INFRARED SPECTROSCOPY OF METHANE DIMER By Abdullah Hamdan A thesis submitted to the Faculty of the Graduate School of Ruhr-Universität Bochum in fulfillment of the requirements for the doctor rer. nat Department of Chemistry December 2005
INFRARED SPECTROSCOPY OF METHANE
DIMER
By
Abdullah Hamdan
A thesis submitted to the
Faculty of the Graduate School of
Ruhr-Universität Bochum in fulfillment of the
requirements for the doctor rer. nat
Department of Chemistry
December 2005
Abstract:
Rotationally resolved infrared spectra of methane dimer complex have been detected for
the first time in the R (0) spectral region of the triply degenerate bending mode of
methane monomer using tunable diode laser spectrometer along with supersonic jet
system. Methane dimer lines were confirmed by scanning the desired wavelength regions
with a mixture of 40% methane in Ar and He-Ne separately and then exclude the CH4-Ar
and CH4-Ne spectral lines. Many dimer lines are observed between 1290 and 1320 cm-1
,
but the lines are found to be more concentrated after the band center of the bending mode
of CH4 monomer. The spectra exhibit well resolved R branch, while the P and Q branches
have been predicted. A Hamiltonian model based on Coriolis coupling model was used to
assign and fit the recorded spectrum to within 20-30 MHz accuracy. The calculated value
of the effective rotational quantum number (j*) conclude that methane molecule is close
to free rotor limit in the complex.
Acknowledgements
Praise be to God (Allah), who gave me the strength and the patience and who bestowed
his boundless mercy on me to accomplish this work.
I would like first to express my profound gratitude and appreciation to Prof. Martina
Havenith Newen for giving me the opportunity to join her very well established research
group and a well equipped laboratory to work for my PhD. It is really fortunate to work
with such group like that. Thank you so much Frau Havenith for the continual
enthusiasm, guidance, encouragement, and all kinds of support that I received over the
years of my study in the RUB-Germany.
I am also extremely grateful to Dr. Gerhard Schwaab who has always been of great and
remarkable assist in all stages of completing my PhD project. My words cannot really
thank him enough for his endless help and his welling always to work out and discuss all
the related issues in the experiment as well as in the writing up this thesis. Thanks a lot
for everything Gerhard.
I would like also to record my thankful to Dr. Erik Bruendermann for the fruitful
discussions on some of the primary results using the previous version of the fitting
program, helping in the lab from time to time and his great computer support.
A lot of thanks to Dr. Guido Gimmler who put a great efforts in training me and
explaining many details about the diode laser spectrometer and the supersonic jet
technique which helped me to be in control of the system.
I want also to thank my friend Rachael for translating some scientific materials which
were of good help for me in this project.
My thanks extend to the secretaries of the department specially Frau Ulla Knieper who
has been welling always to help and advise me in managing all kind of related issues
regarding my legal stay and legal rights during my studying period in Germany.
Thanks to Mr. Christian Fester and Mr. Reinhard Renzewitze for their prompt positive
reaction for seeking their help in the lab.
Finally, I would like to thank all my colleagues who helped me in one way or another to
finish this work.
List of Contents.
1. Introduction. 1 1.1 Molecular Forces
1.2 Origin of Van der Waals Forces
1.3 Importance of Van der Waals Forces
2. The Theory of Intermolecular Interactions. 6
2.1 The Electrostatic energy.
2.2 Dipole-Induced-Dipole Interactions.
2.3 Dispersion Interactions.
2.4 Repulsive Interactions.
3. Methane hydrates-A potential Energy Source of the 21st Century. 11
3.1 Methane Definition
3.2 Methane Production
3.3 Methane Hydrates
3.3.1 Definition
3.3.2 Historical Perspective
3.3.3 Crystal Structure
3.4 Phase Equilibrium
3.5 Occurrence and Locations (Global Distribution)
3.6 Estimated Amounts
3.7 Energy Prospects
4. Experimental Apparatus. 25 4.1 The Tunable Diode Laser Spectrometer.
4.1.1 Laser Source
4.1.2 Cryostat
4.2 Supersonic Molecule Jet Apparatus.
4.2.1 Types of Expansions
4.2.2 Pulsed Slit Nozzles
4.3 Possible Detection Methodes
4.3.1 The Rapid-Scan Method
4.3.2 The Step-Scan Method
4.4 Set-Up and Operation of Pulsed Slit Nozzle
5. The Infrared Spectroscopy of Methane Complexes. 51
6 The Theory of Symmetric Top Rotors. 62 6.1 Symmetric Top Molecules
6.2 Asymmetric Top Molecules
6.3 Selection Rules
6.4 Perturbations
6.5 Degeneracy
7 Coriolis Force
7. Measurements and Discussion. 81 7.1 Symmetry of Tetrahedral Molecules
7.2 Data Analysis
8. References 103
1
___________________________________________________________________________
CHAPTER 1
Introduction
___________________________________________________________________________
Forces between atoms and molecules have attracted research interest for more than a
century especially in the field of spectroscopy. The development of both high
resolution experimental and theoretical techniques in this field particularly in the last
few decades made a big jump in the knowledge and understanding of these forces
possible. Infrared spectroscopy is the most common and versatile spectroscopic
technique used mainly by chemists to study the physical and chemical properties of
different types of all possible materials in the three states of matter. The main
objective is to determine the chemical and dynamical structure of the investigated
samples by targeting the molecular forces present in the molecular sample. Therefore,
this introductory chapter aims to give a brief description in rather simple way on the
different types of molecular forces acting between atoms and molecules, the origin
and the importance of these forces in life.
1.1 Molecular forces
Intramolecular and intermolecular forces are the two types of molecular forces that
are responsible for keeping and holding the atoms and molecules united to form the
different fascinating and splendid shapes of the three states of matter. Intramolecular
refers to the covalent, ionic and metallic bond forces that are acting between the atoms
within the molecule and from which the chemical properties of the matter can be
extracted (1, 2)
. On the other hand, ´intermolecular` refers to the dispersion, induction
and electrostatic bond forces that are acting between molecules. These forces are also
known as Van der Waals forces and can be used to characterize the dynamical
structure of the molecular complexes. These are also the forces that will be considered
and discussed more here and in the following chapter. The Van der Waals complexes
are characterized by extremely weak binding energies which lie in the range of 10-
100cm-1
and hence lead to very long bond lengths in the complex. Hydrogen bonding
2
is also another form of the intermolecular forces but does not belong to the same
category of Van der Waals interactions. They are the strongest intermolecular forces
with bond energies of 100-1000 cm-1
; this range is still at least 1 to 2 orders of
magnitude weaker than the energy of normal chemical bonds which exhibit bond
energies between 104 and 10
5 cm
-1. This is the reason why Van der Waals complexes
are only stable at very low temperatures, when their average thermal energies lie
below their bond energies which leads to long intermolecular bond lengths in the
aggregates of about 3-4 Ao.
1.2 Origin of Van der Waals forces
By the mid of 19th
century, the kinetic theory of gases confirmed the fact that atoms
and molecules are the basic blocks of matter. At the same time it also neglected the
volume and the intermolecular forces of molecules which led to the break down of the
ideal gas law when gases are put under conditions of high pressure and low
temperature. In 1873 the Dutch physicist Johannes van der Waals was the first to
incorporate these ideas along with the results of his experiments on pure gases to
develop an equation that was known later as van der Waals equation which describes
the behavior of real gases compared to ideal gases.
(P + a n2/V
2)(V – n b) = n R T
where P, V, n, R, T are pressure, volume, number of molecules, Boltzmann constant
and temperature respectively. The constant "a" is a correction term for intermolecular
force while "b" is the correction term for the real volume of the gas molecules.
The equation indicates that the actual free volume of the container is reduced by the
molecules occupied volume of the molecules which implies that strong repulsive
forces are effective at short distances. The equation also suggests that the gas pressure
is actually slightly less than it would be without attractive forces which lead to the
conclusion that long range intermolecular forces are effective. These forces were
believed to be of classical electrostatic origins, but the development of the quantum
mechanical methods led to conclude that these forces have also quantum mechanical
character (3)
.
3
Intermolecular interactions originate from the fact that atoms and molecules are
composed of charged particles which interact by the means of Coulomb forces. The
quantum mechanical treatment of these forces can lead to the following
intermolecular interactions; electrostatic, induction, dispersion, and exchange
repulsion. The first order perturbation theory describes the electrostatic and the
exchange repulsion contributions, while the induction and dispersion components
occur in second order perturbation theory. The electrostatic components originate
from classical interaction between static charge distributions of two polar molecules
leading to the dipole-dipole interactions. The induction contributions arise from the
distortion of a particular unpolar molecule by the electric field of all other
neighbouring molecules. The dispersion energy is a result of quantum fluctuation of
the electron distributions on atoms or molecules leading to the induced dipole-induced
dipole interaction even for the most symmetrical systems. The exchange-repulsion
forces are the short-range forces which occur as consequence of strong repulsion
between strongly overlapped and occupied orbitals of neighboring atoms or
molecules. The theory of these forces (interactions) will be discussed in the following
chapter (3)
.
There has been a lot of scientific interest on investigating small weakly bound van der
Waals complexes for the last few decades in order to understand the mechanism of
intermolecular interactions. But in spite of its long history, it was possible to examine
these complexes with high resolution spectroscopic techniques only 20-30 years ago.
The experimental and theoretical advances in this field have been documented in 3
complete editions of “Chemical Reviews” (4, 5, 6)
.
1.3 Importance of Van der Waals forces
Intermolecular forces are feeble; but without them all matter would exist in a gaseous
state, and life as we know it would be impossible. They form the basis of a wide range
of fundamental scientific phenomena in different branches of physics, chemistry and
biology. Many physical and chemical properties (e.g. melting points, boiling points,
heats of fusion and vaporization, surface tension, densities …etc) of molecular
compounds, including crystal structures, depend mainly on intermolecular forces. For
instance, in the gaseous phase they are responsible for the transport properties of
4
gases like diffusion, viscosity and heat capacity, in molecular solids the structure of
molecular crystals depends very much on the anisotropy of the intermolecular
potentials between the individual molecules, and the solvation processes in liquids are
determined only by intermolecular interactions.
They also play a central role in biology and life sciences, being responsible for
holding gigantic molecules like enzymes, proteins, and DNA into their original and
required shapes. The biological relevance of hydrogen bonds is due to their lack of
strength; they are stable at room temperature but can be easily broken by a small
amount of energy input, which allows changes in the stable configuration. This is how
the genetic code on DNA is replicated. It was also reported that intermolecular forces
can act as the mediator of some protein receptor-drug reactions (7, 8, 9)
. Investigation of
Van der Waals complexes started first with the closed-shell molecular species. The
open-shell molecular species started recently to attract more research interest,
especially the ones that form in the atmosphere and the interstellar clouds as many of
their properties are poorly understood. It is well known that these complexes are
expected to have a profound effect on the chemistry of the atmosphere as a result of
relatively low temperature, production of many free radicals and the effects of
radiation (10)
. These examples are just some of many other phenomena that depend on
intermolecular interactions.
From the above examples, one can realize how essential these weak forces are in our
life, and how crucial it is to have an accurate theoretical description of these forces
especially for large systems like DNA molecules. Therefore, it is highly desirable to
obtain a simple and reliable model that can be tested first on relatively small prototype
Van der Waals complexes to help understand the existing interactions. The results can
then be expanded to more complex systems and thus used to develop an exact model
of the Van der Waals forces between the molecules.
Infrared spectroscopic techniques are, in principle, the techniques that are used to
serve in achieving the above goal experimentally. These techniques are used to
measure inter- intermolecular vibrational modes of the weakly bound complexes and
thus determine the positions of their energy levels. Experimental results are then used
to help determine accurate intermolecular potential surfaces. Modeling the potentials
5
from many of these small prototype systems can give a general understanding of the
interactions in larger systems. Moreover, high resolution infrared spectra can also
provide detailed information about the dynamics and structure of molecular
complexes in both ground and electronically excited states. The molecular constants
extracted from the infrared spectra are directly related to the geometrical structures in
both states, giving access to information about intermolecular bond lengths and their
changes upon excitation. For example, if there is tunneling present within the
complex, it becomes apparent in the splitting of the rotation lines in the spectrum.
6
___________________________________________________________________________
CHAPTER 2
Theory of intermolecular forces
___________________________________________________________________________
Investigation of intermolecular forces started practically with the developments of
both high resolution experimental and computational techniques about thirty years
ago. These studies led to numerous advances in the knowledge and understanding of
these forces especially in the last decade where a lot of progress has been achieved in
the construction of reliable intermolecular potential energy surfaces from which
various chemical and physical properties of the molecular system can be extracted.
Intermolecular potentials are also important and necessary for the determination of the
structure, stability, and dynamics of weakly bound clusters and condensed phases.
The molecular interactions described by these potentials depend mainly on the
distance between the involved molecules and their relative orientation to each other.
Intermolecular potentials can not be measured directly from experiments. Different
theoretical computational approaches are used to derive intermolecular potentials.
According to their point of origin, these approaches are grouped into two classes: the
semi-empirical and quantum mechanical potentials. In the first approach the
intermolecular potentials (PES) can be inferred from experimental data from different
experimental sources such as spectroscopic measurements, second virial coefficients
and molecular-beam scattering data, but in this case it is important here to assume
some functional form of the interaction and attempt to vary the parameters of the fit to
reproduce the experimental results. Another approach to the PES is via ab initio
quantum mechanical calculations, where the molecular potentials are calculated
theoretically using the electronic molecular orbital theory. This method has been
lately improved by the growth of the computing power.
The theory of intermolecular interactions and its contributions is described in different
articles and books (3, 11, 12)
. It has four main energy contributions which are classified
as long-range forces; electrostatic, induction, and dispersion and short-range forces
7
like the exchange-repulsion force. These forces will be briefly discussed in this
section
2.1 Electrostatic Energy
Electrostatic forces are the energy contributions that occur between the charged
particles of molecules with permanent dipole moment which can be classically
represented by the Coulomb interaction law between the individual complex-building
molecules. In general, all electrostatic charges in the molecule have to be taken into
account to calculate the Coulomb potential of the system, i.e. all electrons and nuclei
in the molecular system are contributing to the overall potential. The Coulomb
potential energy is given as:
∑=ij ij
ji
electR
qqV
π4 (2.1)
where qi is the charge on the i-th particle and Rij is the distance between the i-th and
the j-th particle.
As a result of their long range behavior, these interactions will have substantial
contribution to the intermolecular potential energy. The consequence of 1/R
dependency leads to conclude that the electrostatic potential will contribute to the
total energy much more than from dispersion energy at larger distances. The
electrostatic components of the intermolecular potential are strictly pair wise additive
and it can be attractive or repulsive depending on the orientation of the two monomers
(3, 13, 14).
The interaction energy of two permanent dipoles depends on the relative orientation
of both dipoles which could be zero if all the orientations are possible. This can be
true if the molecules are completely free to rotate, but in practice molecules are not
totally free to rotate and some orientations are preferred over others. Therefore, the
interaction energy varies as 1/ R6, while the force between the dipoles varies as 1/R
7.
In a solid sample the interaction energy varies as 1/ R3 (3)
.
8
2.2 Induction Energy (Dipole – Induced – Dipole Interaction)
The energy contributions here emerge when a molecule with a permanent dipole
moment induces a dipole moment in a neighboring polarizable molecule. This
interaction creates an attractive atmosphere between the two dipoles. Therefore, the
induction forces are always attractive. The strength of this interaction is a function of
the electric field E of the permanent dipole and the polarizability α of the neighboring
molecule. The induction energy is given by:
2
2
1EV ind α= (2.2)
Induction energy is severely non-additive depending on the direction of the multipole
moment, i.e. when a molecule is surrounded by other neighbor molecules, the electric
fields of the surrounding molecules may reinforce or cancel each other.
The second-order perturbation theory indicates that if one monomer possesses electric
dipole moment, the magnitude of induction energy varies as R-6
, where R is the
distance between the molecules. The induction energy delivers a non-zero
contribution also if only one bond partner has a multipole moment (3)
.
2.3 Dispersion Energy (Induced Dipole – Induced Dipole Interaction)
It is also known as London dispersion forces or Van der Waal force. This contribution
is of a purely quantum mechanical nature and cannot be explained classically. In
principle, all molecules have the possibility to form London forces. But they mainly
occur between nonpolar atoms or molecules such as (noble gases, N2, H2, O2…CH4,
CCl4, BF3…etc). These are the weakest intermolecular forces which arise from the
fluctuations of the charge density distribution in atoms or molecules as a result of
constant motion of electrons leading to a temporary dipole. These transient dipole
moments cancel out each other to zero over a certain period of time, but the molecule
can still interact with neighboring molecules at any time. This in turn can induce an
instantaneous dipole moment in a second molecule and hence result in building up a
net attractive force between the two molecules. The electrons in both molecules then
become correlated which leads to the favored lower energy configuration of the
9
complex. These dipoles depend on the polarizability of the molecule, and vary as
1/R6. Dispersion forces increases with mass, number of atoms or electrons which
reflect on certain properties of materials.
An exact theory of the dispersion interaction naturally includes the higher order
multipole moments. The dispersion energy is obtained from the second order
correction of the perturbation theory and is given as:
Vdis = C6 R-6
+ C8 R-8
+ C10 R-10
+ …. (2.3)
The dispersion energy has much more isotropic properties than the electrostatic or the
induction energy, because the instantaneous multipole moments can orientate
themselves in any direction relative to the static molecular coordination system (3)
.
2.4 Exchange-Repulsion Energy
The Exchange-Repulsion means that the electron motions can extend over either both
atoms or molecules for the exchange part, whereas the repulsion means that in term of
atomic and molecular orbitals an antibonding orbital can be populated. These are
repulsive forces which exist or operate at very short distances where the wave
functions of atoms or molecules are significantly overlapped, i.e. the charge
distributions/densities of neighboring atoms or molecules are strongly overlapped.
This result in a strong repulsion between the tightly bound electrons which in turn
leads to a reduction in the electron density between the nuclei due to the Pauli
principle and the nuclei then repel each other. The simplest representation of the
repulsive force is a single exponential function:
Vrep = A e-βR
(2.4)
with A and b are two adjustable parameters which depend on the angular orientation
of the two monomers (15)
. This functional relationship is included in several semi-
empirical potentials (16-21)
The individual contribution of the above mentioned intermolecular forces to the
intermolecular potential depends mainly on the symmetry of the charge distribution,
10
the spatial distance between the individual interacting partners and their orientation to
one another. It shows whether a potential minimum is formed between the interacting
molecules and if the formation of a complex is at all possible or not.
The construction of a theoretical model for each of these contributions to the
intermolecular interaction -especially the repulsive and dispersive forces- still remains
a challenge for quantum chemistry up to date.
