Infrared spectroscopy of triatomics for space observation

231 Pages·2019·6.131 MB·English

by  NOT AVAILABLE

Infrared Spectroscopy of
Triatomics for Space Observation
Infrared Spectroscopy Set
coordinated by
Pierre Richard Dahoo and Azzedine Lakhlifi
Volume 2
Infrared Spectroscopy of
Triatomics for Space
Observation

Pierre Richard Dahoo
Azzedine Lakhlifi
First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter 1. Symmetry of Triatomic Molecules . . . . . . . . . . . . . . . . . . . 1
1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. The symmetry group of the Hamiltonian of
a triatomic molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3. Symmetry of the nonlinear triatomic molecule (O) . . . . . . . . . . . . . . . 6
3
1.3.1. The nonlinear asymmetric molecule O ( 16O16O18O (668)) . . . . . . . . . 8
3
1.3.2. The nonlinear symmetric molecule O (16O16O16O (666)) . . . . . . . . . . 9
3
1.3.3. Symmetry of eigenstates of a nonlinear molecule . . . . . . . . . . . . . . 11
1.4. Symmetry of the linear triatomic molecule (CO) . . . . . . . . . . . . . . . . 15
2
1.4.1. The linear asymmetric molecule CO (16O12C18O (628)) . . . . . . . . . . 17
2
1.4.2. The linear symmetric molecule CO (16O12C16O (626)) . . . . . . . . . . . 19
2
1.5. Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.1. Symmetry of the eigenstates of a triatomic
molecule taking into account the nuclei spins . . . . . . . . . . . . . . . . . . . . 21
Chapter 2. Energy Levels of Triatomic Molecules in Gaseous Phase . . . . 25
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2. Vibrational–rotational movements of an isolated molecule . . . . . . . . . . . 27
2.3. Vibrational movements of an isolated triatomic molecule . . . . . . . . . . . . 34
2.3.1. Nonlinear triatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.2. Linear triatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.3. Introduction of the perturbative Hamiltonians H, H, H, … . . . . . . . . 37
1 2 3
2.3.4. Transitions between two vibrational levels: selection rules . . . . . . . . . 38
2.4. Rotational movement of an isolated rigid molecule . . . . . . . . . . . . . . . 40
vi Infrared Spectroscopy of Triatomics for Space Observation
2.4.1. Linear triatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4.2. Symmetric top molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4.3. Nonlinear triatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4.4. Transitions between rotational levels . . . . . . . . . . . . . . . . . . . . . 46
2.5. Vibrational–rotational energy levels of an
isolated triatomic molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.6. Rovibrational transitions: selection rules . . . . . . . . . . . . . . . . . . . . . 48
2.6.1. Dipole moment in terms of normal coordinates . . . . . . . . . . . . . . . 50
2.7. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.7.1. Rotational matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.7.2. Perturbative Hamiltonians of vibration
and vibration–rotation coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.7.3. Components of the angular momentum J . . . . . . . . . . . . . . . . . . 60
2.7.4. Rotational Hamiltonian of a symmetric top . . . . . . . . . . . . . . . . . . 60
2.7.5. Elements of the rotational matrix . . . . . . . . . . . . . . . . . . . . . . . 61
2.7.6. Vibrational anharmonic constants . . . . . . . . . . . . . . . . . . . . . . . 62
Chapter 3. Clathrate Nano-Cages . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2. Clathrate structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3. Inclusion model of a triatomic molecule
in a clathrate nano-cage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3.1. Inclusion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3.2. Interaction potential energy . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4. Thermodynamic model of clathrates . . . . . . . . . . . . . . . . . . . . . . . . 73
3.4.1. Occupation fractions and Langmuir constants . . . . . . . . . . . . . . . . 74
3.4.2. Determination of the Langmuir constants . . . . . . . . . . . . . . . . . . . 74
3.4.3. Application to triatomic molecules . . . . . . . . . . . . . . . . . . . . . . 75
3.5. Infrared spectrum of a triatomic in clathrate matrix . . . . . . . . . . . . . . . 79
3.5.1. Infrared absorption coefficient . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5.2. Hamiltonian of the system and separation of movements . . . . . . . . . . 79
3.5.3. Vibrational motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5.4. Orientational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5.5. Translational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.5.6. Bar spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.6. Application to the CO molecule . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2
3.6.1. Vibrational motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.6.2. Orientational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.6.3. Translational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.6.4. Bar spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.7. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Contents vii
