equation with certain boundary conditions, i.e. for bound states, can be used to obtain both the bound and scattering wave functions.
26 27
4. Numerical methods 4.3. LEVEL
R R R to:MIN+RH X MAX
R ∫
RMIN RMAX-RH √
ψ(r) ∝ [V (r, J)−E]1/4exp(− 2µ/2 ) (V (r, J)−E)1/2dr (4.14)
Figure 4.1: A schematic picture of the relevant points when describing For quasibound states the wavefunction at RMAX is defined by the
the intergration performed by LEVEL. slope of an Airy function at the outermost turning point.
4.3 LEVEL
LEVEL (Le Roy [2017]) is a program that can be used to locate a
chosen number of bound and quasibound states by solving the one
dimensional or radial Schrödinger equation.
The biggest part of the calculation is finding the eigenvalues for
which Cooley-Cashion-Zare (Cooley [1961], Zare and Kashion [1963],
Zare [1963]) routine is used. The routine has been modified in order
to be able to locate quasibound states and to obtain their widths.
The integration of the Schrödinger equation is performed between
RMIN and RMAX with the step size RH using the Numerov method.
The first step in finding the eigenvalues is to choose a trial energy.
Thereafter the Schrödinger equation is integrated from RMIN to RX
(see Fig. 4.3) and from from RMAX to RX . The results from these
two integrations are compared at RX and the discontinuety between
them is used to obtain a new trial energy.
Starting the integration one needs to specify two adjacent points
at the beginning (RMIN , RMIN +RH) and the end of the integration
interval (RMAX , RMAX −RH). Both bound and quasibound states
will have the same boundary conditions at RMIN but the wave
function at RMAX will be initialized differently depending on if the
state is bound or unbound. At RMIN the wavefunction is set to zero
since the wavefunctions of diatomic molecules die off quickly at short
internuclear distances due to the steepness of the potential energy
in that area.
For bound states, an arbitrary value of the wavefunction is used
at RMAX . At the adjacent point the wavefunction is proportional
28 29
4. Numerical methods 4.3. LEVEL
R R R to:MIN+RH X MAX
R ∫
RMIN RMAX-RH √
ψ(r) ∝ [V (r, J)−E]1/4exp(− 2µ/2 ) (V (r, J)−E)1/2dr (4.14)
Figure 4.1: A schematic picture of the relevant points when describing For quasibound states the wavefunction at RMAX is defined by the
the intergration performed by LEVEL. slope of an Airy function at the outermost turning point.
4.3 LEVEL
LEVEL (Le Roy [2017]) is a program that can be used to locate a
chosen number of bound and quasibound states by solving the one
dimensional or radial Schrödinger equation.
The biggest part of the calculation is finding the eigenvalues for
which Cooley-Cashion-Zare (Cooley [1961], Zare and Kashion [1963],
Zare [1963]) routine is used. The routine has been modified in order
to be able to locate quasibound states and to obtain their widths.
The integration of the Schrödinger equation is performed between
RMIN and RMAX with the step size RH using the Numerov method.
The first step in finding the eigenvalues is to choose a trial energy.
Thereafter the Schrödinger equation is integrated from RMIN to RX
(see Fig. 4.3) and from from RMAX to RX . The results from these
two integrations are compared at RX and the discontinuety between
them is used to obtain a new trial energy.
Starting the integration one needs to specify two adjacent points
at the beginning (RMIN , RMIN +RH) and the end of the integration
interval (RMAX , RMAX −RH). Both bound and quasibound states
will have the same boundary conditions at RMIN but the wave
function at RMAX will be initialized differently depending on if the
state is bound or unbound. At RMIN the wavefunction is set to zero
since the wavefunctions of diatomic molecules die off quickly at short
internuclear distances due to the steepness of the potential energy
in that area.
For bound states, an arbitrary value of the wavefunction is used
at RMAX . At the adjacent point the wavefunction is proportional
28 29
Chapter 5
Results
Below you can find a summary of the results obtained in the three
papers and the manuscript included in this thesis. I do not to include
all the results from the papers but the ones that I find necessary in
order to be able to explain my research.
5.1 Paper I
In the first paper, formation of CH and CD through RA was studied.
The goal was to obtain reaction cross sections in order to be able to
calculate reaction rate constants. In this section I will only present
the results corresponding to the formation of the CH molecule and,
to see the results for the formation of the CD molecule, see paper I.
The three reaction channels that were treated in the calculations
are:
( ) ( )
C(3P) + H(2S) → CH(X
2Π) → CH
2
(X Π) + ω (5.1)
C(3P) + H(2S) → CH A2∆ → CH X2( ) ( Π )+ ω (5.2)
C(3P) + H(2S) → CH B2Σ− → CH X2Π + ω (5.3)
The first and the third reactions are RA while the second reaction is
inverse predissociation. The potential energy curves corresponding
to the electronic states A2∆, B2Σ− and X2Π can be seen in Figure
5.1.
As mentioned in the introduction, the cross sections consist
of both direct and resonance contribution. In case of diatomic
31
Chapter 5
Results
Below you can find a summary of the results obtained in the three
papers and the manuscript included in this thesis. I do not to include
all the results from the papers but the ones that I find necessary in
order to be able to explain my research.
5.1 Paper I
In the first paper, formation of CH and CD through RA was studied.
The goal was to obtain reaction cross sections in order to be able to
calculate reaction rate constants. In this section I will only present
the results corresponding to the formation of the CH molecule and,
to see the results for the formation of the CD molecule, see paper I.