11
_____________________________________________________________________
CHAPTER 3
Methane Hydrates-a potential energy source of the 21st Century
_____________________________________________________________________
The awareness of methane as a possible energy source began after the discovery of
the natural methane hydrates back in the 1960’s. This discovery triggered many
research groups at the global scale to put more efforts on studying these hydrates
especially in the last two decades. This chapter will briefly discuss the basic concept
as well as the most important and relevant aspects of hydrates concentrating mainly
on methane hydrates.
3.1 Definition
Methane is a colorless, odorless, nontoxic and highly flammable gas with a wide
distribution in nature. At room temperature, methane is lighter than air, melts at –
183°C and boils at –164°C. It belongs to the alkane group of hydrocarbons which are
basically organic compounds that consist only of carbon and hydrogen atoms. These
atoms can combine together in virtually countless ways to make a diversity of
products composing the different groups of hydrocarbons. Methane is the simplest
molecular structure of hydrocarbons with a chemical formula of CH4. It has the
typical tetrahedral shape where the carbon atom is attached or connected to four
hydrogen atoms by single bonds making an angle of 109.5 degrees at room
temperature. Methane constitutes the primary or principal component of natural gas; it
normally makes up 50-90 % of the mixture depending on the source. The balance is a
varying amount of ethane, propane, butane, and other hydrocarbon compounds.
12
3.2 Methane production
Two models are proposed for methane production: thermogenic and biogenic models.
In the thermogenic model, methane is produced by the combined action of heat,
pressure and time on buried organic materials which is mainly the common
mechanism for the production of hydrocarbon gases. In the biogenic model, methane
is formed by anaerobic digestion of certain organic matters (plants, animals,
waste...etc) in areas that are almost oxygen free. The digestion is a two step process
which is mainly done by a special kind of anaerobic bacteria that are usually found in
oxygen poor or oxygen free environments like livestock, landfills and dumps,
wetlands, along side with oil fields inside the earth and shallow sea floor sediments.
The first step is the breakdown of the complex organic waste into simple organic
acidic compounds by a particular group of bacteria, called acid formers. In the second
step, a highly specialized group of bacteria, called methane formers, converts the
acids to methane gas and carbon dioxide. In a properly functioning digester, the two
groups of bacteria must balance so that the methane-formers use just the acids
produced by the acid-formers.
The detailed stages of methane formation have been described by Hesse (22)
, while the
overall of methane production is summarized by Sloan (23)
in the following equation:
(CH2O)106(NH3)16(H2PO4) 53CO2+53CH4+16NH3+H2PO4
The equation summarizes successive stages of oxidation by oxygen and reduction by
nitrates, sulfates, and carbonates.
Methane can also be produced industrially by the destructive distillation of coal or
wood, or by heating certain mixtures like sodium acetate and sodium hydroxide or
carbon and hydrogen, and by the reaction of certain complexes like aluminum carbide
and water.
In addition to the above mentioned different sources of methane production on earth,
methane is also found as a major constituent of the atmospheres of most of the
gaseous planets in our solar system including the earth (24)
.
Since methane is a highly flammable gas and being continuously produced by supply
from biogenic/bacterial emission, methane is used as an alternative source of energy
13
on many different industrial, social, environmental and economical fields of life
worldwide, therefore it is considered to be the most important, versatile, viable,
sustainable and economic molecular gas of hydrocarbons.
On the other hand, methane is an important greenhouse gas with global warming of
25, (i.e, it has 25-30 times the warming ability of carbon dioxide.)
3.3 Methane Hydrates (clathrates)
3.3.1 Definition
A newly discovered source of highly concentrated methane on earth is the so-called
methane hydrates or clathrates. These are a unique class of chemical compounds
where hydrogen bonded water molecules combine to form a cage-like symmetrical
structure that hosts, without chemical bonding, a high concentration of methane
molecules under high pressure and low temperature (25)
as shown in fig (3.1). These
water lattice structures can also be stabilized to form hydrates by other common guest
molecules like nitrogen, carbon dioxide, hydrogen sulfide and larger hydrocarbons
such as ethane, propane isobutane, normal butane, of which methane occurs most
abundantly in nature. These hydrate complexes are kept united and held together in
place by Van der Waals forces; therefore, they are also categorized as Van der Waals
complexes. The accumulation of such ice-like crystalline structures over a long period
of time (thousands of years) form what is currently known as gas hydrates or methane
hydrates.
No hydrates can be formed or stabilized by small molecules such as hydrogen or
helium because they are not large enough to be trapped or to support the cavity
structures. The molecules that are too big to fit the hosting cavities can also not form
hydrates.
14
Fig. 3.1: Hydrate structure showing carbon atom in the center (gray color) attached to
hydrogen atoms (green color) trapped in an ice lattice. (USGS)
3.3.2 Historical Perspective
Molecular hydration was first noticed in laboratory by H. Davy and M. Faraday
almost two centuries ago while experimenting with a chlorine-water mixture (26, 27)
.
Many scientists continued to study and investigate these strange or unusual materials
until the beginning of 20th century. In the 1930s, E.G. Hammerschmidt (28, 29, 30)
determined that hydrates were responsible for plugging natural gas pipelines,
particularly those located in cold environments. This problem was solved by a group
of researchers (29, 31-35)
who studied the physics of various hydrates in order to develop
proper chemical additives (inhibitors) and other methods to inhibit and remediate
hydrate formation in pipelines. Then in the 1960s, naturally formed methane hydrates
were discovered in a giant Siberian gas field (36)
, and soon after in shallow sub-
permafrost sediments on the North Slope of Alaska. This led the scientists to
speculate that the necessary conditions of hydrate formation of high pressure and low
temperature should not only be in permafrost regions but also in other global locations
like deep oceans.
15
The new discovery also encouraged the scientists to continue their investigation on
hydrates, which was then intensified, expanded, and spread out to cover different
types of hydrates along with the newly and continuously developed spectroscopic
techniques. In the massively increasing number of reports, the scientists concentrated
on studying different aspects of these compounds like physical and chemical
properties, formation and decomposition, structure, global distribution, locations and
stability, concentration, and the true energy potential of natural hydrates. Such
information is necessary to develop computer models that can accurately predict the
behavior of hydrates and hydrate-sediment systems under changing conditions. It can
be also a build up foundation of basic knowledge for methane hydrates and other
types of hydrates.
The occurrence of natural methane hydrates has also promoted many countries to
launch different research projects around the globe looking for all possible locations
on earth that have environmental conditions of high pressure and low temperature for
natural hydrates formation.
16
Table 3.1: A brief list on the development of the ongoing research efforts on hydrates
over almost two centuries. Ref. (37)
1810 Discovery of chlorine hydrate
1828-29 Discovery of Bromine & SO2 hydrates
1848-55 Determination and measurement of SO2 hydrate formulas
1877 Deasurement of mixed gas hydrates ( CO2+PH3 & H2S+PH3)
1884,85 Postulation of the upper/lower quadruple points of hydrates
1888 Measurement of different hydrates including methane hydrates
CH4, C2H6, C2H4, C2H2, N2O,....etc.
1888 Pressure and temperature dependence of some hydrates
1890 Determination of the crystal structure of hydrates
1896-1925
Discovery of other different hydrates like Ar, N2, O2, Kr, Xe,......etc,
discovery of double hydrates.....
1934 Blocking of natural gas pipelines by man-made hydrates
1951-58 Identification of the hydrates cavity structures
1959 Proposing the mathematical model to predict hydrates properties
1965 Discovery of hydrates in permafrost (Siberia)
1970-85 Reporting and recovering hydrates from onshore and offshore
locations
1987 Discovery of new hydrate cavity structure (structure H)
1979-92
Initial characterization and quantification of methane hydrate deposits
in deep water. and conducting several research projects (kinetic,
molecular dynamics, calorimetry, phase equlibria, volume estimation,
green house effect...etc) on different types of hydrates.
1993 First international conference on natural gas hydrates
1996 Microscopic studies using Raman spectroscopy
1996 Second international conference on hydrates
2000
Efforts to quantify location and abundance of hydrates begin. Large
scale efforts to exploit hydrates as fuel begin
17
3.3.3 Crystal structure
X-ray diffraction technique has been extensively used to study the crystal structure of
hydrates by Von Stakelberg and coworkers in 1950s (38 - 49)
. The analysis of their
efforts led to the determination of the first two types of crystal structure of hydrates
known as structure I and structure II. These structures represent different
arrangements of water molecules resulting in slightly different shapes, sizes, and
assortments of cavities. The structure formation depends on various aspects of the
available guest molecule. Both structures I and II can be stabilized by filling at least
70 percent of the cavities by a single guest molecule, therefore known as simple
hydrates.
In 1987 Ripmeister and others (50 - 54)
discovered a third type of hydrate structure
named as structure H which requires the cooperation of two guest molecules (one
large and one small) to stabilize, thus known as double hydrate. Structure H hydrates
are rare, but are known to exist in locations where a thermogenic production of heavy
hydrocarbons is common.
The continuous experimental advances and developments in this field may result in
discovering more exotic and complex structures of gas hydrates. Methane is
commonly the dominant component of clathrate gas hydrates formed either in nature
or in industrial processes. Due to its small molecular size, methane can serve as a
guest molecule in all the three known gas hydrate structures I, II, and H.
3.3.3.1 Structure I
Each unit cell of Structure-I gas hydrate consists of 46 water molecules which form
two small dodecahedral voids and six large tetra-decahedral voids. Structure-I gas
hydrates can only hold small gas molecules such as methane and ethane, with
molecular diameters not exceeding 5.2 angstroms. The chemical composition of a
Structure-I gas hydrate can be expressed as 8 (Ar, CH4, H2S, CO2)46H2O or (Ar, CH4,
H2S, CO2)5.7H2O (55)
.
18
512
512
62
3.3.3.2 Structure II
The unit cell of Structure-II gas hydrate consists of 16 small dodecahedral and 8 large
hexakaidecahedral voids formed by 136 water molecules. Structure-II gas hydrates
may contain gases with molecular dimensions in the range of 5.9 to 6.9 angstroms,
such as propane and isobutane. The chemical composition of a Structure-II gas
hydrate can be expressed as 8(C3H8, C4H10, CH2C12, CHCL3)136H2O or (C3H8,
C4H10, CH2C12, CHCL3)17H2O (55)
.
512
512
64
3.3.3.3 Structure H
The unit cell of this double hydrate structure composes of 34 water molecules
producing 3 small cavities, 12 slightly larger cavities, and 1 relatively huge cavity.
The large cavity of structure H allows this hydrate structure to incorporate large
molecules such as butane and lager hydrocarbons leading to the occurrence of smaller
help gases to fill and support the other smaller cavities (50, 54)
20
3.3.4 Phase equilibrium Stability
In general, a combination of low temperature and high pressure is needed to support
methane hydrate formation. In addition to temperature and pressure, the composition
of both the water and the gas are also critical for the fine tuning of gas hydrates
stability, i.e. the type of the used water and natural gas in the experiment (56)
.
Fig (3.2) depicts the phase stability diagram for methane hydrates in permafrost and in
oceans respectively. The broken lines in both figures stand for the geothermal gradient
as a function of the depth. Whereas the solid lines are based on methane hydrates
phase boundary data. The phase stability diagrams of methane hydrates are usually
displayed with the pressure being converted to depth in meters along with the natural
geothermal gradient curve to indicate the expected temperatures as the pressure
(depth) increases. The intersection of both the geothermal gradient and the phase
boundary curves in the figures defines the depth of the Gas Hydrate Stability Zone
(GHSZ).
Fig. 3.2: Envelopes of methane hydrate stability in permafrost (A) and in ocean
sediment (B). Ref. (37)
21
In fig (3.2-A), the phase diagram shows typical conditions in a permafrost region of
the North Pole assuming a permafrost depth of 600 meters. The overlap of both the
phase boundary and temperature gradient curves indicates that the GHSZ should
extend from a depth of about 200 meters to slightly more than 1,000 meters, i.e. when
hydrates are initiated, more nucleation can occur with increasing pressure or
decreasing temperature.
Figure (3.2-B) shows the phase diagram for a typical location on Deep Ocean. A
seafloor depth of 1200 meters is assumed. The temperature steadily decreases with
increasing depth, reaching down to values close to 0°C at the ocean bottom. As one
goes down below the ocean bottom, the temperatures start to constantly increase
again. These settings imply that the top of the GHSZ occurs at roughly 400 meters
while the base of the GHSZ lies at 1500 meters. Therefore, hydrates will only form in
the sediments within this region. However, at very deep sediments, methane hydrates
are not likely to be formed due to the lack of high biological productivity (the bacteria
which are needed to produce the organic matter that is converted to methane) and
rapid sedimentation rates (to eliminate the organic matter) that support hydrate
formation on the continental shelves (37)
.
3.3.5 Occurrence and Locations (global distribution)
The knowledge of methane hydrates was limited on it’s occurrence in chemical
laboratories and natural gas pipelines. However, a series of discoveries started first at
the North Pole and then spread out to deep water regions of all continents indicated
that natural methane hydrates exist on a huge scale.
The existence of natural methane hydrate in many locations is concluded by using
certain geophysical survey techniques or geochemical analyses of sediment samples.
However, the number of locations is continuously increasing where more detailed
information is being collected. This wide range of information can ultimately form a
knowledge basis for natural gas hydrates. Fig (3.3) shows both an onshore and
offshore global map for more than 50 sites of methane hydrates which have been
identified by geophysical and geochemical techniques (57, 58)
.
22
Fig. 3.3: The onshore and offshore global locations of known and inferred hydrate deposites in ocean ( ) and permafrost ( )
23
3.3.6 Estimated amount
There is no available data on the absolute amount of methane hydrate in earth, but it is
generally accepted that the global volume of methane in hydrates is immense and far
exceeding the volume of methane in any other form. However, the estimates of
methane volume compressed in hydrates are widely changing among different
research groups. Table (3.3) lists the estimates of the total volume methane in
hydrates for different groups over two decades of research efforts. These values are
supposed to be the most reliable data. However, researchers have concluded that the
estimated amount of natural methane trapped in hydrates is twice the amount of
methane equivalent to ever known fossil fuels in earth (i.e. gas, oil, coal…etc) (59 - 67)
Table 3.3: Estimates of In Situ Methane Hydrates. Ref. (37)
3.3.7 Energy prospects
The above estimation of methane trapped in hydrates indicates that methane is highly
concentrated in methane hydrates. It was calculated using the ideal gas law under
standard conditions of temperature and pressure (15° C & 1 atm.) that if all the cages
would be 100 percent occupied by methane molecules, then the dissociation of one
cubic meter of solid hydrate can release about 170 m3 of methane gas. However, the
24
maximum occupancy ranges between 70 and 90 percent, therefore one cubic meter of
methane in nature turns out to contain up to 164 m3 of methane. In another reference,
it is stated that each volume of methane hydrates can contain 184 volume of methane.
These figures conclude that methane hydrates are considered as a huge potential
energy source for many applications (57, 58, 63)
.
25
___________________________________________________________________________
CHAPTER 4
Experimental setup and Instrumentation
___________________________________________________________________________
The experimental layout of the computer controlled diode laser spectrometer and the
molecular jet system used in this work is shown in fig. (4.1). This system is used to study
and investigate different species of relatively small and weakly bound molecular
complexes. The setup is constructed of three major parts, the diode laser spectrometer,
the molecular jet and the data acquisition system. A comprehensive introduction on
the tunable diode laser spectroscopy is given in (68-71), and an introduction on the
theory of lead salt diode lasers is also given in (72, 73). The components of
experimental setup will be described briefly in the following sections. More detailed
information about the diode laser spectrometer system can be found in (74).
4.1 The Tunable Diode laser spectrometer
4.1.1 Laser source
Diode lasers are classified as solid state lasers, where the laser or lasing medium is
usually made up of doped solid crystalline materials (e.g.; ruby, Nd-YAG, Nd-Glass,
Nd-YLF, etc). Diode lasers, also called semiconductor lasers, are the smallest ever
made lasers with an active medium of grain size crystal, which is usually cut in a
rectangular shape with cleaved facets to work as the laser resonator. The other facets
are destroyed by using different methods like etching, grounding, ion
implantation....etc. These little tiny crystals are basically a combination of some
doped semiconductor materials or alloys in a p-n junction. The frequency range of a
diode laser is commonly determined by the exact composition of the semiconductor
crystals, which is precisely selected and controlled in the manufacturing phase. The
laser action takes place in these crystals when a voltage is applied on the p-n junction.
Then holes and electrons are generated in the junction, thereby creating a population
inversion within the junction. Electrons and holes then recombine and emit the
26
recombination energy as a laser radiation which covers so far the visible and infrared
regions depending on the composition of the laser medium.
The diode laser spectrometer used in this work is the commercially available model
(Mutek MSD 1100), consisting of the laser diodes, the cryostat and the optics. The
diode lasers are
Fig. 4.1: Experimental setup of the tunable diode laser spectrometer system in our lab.
Ref. (125)
lead salt lasers from Laser Component and Aero Laser companies. The active medium
is a combination of crystalline structure from (PbSe, PbTe, PbEu…etc). Lead salt
diode lasers have a wavelength emission range between 3 and 15 µm or (3300-650)
cm-1
. These lasers provide a typical output power from 100µW to 1 mW with a typical
emission line width of 30-100MHz. Each diode laser has a quasi continuous spectral
coverage over a region of 50-150 cm-1
. More than 50 of these diodes are available
along with the spectrometer in our lab, which cover a tunable wavelength range of
900-2800 cm-1
. The rectangular and extremely small size (50-200) µm laser cavity,
results in a highly divergent (20-40 degrees) beam which suffers from astigmatism
27
and elliptical beam profile. These drawbacks of the laser beam lead to inhomogeneous
broadening of the gain profile which results in multimode laser radiation or emission
across the frequency range of the diode laser. These modes are typically separated by
1 to 4 cm-1
and can be continuously tuned over a frequency range of 0.5 to 2 cm-1
.
Therefore, single mode operation of these lasers is limited to small regions and only
possible in certain cases. In principle, all diode lasers have a similar overall
performance, but each diode laser is a unique device with highly individual
characteristics that depends on the composition of the semiconductor crystal and the
applied current and temperature. Even diode lasers from the same crystal may have
unique beam properties.
The sensitivity of the spectrometer is not limited by the f1 laser noise that shows up
to frequencies of 100MHz, but through etalon structures in the signal. This can be
caused by every pair of reflecting surfaces in the beam path which in our setup are for
example the 12 cm diameter mirrors of the Herriott multi-pass-cell. But the pump
vibrations are transferred to the mirrors of the cell as they are very heavy. These
mirror vibrations destroy the phase coherence of the etalon signals, so that they are
mostly damped and show up rarely. The distinct improvement of the signal-to-noise
ratio is depicted exemplarily in (75).