3.7.1. Non-zero orientation matrix elements used
to calculate the corrections to first-order perturbation energies . . . . . . . . . . 98
3.7.2. Correction to eigenenergies of the orientation Hamiltonian . . . . . . . . . 99
3.7.3. Expressions of the vector components derivatives
of the dipole moment with respect to the normal vibrational coordinates . . . . . 102
3.7.4. Expressions of the orientational transition elements in
the approximation of harmonic librators . . . . . . . . . . . . . . . . . . . . . . . 102
Chapter 4. Nano-Cages of Noble Gas Matrices . . . . . . . . . . . . . . . . . . 107
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2. The theoretical molecule–matrix model . . . . . . . . . . . . . . . . . . . . . . 110
4.2.1. Site inclusion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2.2. 12-6 L-J potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.2.3. Site distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.2.4. Coupling of the molecule–matrix system . . . . . . . . . . . . . . . . . . . 118
4.2.5. Vibrational frequency displacements . . . . . . . . . . . . . . . . . . . . . 119
4.2.6. The calculation of the orientational modes . . . . . . . . . . . . . . . . . . 123
4.2.7. Bar spectra and spectral profiles . . . . . . . . . . . . . . . . . . . . . . . . 124
4.3. Application to triatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.3.1. The triatomic molecule C . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3
4.3.2. The nonlinear triatomic molecule O . . . . . . . . . . . . . . . . . . . . . 135
3
4.4. Appendix: Program for determining the equilibrium configuration
of an O molecule in a noble gas matrix nano-cage . . . . . . . . . . . . . . . . . . 140
3
Chapter 5. Effect of Nano-Cages on Vibration . . . . . . . . . . . . . . . . . . . 145
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.2. The theoretical molecule–matrix model . . . . . . . . . . . . . . . . . . . . . . 146
5.3. Calculation of the shift of vibrational frequencies . . . . . . . . . . . . . . . . 147
5.3.1. Calculation principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.3.2. Application of the MAPLE program . . . . . . . . . . . . . . . . . . . . . 151
5.4. Application to linear triatomic molecules . . . . . . . . . . . . . . . . . . . . . 155
5.4.1. Experimental study of linear triatomic molecules (CO, NO) . . . . . . . 155
2 2
5.4.2. Frequency shift calculation for degenerate mode ν . . . . . . . . . . . . . 156
2
5.4.3. Calculation results for linear triatomic molecules (CO, NO) . . . . . . . 158
2 2
5.5. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.5.1. Transition from Cartesian coordinates to normal coordinates . . . . . . . . 163
5.5.2. MAPLE program for displacement/shifts of vibrational
frequency modes of a CO molecule in a noble gas nano-cage matrix . . . . . . 166
2
viii Infrared Spectroscopy of Triatomics for Space Observation
Chapter 6. Adsorption on a Graphite Substrate . . . . . . . . . . . . . . . . . . 173
6.1. Molecule adsorbed on a graphite substrate (1000) at
low temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.1.1. Astrophysical context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.1.2. Molecule adsorbed onto a graphite substrate . . . . . . . . . . . . . . . . . 175
6.1.3. Graphite substrate–molecule interaction energy . . . . . . . . . . . . . . . 176
6.2. Adsorption observables at low temperature . . . . . . . . . . . . . . . . . . . . 178
6.2.1. Equilibrium configuration and potential energy surface . . . . . . . . . . . 178
6.2.2. Adsorption energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.2.3. Diffusion constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.3. Interaction energy between two molecules . . . . . . . . . . . . . . . . . . . . 183
6.3.1. Electrostatic contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.3.2. Induction contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.3.3. Dispersion–repulsion contribution . . . . . . . . . . . . . . . . . . . . . . . 188
6.4. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.4.1. Expressions of action tensors . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.4.2. Multipolar moments and dipolar polarizability of
a molecule relative to the fixed (absolute) reference frame . . . . . . . . . . . . 191
6.4.3. Code in the FORTRAN language for the
calculation of the interaction potential energy between two molecules . . . . . . 191
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Foreword
Space is an extraordinary laboratory for glimpsing the extent and complexity of
physical phenomena at work in nature. It offers extreme environments that humans
cannot reproduce on Earth. Temperatures reach absolute zero in dense interstellar
clouds and reach trillions of degrees around super massive black holes. The density
of the diffuse nebulae is lower than that produced by the best terrestrial vacuum
generators; the density of the residues of dead stars is so great that the matter
becomes unstructured.