The three reaction channels that were treated in the calculations
are:
( ) ( )
C(3P) + H(2S) → CH(X
2Π) → CH
2
(X Π) + ω (5.1)
C(3P) + H(2S) → CH A2∆ → CH X2( ) ( Π )+ ω (5.2)
C(3P) + H(2S) → CH B2Σ− → CH X2Π + ω (5.3)
The first and the third reactions are RA while the second reaction is
inverse predissociation. The potential energy curves corresponding
to the electronic states A2∆, B2Σ− and X2Π can be seen in Figure
5.1.
As mentioned in the introduction, the cross sections consist
of both direct and resonance contribution. In case of diatomic
31
5. Results 5.1. Paper I
molecules, the the resonances show up due to the barrier on the 7
effective potential energy curve that gives rise to quasi bound states. X 2
B2 -
This is illustrated in Figure 3.1 which shows a plot of the potential 6 A2
C 2 +
energy against the distance between the colliding atoms. At large
separation between the atoms, the potential energy is approaching 5
the asymptotic energy which is the potential energy corresponding
4
to the atoms being at infinite separation. All the energy levels below
the asymptotic energy are bound states. There are some energy 3
levels that are above the asymptotic energy level but below the top
of the barrier. These levels are not bound because they can tunnel 2
out through the barrier. However, the barrier makes it harder to
1
dissociate and these levels are therefore called quasibound.
If an incoming atom has an energy that matches the energy of the 0
0.5 1 1.5 2 2.5 3 3.5
quasibound state, it is probable that it will tunnel through the barrier. r (Aangstrom)
Since the barrier prohibits the atom from dissociating straight away,
it gets a higher probability to emit a photon and transition to one of Figure 5.1: Potential energy curves for the X2Π, B2Σ−, A2∆ and
the bound states. When this happens, a resonance, corresponding C2Σ+ states as functions of the internuclear distance r.
to an increase in probability, will appear in the cross section which
can be seen both in Figures 5.2 and 5.3. The cross sections are
plotted against the energy of the colliding atoms and each resonance
has the position on the energy axis that is equal to the energy of a a transition between two different electronic states and in that case
quasibound state. Inverse predissociation, on the other hand, occurs the SCl method could be used to obtain the cross sections.
when two coupled molecular states intersect, allowing the molecule The cross sections obtained with the SCl and Cl methods do
to transfer between them. In our case, this is only possible for energy not have any resonances. In order to add resonance contribution
levels lying above the asymptotic energy of the X2Π state and below to the cross sections, the quasibound states were calculated using
the one for the A2∆ state. the computational program LEVEL (Le Roy [2017]). As mentioned
As discussed in the ’Computational methods for RA’ chapter, above, the position of the resonances on the energy axis correspond
three different methods were used to obtain the cross sections for the to the energies of the quasibound states. This in turn means that
formation of the CH and CD molecules: the semiclassical method knowing the positions of the resonances, the only thing that is left
(SCl), the classical method (Cl) and the method based on pertur- to calculate is the resonance widths which is done using the Breit-
bation theory (PT). The cross sections were calculated in slightly Wigner (BW) method. After that, the resonances can be added
different ways for the two different reaction channels: X2Π −X2Π to the cross sections obtained by the classical and semiclassical
and B2Σ− − X2Π. Both could be treated with the PT method, methods. The inverse predissociation contribution was calculated
which is shown in the orange curves in Figures 5.2 and 5.3. using lifetimes obtained from literature, for more information see
Since theX2Π−X2Π transition occurs within the same electronic Paper I.
state, it can not be treated with the SCl method and the Cl method When the reaction cross sections have been obtained, the reaction
was therefore used to calculate the cross section (blue curve in Figure rate constants can be calculated using equation 3.1. In Figure 5.4,
5.2). The B2Σ−−X2Π reaction channel, on the other hand, includes the reaction rate constants for the formation of CH through the
32 33
U(r) (eV)
5. Results 5.1. Paper I
molecules, the the resonances show up due to the barrier on the 7
effective potential energy curve that gives rise to quasi bound states. X 2
B2 -
This is illustrated in Figure 3.1 which shows a plot of the potential 6 A2
C 2 +
energy against the distance between the colliding atoms. At large
separation between the atoms, the potential energy is approaching 5
the asymptotic energy which is the potential energy corresponding
4
to the atoms being at infinite separation. All the energy levels below
the asymptotic energy are bound states. There are some energy 3
levels that are above the asymptotic energy level but below the top
of the barrier. These levels are not bound because they can tunnel 2
out through the barrier. However, the barrier makes it harder to
1
dissociate and these levels are therefore called quasibound.
If an incoming atom has an energy that matches the energy of the 0
0.5 1 1.5 2 2.5 3 3.5
quasibound state, it is probable that it will tunnel through the barrier. r (Aangstrom)
Since the barrier prohibits the atom from dissociating straight away,
it gets a higher probability to emit a photon and transition to one of Figure 5.1: Potential energy curves for the X2Π, B2Σ−, A2∆ and
the bound states. When this happens, a resonance, corresponding C2Σ+ states as functions of the internuclear distance r.
to an increase in probability, will appear in the cross section which
can be seen both in Figures 5.2 and 5.3. The cross sections are
plotted against the energy of the colliding atoms and each resonance
has the position on the energy axis that is equal to the energy of a a transition between two different electronic states and in that case
quasibound state. Inverse predissociation, on the other hand, occurs the SCl method could be used to obtain the cross sections.