The wavelength emission of diode lasers is a function of the diode temperature and
the applied current. The coarse tuning is mainly done by varying the diode
temperature, while the fine tuning is achieved by smooth changes of the applied
current; i.e. continuous tuning over a small limited range or across a selected
longitudinal mode. In coarse tuning, the temperature change causes a variation in the
band energy gap and a modification of the cavity length of the diode laser due to the
changing refractive index (n) of the semiconductor crystal. These changes lead to the
so called mode jumping where different modes are generated to fit different cavity
lengths, i.e. one mode is terminated and a second one is generated at another
temperature to fit the new cavity length. The second tuning method of the diode laser
is based on changing the applied current while the temperature being held fixed. This
normally produces a small amount of Joule heating that causes a slight change in the
diode temperature and leads to alteration of the refractive index (n). In this method the
change of refractive index results in a negative tuning rate ∆ν ⁄∆I, while the
28
temperature increase yields a positive tuning rate. However, these changes shift the
laser modes in the same direction as the band gap change, but at a slower rate, thus,
providing a more precisely and controllable way of continuous tuning over limited
ranges, i.e. single mode range. The very short time scale of current tuning ≤ 1µs
compared with the time scale of the temperature tuning 5-30 seconds makes it more
profitable or suitable to use for scanning the diode lasers over their frequency range.
The typical tuning rates are:
Current tuning: ∆ν ⁄ ∆I = 0.2-3 GHz ⁄ mA
Temperature tuning: ∆ν ⁄ ∆T = 10-100 MHz ⁄ mK
4.1.2 Cryostat
Lead salt diode lasers operate at cryogenic temperatures, i.e. < 80 K. A closed-cycle
helium cooler from Leybold is used to cool the laser diodes down to 20 K; it also has
a precise temperature control over the working range of the diode laser between 20
and 70 K. It is a long term, maintenance free system with a water-cooled compressor.
The cryostat is mechanically isolated against the vibration of the helium cooler. Four
different diode lasers can be accommodated and simultaneously cooled down in the
cryostat chamber. The desired temperature of the diode laser is achieved by resistive
heating, i.e. by changing the current through a heating coil plugged to the cold finger
of the copper cold head which hosts the diode lasers. A special temperature controller
with the required accuracy over the whole range (10-200K) was developed in our
group, since such a controller was not commercially available; it utilizes a platinum
resistor (Pt1000) as a temperature sensor which guarantees high stability, absolute
accuracy (± 0.5 K), excellent reproducibility (0.05-0.1 K) and quick response (76)
. The
actual control is achieved by using a precision analog PID (Potential-Integration-
Differential) controller designed also in our group to produces an out put voltage
which drives the heater current of the diode laser. The current can be adjusted
between 0 and 900 mA with smallest step of 0.3 µA.
A set of compensated mirror optics consisting of two ellipsoidal mirrors with foci of
40mm and 140mm, one toroidal mirror with (f = 110mm) and some plane mirrors are
29
used to both select one of the four laser diodes in the cryostat and to collimate the
diffraction broadened laser beam which exits the cryostat via a tilted CaF2 plane
window. The collimated output laser beam is then passed through a telescope
arrangement (two mirrors with f = 60 cm and f = 10 cm) to reduce the beam diameter
from 14 mm to 3mm. This is the optimal beam size required for coupling into the
multi-pass cell located inside the vacuum chamber to exclusively probe the expansion
zone of the slit nozzle. This extends a few cm vertical to the expansion direction,
thereby increasing the signal to noise ratio. Purely reflective optical elements are used
in the optical path of the laser beam in order to minimize the feedback to the diode
lasers. No lenses are used in the optical setup as they act as a source of small back
reflection in the laser cavity. The optimal laser beam is then guided to enter the
vacuum chamber through a CaF2 window, where it is coupled in a Herriott multi-pass
cell. The cell arrangement consists of two spherical gold plated and identical concave
mirrors separated by a distance of their radius of curvature; both mirrors have a
diameter of 12 cm and a focal length of 50 mm. The design is made up to couple the
optical beam into the cell through a hole in the first concave mirror. A correct
alignment of the optical beam in the cell results in an elliptical beam spot pattern on
both mirror surfaces, a maximum number of 40 spots can be achieved between the
two mirrors before the beam emerges out of the same coupling hole as it entered. The
number of spots usually determines the number of the beam reflections within the two
mirrors and also specifies the optical absorption length in the cell which is between 80
and 160 cm for this design. This cell was built and integrated into the apparatus in
context of a research Master degree project done in our research group (77)
. This
design enables all reflections of the ellipse to be utilized, whereas in the old White
cell design developed by König (70)
, only half of the ellipse could be used as shown in
the fig. (4.2). The new cell has brought a further positive aspect for spectroscopy,
apart from the increased number of passes and therefore the increased absorption
length from 80 to 160 cm as shown in fig. (4.3) (77)
.
30
Fig. 4.2: Herriott cell design from König, Ref. (125)
The slit nozzle is usually aligned in the middle of the Herriot type cell to ensure that
the molecular jet is generated 5 to 7mm perpendicular to the laser beam which crosses
the narrow expansion zone at each path. As leaving the vacuum chamber, the laser
beam is directed into a monochromator to separate and select the desired mode from
the multimode emission of the diode laser. The monochromator employed in our
diode laser spectrometer system is a 0.5 m Czerny Turner type (Mutek MDS1200)
from Mutek Company with a frequency resolution of ~ 1 cm-1
. A grating of 30 lines /
mm with blaze wavelength of 25 µm
31
Fig. 4.3: New Herriott Cell design from Lehnig, Ref. (125)
is used in this monochromator which allows coverage of the whole pertinent
wavelength range from ~ 800-3000 cm-1
by scanning over the different grating orders.
Absolute wavelength calibration of the monochromator is done by using a He-Ne
laser. The absorption lines of CH4 monomer gas have been used to provide absolute
frequency calibration of the spectra with an accuracy of 0.001 cm-1
. Spectral
frequency calibration is achieved by deflecting a (~ 70%) fraction of the laser
radiation using a ZnSe beam splitter and send it through a highly stable confocal
etalon with a free spectral range of 0.01cm-1
(300 MHz). The etalon transmission is
then used by the computer control program to determine the tuning rate with an
accuracy of better than 1% and readjust the grating accordingly. The two portions of
the beam are focussed onto HgCdTe-detectors; the signals are then amplified and
detected by means of Stanford Research phase-sensitive lock-in amplifiers.
The reference frequency is generated by modulation of the diode laser current with a
frequency of 7 kHz and amplitude between 0 and 1 mA, while the laser frequency is
increased. The line width of the emission of a typical diode laser is between 50
and100 MHz. The modulation is thus adjusted in a way so that the spectral lines have
an optimal intensity, without being significantly broadened due to the modulation
32
frequency. As a result of frequency modulation of the diode laser, one obtains a 1f
derivative of the line profile, after demodulation in the lock-in amplifier. In order to
avoid the difficulty of the frequency determination of the spectral lines by certain
fluctuations of the central point of this derivative, one should take the second
harmonic of the demodulated signal at 14 kHz. This procedure will differentiate the
signal once again, so that the line frequency corresponds to the maximum of the line-
shape once more. This new differentiation of the signal also suppresses background
fluctuations with small gradients efficiently. The calculated line widths using the
second harmonic, depending on the modulation, lie in the region of 30-100 MHz.
Despite the fact that absorption spectroscopy is a relatively simple technique,
sensitivities of as low as ∆I/I = 10-5
– 10-6
can be achieved (78, 79)
.
The spectrometer is controlled by means of a computer, programmed with LabVIEW
(71), which also enables, besides the spectroscopic measurements, a characterization of
the laser diode by measurement of the mode chart.
4.2 Supersonic Molecular Jet Apparatus
In principle, a large number of vibrational and the associated rotational levels of
atomic and molecular structures are highly populated at room temperature. The
spectroscopy of such systems usually leads to very complex and congested spectra
which can show several hundreds of overlapping lines that are very difficult or even
impossible to resolve and analyze. Therefore, cooling of atoms and molecules has
been a very important issue in spectroscopy for the last few decades. The aim was
always to look for cooling methods that can dramatically decrease the internal
temperature of the investigated samples. Consequently, very few vibrational and
rotational levels of the ground electronic state of the analyte sample will be populated,
which in turn leads to significantly simplified spectra that are possible to resolve and
analyze. Different types of cooling methods have been developed and used in order to
achieve very low sample temperatures: One method is to cool down the gas in the
spectroscopic cell by surrounding it with liquids at low temperature (e.g. liquid
nitrogen), but this can cause a rapid decrease in the vapour pressure to be too low for
use. The other method is the cryogenic cooling equipments which are very bulky and
expensive to use. A third method is the so-called supersonic jet molecular beam
33
source (supersonic jet expansion technique) that was first described and introduced by
Kantrowitz and Grey in 1951 (80)
. This type of beam source results in remarkably
higher sample density (~ 75 times) than effusive beam sources that were used to
produce atomic and molecular beams as a sample source in various experiments since
the 1920's (81-83)
. These sources played a key roll in the field of chemical physics for
many years (84-86)
. A good description of the early history and development of
supersonic nozzle beams is given by Anderson (87-89)
.
The supersonic jet expansion is a beam source of collision-free atoms and molecules
which are characterized by a very narrow velocity distribution due to negligible
Doppler width and by extremely low translational, vibrational and rotational
temperatures. The rotational temperature can reach down to 1 K while keeping the
sample in the gas phase Fig (4.4). The supersonic jet expansion can as well be used to
intensely produce exotic and transient species (complexes) that normally don't exist at
room temperature. At the primary jet expansion, many complexes or clusters are
formed. As a result of the extremely cold sample beam, the weakly-bound molecular
species such as hydrogen-bonded complexes, Van der Waal complexes or metal
clusters don't decompose due to their very weak binding energies (10-100) cm-1
and
the collision-free condition. Unstable species such as free radicals and ions can also
be produced by the supersonic expansion technique.
34
Fig. 4.4: Adiabatic expansion of the supersonic jet expansion, Ref. (125)
The unique properties of the supersonic jet technique have encouraged many scientists
to employ it as an intense sample source in various experiments such as scattering
experiments, low-temperature kinetics, molecular spectroscopy and photochemical
dynamics (89-90)
. However, in the last two decades, the supersonic beams have been
used more and more in the fields of molecular spectroscopy and photochemical
dynamics where chemists and physicists have applied these techniques to generate
and investigate the weakly bound molecular complexes and clusters. As a result,
molecular complexes have been a source of valuable information on intermolecular
forces where great deals of information about the dynamical structures in both ground
and excited electronic states of the investigated sample can be concluded.
The theory of supersonic jet and cluster generation has been elucidated in many
theoretical and experimental reports or articles (91-95)
. In this section, I will briefly
cover the basic principles of the supersonic jet expansion technique. A very detailed
description of supersonic jet beams can be found in (94, 95)
.
A supersonic jet expansion can be achieved by expanding the gas of interest from a
reservoir at high stagnation pressure P0 and starting temperature T0 through a small
35
orifice or nozzle with diameter greater than the mean free path into a chamber at much
lower back ground pressure Pb. The chamber is usually evacuated either by a
mechanical or oil diffusion pump to keep the background pressure at Pb. A schematic
diagram of supersonic free jet expansion is shown in fig. (4.5) In general, supersonic
jet beams can be distinguished from the supersonic free jet by collimation skimmers
or baffles placed downstream the nozzle. No skimmers or baffles are used for free
jets.
Fig. 4.5: A diagram of the molecular beam expansion. Ref (85, 86)
The adiabatic jet expansion is accelerated and conserved by the pressure difference
(P0 - Pb). At the same time it causes faster molecules to collide with slower
background molecules in the expansion chamber. This leads to a redistribution of the
thermal energy of the different degrees of freedom (translational, vibrational and
rotational) of the investigated sample into the kinetic energy of the free jet expansion.
This results in a narrower velocity distribution and therefore cools down the jet beam
to very low temperatures. In other words, if the pressure ratio P0 / Pb is higher than 2,
we have a supersonic flow and the Mach number is larger than one (M > 1). The
Mach number is defined as the ratio of the mean flow velocity, V, at a given point to
the sonic speed at that point. Supersonic expansions are characterized by
hydrodynamic flow conditions compared to effusive flow (96)
. In hydrodynamic flow
the gas molecules experience more collisions with each other as they pass through the
36
nozzle and at some distance downstream; whereas atoms or molecules don’t likely
experience collisions with each others in effusive flow. The conditions are well
described by the Knudsen number
D
Kf
n
λ= (4.1)
Where λf is the mean free path of the molecules in the reservoir and D is the nozzle
diameter. The situation is evaluated as either Kn >> 1 for effusive flow where atoms
or molecules don’t interact, or Kn << 1 for hydrodynamic flow where sample particles
have higher collision rate. In this case the sample particles are more concentrated
about the jet axis and therefore the beam source produces much higher flux. Another
important aspect of supersonic flow is that as a result of nonzero background pressure
in the chamber and as the expansion proceeds in the chamber, the adiabatic expansion
pushes on the gas in the chamber which results in standing shock waves that enclose
the jet expansion to satisfy the boundary conditions exposed by the background
pressure: A symmetric shock wave around the jet called a barrel shock and a disk
shape shock wave far downstream called Mach disc as shown in the figure. The
higher the ratio of P0 / Pb the longer is the distance will be from the nozzle and the
Mach disk. In this case, the supersonic expansion is unable to sense the downstream
boundary conditions and the system of shock waves at the free-jet boundary
conditions compress the gas in the chamber creating regions of high density, pressure,
temperature, and velocity gradients to meet the boundary conditions. The expansion
core is not affected by any external conditions hence the flow is isentropic and is
independent of the background pressure Pb, this region is then called the zone of
silence. At this distance, the regions of high density, pressure, temperature, and
velocity gradients cause numerous collisions between atoms and molecules which
lead to change of flow direction, reduction of Mach number and hence thermalization
of the beam, i.e. the beam is no longer cold and the clusters are destroyed. Therefore,
the measurements should be always taken at few millimetres from the nozzle, or
within the isentropic expansion area (97, 98)
enclosed by the shock waves (zone of
silence). The kinetic parameters pressure P, temperature T, density n, and velocity V
at any point within the zone of silence can be characterized by the Mach number M
which is the ratio of the flow velocity, V, to the local speed of sound, a:
37
a
VM = , WRTa /γ= (4.2)
Where γ = Cp / Cv is the heat capacity ratio of the gas, R is the gas constant and W is
the molecular weight.
The mean flow velocity V can be calculated by using the conservation of energy as
)(22 0
20
TTCdTCV p
T
T
p −== ∫ (4.3)
Therefore, the maximum velocity is given as
)1(/22 00max −== γγ WTRTCV p (4.4)
Where Cp is the heat capacity of the gas
The above equations can be used to calculate the dependency of kinetic parameters, P,
T, and n relative to the stagnation conditions, Po, To, and no, on the Mach number as
follows:
[ ] 12
)1(
0
/)1(
00
2/)1(1−
−−
−+=
=
= M
n
n
P
P
T
Tγ
γγγ
(4.5)
The above equations implies that the flow velocity increases very rapidly with the
Mach number M and then approaches a constant value Vmax compared to the terminal
Mach number Mt where the jet beam gets weaker and the flow is no longer
hydrodynamic. In contrast, the other parameters P, T, N will continue to decrease with
increasing Mach number (91)
.
The Mach number for the regions around the jet axis is given by Levy (98)
as
)1()/( −= γDXAM (4.6)
Where A is constant, X is the distance from the nozzle, and D is the nozzle diameter.
38
The Mach number does not increase with X/D far from the nozzle because the jet gets
weaker and the flow will not be hydrodynamic at long distance from the nozzle and M
will approach a finite terminal value called terminal Mach number Mt.
The Mach disc location can also be calculated in terms of nozzle diameter D by
2/1
0 )/(67.0 bm PPDX = (4.7)
The production of weakly bound complexes is a many body process, i.e. a third
collision partner is needed to form the cluster or molecular complex (dimer, trimer,
etc) and to carry away the excess energy. The collision partner can be either a third
molecule or the nozzle wall. However, the amount of kinetic energy resulting from
complex formation heats up the jet beam again and therefore reduces the adiabatic
cooling in the jet. To overcome this problem as much as possible, the gas from which
the complex is formed can be mixed with a high proportion of either a noble gas ( Ar,
He, Kr,…etc). Helium is found to be the optimum gas for this purpose, because of the
extremely low formation energy of (He)2 which is ≈ 0.0007(2) cm-1
that cannot be
achieved in the jet1. But unfortunately as helium is poorly pumped out by the attached
pumps to the jet beam apparatus, a large background pressure is created in the vacuum
chamber. In order to avoid this problem, and the high cost, argon is usually used as
“carrier-gas” in the molecular jet system.
To achieve the lowest possible background pressure in the vacuum chamber, in spite
of the large gas volume, it is continuously pumped by means of a “three-step” pump-
system, which consists of an Edwards 2600EH-“Root” pump with pump capacity of
2600 m3/h, a Leybold Ruvac 501-“Root” pump with pump capacity 500 m
3/h and a
Leybold S65B pump with pump capacity 65 m3/h. Therefore the background pressure
in the vacuum chamber can be kept within the lower 10-1
mbar region during the
measurements, as long as the stagnation pressure in front of the nozzle does not
exceed 1 bar.
____________________________________________________________________________________________
[1 The depth of the (He)2 potential well measures 7.60(4) cm-1; however the first and only bound state is just
0.0007 cm-1 beneath the dissociation barrier.]
39
4.2.1 Types of Expansions
Two types of nozzles are used in the course of this work; the continuous and the
pulsed slit supersonic nozzles. A short introduction describing these two nozzle types
along with brief introduction on point nozzles will be presented in the following
section.
Pulsed supersonic jets are typically generated from circular (pinhole) nozzles which
produce an axially symmetric expansion, also called point nozzles. The first pulsed
nozzles used in IR-Spectroscopy were “point” nozzles, whose early development is
described by Gentry (100)
. The predominantly employed construction consisted of a
commercial magnetic valve with an aperture ranging from a few tens to several 100
µm in diameter. This allowed pulse lengths of 200-500 µs and modulation frequencies
of up to several kHz to be reached.
Expansions from slit nozzles are known as planar expansions which can be either
continuous or pulse slit nozzles. Normally, the slit has a certain width (d) that can be
also adjustable and infinitely long. However, the length of continuous slits is limited
(4-7) cm, in order to reduce the gas load on the vacuum pumps. The continuous slit
nozzle used in this work is shown in fig. (4.6); the slit length is 5 cm with a typical
width of 50-100 µm. Both pinhole and slit nozzle geometries are commonly used in
spectroscopy, with the slit nozzles having better expansion properties over point
nozzles for several reasons. First, the expansion density falls off as 1/D for slit nozzles
(D is the distance from the nozzle) compared to 1/D2 for point nozzles and thus yields
slower adiabatic cooling. In addition to slower cooling, the slit expansion provides a
higher molecular density per quantum state in the interaction region which increases
the total number of two and three body collisions and therefore greatly enhances the
weakly bound cluster formation. Second, the translational cooling of the slit design
results in a higher collimation of the jet beam (lower velocity spread or small velocity
dispersion) along the slit axes leading to reduced Doppler broadening of the observed
spectral lines i.e. higher experimental resolution. Third, the absorption path length is
much larger in the slit geometry as compared to the point nozzles (101, 102)
.