Even if these laboratories are mostly inaccessible to humans and even to space
probes (except for a few stars in the solar system), a colossal amount of information
is contained in the light that passes through space. When light rays meet the mirrors
of our telescopes, they are focused on increasingly powerful detectors and analytical
instruments.
The spectral analysis of light, split into an optical prism, is particularly rich in
information about the physicochemical nature of stars and their environment. Like a
fingerprint, each of the chemical elements leaves a unique signature in the spectrum
of light, making it possible to specify the chemical composition of stars. The shape
of these spectral lines also testifies to the physical conditions that reign at the source
of this light.
Nevertheless, according to the 19th-Century positivist philosopher Auguste
Comte, all the chemical elements in the solar system – including those found in
living beings – have a cosmic origin. The elements can thus be classified into a
small number of families, which are defined by the process that created them: the
Big Bang (hydrogen and helium), nuclear reactions in stars (carbon, nitrogen, etc.),
explosions of supernovae (oxygen, phosphorus, sulfur, iron, etc.), fusion of neutron
stars (francium, uranium, thorium, etc.) and the spallation of cosmic rays (boron,
beryllium).
x Infrared Spectroscopy of Triatomics for Space Observation
“The surface of the Earth is the shore of the cosmic ocean […] We’re made of
star stuff. We are a way for the cosmos to know itself.” Expressed in a poetic way by
the astrophysicist Carl Sagan, it seems that our cosmic origin and the detailed
understanding of atoms and molecules take on a meaning that surpasses us.
In a famous analogy, Richard Feynman likens the physicist to an insect floating
in the corner of a pool, rising and falling with the waves, and trying to reconstruct
what is happening in this pool simply by measuring the height of the
electromagnetic waves. The astrophysicist is also striving towards the goal of
discovering and understanding what is happening in the cosmic ocean, by the mere
observation of electromagnetic waves that reach the shore.
Spectroscopy in all its facets (instruments, theoretical frameworks, analyses and
techniques) is today a vast field with multiple ramifications. It is without a doubt the
most powerful, the finest and most universally used tool available to translate these
waves into a coherent vision of the universe on all scales, from the infinitely small to
the infinitely large.
An entity is more than just the sum of its parts, as Aristotle had already
formulated several millennia ago.
The authors, specialists in modeling and spectroscopy, show us the theoretical
models of triatomic molecules and their infrared spectra in different environments of
space. This book adds a string to the bow, adding to our understanding of this part of
the entity.
Céline REYLÉ
Astrophysicist at Institut UTINAM
Science Observatory at Univers Franche-Comté Bourgogne
Preface
In the preface to Volume 1 [DAH 17], the importance of spectroscopy was
emphasized, both from a theoretical and an instrumental point of view, for the
analysis of observations of chemical species, molecules, radicals and ions. In the
infrared (IR), using various types of spatial observation instruments, it is possible to
detect molecules or chemical species (ions, radicals, macromolecules, nano-cages,
etc.) present in the atmospheres of planets, Earth included, and their satellites, in
interstellar media, comets or exoplanets, for example.
One of the most striking observations using ground-based instruments or
embedded in space probes or telescopes was listed to show the diversity
of discoveries that can lead to advances in the field of astrophysics or cosmology.
Note, in particular, the observations mentioned in the preface to Volume 1
[DAH 17], that is:
And very recently, on September 14, 2015, the LIGO (Laser
Interferometer Gravitational-Wave Observatory) detects for the first
time, the distortions caused by gravitational waves in space-time,
predicted by Einstein’s theory of general relativity and generated by
two black holes that collide nearly 1.3 billion light-years away.
This earned its authors, Barry C. Barish, Kip S. Thorne and Rainer Weiss, the
Nobel Prize in Physics in 2017.
Advances in modern detection systems (Planck and Hubble telescopes) and large
telescopes that are continually improved and programmed to be sent into space
(NASA James Webb Space Telescope (2020), European Extremely Large Telescope
(E-ELT) (2024)) can probe the universe to better understand its origin and what it
comprises (less than 5% of visible matter, about 25% of dark matter and the rest of
dark energy (70%) responsible for a force that repels gravity), to observe exoplanets