when two coupled molecular states intersect, allowing the molecule The cross sections obtained with the SCl and Cl methods do
to transfer between them. In our case, this is only possible for energy not have any resonances. In order to add resonance contribution
levels lying above the asymptotic energy of the X2Π state and below to the cross sections, the quasibound states were calculated using
the one for the A2∆ state. the computational program LEVEL (Le Roy [2017]). As mentioned
As discussed in the ’Computational methods for RA’ chapter, above, the position of the resonances on the energy axis correspond
three different methods were used to obtain the cross sections for the to the energies of the quasibound states. This in turn means that
formation of the CH and CD molecules: the semiclassical method knowing the positions of the resonances, the only thing that is left
(SCl), the classical method (Cl) and the method based on pertur- to calculate is the resonance widths which is done using the Breit-
bation theory (PT). The cross sections were calculated in slightly Wigner (BW) method. After that, the resonances can be added
different ways for the two different reaction channels: X2Π −X2Π to the cross sections obtained by the classical and semiclassical
and B2Σ− − X2Π. Both could be treated with the PT method, methods. The inverse predissociation contribution was calculated
which is shown in the orange curves in Figures 5.2 and 5.3. using lifetimes obtained from literature, for more information see
Since theX2Π−X2Π transition occurs within the same electronic Paper I.
state, it can not be treated with the SCl method and the Cl method When the reaction cross sections have been obtained, the reaction
was therefore used to calculate the cross section (blue curve in Figure rate constants can be calculated using equation 3.1. In Figure 5.4,
5.2). The B2Σ−−X2Π reaction channel, on the other hand, includes the reaction rate constants for the formation of CH through the
32 33
U(r) (eV)
5. Results 5.1. Paper I
Figure 5.2: The cross sections for the transitions between the X2Π−
X2Π states of the CH molecule as functions of the collision energy.
The blue curve corresponds to the cross section calculated using the
classical method (complemented by Breit-Wigner) while the blue
orange shows cross sections obtained using perturbation theory.
RA process are plotted against temperature. Figure 5.4 shows the
Figure 5.3: The cross sections for the transitions between the B2Σ−−
reaction rate constants for 1) inverse predissociation through the 2
A2∆ state, 2) the B2Σ− → X2 X Π states of the CH molecule as functions of the collision energy.Π channel, 3) the X2Π → X2Π
The blue curve corresponds to the cross section calculated using
channel and 4) the total, which is the sum of the other three. The
the semiclassical method (complemented by Breit-Wigner) while
contribution from inverse predissociation dominates up to 570 K
while at higher temperatures the B2Σ− → 2 the orange curve shows cross sections obtained using perturbationX Π channel plays the
theory.
biggest role. The total reaction rate constant is 3.5 · 10−17cm3/s at
100 K and peaks at 20 K where it is equal to 8.0 · 10−17cm3/s. We
have compared the reaction rate constant obtained in this study
with previously calculated values, and although ours is higher, it is
still not large enough to significantly contribute to the formation of
CH.
34 35
5. Results 5.1. Paper I
Figure 5.2: The cross sections for the transitions between the X2Π−
X2Π states of the CH molecule as functions of the collision energy.
The blue curve corresponds to the cross section calculated using the
classical method (complemented by Breit-Wigner) while the blue
orange shows cross sections obtained using perturbation theory.
RA process are plotted against temperature. Figure 5.4 shows the
Figure 5.3: The cross sections for the transitions between the B2Σ−−
reaction rate constants for 1) inverse predissociation through the 2
A2∆ state, 2) the B2Σ− → X2 X Π states of the CH molecule as functions of the collision energy.Π channel, 3) the X2Π → X2Π
The blue curve corresponds to the cross section calculated using
channel and 4) the total, which is the sum of the other three. The
the semiclassical method (complemented by Breit-Wigner) while
contribution from inverse predissociation dominates up to 570 K
while at higher temperatures the B2Σ− → 2 the orange curve shows cross sections obtained using perturbationX Π channel plays the
theory.
biggest role. The total reaction rate constant is 3.5 · 10−17cm3/s at
100 K and peaks at 20 K where it is equal to 8.0 · 10−17cm3/s. We
have compared the reaction rate constant obtained in this study
with previously calculated values, and although ours is higher, it is
still not large enough to significantly contribute to the formation of
CH.
34 35
5. Results 5.2. Paper II
10-16
Figure 5.5: Jacobi coordinates for the Na+ +H2 system
10-18
5.2 Paper II
-20 The second paper was dedicated to the formation of Na
+ +H2 and10
102
+
103 Na + D2 through RA in the interstellar medium. Cross sections
Temperature (K) were obtained through the driven equations method described earlier
B2 -  X2 Bates in this thesis, using potential energy curves obtained from literature.
X2  X2 Smith et al.
Inverse predissociation Lutz In a triatomic case, there are two more coordinates to consider
Total Bain & Bardsley
Julienne & Krauss Brzozowski et al. compared to the diatomic case. The potential energy depends on the
Brooks & Smith distance between the atom and the center of mass of the diatom (R),
the bond length of the diatom (r) as well as the angle between these
Figure 5.4: The reaction rate constants for the formation of the two fragments (θ). The coordinate system used in the calculations
CH molecule. The three curves show the reaction rate constants can be seen in Figure 5.5.
obtained by including the contribution of the three reaction channels For the formation of the Na+ +H2 molecule only one potential
separately (blue, orange and yellow curves) and a sum of both energy surface was taken into account, since excited states lie at
contributions (purple curve). energies that are too high to matter in the temperature interval of
interest. The potential energy surface can be seen in Figure 5.6.
Reaction cross sections for the RA of Na+ +H2 are depicted in
Figure 5.7. The two nuclear spin states of the hydrogen molecule
(oH2 and pH2) are treated separately and j is the rotational angular
momentum quantum number of the diatom. The largest value of j
that was included in the calculations was 18 for oH2 and 19 for pH2.