The supersonic pulsed slit design is the other type of the planar expansion geometries.
The development of both continuous and pulsed planar expansions has been of great
40
importance in spectroscopy, especially in IR spectroscopy, where many weakly bound
complexes have been thoroughly investigated (103, 104)
. The pulsed slit design is of
particular relevance as compared to continuous slit nozzles. The use of a pulsed slit
nozzle reduces the gas flow in the vacuum chamber as a result of low duty cycle2,
while keeping the same level of background pressure as would result from the use of
the continuous nozzle, but at much higher stagnation pressures in front of the nozzle
using the same pump capacity. The high ratio of stagnation pressure to the
background pressure in the pulsed slit expansion design is of twofold advantage. First,
it produces higher beam densities and therefore increases the rate of two and three
body collisions in the interaction region of the jet expansion which consequently
enhances the production of molecular complexes. Second, it also causes a definite
decrease in the translational temperature of sample gas in the jet expansion leading to
significantly simplified spectra as a result of the low number of populated energy
levels. The absorption path length can be further increased by using the pulsed slit
expansions, e.g. the slit length of the continuous nozzle used in this work was fixed to
5 cm in order to avoid unacceptable increase in the background pressure in the
vacuum chamber when using longer slits, while for lower gas consumption much
longer slits can be used. The pulsed slit nozzle used in this work has a slit length of
11.4 cm. This shows that the absorption length is more than doubled as compared to
the continuous slit nozzle which improves the signal to noise ratio.
_______________________________________________________________________________________________________
2 Duty cycle is defined as the ratio of the opening duration of the nozzle to the total measurement time.
41
Fig. 4.6: Continuous slit nozzle used in this work. Ref. (91)
Further more; it is possible now to use helium as a carrier gas in the pulsed slit
nozzles for the production of molecular complexes. Helium, as described above, is
found to be the ideal carrier gas, due to the very low bond energy of (He)2. Based on
the above advantages, the pulsed slit design is therefore used to improve the detection
sensitivity of the tunable diode laser spectrometer system and other IR spectroscopic
techniques.
4.2.2 Pulsed Slit-Nozzles
Different forms of pulsed nozzles have been employed in IR spectroscopy with regard
to their construction and the measurement techniques. The principle of operation and
the construction of selected ones used by other groups will be mentioned here in brief.
The first slit-nozzle reported in literature is the one demonstrated and employed by
Amirav et al (105)
in the UV-absorption spectroscopy with 7 cm path length. The
pulses were created by two spinning and concentric cylinders each with a slit width of
200 µm and a length of 35 up to 90 mm, which rotate inside each other and are sealed
42
up against each other. This enabled a repetition rate of 12 Hz and pulse durations of
150 µs to be obtained.
A pulsed slit-nozzle from Lovejoy and Nesbitt was constructed and used to produce
Van der Waals and hydrogen bonded complexes (106)
. They used a slit length of 1.2
cm with 75 or 125 µm width designed within the nozzle holder with a knife-edge end
projecting on the back side and a mount for interchangeable cutting edge slit nozzles
(blades) on the front side. An elastomer3 seal connected to the solenoid actuator
through a small rod is placed on the knife-edge end of the nozzle holder by a leaf
spring. The valve operates by rapidly pushing and lifting the seal assembly against the
slot of nozzle holder through the applied voltage. A pulse length of 150-600 µs and
repetition rates of up to 60 Hz were achieved with this valve design. A very similar
design was employed by Sharpe et al (107)
and Piante et al (108)
.
Further nozzles can be utilized to increase the absorption path length. The nozzle
holder has then to be modified in order to host the new additional valves. In the design
of Liu et al, three commercially modified and synchronously triggered solenoid valves
from General valve corporation, Series 9 have been employed as shown in fig.
(4.7)(109)
. The use of three valves guarantees a fast homogenous distribution of the jet
gas expansion in front of the slit. This design enables us to increase the slit length to
more than 10 cm. The slit dimensions in Liu et al design is101.6 x 0.127 mm2 with
pulse durations of 0.5-1 ms and a repetition rate up to 80 Hz. Hu et al employed a slit
nozzle of 12 cm x 100 µm with pulse duration of 2ms and a frequency of 3 Hz (110)
.
Both nozzles are adapted to heat them up to 230 C which gives the chance to study
and investigate non-volatile substances like nucleotide basis under standard
conditions. However, a continuous operation with free maintenance for these valves is
only possible for about a 12 hour period
3 In most constructions, it consisted of an o-ring, which was cut open.
43
Fig. 4.7: A schematic diagram of the pulsed slit nozzle design of Liu et al Ref. (109)
The disadvantage of the above slit-nozzles is the high maintenance needs, due to the
complicated mechanism of the involved components. This means that the nozzle has
to be dismantled and reinstalled several times, which requires a new calibration and
alignment every time. This disadvantage can, however, be eliminated by using
another design concept, which was first demonstrated by Veeken and Reuss (111)
. The
design is based on two step expansions; the gas fills the small volume (chamber)
behind the point nozzle in the pulsed valve and then expands into a channel in front of
the actual slit-nozzle. By using this expansion technique, one can avoid the problem
of sealing up the entire slit. Bethardy et al constructed this type of nozzle with a slit
width of 10 µm and a length of 2 cm (112)
. The nozzle produces a pulse length of 400
µs with a repetition rate of 33.3 Hz. Brooks et al, and Xia et al also used the same
principle to construct a 0.1 x 20 mm slit-nozzle with pulse durations of 2–3 ms and a
44
repetition rate of 3 – 4 Hz (113-116)
. Likewise, Pak et al used different dimensions of slit
nozzles with lengths ranging between 7-40 mm and width of 15 µm which produced
pulse durations of 2.0-2.5 ms at a repetition rate of 80 Hz (117)
. The pulse slit nozzle
used in this work is based on the same principle of operation.
4.3 Possible Detection Methods
In the pulsed jet operational mode, the absorption line can be recorded by either a 2f
phase sensitive detection method using lock-in amplifier or by a gated detection
method using a boxcar gate integrator triggered by the jet expansion repetition rate
which has a maximum frequency of 100 Hz. This is the so-called jet modulation
technique or "concentration modulation". The sensitivity of this technique is limited
by the 1/f low frequency excess noise of the TDL caused mainly by mechanical
vibrations from closed cycle helium cooler (cryostat), compressor, pumps and other
sources. The effect of this laser noise will be more pronounced and significant when
the lock-in amplifier triggered by the jet frequencies (~ 100 Hz) is used. The laser
excess noise can be noticeably reduced by using high repetition pulsed valve
producing short duration pulses of few 100 µs. However, these frequencies are still
too small to overcome the detector-preamplifier thermal noise. For this reason, two
different detection methods have been established, using a combination of pulsed slit-
nozzle and tunable diode laser spectrometer, which enables the detection of
absorption signals as small as 10-4
- 10-5
at frequencies of few 10 kHz. These methods
will be introduced in the following
two sections.
4.3.1 The Rapid-Scan Method
This method is commonly used in slit nozzle experiments with low repetition rates of
few hertz and pulse duration in the millisecond range. In this method, a selected
wavelength range (one laser mode) is measured and recorded over the pulse duration
associated with a rapid scan of the laser current. In addition to this open valve
measurement, a second measurement is also recorded with closed valve. Both scans
are then subtracted from each other to scale down the background noise. A high signal
to noise ratio can be achieved if several 100 to several 1000 of the pulses are averaged
45
and added together. De Piante et al demonstrated the use of the rapid-scan method on
the spectra of the Ar-CO-complex (108)
, Sharpe et al (118, 119)
on HX-CO2 and Ar-CO2
and Dutton et al (120)
on CO2-N2O. Hu et al (110)
observed the Van der Waals complex
N2O-Ar using this technique, whilst Brooks et al and Xia et al used this method to
spectroscopically investigate (CO)2,CO-H2O and CO-N2 (113-116)
. The most serious
problem of this method is the temperature drift of the laser frequency. This means that
each individual scan begins at a slightly different frequency, which inevitably leads to
a broadening of the spectral lines as soon as the scans are averaged. To overcome this
problem, Hu et al developed a fast electronics to stabilize the drift in the laser
frequency. The fast electronics work as a feedback circuit to add the change in the
etalon signal due the thermal drift to the laser current as an error signal. Hu et al
developed this laser electronics because the widening of spectral lines started to be
more and more significant after averaging over 1000 scans at a repetition rate of 3 Hz.
This type of averaging needs an exceptional temperature controller cryostat with large
cooling power and a rapid heating system. Such disadvantage does not apply in our
experimental setup because the laser needs a few seconds to return from the end to the
beginning of a mode, during which the temperature stabilizes itself. However, this fact
would limit the highest possible repetition rate significantly. Another disadvantage of
this method is that it only functions with the diodes with single-mode operation,
whereas it is impossible to scan the monochromator over several wave numbers
within a millisecond range. This method also shows more laser intensity fluctuation in
the base line of the spectrum, which makes the use of at least a 12 bit analog-digital-
converter essential in order to achieve the required resolution of the absorption lines.
These disadvantages can be eradicated using the so-called step-scan method, which
was employed for the nozzles used in this work.
4.3.2 The Step-Scan Method
The step-scan procedure is based on increasing the laser current in single steps,
whereas many measurements are taken at every opening and closing time of the
nozzles. The modulation of the laser frequency is kept the same as used in the
continuous slit nozzle which was mentioned in section 4.1. The use of modulation
frequencies in the region of a few 10 kHz effectively reduces the 1/f-laser noise. The
nozzle was operated with opening durations of 2-3 ms and repetition rates of 40 Hz. If
46
the laser frequency is tuned to an absorption line, then the detector signal consists of
two components in addition to the laser modulation frequency, which corresponds to
the addition and difference of the laser modulation and pulse frequencies. A two-step
demodulation process is then required in order to extract the signal components. The
process starts by multiplying the input signal of the lock-in amplifier with the
reference frequency produced by the lock-in amplifier. The resultant signals constitute
the addition and the difference frequencies of the signal components along with the
reference frequency. For example, if the laser modulation frequency lies at 10 kHz
and the pulse frequency at 100 Hz, then the signal in front of the lock-in amplifier
contains frequency components at 9.9, 10 and 10.1 kHz. While in 2f technique the
multiplication produces frequencies of 0, 0.1, 19.9, 20 and 20.1 kHz. In this case,
since the acquired signal is contained in the 100 Hz components, the lock-in amplifier
generates the relevant signal (different from the usual DC components) to work as a
band pass filter. In principle, the lock-in amplifiers direct the multiplied signal to a
low pass filter, which normally damps all AC-components above the threshold value
of some cut-off frequencies in the signal and pass only the lower DC frequency
components. But since this is not desired here, the bandwidth of the low pass filter
which increases with decreasing time constant of the lock-in amplifier, must be set
high enough so that the signal components of the pulse frequency (in the above
example the component at 100 Hz) can pass through the low pass unaltered. This is
the case for time constants smaller than 1ms. In the case of the gated detection method
using boxcar-integrators, one can integrate over two time intervals one before the
pulse and one during the pulse and then average over a certain number of pulses at
each laser frequency. A second lock-in amplifier, which operates with the pulse
frequency as reference frequency, can be used instead of the Boxcar-integrators. This
method was used initially by Sharpe et al in the investigation of the CO2-Ar complex
(107). Quian et al also used it in conjunction with a pulsed point-nozzle to investigate
the (N2O) 2 (121)
and N2O-noble gas complexes (122)
, whereas Pak et al demonstrated
this technique on Ar-CO (123)
and used it to study CH4-Ar and CH2-Kr complexes (117)
.
Qian et al employed both rapid-scan and step-scan methods to investigate N2O-CO
complexes (124)
. The above discussion showed that the step-scan procedure leads to a
better resolution as well as a higher productivity of the spectrometer.
47
4.4 Set-Up and Operation of the Pulsed slit Nozzle
The pulsed slit nozzle used in this work was developed in the framework of an earlier
PhD thesis (125)
carried out in our group and based on the facts mentioned in section
4.2.2. The design is very similar to the one adapted by Pak et al. (117)
. The slit nozzle is
mainly constructed of three point nozzles set next to each other on a stainless steel
block to increase the slit length as shown in fig. (4.8) and fig. (4.9). To do this, three
commercial point-nozzles (Series 9 from the company Parker Hannifin, formerly
General Valve) with an aperture diameter of 800 µm were mounted next to each other
on a stainless steel block.
Three holes of 2mm in diameter are made in the block. These holes are exactly
aligned with the position of the three apertures of the point nozzles on the block to
transport the jet gas into a long channel of 11.4 cm with a cross sectional area of 2 x 2
mm2. Two adjustable stainless steel cutting edge blades are then fixed over this
channel on the bottom of the block; the blades can be adjusted by using a microscope
to form a slit of 50-100 µm along the whole channel on the bottom of the block. In
addition to the above mentioned advantages, this design makes the slit nozzle almost
free of maintenance. Three sealing puppets or plungers have only to be replaced when
they leak out while being in the closing position. The point nozzles are driven by a
commercial pulse driver (IOTA ONE from Parker Hannifin) which can
synchronically trigger up to four valves simultaneously. Pulse durations as low as
microsecond with a repetition rate of up to 50 kHz can be achieved by this driver. The
required pulse parameters for the installed valves (opening and closing time,
triggering mode...etc) are set and executed through the two on-time and off-time
functions within a small window on the front panel of the valve driver. The valve
driver uses these parameters to calculate the frequency to be f = 1/(on-time + off-
time). The set parameters are also used by the valve driver to generate a continuous
series of pulses along with the same number of TTL signals which trigger the data
acquisition inside the PC.
In order to substantially increase the number of the collected data points during one
pulse, a high resolution 8 bit rapid digital-analog converter card (NI 5102 from
National Instruments) with a maximum sampling rate of 20 M sample/s (mega
samples per second) was used. This card replaces the Boxcar-integrators described in
48
section 4.3.2. The schematic diagram of the measurement principle using the pulsed
slit nozzle is shown in figure (4.10). During the spectral measurement (data
collection), the laser frequency increases gradually as the current of the diode laser
increases. The detector output of the signal channel is demodulated by a lock-in
amplifier (Stanford Research, Model 830) at 14 kHz using the 2f-technique. The
output signal of the lock-in amplifier is then recorded by the rapid DA-converter card,
which is triggered by the valve driver. Therefore, the user has the opportunity to select
how many data points are recorded before and after the trigger. Consequently, it is
possible to digitize not only the pulse itself but also a time span before the pulse. If
the laser frequency coincides with an absorption line, the signals recorded by the DA
converter card contain information about the detector signal before and after the pulse.
In order to improve the signal to noise ratio, each individual frequency can be
averaged over an adjustable amount of pulses. Two adjustable time frames are set-up
on the PC by using the data from the averaged pulse, one before the
Fig. 4.8: A drawing scheme of the pulsed slit nozzle. Ref. (125)
49
Fig. 4.9: A photograph of mounted pulsed slit nozzle in our lab. Ref. (125)
pulse and one during the pulse, while the data in these frames are also averaged.
Subtraction of these averaged pulses leads to the absorption signal at the actual
frequency. The control program for the continuous nozzle (written in Lab View) was
extended with two new options to work in the pulsed mode. On one hand, the
measurement process just described above is implemented and, on the other, a
program section was included that directly displays the pulses at a certain laser
frequency, so that both time frames can be created by means of four cursor positions.
The signals of the etalon-channel and the reference gas channel are recorded with the
DA-converter card (AT-MIO-16-XE10 from National Instruments) used beforehand.
Additionally, no changes had to be made to the program sections described in Gim et
al (68)
concerning the control of the current, temperature and monochromator position.
51
___________________________________________________
CHAPTER 5
Infrared Spectroscopy of Methane Complexes
___________________________________________________________________________
Spectroscopy has been the most powerful tool to study and investigate the interaction
of electro-magnetic radiation with the three states of matter (gas, liquid and solid) for
more than a century. The interaction can be absorption, emission or scattering of the
electromagnetic radiation by the atoms or molecules of the sample matter. These
interactions lead to different types of spectroscopy that have been classified based on
the spectral region of the electromagnetic radiation i.e. gamma-ray, X-ray, UV, Vis,
microwave, infrared…etc. In absorption spectroscopy, the type of the involved
transition between energy levels in the studied sample can define the frequency range
of the electromagnetic radiation. For example, if the absorption is associated with a
transition from one molecular rotational level to another, then the radiation belongs to
the microwave region of the EM spectrum and the technique is known as microwave
spectroscopy. While in ultraviolet-visible or electronic absorption spectroscopy, the
involved transition takes place among the valence electrons in atoms or molecules.
But if the transition is from one vibrational level to another level, then the radiation
belongs to the infrared region and the technique is called infrared spectroscopy.
Infrared spectroscopy has a long history, but it has been a well established and
effectively used technique since the 1930’s. Infrared spectroscopy is the most
common and popular spectroscopic technique used mainly by chemists to study a
broad band of atomic and molecular species. It yields a great deal of information on
substance and compound identification and the determination of various
characteristics of their structures. The versatility of infrared spectroscopy along with
the development of a wide variety of new laser techniques encouraged the researchers
to use IR spectroscopy to study the molecular complexes produced by supersonic
molecular beam techniques.
52
Early infrared studies of molecular complexes started back in the beginning of the
1980’s where a long path cell cooled down to low temperatures was used (126)
,
whereas the first set of rotationally resolved infrared studies of molecular complexes
in molecular beams appeared in the late 1980’s (127 -131)
.
The Van der Waals molecular complexes have been of considerable interest both
experimentally and theoretically for a long time. Therefore, in the last two decades
this technique (IR) is continued to be used by an increasing number of research
groups to investigate a variety of Van der Waals molecular complexes. The rare-gas
Van der Waals molecular complexes involving symmetric, asymmetric and spherical
top molecules are of great interest in the field of molecular spectroscopy because
these complexes play an important role in understanding the anisotropic behavior of
the Van der Waals interactions. Methane is a spherical top molecule which exists in
huge quantities and different forms on earth; it is also an active constituent of the
atmospheres of earth and the outer planets of the solar system. The methane Van der
Waal complexes have a relatively simple structure. Therefore, the methane related
phenomena can be accurately monitored by studying and investigating the
spectroscopic and collisional processes of all possible methane complexes. These
studies are highly desirable to produce improved models which can be extended to
more complicated molecular systems.
The spectroscopic studies of Van der Waals complexes allows one to determine the
geometrical structure of the complex, the characteristic of the internal motion of the
molecule relative to the atom inside the complex, and to develop an accurate
intermolecular potential energy surface from which various chemical and physical
properties of the molecular system can be extracted. The potential energy surfaces of
these complexes are far from isotropic; i.e. different mutual orientations result in
minima, maxima, saddle points and other features which are essential to their
structural and dynamical properties. Over more than two decades great experimental
and theoretical progress has been made in the understanding of the properties of Van
der Waals complexes. The work on methane Van der Waals complexes started first by
investigating the methane-argon complex using conventional techniques. There has
been a considerable amount of experimental work and measurements on both the bulk
and transport properties of the system (viscosities (132-134)
, diffusion coefficients (135 -
53
137), second virial coefficients
(138 - 145), and thermal diffusion factors
(136, 137, 146 - 150).
These data are not sensitive enough to provide the detailed information of the features
of the multidimensional potential. The data always showed simple isotropic potential.
After the advent of modern spectroscopic techniques a large number of experimental
studies have been devoted to examine the collisional processes involving methane and
argon systems (151-158)
. The experiments were mainly concerned with the studies of
rotational relaxation processes and integral and differential cross sections of rotational
excitation (151, 159)
. In the late seventies Buck et al, (153)
measured the total differential
cross-sections in a crossed molecular beam experiment. The results were used to
develop an empirical potential that showed a cross-sectional rainbow structure
sensitive to the depth of the potential. In another study Nesbitt et al, (160)
measured the
state-to-state integral cross-sections for rotational excitation of CH4 in collision with
Ar atoms using crossed molecular beams. The agreement with the empirical potential
developed by Buck et al is quite reasonable.
The Rg-CH4 complex has also been the subject of several high resolution
spectroscopic studies aiming to provide a better understanding about the nature of
internal motions of methane in the complex (161 - 163)
. The first high resolution studies
are the infrared spectra of Ar-CH4, Kr-CH4 and Ne-CH4 that were recorded by
McKellar et al (161)
using a Fourier transform infrared spectrometer with a long path
glass cell cooled down to low temperatures. Strong transitions correlated to the R(0)
transition of the triply degenerate ν3 stretching vibrational band of the CH4 monomer
were measured for the Ar-CH4 and Kr-CH4, while weak features were observed for
Ne-CH4. Jet spectra for the same spectral region of Ar-CH4 at lower temperature were
also recorded by Lovejoy and Nesbitt et al (163)
using a diode laser spectrometer
system along with supersonic slit expansion. Further lower temperature spectra for
Ar-CH4 were detected by Block and Miller et al (162)
, and Howard with co workers
(164). Although the spectra showed partially resolved rotational structures it was not
possible to securely assign the spectra because of the limited resolution in the
McKellar measurements (0.01 cm-1
) and the line broadening due to the fast
predissociation in the other measurements. However the large spacing between the
sub-bands in the spectra indicated that the methane molecule is freely rotating within
the complex. The spectra were assigned a few years later using an ab initio dynamical
calculation where infrared spectra have been calculated and a potential energy surface
54
has been also developed for this system (165-167)
. The calculated spectra showed a good
qualitative agreement with the recorded ones and contributed in the assignment of the
most other transitions. In more recent spectroscopic studies, the infrared spectra of
Ar-CH4, Kr-CH4 and Ne-CH4 complexes in the 7µm region correlating to the ν4 triply
degenerate bending mode of the methane monomer have been measured by Pak et al
(168-170, 236) using a diode laser spectrometer system. The spectra were later reassigned
as P, Q, and R branches corresponding to the R (0) transition of the methane monomer
employing a model initiated first by Randall et al and developed later by Brook et al
(171) for the Ne-SiH4 complex. A Coriolis term is introduced in the Hamiltonian model
to be able to fit the recoded spectra.
Despite a lot of interest, a limited number of ab initio studies have been reported so
far on Rg-CH4 complexes. In the first study, Fowler et al (172)
calculated the long
range dispersion coefficients of Ar-CH4. In a recent work, Szczesniak et al, (173)
reported a few cuts through the interaction potential of Ar-CH4 using MP2 calculation
methods with a relatively small basis set. The calculation predicted an equilibrium
structure with a face configuration where the rare gas atom sits on one of the C3 axes
of CH4 monomer and approaches the face of the CH4 tetrahedron. The position and
the depth of the minimum were determined to be 7.5 bohr and -113 cm-1
respectively.
They also stated that the minimum is overestimated by 0.5 bohr while the depth is
underestimated by 25% respectively due to the applied low level theory and small
basis set. In the most recent study, Hijmann et al, (174)
employed the symmetry adapted
perturbation theory (SAPT) to compute enough data points on the surface to
determine the ab initio intermolecular potential energy surface of the Ar-CH4
complex. This potential is in a good agreement with the previous theoretical study
(175), it also displays a face configuration but with a well depth of -144.3 cm
-1 and a
position of 7 bohr. The SAPT potential is shown to reproduce most of the
experimentally observed data. It is also used to calculate the IR spectrum which
assisted in assigning most the of other transitions; it is probably the best available
potential for the Ar-CH4 complex.
The methane-water complex (CH4-H2O) is another example of the Van der Waal
methane complexes which has attracted special research interest. The specialty of this
complex comes from the fact that methane hydrates are a combination of a methane
55
molecule locked in a cavity of hydrogen bonded water molecules. Therefore the
ability to develop a modeling technique that can predict the behavior of methane
hydrates would be very important for the development of production and transmission
operation of conventional methane hydrates. On the other hand, a proper
determination of the intermolecular interaction potential of the complex is also
essential for both computing the thermodynamic properties and performing classical
simulation of the kinetic phenomena of hydrates such as formation and dissociation.
The primary modeling efforts started by applying the Van der Waals and Platteeuw
statistical mechanical model with Lennard-Jones and Devonshire LJD potential
approximation (176, 177 )
. This approximation was shown later to be inadequate (178, 179)
.
The inadequacy is based on the fact that the potential parameters calculated from the
hydrate phase data by this approximation don’t match the calculated parameters from
other experimental data (179, 180, 141)
.
The alternative approach used to derive the potential energy of the complex is the ab
initio calculation methods. This approach provides a direct route to determine the
intermolecular potential that can be corroborated using experimental data. The early
ab initio calculations on the CH4 - H2O complex were aimed to study and characterize
the C-H…O interaction energy (181 - 190)
. These studies pointed out that CH4 - H2O is
bound with predicted binding energies ranging from 0.5-2.3 kcal/mol depending on
the employed basis sets, but because of the low flexibility of the used basis sets the
interaction energies were not corrected for the basis set superposition error (BSSE).
More ab initio studies on the CH4-H2O complex have been carried out later on but
with serious discrepancies among the results. In his study, Novoa et al, performed ab
initio calculations on the methane-water complex at the self-consistent-field molecular
orbital (SCF-MO) and MP2 level with various basis sets along with the near Hartree-
Fock limit (191)
. His calculations provided the first reliable interaction potential for the
C-H…O contact with an estimated binding energy to be 0.59 +/- 0.05 kcal. In another
study, Woon et al (190)
, reported a shallower minimum of 0.5 kcal for the same
configuration, he also stated that the C-H...O contact is more stable than the C...H-O
contact. Few years later Szczesniak et al explored more possible configurations in
methane-water complexes using fourth-order Moller-Plesset perturbation theory with
6-31++G(2d2p) basis set (192)
. They found that the global minimum occurs at the
C…H-O geometry which is inconsistent with the Novoa et al results.
56
Very limited experimental work has been done on the methane-water complex. The
first high resolution spectra of the CH4 - H2O complex have been recorded by using
tunable far-infrared (FIR) laser technique combined with a cw supersonic jet
expansion (193)
. Thirteen VRT bands have been measured and rotationally assigned in
the spectral region from 18 to 35.5 cm-1
. In the same work, an approximate ab initio
calculation using the site-site potential energy surface of Woon et al, have been
carried out to find the bending VRT levels of the complex. A comparison between the
theoretical and experimental results showed that the eigenvalues have almost the same
pattern compared to the observed spectra but are not in quantitative agreement. This
indicates that either a less approximate method or a more reliable potential or both
will be required to obtain a quantitative agreement between theory and experiment.
The second spectroscopic study was almost accomplished in the same time as the first
one using Fourier transform microwave spectrometer technique (194)
. The observed
data in this study are in a good agreement with and support the bands assignment in
the first study. However, the authors in reference (195) stated that the relation of the
observed spectra and the intermolecular potential is still not clear and represents a
challenging task for future studies.
The above studies on the methane-water (CH4 - H2O) complex show that the potential
energy surface of this complex has not been properly described yet and there is a need
for a full characterization of this potential.
In a series of studies, Legon and co-workers investigated the rotational spectra of
CH4-HX (X=CN, Cl, Br, and F) using Fourier transform microwave technique (196 -
199). The recorded spectra of these complexes are complicated by the motion of
methane within the complex and also showed different patterns for different
complexes. These studies indicated that methane acts as a proton acceptor. Few more
studies appeared on other methane complexes in the microwave region for CH4-O3
(200) and in the infrared region for CH4-para H2 and CH4-CO
(201, 202). The spectra of
these complexes showed more complications.
Interactions between methane molecules can also result in additional products of Van
der Waals methane complexes like methane dimers, trimers, tetramers, pentamers.
These complexes have been the subject of interest for many researchers in different
branches of chemistry. As this work here is concerned with the IR spectra of methane
57
dimer, I will concentrate on giving a brief introduction about both experimental and
theoretical efforts that have been achieved on this complex. Experimentally, the
growing interest in reaching an exact description of methane-methane interaction led
to number of spectroscopic studies from which they tried to develop a proper
intermolecular potential for this complex (203-205)
. These studies were mainly based on
a large body of experimental data on the bulk and transport properties of methane
such as (spectroscopic measurements, second virial coefficients, molecular-beam
scattering data, viscosities …etc) often measured at a narrow range of interaction
energies. These measurements normally lead to a semi-empirical isotropic potential
that can only predict one or at most two of the interaction properties of the methane-
methane complex. The results of these studies are consequently not consistent.
On the other hand, theoretical efforts on methane dimer started only about two
decades ago. Various methods of ab initio calculation have been used in many studies
to develop a potential energy surface for methane dimer. In one study, Szczesniak et
al (206)
, applied the newly proposed combination of intermolecular Moller-Plesset
perturbation theory (IMPPT) with the super-molecular Moller-Plesset perturbation
theory (SMPPT) to generate a potential energy surface for (CH4)2. This coupling
potential consists of two major interaction components: the repulsive Heilter-London
(HL) exchange energy and the dispersion attractive force. The former contribution is
responsible for the main anisotropy in the potential surface, while the dispersion
energy represents the dominating attractive force in the complex. Both contributions
show orientation dependence of the hydrogen atoms on both methane monomers.
These results are almost in a good agreement with the early attempts of ab initio
studies on methane dimer using SCF methods (207)
. Novoa et al (208)
, also used the ab
initio MPn (n = 2-4) method with small and moderate size basis sets to determine the
dissociation energies and the equilibrium distances of all possible CH…HC contacts
within the several orientations of two methane molecules. Although a previous study
(209) showed that the methane dimer (CH4)2 is not bound, Novoa et al
(208) found that
the methane dimer (CH4)2 is bound in all possible orientations of the two methane
monomers with all used basis sets, and the arrangements with more than one
CH…CH contact give more stabilization than the arrangement of one CH…CH
contact. This method along with 6-311G (2d, 2p) basis set shows a quantitative
agreement with the experimentally deduced isotropic potential. In another study,
58
Ferguson et al (210)
, used three molecular mechanics and three semi-empirical
parameter sets along with 6-311G (2d,2p) basis set at MP2 level to examine the
interaction energies for four different orientations of the methane-methane complex.
The results indicated that the molecular mechanics models are consistent with the ab
initio calculations while the semi-empirical models produced a diversity of results.
The atomic probe approach has been employed by Hill et al (211)
, to show that it can be
used to derive a reliable intermolecular potential based on ab initio calculations. He
calculated the ab initio Counterpoise-corrected CPC interaction energies for different
orientations of Ne-CH4 using aug-cc-PVTZ basis set at the MP2 level in order to
produce the Lennard-Jones LJ type of analytical potential. The ab initio CPC
interaction energies have been also calculated for Ne-C, Ne-H, and Ne-Ne using the
LJ type parameters. The potential parameters were in a good agreement with the
empirical values and properties in MD calculations. This model (atomic probe) has
been extended by Stone et al (212)
to study methane dimer in a trail to give an accurate
anisotropic potential in terms of atomic parameters. The results demonstrated that the
atom probe model, when used for two pairs of molecules, can be useful in exploring
some functional forms of the intermolecular potential, but at the same time does not
produce a good fit for the real potential and has to be refined with more calculations.
In a relatively recent study, Rowley et al (213)
, used the Counterpoise-corrected (CPC)
ab initio model with a 6-311G (2d f,2pd) basis set to compute the interaction energies
of eleven orientations of two methane molecules as a function of C-C separation
distance. These energies were then used to derive an analytical site-site potential
consistent with the models from the MD simulation. The C-C, C-H, and H-H
interactions were directly extracted from the calculated ab initio potential energies.
This model suggests that the C-H interaction energies are the dominant energy
contributions in the weakly bound methane dimer.
In spite of the above research interest, there is still no reliable ab initio potential
surface that can predict all the experimental and theoretical properties (parameters)
available for methane-methane complex.
Table (5.1) summarizes the above theoretical efforts on methane dimer. The table
shows all the possible orientations between two methane molecules along with the
available basic information on each orientation. The A, B, and C rotational constants
in the table have been calculated in this work for some selected R (intercarbon
59
distance) values predicted by theses studies. The notation used in the category column
refers to (F) face, (E) edge, and (V) vertex for a regular tetrahedron monomer with the
carbon atom in the center and hydrogen atom connected to each vertex. The staggered
(St) and eclipsed (Ec) positions represent the rotation of one methane molecule
relative to the other molecule in the complex around the rotational axis.
60
Table 5.1: Possible orientations of two methane molecules in methane dimer
complex.
Rotational Constants
[cm-1
] Category Dimer Geometry
A B C
RC-C
[o
A ]
RH-H
[o
A ]
Binding
Energy
[kcal
/mol]
References
2.6677 0.1242 0.1223 4.042 3.314 0.29 Ref.208
FF-Ec
- - - ~ 3.8 - ~ 0.3 Ref.213
-563µ? Ref.206 2.7049 0.1288 0.1268 3.968 -
-596µ? Ref.206
- - - ~
3.76 - ~ 0.33 Ref.213
- - - 3.928 3.447 0.31 Ref.208
- - - 3.765 3.206 0.43 Ref.208
FF-St
- - - 3.7 - 0.56 Ref.210
FE-Ec
2.2935 0.1279 0.1258 ~ 4.0 - ~ 0.29 Ref.213
FE-St
2.3567 0.1288 0.1277 3.968 - -498µ? Ref.206
EE-Ec
2.0008 0.1128 0,1128 ~
4.25 - ~ 0.2 Ref.213
61
Rotational Constants
[cm-1
] Category Dimer Geometry
A B C
RC-C
[o
A ]
RH-H
[o
A ]
Binding
Energy
[kcal
/mol]
References
- - - 4.232 - -370µ? Ref.206
- - - ~ 4.2 - ~ 3.2 Ref.213
- - - 3.812 2.847 0.38 Ref208 EE-St
2.0008 0.11735 0,11735 4.256 3.25 0.29 Ref.208
EE-Ec
2.0008 0.1304 0.1304 3.942 3.122 0.32 Ref.208
- - - 4.232 - -323µ? Ref.206
- - - ~
4.37 - ~ 0.2 Ref.213
VE-Ec
2.3024 0.1596 0.1596 3.631 3.131 0.25 Ref.208
2.287 0.1128 0.1113 ~
4.25 - 0.15 Ref.213
VF-Ec
- - - 3.8 2.53 0.84 Ref.210
VF-St
3.4712 0.0905 0.0889 4.232 - -469µ? Ref.206
VV-St
2.6677 0.0867 0.0858 4.868 2.686 0.15 Ref.208
VV-Ec
3.4712 0.0905 0.0889 4.761 - -205µ? Ref.206
62
_____________________________________________________________________
CHAPTER 6
Theory of symmetric top molecules
_____________________________________________________________________
The different categories of molecules along with the most relevant theoretical
concepts to this work will be briefly covered in this chapter. More detailed discussion
of the following subjects can be found in (214-220).
6.1 Molecular Categories
In general, molecules can be classified based on the relative values of their three
principal moments of inertia labeled as IA, IB, and IC into four different groups as
follows:
– Linear molecules
These are mainly the molecules, in which all the atoms are aligned in a straight line,
so that the moment of inertia about the intermolecular axis is very small or zero.
Therefore, the moments of inertia are given as
IB
= IC and IA = 0 , e.g. HCN, HCl, OCS…
– Spherical top molecules
Molecules which have three identical moments of inertia
IA = I
B = I
C , e.g. SF6, SiH4, CH4 …
– Symmetric top molecules
Symmetric tops are the molecules which have two equal moments of inertia while the
third one is different. This group is divided into two classes:
63
• Oblate symmetric top molecules: (saucer or pancake shaped)
IA= I
B < I
C , e.g. BF3
• Prolate symmetric top molecules: (cigar shaped)
IB = I
C > I
A , e.g. CH3Cl…
– Asymmetric top molecules
These are the molecules to which the majority of substances belong and which have
three different moments of inertia given as
IA< I
B < I
C , e.g. H2O, CH2O, CH3OH
Where IA = Ixx, IB = Iyy, and IC = Izz in terms of the Cartesian coordinate system.
Upon the interaction with electromagnetic radiation, these molecules may experience
different types of motion: translational motion, rotational motion and vibrational
motion. Each atom in a molecule has three degrees of freedom, i.e. a molecule with N
atoms has 3N degrees of freedom. In translational motion (center of mass motion) the
entire molecule moves through space with a certain velocity and specific direction.
The velocity can be resolved into three different components in the Cartesian
coordinate system which means the total translational kinetic energy of the molecule
has also three components in the Cartesian coordinate system with the form
2222
2
1
2
1
2
1
2
1zyx mvmvmvmv ++= (6.1)
where v is the velocity and m is the mass of the molecule.
The molecule may rotate about some internal axis. The rotational motion can also be
resolved into three components in the x, y, z axes of the Cartesian coordinate system.
The rotational kinetic energy of the molecule consists then of three components in x,
y, z axis
64
222
2
1
2
1
2
1zzyyxxrot IIIE ωωω ++= (6.2)
where I is the moment of inertia and ω is the angular velocity. This equation indicates
that the molecule can have 3 or 2 different rotational degrees of freedom depending
on the type of the investigated molecule.
Finally, the molecule may also vibrate; in this case the number of vibrational degrees
of freedom (known also as vibrational modes) within the vibrating molecule is then
given by the remaining 3N-6 and 3N-5 degrees of freedom for nonlinear and linear
molecules respectively.