For the ortho spin isomer of hydrogen only the odd values of
j are allowed (Atkins and Paula [2006]) while para hydrogen only
36 37
Rate constant (cm3/s)
5. Results 5.2. Paper II
10-16
Figure 5.5: Jacobi coordinates for the Na+ +H2 system
10-18
5.2 Paper II
-20 The second paper was dedicated to the formation of Na
+ +H2 and10
102
+
103 Na + D2 through RA in the interstellar medium. Cross sections
Temperature (K) were obtained through the driven equations method described earlier
B2 -  X2 Bates in this thesis, using potential energy curves obtained from literature.
X2  X2 Smith et al.
Inverse predissociation Lutz In a triatomic case, there are two more coordinates to consider
Total Bain & Bardsley
Julienne & Krauss Brzozowski et al. compared to the diatomic case. The potential energy depends on the
Brooks & Smith distance between the atom and the center of mass of the diatom (R),
the bond length of the diatom (r) as well as the angle between these
Figure 5.4: The reaction rate constants for the formation of the two fragments (θ). The coordinate system used in the calculations
CH molecule. The three curves show the reaction rate constants can be seen in Figure 5.5.
obtained by including the contribution of the three reaction channels For the formation of the Na+ +H2 molecule only one potential
separately (blue, orange and yellow curves) and a sum of both energy surface was taken into account, since excited states lie at
contributions (purple curve). energies that are too high to matter in the temperature interval of
interest. The potential energy surface can be seen in Figure 5.6.
Reaction cross sections for the RA of Na+ +H2 are depicted in
Figure 5.7. The two nuclear spin states of the hydrogen molecule
(oH2 and pH2) are treated separately and j is the rotational angular
momentum quantum number of the diatom. The largest value of j
that was included in the calculations was 18 for oH2 and 19 for pH2.
For the ortho spin isomer of hydrogen only the odd values of
j are allowed (Atkins and Paula [2006]) while para hydrogen only
36 37
Rate constant (cm3/s)
5. Results 5.2. Paper II
 180
-15 18 14
20 12
 160 -250 -8-17 410 10 6 2-20 8 80 4
-350 4-7 0
 140 -19 2-9 -7 -3 -2-5 -16 -5 0
-400 -10 -5
0
-800 -4 -1 -1
 120 -18 -11 -6 -2-8 -3 -2
--9150000 -200 -12
-1 -4
 100 6 10-450 -15
 80 j=1-700 -13 -7
2
-50 -17 -6-13
-300 -5
-4
-2 10 60 -850 -1-500 -20 -3
-100 -10 -6 -3
-600 -14 -9 -1 0
 40 -19 -8 -2 -2
-1 0 j=3
-500
-18 -4 0 2
-8
-150
-3 -3 2
4 4
6 100 6
 20 -15 10
-17 8 16 8 2
12 14 j=5
 0 22 -1010
 5  10  15  20  25  30  35  40
R (Bohr)
-12
10
+
-14 Na  + oH (j)
10 2
Figure 5.6: The potential energy surface of Na+ + H2 with the inter
atomic distance on the x axis and the angle between the atom and -1610
-3 -2 -1 0 1 2 3
10 10 10 10 10 10 10
the diatom on the y axis. -1
E/cm
-2
10
-4 j=2
supports even values of j. In the upper panel of the figure, the 10
cross sections corresponding to the ortho spin state of the hydrogen -610 j=0
molecule are shown while the lower panel depicts the cross sections
-8
corresponding to the para spin state. 10 j=4
The cross sections were calculated separately for different values -10
10
of j, as shown in Figure 5.7, and then added together to form total
+
cross sections for Na+ + oH2 and Na
+ + pH . -1210 Na  + pH2 2(j) 
In Figure 5.8 the reaction rate coefficients for each one of the -14
10
cross sections shown in Figure 5.7 have been calculated for oH2 in
-3 -2 -1 0 1 2 3
10 10 10 10 10 10 10
-1
the upper panel and for pH2 in the lower panel. The black curves
E/cm
in the plots are the Boltzmann averaged cross sections. It is worth Figure 5.7: The upper and the lower panels of this figure depict the
to note that at low temperatures, the Boltzmann averaged cross reaction cross sections for oH2 and pH2, respectively.
sections follow the cross section curve corresponding to the lowest j
value for that spin isomer.
In Figure 5.9, contribution from reaction rate coefficients corre-
sponding to all allowed values of j are summed up for oH2 in the blue
curve. The green curve shows the same summation for pH2 and the
black curve is the total reaction rate coefficient including both spin
isomers. When adding the reaction rate coefficients corresponding
38 39
θ (degree)
2 2
σ (Bohr  ) σ (bohr )
5. Results 5.2. Paper II
 180
-15 18 14
20 12
 160 -250 -8-17 410 10 6 2-20 8 80 4
-350 4-7 0
 140 -19 2-9 -7 -3 -2-5 -16 -5 0
-400 -10 -5
0
-800 -4 -1 -1
 120 -18 -11 -6 -2-8 -3 -2
--9150000 -200 -12
-1 -4
 100 6 10-450 -15
 80 j=1-700 -13 -7
2
-50 -17 -6-13
-300 -5
-4
-2 10 60 -850 -1-500 -20 -3
-100 -10 -6 -3
-600 -14 -9 -1 0
 40 -19 -8 -2 -2
-1 0 j=3
-500
-18 -4 0 2
-8
-150
-3 -3 2
4 4
6 100 6
 20 -15 10
-17 8 16 8 2
12 14 j=5
 0 22 -1010
 5  10  15  20  25  30  35  40
R (Bohr)
-12
10
+
-14 Na  + oH (j)
10 2
Figure 5.6: The potential energy surface of Na+ + H2 with the inter
atomic distance on the x axis and the angle between the atom and -1610
-3 -2 -1 0 1 2 3
10 10 10 10 10 10 10
the diatom on the y axis. -1
E/cm
-2
10
-4 j=2
supports even values of j. In the upper panel of the figure, the 10
cross sections corresponding to the ortho spin state of the hydrogen -610 j=0
molecule are shown while the lower panel depicts the cross sections
-8
corresponding to the para spin state. 10 j=4
The cross sections were calculated separately for different values -10
10
of j, as shown in Figure 5.7, and then added together to form total
+
cross sections for Na+ + oH2 and Na
+ + pH . -1210 Na  + pH2 2(j) 
In Figure 5.8 the reaction rate coefficients for each one of the -14
10
cross sections shown in Figure 5.7 have been calculated for oH2 in
-3 -2 -1 0 1 2 3
10 10 10 10 10 10 10
-1
the upper panel and for pH2 in the lower panel. The black curves
E/cm
in the plots are the Boltzmann averaged cross sections. It is worth Figure 5.7: The upper and the lower panels of this figure depict the
to note that at low temperatures, the Boltzmann averaged cross reaction cross sections for oH2 and pH2, respectively.