These motions can lead up to different types of molecular spectra depending on the
type of the molecule. For example, the motions that produce a net change in the dipole
moment of the molecule result in MW or IR spectra, whereas the motions that cause a
change in the polarizability of the molecule result in Raman spectra. The energies of
each category of molecules can be described by using Hamiltonians that should
include all the effective parameters of the molecular complex. The Hamiltonian can
then be solved by Schrödinger equation to determine the allowed energy levels of the
molecular complex.
In the present work, a high resolution infrared diode laser spectrometer is used to
investigate the spectra of the methane dimer complex (CH4)2 that can be classified as
a nearly symmetric top molecule. Therefore, it is expected to have a spectrum similar
to one of the limiting cases of symmetric top molecules. To achieve a better
understanding of the measured spectra, a short introduction on the theory of
symmetric top rotors will be given in this chapter. A more detailed discussion can be
found in (214-219).
6.2 Symmetric top molecules
The classical form of the rotational kinetic energy for a rigid symmetric top molecule
is
65
C
c
B
b
A
a
rotI
J
I
J
I
JE
222
222
++= (6.3)
where J is the angular momentum, and J2 for a three dimensional rotor is given by
2222
cba JJJJ ++= (6.4)
The starting Hamiltonian for the symmetric top rotor is the same as the one for the
general case of free rotating object in three dimensions written as
C
c
B
b
A
a
rotI
J
I
J
I
JH
2
ˆ
2
ˆ
2
ˆˆ
222
++= (6.5)
where (IA< I
B= I
C) and (I
A= I
B < I
C) are the two limiting cases of the prolate and
oblate symmetric tops.
The energy level pattern for the rotation of a rigid symmetric top, can be found by
solving the Schrödinger equation for Hamiltonian of the two limiting cases of the
prolate symmetric top (IA< I
B = I
C) and the oblate symmetric top (IA= IB < IC). This
consequently leads to the following two expressions that describe the possible energy
levels of the symmetric top rotors.
EJK
= BJ (J + 1) + (A – B) K2 for prolate top (6.6)
EJK
= BJ (J + 1) + (C – B) K2 for oblate top (6.7)
The A, B, and C are the rotational constants of the molecule which are defined in
terms of the energy units of wavenumbers (cm-1
) as
AcI
hA
28π= ,
BcI
hB
28π= ,
CcI
hC
28π= (6.8)
J is the total angular momentum quantum number, and K is the component of the
angular momentum on the principal rotational axis of the molecule which is the a-axis
for a prolate top or the c-axis for an oblate top.
66
Each rotational level has a (2J + 1) - fold degeneracy for the levels with K ≠ 0. Fig
(6.1) displays the energy level diagram for the prolate and the oblate classes of the
symmetric tops. The figure shows that for every K value there is a series of varying
energy levels of J. It also shows that the energy of a given J level in the prolate top is
increasing with increasing K, while it decreases with increasing K in the oblate top.
The selection rules for rotational transitions in symmetric top molecules are
∆ J = 0, ± 1 and ∆ K = 0 (6.9)
Prolate symmetric top Oblate symmetric top
Fig. 6.1: Rotational energy levels of the prolate and oblate symmetric top. Ref. (229)
The above description of molecular rotations is based on assuming that molecules are
rigid rotors. However, it has been identified by microwave spectroscopy that
molecules are not rigid and the atoms in the rotating molecules experience a
centrifugal force which slightly distorts their bond lengths depending on the type of
the molecule. Thus, the energy level expression for a non-rigid symmetric top
molecule is given as
F (J, K) = BJ (J + 1) + (A – B) K2 – DJ J
2 (J + 1)
2 – DJK (J + 1) K
2 – DK K
4 (6.10)
where DJ, DK, and DJK are the three centrifugal force constants.
67
6.3 Asymmetric top molecules
Asymmetric top molecules comprise of the large majority of polyatomic molecules
which have no C3 axis of symmetry and with different moments of inertia given as
IA< IB < IC. The rotational spectra of this group of molecules are usually very
complex. The Schrödinger equation has no analytical solutions for the energy levels
of asymmetric tops analogous to that for the linear and the symmetric top molecules.
Therefore, the best approach is to consider the asymmetric tops as a case lying
between the prolate and the oblate limits of symmetric tops which leads to a relatively
good approximation of the energy levels of the asymmetric top molecule. This
indicates that the asymmetric rotors may include some symmetric top properties.
Considering the general case of a rigid rotor in three dimensions leads to the fact that
the angular momentum J can be represented as in equ. (6.4)
2222
cba JJJJ ++=
But the general case (IA ≠ IB ≠ IC) is practically the asymmetric top rotor, where the
quantum mechanical Hamiltonian operator of the asymmetric top is stated as in equ.
(6.5)
The Schrödinger equation of this Hamiltonian leads to the overall rotational energy of
the system along the three principal axes of inertia a, b, and c. These energy
eigenvalues are expressed in the form of
Ĥrot = A Ĵ
2
a + B Ĵ2
b + C Ĵ2
c (6.11)
where A, B, and C are rotational constants of the molecular system defined previously
in equ. (6.8) for symmetric top molecules.
AcI
hA
28π= ,
BcI
hB
28π= ,
CcI
hC
28π=
C
c
B
b
A
arot
I
J
I
J
I
JH
2
ˆ
2
ˆ
2
ˆˆ
222
++=
68
The most useful and commonly used method to label the energy levels of the
asymmetric top molecules is to incorporate the wavefunctions of the prolate (IB = IC)
and the oblate (IA= IB) limiting cases of the symmetric top molecules. The energy
levels of the prolate and the oblate limits are given before as
EJK
= BJ (J + 1) + (A – B) K2 for prolate top
EJK
= BJ (J + 1) + (C – B) K2 for oblate top
It is also possible to write the matrix form of the Hamiltonian operator in polynomial
equation for each value of J as follows
)(4
ˆ2
)(
2ˆ 2222
−+ +×
−+
+−
+×
+= JJ
CBJ
CBAJ
CBH zrot (6.12)
The description of the rotational levels of the asymmetric tops is possible by using the
matrix form of the Hamiltonian operator which employs the eigenfunctions of the
symmetric top rotors. The matrix elements of the Hamiltonian can be written in the
following form
)1(,,ˆ,, 2 += JJMKJJMKJ (6.13)
22,,ˆ,, KMKJJMKJ z = (6.14)
)1()1(,,ˆ,1, −−+=+ − KKJJMKJJMKJ (6.15)
)1()1(,,ˆ,1, +−+=− + KKJJMKJJMKJ (6.16)
where Ĵ+ and Ĵ- are the lowering and the raising operators and Ĵz is the z-component of
the angular momentum.
The first two terms constitute the main diagonal matrix elements of energy operator,
while the third term describes the off diagonal components of the operator:
69
2
2
)()1(
2,, K
CBAJJ
CBKJHKJ rot
+−
++×+
= (6.17)
[ ] [ ])2)(1()1()1()1(4
,2, ±±−+×±−+×−
=± KKKJJKKJJCB
KJHKJ rot
(6.18)
These diagonal components represent the eigenfunctions of the asymmetric top rotors,
with the exception that B is replaced by (B + C) / 2 for the prolate symmetric top and
A is replaced by (A + B) / 2 for oblate symmetric top. In asymmetric rotors, each
value of J has a (2J + 1) × (2J + 1) dimensional matrix from which the energy
eigenvalues are obtained by matrix diagonalization.
Fig 6.2 depicts the energy levels diagram of an asymmetric top rotor. The two
extreme cases of the right and left in the figure show the energy levels of the slightly
deviated prolate and oblate symmetric top from their original position (shape), i.e.
when IB = IC of the prolate top decreases gradually to IB = I
A and when I
B = IA of the
oblate top gradually moves to IB = I
C. The degenerate levels of the symmetric rotor for
K ≠ 0 don’t appear in the figure. The energy levels of the asymmetric top rotors,
where levels with K ≠ 0 split into two components are displayed in the middle way of
the two extreme cases of left and right in the figure. The energy levels are obtained by
simply connecting the lowest / highest level of a certain J in the prolate symmetric top
with the lowest / highest level of the same J on the oblate symmetric top giving rise to
J-J level, then the levels in the next lowest / highest to the levels in the next lowest /
highest giving rise to J-J+1 level and so on. These levels are labeled as Jkakc where J is a
good quantum number and ka and kc are just labels for the asymmetric tops.
70
Prolate symmetric top Oblate symmetric top
Fig. 6.2: Rotational energy levels of the asymmetric top rotors. Ref. (215)
In a second notation, the energy levels of asymmetric rotors are described in terms of
the Ray’s asymmetry parameter, “κ”, used to measures the degree of asymmetry in the
asymmetric rotor and also defined as
CA
CAB
−
−−=
2κ (6.19)
with A, B, and C as rotational constants along the axes of inertia a, b, and c. The
limiting values of κ which ranges from -1 to +1 are equivalent to the prolate and
oblate limiting cases of symmetric tops respectively. The highest degree of
asymmetric top has a value of κ = 0. The energy levels of asymmetric tops are also
obtained by connecting the lowest / highest K levels for a given J value of the prolate
symmetric top with the lowest / highest K levels for the same J of the oblate
symmetric top giving rise to JK-1 equivalent to asymmetric top level as κ approach -1,
and then the next lowest / highest to the next lowest / highest giving rise to JK+1
equivalent to asymmetric top level as κ approach +1, and so on. Therefore, these
levels are labeled here as JK-1 K+1, which is identical to Jkakc
label in the first notation
where J is always a good quantum number and K values are just labels for the
asymmetric top levels. Fig (6.2) points out that for a given value of J, the asymmetry
71
splitting of K levels increase in energy as J increases for prolate top, while it
decreases in energy as K increase for oblate top.
6.4 Selection rules
The transitions in asymmetric top rotors are more complicated than the ones in linear
or symmetric top rotors. These transitions depend mainly on the symmetry of the
dipole moment components µa, µ
b and µ
c along their principal axis of inertia. Any
effective component of the dipole moments (µa, µ
b or µ
c ≠ 0) results in a set of certain
transitions in the molecule leading to different types of spectra. These spectra are
named as a-type, b-type and c-type spectra based on the individual dipole
components. The selection rules for these transitions are
a - Type: ∆ Ka = 0, (± 2, ± 4,…) and ∆ Kc = ± 1, (± 3, ± 5,…)
b – Type: ∆ Ka = ± 1, (± 3, ± 5,…) and ∆ Kc = ± 1, (± 3, ± 5,…)
c – Type: ∆ Ka = ± 1, (± 3, ± 5,…) and ∆ Kc = 0, (± 2, ± 4,…)
The selection rule for the total angular momentum J is ∆J = 0, ±1. An example for the
different types of transitions for the lowest Ka: 1 ← 0 transition is presented in Fig.
(6.3). The transitions for which the ∆ Ka and ∆ Kc values lie within the brackets are
only weakly allowed compared to the transitions corresponding to the values outside
the brackets.
72
Fig. 6.3: The different types of transitions in asymmetric tops. Ref. (215)
6.5 Perturbations
The above treatment of rotating molecules is based on assuming a rigid rotor model.
This assumption was just an approximation because it was confirmed experimentally
that molecules are not rigid systems, i.e. all the bonds in molecular systems, even
molecules in condensed matter, are elastic to some extent. Therefore, the rigid rotor
model is not sufficient to describe the measured spectra and to exactly predict the
energy levels of the molecular structure. This is due to the fact that the molecular
energy levels are influenced by various perturbations as a result of the rotational-
vibrational interactions and the force of centrifugal distortion. The effect of these
perturbations can be mostly observed in the measured spectra as frequency shifts,
intensity variation, broadening, or splitting of the spectral lines. The deviations of the
spectral lines from the rigid rotor positions are usually very small, and the lower J
levels are commonly the least effected.
There are two categories of perturbation theory: the time-independent and time-
dependent perturbations. In this work, we are interested in the time-independent
perturbation theory, invented by Erwin Schrödinger in 1926, which deals with static
perturbations in the molecular energy levels (220)
.
73
To give a brief idea about the time-independent perturbation theory, we start by
considering an unperturbed Hamiltonian H0 which possesses no time dependence and
whose eigenfunctions ψn0
and eigenvalues En0
are known from the time-independent
Schrödinger equation
000
0 nnn EH ψψ = n = 0, 1, 2,… (6.20)
It is assumed here that En0 are discrete and non-degenerate. The 0 subscript indicates
that energies, wave functions and the Hamiltonian operators are of the unperturbed
system.
Now if we introduce a small perturbation to the system e.g. a weak physical
disturbance V like potential energy in an external field, then the resultant
modifications on ψn0 and En
0 can be determined by the perturbation theory. The
perturbed Hamiltonian is given by
H = H0 + λ V (6.21)
where λ is a dimensionless parameter that can take continuous values from 0
(perturbation off) to 1 (perturbation on). The eigenvalues and eigenstates of the
perturbed Hamiltonian are given by Schrödinger equation as
nnn EVH ψψλ =+ )( 0 (6.22)
If the perturbation is sufficiently weak, then the eigenvalues En and the eigenstates ψn
can be expressed in terms of a power series in λ:
En = En
0 + λ En
1 + λ
2 En
2 + ... (6.23)
...2210
+++= nnnn ψλψλψψ (6.24)
Substitution of the power series into the Schrödinger equation yields
( ) ( )......)(...)(20221010
0 +++++=+++ nnnnnnn EEEVH ψλψλλψλψλ (6.25)
74
It is evident that when λ = 0, all the values lead to the original values of the
unperturbed eigenvalues and eigenfunctions. As a result of small perturbation when λ
≠ 0, the energy levels and the eigenstates will not deviate much from their original
unperturbed values, therefore, the higher order terms will rapidly become smaller.
Consequently, the zeroth-order equation simply represents the Schrödinger equation
for the unperturbed system, while the first and the second order equations are
011001
0 nnnnnn EEVH ψψψψ +=+ (6.26)
02112012
0 nnnnnnnn EEEVH ψψψψψ ++=+ (6.27)
These can be used to derive the first order and the second order perturbations of the
system considering that H0, En0, ψn
0, and V are known values. The first order
perturbation theory is used to determine En1 and ψn
1.
Since ψn
1 in the first order
perturbation is not a known wave function, it has to be expressed as a series in terms
of complete set of orthogonal eigenfunctions
01
j
j
jn a ψψ ∑= (6.28)
ψj
0 is a complete orthogonal set of wave functions. Therefore, to determine the
perturbed eigenfunction ψn
1, the aj coefficients must be also determined. Substituting
this equation in the first order perturbation equation yields
010000
0 nnj
j
jnjj
j
j EaEVaH ψψψψ +=+ ∑∑ (6.29)
Taking a particular wave function ψm
0 out of the orthogonal set of wave functions ψj
0
and by using the orthogonality property of the ψj
0, where ‹ ψm
0 | ψj
0 › = 0 for m ≠ j
and ‹ ψm
0 | ψj
0 › = 1 for m = j, which can be expressed by Kronecker delta ‹ ψm
0 | ψj
0
› = δmj or ‹ ψm
0| ψj
0 › = δmn, one can get the first order correction term of the energy
for the perturbed system for m = n as follows
75
001
nnn VE ψψ= (6.30)
Therefore, the first order perturbation of the nth
state of a perturbed system is given by
000
nnnn VEE ψψλ+= (6.31)
For m ≠ n, the am coefficients are defined as
00
00
mn
nm
mEE
Va
−=
ψψ (6.32)
For m = n, the value of am =n can be calculated by using the normalization condition,
i.e. ‹ ψn
0 + λ ψn
1 | ψn
0 + λ ψn
1 › = 1 which gives the first order perturbation for the
wave function of the nth
state of the system as
0
00
00
0
m
nm mn
nm
nnEE
Vψ
ψψλψψ ×
−+= ∑
≠
(6.33)
The same analogy is used to obtain the second order perturbation for En2 and ψn
2,
where ψn2 is expressed in terms of a complete set of orthogonal wave functions of the
perturbed system as follows
02
j
j
jn b ψψ ∑= (6.34)
Now, substituting ψn
1and ψn
2 in the second order perturbation equation (6.27), leads to
021100000
nnj
j
nj
j
jnj
j j
jjjjj EEaEbaVEb ψψψψψ ++=+ ∑∑∑ ∑ (6.35)
Again, considering one specific wave function ψm0 of the complete orthogonal set of
the wave functions and then using the orthogonality function of ψj0 in terms of
Kronecker delta δmn leads to the second order correction term of the energy of the
perturbed system for m = n as
76
nnjn
j
jn aEVaE1002
−=∑ ψψ
= nnnnnjn
nj
j aEVaVa10000
−+∑≠
ψψψψ (6.36)
Using the equations of En1 and aj (am) in the first order perturbation, the second order
correction term of the energy of the perturbed system is then given by
∑≠ −
=nj jn
jn
nEE
VE
00
200
2ψψ
(6.37)
This equation indicates that the second order term increases drastically for closely
spaced energy levels, i.e. for small En0 – Ej
0.
For m ≠ n, the bm (bj) coefficients are expressed as
200
0000
0000
0000
)())(( jn
nmnn
nj jmjn
jmnj
mEE
VV
EEEE
VVb
−−
−−=∑
≠
ψψψψψψψψ (3.38)
For m = n, the value of bm = n can be found by using the normalization condition as
done in the first order which leads to
∑∑−
==200
200
2
)(2
1
2
1
jn
nj
j
jmEE
Vab
ψψ (6.39)
Therefore, the second order correction for the eigenfunction is given by
∑ ∑≠ ≠
−−
−−
−−=
nm nj
m
mn
nm
m
mn
nmnn
mnjn
jmnj
n
EE
V
EE
VV
EEEE
VV
0
200
200
0
200
0000
0000
0000
2
)(2
)())((
ψψψ
ψψψψψψψψψ
ψ (6.40)
77
6.6 Degeneracy
In the previous discussion, the first order correction of the wave function is obtained
based on the assumption that there is no degeneracy between the involved states n and
m, i.e. En
0 ≠ Em
0 in equation (6.33). If these states are degenerate in the original
unperturbed Hamiltonian, then the perturbation theory breaks down for these states
because the energy difference in the denominator (En
0 - Em
0) goes to zero and the
associated terms in the sum becomes infinite. However, this problem can be solved by
treating all the degenerate states as a linear combination of eigenstates of H0 with the
same eigenvalues. This is achieved by stating that n is M-fold degenerate with energy
En0 (220)
. The components of the unperturbed eigenfunctions are then denoted as ψni0
with i = 1, 2 …M, which share the same eigenvalues, i.e.
Edeg = En,1
0 = En,2
0 = En,3
0 = …= En,M
0 (6.41)
It is also possible to select these degenerate eigenstates to be orthogonal linear
combination of states which still have the same energy En0. Therefore, one can chose
a new set of M degenerate combinations as
∑=
=ΦM
j
njijnj b1
00 ψ (6.42)
The bij are the numerical coefficients, and the orthogonality condition leads to
ijnjni δ=ΦΦ00
(6.43)
The expansion for the jth
components of the degenerate set of eigenstates Φn,j0 and
then comparing the coefficients leads to
ijnjni EV δdeg
00=ΦΦ (6.44)
The selected degenerate states satisfy this equation and still remain eigenstates of H0.
This means that the perturbation does not mix the eigenstate members of the M
combination Φn,j0. Therefore, the first order wavefunction for any component of the
degenerate set is given by
78
0
00
00
1
i
i ni
nji
njEE
Vψ
ψψ ×
−
Φ−= ∑ (6.45)
Similarly, the second order energy can be expressed as
∑−
Φ−=
i ni
nji
njEE
VE
00
200
0ψ
(6.46)
The centrifugal distortion and Coriolis coupling are two examples of the time-
independent perturbation theory. These forces are present only when considering a
non-rigid rotor motion (rotating-vibrating molecule) with respect to a uniformly
rotating frame of reference. These forces result in a detectable effect on the rotational
energy levels which can be observed in the rotational spectra of the investigated
molecule. This makes the rigid rotor model a good basis for the treatment of these
interactions. However, for methane dimer, the experimental results show clear
evidence of coriolis interaction between the rovibronic levels. Therefore, the next
section will discuss the related aspects of coriolis interaction in non-rigid rotating
molecules
6.7 Coriolis Force
Coriolis effect is the apparent deviation of a freely moving object from its main
course as seen from a rotating frame of reference. This effect is referred to the
introduction of the Coriolis force which balances the equation of motion. The Coriolis
deflection is responsible for the counterclockwise rotation of tornados and hurricanes
in the northern hemisphere. This is because the earth's surface is rotating eastward at
greater speed near the equator than near the poles which makes the wind to shift to the
right in the northern hemisphere. Coriolis force is also very important in non-rigid
molecules. It is only effective in rotating and vibrating molecules where the motion
can be described in terms of a rigid rotor and the internal atomic vibrations about their
equilibrium positions. The atomic vibrations cause the atoms to move with respect to
the coordinate system of the rotating molecule. These atomic and molecular motions
produce a Coriolis effect which induces the atoms to move in a vertical direction to
the original oscillation leading to mixing between the rotational and vibrational
79
energy levels of the molecular system. The equation of motion describing the Coriolis
force is
( )ωξˆˆ2 ×= vmF (6.47)
where v is the velocity of the molecule relative to the molecule-fixed axis system (the
rotating system) and ω is the angular velocity of the molecule-fixed axis system (the
rotation rate). The formula indicates that the Coriolis force is perpendicular to both
the velocity of the moving molecular mass and the rotational axis. The physical
concepts of the Coriolis force as well as centrifugal force can be acquired by referring
to the classical representation of these forces presented in many reference books (214,
215, 218). A large number of experimental and theoretical work has been reported on
studying the different aspects of vibrational-rotational interactions in linear,
symmetric, and asymmetric top molecules. The detailed reviews of Nielsen (221)
and
Amat and Nielsen (222-224)
have a complete analysis of the non-rigid polyatomic
molecules where the Hamiltonian includes all possible interaction terms.
Under proper conditions, the Coriolis force can lead to considerable coupling between
rotational and vibrational motions of the molecular system. This in turn imposes
significant changes on the rigid rotor energy levels which may occur in either splitting
or shifting of the energy levels due to removal of degeneracy or to near degeneracy
respectively. For a given molecule, the vibrational-rotational interaction can be
described by the matrix element of the Coriolis coupling constant ζij(α)
which
represents the coupling of the rotation of two normal modes of vibration. This
indicates that the Coriolis coupling constants can be determined by the vibrational-
rotational constants α which are given by
αi = αi (harmonic)
+ αi (anharmonic)
+ αi (coriolis)
(6.48)
The Coriolis coupling has an observed effect on the spectra of linear, symmetric and
asymmetric top molecules with being more pronounced in symmetric top molecules
where first order effect is more common or possible. The vibrational-rotational bands
in the infrared region are those which experience pronounced deviations more often
due to Coriolis interactions. This effect on the rotational spectra of Van der Waal
weakly bound complexes will be discussed in this section taking the spectra of
80
methane argon (CH4-Ar) complex on the triply degenerate region ν4 as an example to
discuss our data of methane dimer (CH4)2 recorded in the same spectral region. The
rare gas-diatomic complexes (e.g. Ar-H2, and Ar-HF) have served as good models for
studying the dynamics and intermolecular potential in the weakly bound complexes
(225-228). The dynamics of these systems are very well understood and the most
accurate intermolecular potentials have been derived for these systems from which the
spectroscopic predictions appear to be in very good agreement with the latest
experimental data. The models of the rare gas-diatomic complexes can be extended to
study complexes of rare gas with other molecules such as rare gas-symmetrical,
spherical or asymmetric top complexes. The Hamiltonian for the rare gas-tetrahedral
molecular complexes can be formulated using the same method as for the rare gas-
diatomic complexes:
( )
),,(/ˆˆ
2
22
2
22
φθµ
RVHRR
jJH mon ++
∂∂−
−=h
(6.49)
where µ is the reduced mass of the complex, R is the distance between the center of
mass of the monomer molecule and the rare gas atom, Ĵ is the total angular
momentum of the complex, and j is the angular momentum of the monomer molecule.
The projections of ĵ on the molecules fixed axis z and on the body fixed axis Z are
designated as k and m respectively, but since the Ĵz is equivalent to ĵz, (Ĵz ≡ jz), then it
81
___________________________________________________________________________
CHAPTER 7
Measurements and Discussion
___________________________________________________________________________
In addition to presenting, discussing and analyzing the measured data of methane
dimer complex, this chapter will also cover a short summery on the symmetry of
tetrahedral molecules in order to have a better understanding of the energy levels of
such complexes. Comprehensive discussion of symmetry and the related topics in this
chapter can be found in references (214-219, 231).
7.1 Symmetry of Tetrahedral Molecules
In general, symmetry is a property of an object that is not influenced by the action of
certain movements or operations applied on the object such as rotation, reflection, and
translation, i.e. an object is said to be symmetrical if the object, when subjected to a
certain operation, appears exactly the same as before the operation applied with
respect to one or more of the geometrical symmetry elements, a point, a line, or a
plane. Symmetry has a fundamental contribution in understanding wide areas of
modern science especially in spectroscopy where it plays a central role in studying the
dynamical structure of a wide range of model systems. Symmetry has also been an
essential tool in studying theory and applications of dynamical systems and in orbital
theory calculations.
Molecules, like any other geometrical figure or object, may have one or more
symmetry elements. For example, a molecule may have an axis of symmetry around
which the molecule rotates leading to a configuration indistinguishable from the
original molecule before the rotation. The other symmetry elements, i.e. the center of
symmetry and the plane of symmetry, can also lead to the same result. On the other
hand, four different symmetry operations can also be applied to the molecule
(reflections σ, rotations Ĉn, rotation-reflections Ŝn and inversion ΐ). Each operation
transforms the molecule to an identical position of the original molecule before the
operation. Therefore, the use of symmetry in molecules is essential in understanding
82
the structures and properties of organic compounds and explaining many other
phenomena in chemistry and physics.
As a result of their natural high symmetry, tetrahedral molecules have been of special
interest to scientists in many research areas such as spectroscopy, crystal engineering,
polymers…etc. A detailed knowledge of these molecules can result in a better
understanding of molecular complexes involving tetrahedral molecules and how these
molecules behave and move within the molecular complexes. The symmetry
properties of tetrahedral molecules have been very useful in building up the
Hamiltonian of this group of molecules. The properties of highly symmetrical systems
don’t apply to the molecular complexes of tetrahedral molecules with other systems
(e.g. rare-gas tetrahedral complex), but most of these properties are preserved by the
tetrahedral molecule within the complex. This implies that the properties of the
complex must be invariant with respect to all possible internal rotations
(permutations) of the tetrahedral molecule within the complex which is equivalent
(isomorphic) to the point group T, where T is a subgroup of the full tetrahedral point
group Td that consists only of pure rotations of Td. The energy levels of the atom-
tetrahedral molecular complex can then be described in terms of having a good and
proper knowledge of the energy levels of the tetrahedral molecule in the complex.
Therefore, the following will be a short outline on the energy levels of the tetrahedral
molecules which is also extendable to other spherical top molecules that belong to Oh
and Ih point groups.
As classified among non-linear molecules, tetrahedral molecules may have 3N – 6
possible vibrational degrees of freedom. The most common and relatively simple
tetrahedral molecules of XY4 type have then (3N – 6) = 9 vibrational modes of
freedom. These modes, according to irreducible representations, are transformed to
A1, E, and two T2 of the Td point group. These species are designated conventionally
as ν1 (A1), ν2 (E), ν3 (T2), and ν4 (T2), where ν1 represent the symmetric stretching
mode, ν2 is the symmetric bending mode, ν3 is the asymmetric stretching mode, and ν4
represents the asymmetric bending mode. The ν3 and ν4 triply degenerate bands are the
only infrared active fundamental bands, i.e. bands that can produce infrared spectra as
a result of possessing electric dipole moment. The other two bands (ν1 and ν2) are low
intensity bands and can be induced through Coriolis coupling by the fundamental
83
bands. In tetrahedral molecules, each rotational level of the vibrational ground state
shows (2j + 1)-fold degeneracy for the j projections in both coordinates. The
rotational levels of this group of molecules can be classified as symmetry species in
the Td point group. The ground vibrational level with all υi = 0 is totally symmetric
which indicates that the overall symmetry of the rovibrational wavefunction has the
same symmetry as the rotational wavefunction. Table (7.1) displays the symmetries of
the first six rovibrational levels of the ground state in a tetrahedral molecule. For the
excited vibrational levels where υi > 0, the pattern of the rotational energy levels is
complicated due to the Coriolis interaction between the rotational and vibrational
angular momentum.
Table 7.1 : Symmetry labels of the rotational levels in the ground vibrational state of
a tetrahedral molecule. Ref. (231)
_____________________________________________________________________
j Symmetry species
0 A1
1 T1
2 E + T2
3 A2 + T1 + T2
4 A1 + E + T1 + T2
5 E + 2T1 + T2
_____________________________________________________________________
Considering the tetrahedral molecule of XY4 type, the quantum mechanical
Hamiltonian of the atom-tetrahedral complex is given by equation (6.49) in the
previous chapter.
The described system in that Hamiltonian is a relatively smaller and simpler complex
as compared to higher order molecular complexes such as methane dimer (CH4)2
which is the topic of this thesis and where two methane molecules have more than ten
possible coupling orientations to form the dimer structure (CH4)2. The rotational
spectra of such molecular systems are usually more complicated than the ones for the
atom-tetrahedral complex due to the fact that their Hamiltonian includes additional
84
rotational terms to properly describe the system. The Hamiltonian for an atom-
tetrahedral complex consists of one rotational and one stretching term motions of the
complex, whereas the Hamiltonian of the two tetrahedral methane molecules (CH4)2 is
composed of two rotational motions in addition to the stretching motions of the
complex. Consequently, more effective Coriolis interaction takes place between the
rotational and vibrational angular momenta of the dimer complex resulting in
complicated spectra. These rotational spectra display a large number of mixed
rotational lines which appear as result of all possible rotational motions within the
complex. The quantum mechanical Hamiltonian for such system can be written as
( ) ( )
),,(
/ˆˆˆ2ˆˆˆ
2
)(
)(
22
2
222
BABmon
Amon
BABA
AB
BA
RVH
HRR
JjjjjJTTH
ωω
µ
++
+
∂∂−
+−++++=
h
(7.1)
where Ĵ is the total angular momentum of the complex, ĵA is the angular momentum
of the first tetrahedral molecule and ĵB is the angular momentum of the second
tetrahedral molecule in the complex, TA and T
B are the kinetic energies of the two
tetrahedral molecules forming the complex and V is the intermolecular potential
which is a function of the intermolecular separation R and the orientation of the
tetrahedral methane molecules with respect to the dimer frame defined by a set of
Euler angles ωA and ωB with ω = (ω1, ω2, ω3). The Hamiltonian consists of all the
possible interaction terms in the complex, where the first two terms describe the
rotational motion of the two tetrahedral methane molecules within the complex, while
the third term represent the stretching motions of the complex. This Hamiltonian
shows that, in addition to the stretching motion of the complex, both methane
molecules can rotate in the complex (230)
.
This pattern is actually found in the infrared spectra of methane dimer (CH4)2
investigated in this work where many dimer lines have been identified over the
spectral regions of P (1), Q (1), R (0), R (1), and R (2) correlated respectively to j =
0← 1, j = 1← 1, j = 1← 0, j = 2← 1 and j = 3← 2 transitions of the triply degenerate
vibrational mode ν4 of methane monomer. The analysis and assignment of all these
lines far exceeds the time frame of this thesis. Therefore, we start this work here by
85
considering first the R (0) spectral region correlating to j = 1← 0 transitions of (CH4)
monomer which is very similar to R (0) spectral region in methane-Argon complex
(Ar-CH4). The work on other spectral regions is in progress. This leads to assume that
only one of the two methane molecules is rotating in the dimer complex, (i.e. ĵAor ĵB =
0). This assumption reduces and simplifies the above Hamiltonian to the Hamiltonian
form in equation (6.49) describing the atom-tetrahedral complexes discussed in the
previous chapter. This Hamiltonian was successfully used by Pak et al (168)
to analyze
and assign the energy levels of (Ar-CH4) complex which is taken as a reference model
in this work. The Hamiltonian of methane-argon (Ar-CH4) -atom-tetrahedral-
complex is given as
( )
),,(/ˆˆ
2
22
2
22
φθµ
RVHRR
jJH mon ++
∂∂−
−=h
(7.2)
where µ is the reduced mass of the complex, R is the distance between the center of
mass of the monomer molecule and the rare gas atom, Ĵ is the total angular
momentum of the complex, and ĵ is the angular momentum of the tetrahedral
molecule. The Hamiltonian consists of all the possible interaction terms of the
complex. The first two terms describe the rotational and the stretching motions of the
complex, while Hmon is the Hamiltonian of the isolated spherical top molecule and V
represents the intermolecular potential that hinders the free rotation of the tetrahedral
molecule which is a function of the intermolecular separation, R, and the orientation
of the tetrahedral molecule with respect to the intermolecular axes defined by θ and φ.
This Hamiltonian, which was used successfully to analyse and assign the rotational
energy levels of the methane-argon complex will be also used here to analyse and
assign the energy levels of methane dimer (CH4)2 complex in the R (0) spectral region
of the triply degenerate bending vibration ν4 of methane monomer.
In the present work, the infrared spectra of methane dimer (CH4)2 have been measured
in the triply degenerate bending mode ν4 of methane monomer using the tunable
diode laser (TDL) spectrometer with 40-60 MHz resolution combined with a
supersonic jet expansion technique. The spectral regions of R (0) and R (1) correlated
to j = 1← 0, j = 2← 1 transitions of methane monomer have been measured by both
continuous and pulsed slit nozzles, whereas the P (1), Q (1) and R (2) spectral regions
86
correlated to j = 0← 1, j = 1← 1 and j = 3← 2 transitions of methane monomer were
measured by continuous slit nozzle only. A large number of dimer lines have been
identified over the above regions. The dimer lines are more concentrated after the
band center of the ν4 vibrational mode of methane (1306.25 cm-1
) which covers the R
(0) and R (1) spectral regions. A lower number of dimer lines were identified in the P
(1), Q (1), and R (2) regions. As an example of the measured spectra from different
regions, fig (7.1-A) displays the spectra of methane in helium and in argon for part of
the R (0) spectral region, while fig. (7.1-B) represents the spectra for part of the R (1)
spectral region. The spectral gap between the two figures is mainly due to the mode
jumps in the radiation of the diode laser system. More than one laser diode laser have
been used to cover all spectral regions in this work which also can cause spectral gaps
in between the collectively measured spectra. The intensity of the dimer lines is not
absolute due to the laser power fluctuation across the radiation of a single mode and
between the different modes. Figure (7.2) is also another example showing two
spectra of the same spectral region measured with both continuous and pulsed slit
nozzles using methane in helium for the two scans.
87
1311,5 1312,0 1312,5 1313,0
-2
-1
0
1
2
3
4
5
6
7
8
9
10
A
CH
4 m
onom
er
He-CH4
Ar-CH4
CH
4 m
onom
er
CH
4 m
onom
er
Inte
nsity (arb
.units)
Wavenumbers(cm-1)
1315,0 1315,5 1316,0 1316,5 1317,0
-2
0
2
4
6
8
10
B
CH
4 m
onom
er
CH
4 m
onom
er
CH
4 m
onom
er
He-CH4
Ar-CH4
CH
4 m
onom
er
Inte
nsity (arb
.units)
Wavenumbers(cm-1)
Fig. 7.1: Sample scans of methane in helium and argon for the R (0) spectral region in
(A), and for the R (1) spectral region in (B).
88
1309,5 1310,0 1310,5 1311,0
-10
-5
0
5
10
15
20B
ACH
4-m
onomer
CH
4-m
onomer
CH
4-m
onomer
Inte
nsity (arb
,units)
Wavenumbers(cm-1)
Fig. (7.2): Scan traces of both continuous (A) and pulsed (B) slit nozzles using
methane in helium for the two scans.
The dimer lines were confirmed by scanning the desired wavelength regions with a
mixture of ~ 40 % methane in argon and in helium-neon separately and then by
excluding all the lines which don’t appear in both scans like Ar-CH4 and Ne-CH4
cluster lines and other possible lines that may occur at high concentration of methane
in the mixture. These complexes have been investigated in two separate studies
reported by Pak et al (168, 236)
, where the cluster lines of Ar-CH4 and Ne-CH4 were
observed at a lower concentration of methane in the mixture as compared to methane
dimer lines. In both studies, a gas mixture of 5-10 % of methane in argon and helium-
neon was used to produce these complexes through a pulsed supersonic jet expansion.
In search for the optimal concentration for the methane dimer complex, the same set
of Ar-CH4 lines have been also identified in this work at the same concentration
reported by Pak et al. The optimum concentration for the production of methane
dimer complex in both continuous and pulsed slit nozzles is obtained by selecting first
a short spectral region after the band center of the ν4 vibrational mode of methane
expected to have strong transitions of methane dimer lines, then start changing the
mixing ratio of argon and methane components in the mixture while monitoring the
development of the dimer lines for each single change in the mixing ratio. This has
89
been done here by fixing the methane pressure at for example 100 mbar with 500
mbar as starting values for the argon pressure, then start increasing the argon pressure
in steps of 100 mbar until we reach a pressure value of 2000-2500 mbar where no
appreciable changes were noticed on the cluster lines. The Ar-CH4 lines reported by
Pak et al have been observed in this step with relatively good signal to noise ratio.