sections follow the cross section curve corresponding to the lowest j
value for that spin isomer.
In Figure 5.9, contribution from reaction rate coefficients corre-
sponding to all allowed values of j are summed up for oH2 in the blue
curve. The green curve shows the same summation for pH2 and the
black curve is the total reaction rate coefficient including both spin
isomers. When adding the reaction rate coefficients corresponding
38 39
θ (degree)
2 2
σ (Bohr  ) σ (bohr )
-17
10
ortho
global
-18
10
para
-19
10
-20
10
+
Na  + D2 
-21
10
-1 0 1 2 3
10 10 10 10 10
5. Results T/K 5.2. Paper II
-18
10
Ortho
-19
10
global
-18
10
BA Para
-19
10 -20
j=1 10
+
-20
10 Na  + H  j=3 2
-21
10 -21
10
+ -1 0 1 2 3
Na  + oH 10 10 10 10 10
-22 2
10 T/K
j=5
-23
10 Figure 5.9: Reaction rate coefficients for the oH2 and pH2 together
with a total reaction rate coefficient.
-24
10
-1 0 1 2 3
10 10 10 10 10
Temp (K)
-17
10 to oH2 and pH2 spin states of the hydrogen, one needs to take into
j=0 account the ratio of the two spin isomers which in this case is taken
-18
10 j=2 BA to be 3:1.
The need to calculate the reaction rate coefficient for the RA of
-19
10 Na+ +H2 was motivated by the fact that previous investigations of
this reaction have resulted in reaction rate coefficients that differed
-20
10 +
Na  + pH2 by four orders of magnitude. Smith et al. [1983] obtained a reaction
rate coefficient that was independent of temperature and equal to
-21
10 j=4 4 · 10−19 cm3molecule−1s−1, but a later study by Petrie and Dunbar
-22 [2000] yielded a reaction rate coefficient equal to 8.7 · 10−23 at 30 K
10
-1 0 1 2 3
10 10 10 10 10 which is small enough to exclude this reaction from astrochemistry.
Temp (K) The reaction rate coefficients obtained in our study are significantly
Figure 5.8: The upper and the lower panels of this figure depict the closer to the ones obtained by Smith et al. [1983], supporting the
+
reaction rate coefficients for the oH and pH , respectively. need to include the RA process of Na +H2 in astrochemical models.2 2
40 41
Rate (cm3molecule-1s-1)
Rate (cm3molecule-1s-1)
3 -1 -1 3 -1 -1
k(T) (cm molecule s ) k(T) (cm molecule s )
-17
10
ortho
global
-18
10
para
-19
10
-20
10
+
Na  + D2 
-21
10
-1 0 1 2 3
10 10 10 10 10
5. Results T/K 5.2. Paper II
-18
10
Ortho
-19
10
global
-18
10
BA Para
-19
10 -20
j=1 10
+
-20
10 Na  + H  j=3 2
-21
10 -21
10
+ -1 0 1 2 3
Na  + oH 10 10 10 10 10
-22 2
10 T/K
j=5
-23
10 Figure 5.9: Reaction rate coefficients for the oH2 and pH2 together
with a total reaction rate coefficient.
-24
10
-1 0 1 2 3
10 10 10 10 10
Temp (K)
-17
10 to oH2 and pH2 spin states of the hydrogen, one needs to take into
j=0 account the ratio of the two spin isomers which in this case is taken
-18
10 j=2 BA to be 3:1.
The need to calculate the reaction rate coefficient for the RA of
-19
10 Na+ +H2 was motivated by the fact that previous investigations of
this reaction have resulted in reaction rate coefficients that differed
-20
10 +
Na  + pH2 by four orders of magnitude. Smith et al. [1983] obtained a reaction
rate coefficient that was independent of temperature and equal to
-21
10 j=4 4 · 10−19 cm3molecule−1s−1, but a later study by Petrie and Dunbar
-22 [2000] yielded a reaction rate coefficient equal to 8.7 · 10−23 at 30 K
10
-1 0 1 2 3
10 10 10 10 10 which is small enough to exclude this reaction from astrochemistry.
Temp (K) The reaction rate coefficients obtained in our study are significantly
Figure 5.8: The upper and the lower panels of this figure depict the closer to the ones obtained by Smith et al. [1983], supporting the
+
reaction rate coefficients for the oH and pH , respectively. need to include the RA process of Na +H2 in astrochemical models.2 2
40 41
Rate (cm3molecule-1s-1)
Rate (cm3molecule-1s-1)
3 -1 -1 3 -1 -1
k(T) (cm molecule s ) k(T) (cm molecule s )
5. Results 5.3. Paper III
10-2
j=0
+
Contour Plot of Al  + H2 for RH =Re Bohr j=12
 180 -50 -4 j=2
10 -100
10
-17 -5 4 j=3
350
 160 351--601884 -20 j=4-7-2 6 8
200-219 -150 -15 -1-39
-250 -11 -4
0 2
0 j=5
 140 14-120-322 -300 -18 -8
-3
-350 10-622 -14 --10-400 12
48 -450 -19
 120 865000-11400106
2-580
 100 -150 10-8-250
-17
 80 450 -600
3-544
 60 9530 -500
-20
700014
0
5-2000 -15
-10
20 -200 -18 -1-13
10
1 -9 -7
 40 -4165 -4 -2--72 -14-16-10 -6 -1 0
-1 -19 -12 8 6
 20 4 10-16 12 -12
550 -6
-5 14 16 10
 0 30 -17
 4  6  8  10  12  14
R (Bohr) 10-14
Al+ + H (j)
10-16 2
Figure 5.10: Potential energy surface of Al+ − H2 obtained by
Emmeluth et al. for r = 1.4097a0. The energy is given in cm
−1. 10-18
10-3 10-2 10-1 100 101 102 103
Energy (cm-1 )
5.3 Paper III Figure 5.11: RA cross section of Al+ − H2 as a function of collision
energy depicted for different values of j.
In this paper, we used the same methodology as in paper II to
calculate the reaction rate constant for the formation of Al+ + H2
through RA. The potential energy surface, seen in Figure 5.10, and
by combining the ortho and para H /D reaction rate constants
dipole moment were obtained from literature. 2 2
using spin statistics. A ratio of 3:1 was used when combining the
The Al+-H2 potential energy surface has a shallower well than ortho H2 and para H2 rate coefficients, and 2:1 when combining the
the Na+-H2 complex treated in the previous paper, resulting in a ortho D2 and para D2 rate coefficients.
significant difference in the energy of the lowest bound state; while
the lowest energy of the Na+-H lies at -844.37cm−1, the one for The Boltzmann averaged rate coefficients are seen to be almost2
Al+-H is positioned at -461.81cm−12 . This has a large effect on the flat at low temperature before reaching a maximum close to 87 K
+
cross sections as they are proportional to ω3, and it is therefore not for Al -H2 formation while minimums are visible at 5 and 43 K for
+
surprising that the cross sections for Al+-H /D , depicted in Figures the Al -D2 formation. Thereafter, both rate coefficients decrease2 2 −1
5.11 and 5.12, are smaller than the cross sections for Na+-H /D . monotonously and follow the expected T law predicted by modified2 2
This trend can also be observed for the reaction rate constants, where thermal theory for radiative association of an atom with a linear
the ones for Al+-H2 are smaller than the ones obtained for Na
+-H . molecule with no dipole moment (Bates [1979], Herbst [1980]).2
Figures 5.13 and 5.14 show the j-resolved RA rate coefficients for The goal of this study was to assess whether the RA of Al+-H2 is
the Al+-H2 and Al
+-D2 molecules, respectively. A total Boltzmann fast enough to play a role in interstellar chemistry of dense clouds. As
averaged reaction rate constant can be seen in both figures, obtained RA is competing with radiative recombination (RR) in these regions,
42 43
θ (degree)
Reaction cross section (bohr 2)
5. Results 5.3. Paper III
10-2
j=0
+
Contour Plot of Al  + H2 for RH =Re Bohr j=12
 180 -50 -4 j=2
10 -100
10
-17 -5 4 j=3
350
 160 531--061884 -20 j=4-7-2 6 8
200-219 -150 -15 -1-39
-250 -11 -4
0 2
0 j=5
 140 14-120-322 -300 -18 -8
-3
-350 10-622 -14 --10-400 12
48 -450 -19
 120 865000-11400106
2-580
 100 -150 10-8-250
-17
 80 450 -600
3-544
 60 9530 -500
-20
700014
0
5-2000 -15
-10
20 -200 -18 -1-13
10
1 -9 -7
 40 -4165 -4 -2--72 -14-16-10 -6 -1 0
-1 -19 -12 8 6
 20 4 10-16 12 -12
550 -6
-5 14 16 10
 0 30 -17
 4  6  8  10  12  14
R (Bohr) 10-14
Al+ + H (j)
10-16 2
Figure 5.10: Potential energy surface of Al+ − H2 obtained by
Emmeluth et al. for r = 1.4097a0. The energy is given in cm
−1. 10-18
10-3 10-2 10-1 100 101 102 103
Energy (cm-1 )
5.3 Paper III Figure 5.11: RA cross section of Al+ − H2 as a function of collision
energy depicted for different values of j.
In this paper, we used the same methodology as in paper II to
calculate the reaction rate constant for the formation of Al+ + H2
through RA. The potential energy surface, seen in Figure 5.10, and
by combining the ortho and para H /D reaction rate constants
dipole moment were obtained from literature. 2 2
using spin statistics. A ratio of 3:1 was used when combining the
The Al+-H2 potential energy surface has a shallower well than ortho H2 and para H2 rate coefficients, and 2:1 when combining the
the Na+-H2 complex treated in the previous paper, resulting in a ortho D2 and para D2 rate coefficients.
significant difference in the energy of the lowest bound state; while
the lowest energy of the Na+-H lies at -844.37cm−1, the one for The Boltzmann averaged rate coefficients are seen to be almost2
Al+-H is positioned at -461.81cm−12 . This has a large effect on the flat at low temperature before reaching a maximum close to 87 K
+
cross sections as they are proportional to ω3, and it is therefore not for Al -H2 formation while minimums are visible at 5 and 43 K for
+
surprising that the cross sections for Al+-H /D , depicted in Figures the Al -D2 formation. Thereafter, both rate coefficients decrease2 2 −1
5.11 and 5.12, are smaller than the cross sections for Na+-H /D . monotonously and follow the expected T law predicted by modified2 2
This trend can also be observed for the reaction rate constants, where thermal theory for radiative association of an atom with a linear
the ones for Al+-H2 are smaller than the ones obtained for Na
+-H . molecule with no dipole moment (Bates [1979], Herbst [1980]).2
Figures 5.13 and 5.14 show the j-resolved RA rate coefficients for The goal of this study was to assess whether the RA of Al+-H2 is
the Al+-H2 and Al
+-D2 molecules, respectively. A total Boltzmann fast enough to play a role in interstellar chemistry of dense clouds. As
averaged reaction rate constant can be seen in both figures, obtained RA is competing with radiative recombination (RR) in these regions,
42 43
θ (degree)
Reaction cross section (bohr 2)
5. Results 5.3. Paper III
10-2
j=0
j=1
10-4 j=2j=3
j=4
j=5
10-6
10-8
10-10 10-19
j=1
-12 j=210
10-20 BA
j=0
10-14 +
+ 10-21 Al  + H (j)Al  + D (j) 2
10-16 2
10-22 j=3
10-18
10-3 10-2 10-1 100 101 102 103 j=4
Energy (cm-1 ) 10-23
-24
Figure 5.12: RA cross section of Al+ −D2 as a function of collision 10 j=5
energy depicted for different values of j.
10-25
the RA reaction rate constant would need to fulfil this condition: 10-26
10-1 100 101 102 103
H  e−k 2 k · f (5.4) Temperature (K)RA RR e
+
After evaluating this condition, we found out that the fe for which Figure 5.13: RA rate coefficient for forming Al − H2 as a function
RA of Al+-H2 matters is considerably smaller than the value reported of temperature depicted for different values of j.
for dense clouds. We therefore came to the conclusion that RA of
Al+-H2 is too slow to play a role in dense clouds.
44 45
Reaction cross section (bohr 2)
Reaction Rate constant (cm3/molecule s)
5. Results 5.3. Paper III
10-2
j=0
j=1
10-4 j=2j=3
j=4
j=5
10-6
10-8
10-10 10-19
j=1
-12 j=210
10-20 BA
j=0
10-14 +
+ 10-21 Al  + H (j)Al  + D (j) 2
10-16 2
10-22 j=3
10-18
10-3 10-2 10-1 100 101 102 103 j=4
Energy (cm-1 ) 10-23
-24
Figure 5.12: RA cross section of Al+ −D2 as a function of collision 10 j=5
energy depicted for different values of j.
10-25
the RA reaction rate constant would need to fulfil this condition: 10-26
10-1 100 101 102 103
H  e−k 2 k · f (5.4) Temperature (K)RA RR e
+
After evaluating this condition, we found out that the fe for which Figure 5.13: RA rate coefficient for forming Al − H2 as a function
RA of Al+-H2 matters is considerably smaller than the value reported of temperature depicted for different values of j.
for dense clouds. We therefore came to the conclusion that RA of
Al+-H2 is too slow to play a role in dense clouds.
44 45
Reaction cross section (bohr 2)
Reaction Rate constant (cm3/molecule s)
5. Results 5.4. Paper IV
5.4 Paper IV
This manuscript is dedicated to RA of the Al and Cl atoms, forming
AlCl. Same methodology as in Paper I was used, and the potential
energy surfaces, dipole moment and transition dipole moment were
obtained from literature. The reaction rate constant for the formation
of AlCl though RA has previously been calculated by Andreazza et al.
[2018a] and the conclusion was that this reaction was too slow to
matter in the chemistry of interstellar clouds where the temperature
10-19 ranges between 2000 and 4000 K. That study only treated theA1Π →
j=2 X1Σ+ reaction channel without including resonance contribution.
j=1
10-20
BA We therefore decided to see if adding resonance contribution to the
j=0 A1Π → X1Σ+ reaction channel, as well as including the X1Σ+ →
10-21
1 +
Al+ + D (j) X Σ reaction pathway, would increase the reaction rate constant2 enough for it to play a role under the above-mentioned conditions.
j=3
10-22 Cross sections for the A
1Π → X1Σ+ and the X1Σ+ → X1Σ+
channels obtained in this study are shown in Figures 5.15 and 5.16,
-23 j=4 respectively. Both figures include cross sections obtained using PT as10
well as the Cl+BW/SCl+BW. The A1Π potential energy curve has
a barrier of 0.76 eV, which makes dissociation harder and therefore
10-24 j=5 gives rise to an increase in the baseline of the cross sections in Figure
5.15. The X1Σ+ potential energy curve does not have a barrier and
10-25 the X1Σ+ → X1Σ+ cross sections are therefore slowly decreasing
with increased energy, as shown in Figure 5.16.
10-26 Similarly to the study involving CH in Paper I, we have used
10-1 100 101 102 103
Temperature (K) the cross sections obtained using the SCl+BW/Cl+BW to calculate
the reaction rate constants which are depicted in Figure 5.17. The
Figure 5.14: RA rate coefficient for forming Al+ −D as a function major contribution comes from the A1Π → X1Σ+ reaction, aside2
of temperature depicted for different values of j. from the lowest temperatures where the small barrier on the A
1Π
potential energy curve has a major effect on this reaction channel.
The figure also shows a comparison between the results obtained in
this study with ones from Andreazza et al.. The differences between
the two studies can be explained by the fact that different ab initio
data was used and the addition of the resonance contribution to the
A1Π → X1Σ+ reaction channel.
Although our calculated reaction rates are two to four times
larger, this difference is not significant enough to suggest that a large
portion of AlCl would be created through radiative association of
46 47
Reaction Rate constant (cm3/molecule s)
5. Results 5.4. Paper IV
5.4 Paper IV
This manuscript is dedicated to RA of the Al and Cl atoms, forming
AlCl. Same methodology as in Paper I was used, and the potential
energy surfaces, dipole moment and transition dipole moment were
obtained from literature. The reaction rate constant for the formation
of AlCl though RA has previously been calculated by Andreazza et al.
[2018a] and the conclusion was that this reaction was too slow to
matter in the chemistry of interstellar clouds where the temperature
10-19 ranges between 2000 and 4000 K. That study only treated theA1Π →
j=2 X1Σ+ reaction channel without including resonance contribution.
j=1
10-20
BA We therefore decided to see if adding resonance contribution to the
j=0 A1Π → X1Σ+ reaction channel, as well as including the X1Σ+ →
10-21
1 +
Al+ + D (j) X Σ reaction pathway, would increase the reaction rate constant2 enough for it to play a role under the above-mentioned conditions.
j=3
10-22 Cross sections for the A
1Π → X1Σ+ and the X1Σ+ → X1Σ+
channels obtained in this study are shown in Figures 5.15 and 5.16,
-23 j=4 respectively. Both figures include cross sections obtained using PT as10
well as the Cl+BW/SCl+BW. The A1Π potential energy curve has
a barrier of 0.76 eV, which makes dissociation harder and therefore
10-24 j=5 gives rise to an increase in the baseline of the cross sections in Figure
5.15. The X1Σ+ potential energy curve does not have a barrier and
10-25 the X1Σ+ → X1Σ+ cross sections are therefore slowly decreasing
with increased energy, as shown in Figure 5.16.
10-26 Similarly to the study involving CH in Paper I, we have used
10-1 100 101 102 103
Temperature (K) the cross sections obtained using the SCl+BW/Cl+BW to calculate
the reaction rate constants which are depicted in Figure 5.17. The
Figure 5.14: RA rate coefficient for forming Al+ −D as a function major contribution comes from the A1Π → X1Σ+ reaction, aside2
of temperature depicted for different values of j. from the lowest temperatures where the small barrier on the A
1Π
potential energy curve has a major effect on this reaction channel.
The figure also shows a comparison between the results obtained in
this study with ones from Andreazza et al.. The differences between
the two studies can be explained by the fact that different ab initio
data was used and the addition of the resonance contribution to the
A1Π → X1Σ+ reaction channel.
Although our calculated reaction rates are two to four times
larger, this difference is not significant enough to suggest that a large
portion of AlCl would be created through radiative association of
46 47
Reaction Rate constant (cm3/molecule s)
5. Results 5.4. Paper IV
A1   X1 +
10-5
SCl+BW
PT
10-10 -5  X
1 +  X1 +
10
Cl+BW
PT
10-15
10-2 10-1 100 -10
Collision energy (eV) 10
Figure 5.15: Cross sections for the A1Π → X1Σ+ channel for form-
ing the AlCl molecule as functions of collision energy. The blue
curve corresponds to cross sections calculated using the semiclassical 10-15
method (SCl) complemented by Breit-Wigner theory (BW) , while 10-3 10-2 10-1 100 101
the red curve shows cross sections obtained using perturbation theory Collision energy (eV)
(PT).
Figure 5.16: Cross sections for the X1Σ+ → X1Σ+ channel for
forming the AlCl molecule as functions of collision energy. The blue
curve corresponds to cross sections calculated using the classical
the Al and Cl molecules. We therefore came to the same conclusion method (Cl) complemented by Breit-Wigner theory (BW), while the
as Andreazza et al., namely that only a small amount of AlCl can red curve shows cross sections obtained using perturbation theory
be created through RA in the inner regions of evolved stars with the (PT).
temperatures ranging between 2000 and 4000 K.
48 49
Cross section (bohr2)
Cross section (bohr2)
5. Results 5.4. Paper IV
A1   X1 +
10-5
SCl+BW
PT
10-10 -5  X
1 +  X1 +
10
Cl+BW
PT
10-15
10-2 10-1 100 -10
Collision energy (eV) 10
Figure 5.15: Cross sections for the A1Π → X1Σ+ channel for form-
ing the AlCl molecule as functions of collision energy. The blue
curve corresponds to cross sections calculated using the semiclassical 10-15
method (SCl) complemented by Breit-Wigner theory (BW) , while 10-3 10-2 10-1 100 101
the red curve shows cross sections obtained using perturbation theory Collision energy (eV)
(PT).
Figure 5.16: Cross sections for the X1Σ+ → X1Σ+ channel for
forming the AlCl molecule as functions of collision energy. The blue
curve corresponds to cross sections calculated using the classical
the Al and Cl molecules. We therefore came to the same conclusion method (Cl) complemented by Breit-Wigner theory (BW), while the
as Andreazza et al., namely that only a small amount of AlCl can red curve shows cross sections obtained using perturbation theory
be created through RA in the inner regions of evolved stars with the (PT).
temperatures ranging between 2000 and 4000 K.
48 49
Cross section (bohr2)
Cross section (bohr2)
5. Results
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10-24
3 4 P. Atkins and J. Paula. PhysicalPC. hAetmkisntsrya.ndWJ.H. P.Farueleam. aPnh,y2s0ic0a6l.Chemistry. W.H.Freeman, 2006.10 10
Temperature (K)
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L. M. Ziurys, A. J. Apponi, and T. G. Phillips. Ap. J., 433:729,
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medium. Cambridge university press, 2005.
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Phys., 41:25250–25257, 2022.
54 55
Part II
Papers
57
Part II Part II
Papers Papers
57 57