After that, start increasing methane pressure in steps of 100 mbar and change the
argon pressure as done in the first step. After several steps, the optimum mixing ratio
of methane in argon or helium to produce methane dimer complex (CH4)2 is found to
be 1000 mbar of methane in 2500 mbar of argon or helium which corresponds to 40%
of methane in argon. The backing pressure was in the range of 10-1
mbar. The
optimum mixing ratio for methane dimer in a pulsed slit nozzle was found to be 3000
mbar of methane in 6500 mbar of argon or helium which is also about 45% of
methane in argon, with a backing pressure in the range of 10-2
- 10-1
mbar. Therefore,
the ratio is about 40% ± 1% of methane in argon or helium.
Fig (7.3) depicts the diode laser spectrum of the Van der Waals methane dimer
complex (CH4)2 measured in the wavelength region of R (0) covering a range of
1309.5 - 1311.0 cm-1
and correlated to j = 1← 0 transition of the ν4 vibration of
methane monomer.
1309,5 1310,0 1310,5 1311,0
-10
-5
0
5
10
15
20
25
30
35
40
45
50
B
A
CH
4 M
onom
er
CH
4 M
onom
er
CH
4 M
onom
er
Inte
nsity (arb
.units)
Wavenumbers(cm-1)
Fig. 7.3: TDL spectra of 40% methane in argon (A) and in helium (B)
90
A close inspection of the spectrum shows no line spacing corresponding to 2B,
(B+C)/2, or B-C which suggests that an effective perturbation is taking place in this
system. The features of the line spacing also indicate that Coriolis coupling is the type
of perturbation that is taking place in the methane dimer complex. Therefore, based on
assuming that one methane molecule is rotating in the complex as mentioned in the
above discussion, the spectrum can be best described in terms of the atom-spherical
top Hamiltonian developed first by Randall et al (231)
and used later by Brook et al (232,
233). This Hamiltonian incorporates Coriolis interaction between the angular
momentum of the monomer molecule and the rotation of the whole complex.
Hence, the Hamiltonian of the methane dimer complex can be written as
( )
),,(/ˆˆ
2
22
2
22
φθµ
RVHRR
jJH mon ++
∂∂−
−=h
(7.3)
This Hamiltonian has been successfully used to determine and assign the energy
levels of Ar-SiH4 (234)
, N-SiH4 (232, 233)
, and Ar-CH4 (168)
complexes. Therefore, to
determine and assign the energy levels of methane dimer complex we follow the same
analytical approach of solving this Hamiltonian for methane-argon complex (Ar-CH4)
and then apply the calculated parameters obtained from methane dimer spectra.
A couple of approximations should be considered here in order to simplify the above
Hamiltonian and then calculate the energy levels of the rare gas-tetrahedral complex.
The first approximation is to assume that the tetrahedral molecule maintains its
integrity upon complexation. Consequently, the symmetry of the spherical top
molecule is preserved giving rise to exactly the same Hamiltonian within the complex
as if it is a free molecule. It was also shown that the intermolecular potential displays
the properties of the monomer molecule itself, suggesting that a fairly simple
Hamiltonian can be used to represent the energy levels of the monomer molecule. The
second approximation is that the rotational and intramolecular stretching motions of
the complex can be separated from the rest of the Hamiltonian in an adiabatic sense.
This means that the perturbations to the energy levels of the monomer molecule are
mainly caused by the intermolecular potential. It also means that the end over end
rotational levels and the intramolecular vibrational levels of the complex can be built
up on the top of these levels as if the perturbed rotational levels of the monomer are
91
separate librational or hindered rotational levels. Therefore, the eigenfunctions
(rotational energy levels) can be stated or expressed as a simple product of the ones
for the monomer molecule and those for the rotation of the whole complex given as
jKkJKMjKkJKM υυ =;; (7.4)
The rotational function (Ĵ – ĵ)2 in the original Hamiltonian can be expanded as
(Ĵ – ĵ) 2 = Ĵ
2 – 2Ĵ. ĵ + ĵ
2 = Ĵ
2 – 2Ĵz ĵz – 2Ĵx ĵx – 2Ĵy ĵy + ĵ
2
= Ĵ 2 – 2Ĵz
2 + ĵ 2 – 2( Ĵx ĵx + Ĵy ĵy )
= Ĵ 2 – 2Ĵz
2 + ĵ 2 – ( Ĵ+ ĵ+ + Ĵ- ĵ- ) (7.5)
The expanded function indicates that the equivalence condition of Ĵz and ĵz mentioned
before is used in the expansion and the Ĵ± and ĵ± are the lowering and the raising
operators which are given by
Ĵ± = Ĵx m iĴy , ĵ± = ĵx ± iĵy
The separation of the end over end rotation of the whole complex from the rotation of
the monomer molecule within the complex can be carried out by setting J = 0. This
also leads to neglect the Coriolis coupling term (Ĵ+ ĵ+ + Ĵ- ĵ-) from the original
Hamiltonian which practically has the effect to mix the different adjacent K levels and
thereby removing the degeneracy of ± K levels. This is corresponding to the helicity
decoupling approximation methods used in rare gas-hydrogen halide complexes (235)
which causes the decoupling of the angular momentum j from the intermolecular axis.
In this case, K remains to be an approximately good quantum number as long as the
K-splitting due to the anisotropy of the potential is larger than the matrix elements of
the Coriolis term. The matrix elements of the Coriolis coupling term are given by
[ ]22 )1()1( KjjKJJBJjkKHJjkK cor −++−+= (7.6)
92
where Hcor represent the rotational term in the original Hamiltonian [(Ĵ – ĵ)2
× ħ2 / (2µ
R2 )], and B = ħ
2 / (2µ R
2 ) is the rotational constant of the complex.
The rotational energy levels of the complex can be expressed as a power series in [J (J
+ 1) – K2] rather than the usual power series in [J (J + 1)]. The obvious difference
between the two series is an energy shift of -B K2 in each energy level of the complex
leading to a shift in the band origin of the transition between two K states. On the
other hand, the effective rotational constant of the tetrahedral molecule is also affected
by the rotational constant of the whole complex B, (i.e. the monomer rotational
constant b becomes B + b). Therefore, this energy deviation of the level system
requires an explicit incorporation of the Coriolis coupling terms and to diagonalise a
proper 2×2 matrix for low j values. The main effect of Coriolis coupling terms is to
remove the degeneracy of the K = ± 1, ± 2, ± 3…levels for j = 1, 2, 3…etc. e.g. for j =
1 state, the K = 0 level couples to a symmetric combination of eigenfunctions with K
= 1 and K = -1, these states are
{ }112
11 −++=+ (7.7)
{ }112
11 −−+=− (7.8)
The K = 1 and K = -1 degeneracy of states is removed by the Coriolis interaction
because it only couples the K = 0 state with the symmetric combination | 1+ ›. The
Coriolis matrix element is then given by
)1()1(210 ** +×+=+ jjJJBH cor (7.9)
where j* is an empirical parameter equivalent to the effective value of j quantum
number.
This interaction has the influence to separate the | 0 › and | 1+ › levels apart from each
other while keeping the | 1- › level unaffected. Consequently, the rotational energy
levels associated with | 1+› state seem to have more energy separation than for K = 0
93
state which appear to be closer together, i.e. K = 0 is lower in energy than K = 1. The
resultant energy terms from the Hamiltonian can also be expressed in terms of the
splitting parameter (α) which also incorporates the effect of the intermolecular
potential and any vibrational terms, i.e. α measures the energy separation between K
levels. Therefore, by taking the energy of K = 0 and K = ± 1 as υE – α and υE + α
respectively, the rotational energy levels relative to the vibrational origins are given
by the eigenvalues of simple 2 × 2 matrix originated from the Coriolis matrix
elements in equ. (7.9)
+×+−
−+
=
=
)1()1(2
)1(
**
0
jjJJB
JJB
H
K α
++
+×+−
= α)1(
)1()1(2
1
**
JJB
jjJJB
K
(7.10)
where BK=0 and B
K=1 are the rotational constants for K = 0 and K = 1 levels
respectively, while B is taken as the effective mean value of these two constants. The
expression of the rotational energy levels is then given by diagonalizing the above 2 ×
2 matrix which results in the following eigenvalues
{ } 2/1**22222
10
)1()1(8)1()1(442
1
)1(2
++++∆++∆+±
+
++= ==
JJjjBJJBJBJ
JJBB
EE KK
J
αα
υ
(7.11)
where Eν is the vibrational energy and ∆B = B
K=1 – BK=0 which is highly correlated
with Coriolis terms leading to insufficient information about this factor. In this case, it
is adequate to constrain ∆B to zero in order to simplify the above equation to
)1()1(2)1( **22 +++±++= JJjjBJBJEEJ αυ (7.12)
In the vibrational excited state, the energy expression of the | 1- › level is given by
EJ = E
ν + α + B
K=1 J (J+1) (7.13)
For the states correlating to j = 0, the selection rules of the total angular momentum J
are
94
∆J = ± 1 for transitions to | 0 › and | 1+ › levels
∆J = 0 for transitions to | 1- › levels
The infrared spectroscopy of methane-argon complex (Ar-CH4) in the 7 micron region
investigated by Pak et al (168)
is a good example to get better understanding of the
above treatment for the atom-tetrahedral complex. The energy level diagram of
methane-argon is represented in Fig. (7.4), the left part of the figure shows the
rotational energy levels in the ground and excited vibrational states of methane, while
the rotational levels of the end over end rotation of the complex which are built on the
j, n levels of the methane part are located next to the left part and are defined by the
total angular momentum of the whole complex J. The right side of the figure displays
the splitting of the J levels in the upper vibrational state into its three components |1+›,
|1-›, and | 0›.
The recorded spectra were fitted using the above formulas for the ground and the
excited states where the strong transitions of R and P branches terminate in the lower |
0 › and upper |1+› Coriolis coupled levels respectively as shows in the figure.
However, it was difficult to precisely determine the rotational constant of these states
due to high correlation between these constants as a result of Coriolis coupling, i.e.
the dominated Coriolis terms cause mixing and splitting of the states of different K
producing the highly correlated rotational constants and dramatic change in the energy
of these levels. Then the transitions of the R and P branches can be expressed as
)1('')2/3()1(2)2)(1()( **
0 +−+×+×−+++= JJBJjjBJJBJJR υυ
( ){ } 2****
0 )1)(''()1()1(2'')1(2
1+−+++−+++−= JBBJjjBBBjjBυ (7.14)
)1('')2/1()1(2)1()( **
0 +−−×+×+−+= JJBJjjBJBJJP υυ
( ){ } 2****
0 )''()1(2'')1(2
1JBBJjjBBBjjB −++−+−+−= υ (7.15)
95
Since these transitions depend on the same constants (B, B", and ν0), and in order to
precisely determine the values of parameters in the formulas, Pak et al considered the
approach of the nearly forbidden transitions along with the combination differences
which can give some information about the ground state rotational constants. After
fitting the main R and P branches of the Ar-CH4 complex, they predicted the nearly
forbidden transitions of the complex. The predicted lines were then included in the
final fit which resulted in relatively small change in the ground state rotational
constant B". The B rotational constants for all the levels were then calculated
accurately. The strong Coriolis mixing of different K levels puts limits to consider K
as a good quantum number. It also leads to a new quantum number N where N = J – j
and the selection rule ∆N = 0 result in the appearance of Q branch in the spectrum (10-
14).
Fig. 7.4: The energy level diagram of methane-argon complex (Ar-CH4). Ref. (168)
96
7.2 Data Analysis
Fig (7.3) displays the spectrum of methane dimer recorded in the R (0) spectral region
correlating to j = 1← 0 transition of the triply degenerate mode ν4 of methane
monomer. The spectrum shows a strong and well resolved R branch starting at 1309.5
cm-1
, whereas the starting positions of the P and Q branch regions are estimated to be
around 1310.35 cm-1
and 1310.55 cm-1
respectively which exhibit very dense and
strong transition lines but with no obvious or resolved branch due to complex pattern
system for both regions. The intensity of the dimer lines is not absolute as the laser
power was changing across one single mode and between different modes. In the light
of the previous discussion, this spectrum can therefore be described and analyzed by
the Hamiltonian defined in equ. (7.3)
( )),,(/
ˆˆ
2
22
2
22
φθµ
RVHRR
jJH mon ++
∂∂−
−=h
This model incorporates Coriolis coupling between the hindered rotation of one
methane molecule and the rotation of the dimer complex. The perturbed rotational
energy levels of methane dimer are obtained by solving the Schrödinger equation of
this Hamiltonian. These states are represented in equations (7.12) and (7.13)
)1()1(2)1( **22 +++±++= JJjjBJBJEEJ αυ
EJ = E
ν + α + B
K=1 J (J+1)
These formulas are the basis of the fitting program developed and used in this work to
fit the transitions of methane dimer lines shown in Fig. (7.5) taking in account that the
strong transitions of the R and P branch terminate in the lower | 0 › and upper | 1+
›
Coriolis coupled levels respectively, while the transitions of the Q branch terminate in
the | 1- › level as shown in Fig. (7.4). These transitions satisfy the selection rules for
the states correlating to low j values (j = 0) where the total angular momentum ∆J = ±
1 for the | 0 › and | 1+ › states and ∆J = 0 for the | 1
- › states. Knowing and confirming
the position of R branch region from the fit leads us to estimate the positions of the P
97
and Q branch regions which are stated in the beginning of this section. This can be
used to calculate the lower and upper energy levels of the R, P, and Q branches using
equation (7.12). The result is a set of equations that can be solved by the fitting
program to give estimate values of the fitting parameters, i.e. the vibrational
frequency ( υE ), the Coriolis constant (C), and the splitting parameter (α). These
values are then used to predict the structure of methane dimer complex (CH4)2 and
compare it with the results of the ab initio theoretical predictions of the different
orientations of two methane molecules in a dimer complex list in chapter five.
Therefore, the lower and upper energy levels of R, P, and Q, branches of the dimer
complex are calculated as follows:
In R branch region, the energy of the lower level (j = 0, J = 0) is given as
00 =′′=JE
The energy of the upper level (J = 1) can be expressed as
22
1 22 CBEEJ +−+=′= αυ (7.16)
The line frequency is expressed as; ν = upper J – lower J
ν = E´J=1 – E´´
J=0
22 22 CBE +−+= αυ (7.17)
For the P branch, the energy of the lower level (J = 2) can be stated as
BE J 62 =′′= (7.18)
The energy of the upper level (J = 1) is given by
22
1 22 CBEEJ +++=′= αυ (7.19)
98
The line frequency can also be expressed as
ν = E´J=1 – E´´
J=2
22 24 CBE ++−= αυ (7.21)
For the Q branch region, the energy of the lower level (J = 1) is
BEJ 21 =′′= (7.22)
The energy of the upper level (J = 1) is given as
BEEJ 21 ++=′= αυ (7.23)
The line frequency is then give by
ν = E´J=1 – E´´
J=1
= Eν + α (7.24)
Solving these equations results in giving the estimated values of the unknown
parameters Eν, C, and α in the equations. The calculated values of these parameters,
taking B = 0.12 cm-1
, are given as:
Eν = 1310.05 cm-1
, C = 0.264721, α = 0.6985
These values are then used as starting input values in the fitting program in order to fit
the transition lines of methane dimer in Fig. (7.5). The outcome is a set of fitting
parameters shown in table (7.2), and the fitting results of the positions of the observed
dimer lines with respect to calculated positions are shown in table (7.3)
The absence of clear and well resolved P and Q branches in the spectrum led us to
start the fitting with R branch first and then follow the approach of line-by-line fitting.
This is mainly based on predicting the first line of the P or Q branch, the second line,
the third line and so on. Then make a fit of the predicted lines of each branch along
with the main fitted R branch in the spectrum and then do a final fit of the three
99
branches together. The fitting program has a total number of 16 parameters from
which one can select to use certain parameters depending on the investigated
molecular system since most of the parameters are related to each other. Six
parameters were sufficient to fit the observed transitions of the methane dimer shown
in Fig. (7.5).
1309,5 1310,0 1310,5 1311,0
10
5
0
-5
-10
Q-branch
P-branch
R-branch
J=1
J=2
J=0
Monom
er
Monom
er
Monom
er
Inte
nsity (
arb
.un
its)
Wavenumbers(cm-1)
Fig.7.5: The spectrum of 40% methane in helium shows the fitting results of the R, P,
and Q transitions lines of methane dimer complex (CH4)2.
100
Table 7.2: Fitted parameters of methane dimer
All values are in cm-1
J = 1 ← 0 transition of (CH4)2
ν0 1309.96
B`` 0.105874
B` 0.108376
α 0.615426
Dj 0.000234919
C 0.206739
Table 7.3: The position of the observed lines and difference between the observed
and the calculated positions of the j = 1← 0 transition of methane dimer
(CH4)2.
Q-Branch R-Branch P-Branch
J`` OBS O – C OBS O – C OBS O – C
cm-1
cm1
cm-1
cm-1
cm-1
cm-1
0 1309.4945 0.0024
1 1310.5763 - 0.0000 1309.5951 - 0.0024
2 1310.5890 0.0026 1309.6723 - 0.0028 1310.2212 0.0011
3 1310.6050 0.0036 1309.7361 - 0.0009 1310.1359 0.0001
4 1310.6215 0.0001 1309.7914 0.0014 1310.0869 -0.0030
5 1310.6472 0.0007 1319.8389 0.0013 1310.0700 -0.0004
6 1310.6736 -0.0029 1309.8845 0.0027 1310.0700 -0.0004
7 1310.7098 - 0.0018 1309.9242 0.0005 1310.0869 0.0006
8 1310.7521 0.0004 1309.9632 -0.0004 1310.1180 0.0019
9 1310.8000 0.0032 1310.0000 - 0.0016
10 1310.8468 - 0.0001 1310.0361 - 0.0014
11 1310.9000 -0.0020 1310.0705 - 0.0007
12 1310.1042 0.0020
13
14
15
101
7.3 Discussion
Comparing the rotational constants of the fitted methane dimer in table (7.2) with the
calculated rotational constants of the theoretically predicted orientations of methane
dimer in table (5.1), we find that the value of B = 0.108376 cm-1
lies in between two
sets of orientations. The first set is EE-Ec and VF-Ec orientations which have a
rotational constant of 0.1128 cm-1
and the second set of orientations is VF-St and VV-
Ec which have a rotational constant of 0.905 cm-1
. These figures show that the
orientation of the two methane molecules in a dimer is not even close to the
equilibrium structure suggested by theoretical studies. The pattern of these values
indicates that the fitted B value of 0.108376 cm-1
is an average value of different
orientations of methane monomer in the complex which suggest the presence of large
amplitude motions. The fitted rotational constants can also be used to determine the
intermolecular distance R between the two methane molecules in the complex which
is calculated and found for j = 1 to be 4.46 Ao. The difference in the values of R in the
complex usually reflects the difference in the size of the complex.
Compared with other methane complexes such as Ar-CH4, Kr- CH4, and Ne-CH4, the
Coriolis constant C in methane dimer is higher than in Ar-CH4 and Kr- CH4, whereas
it is close to the value of Ne-CH4. This is due to the high value of rotational constant
B for (CH4)2 and Ne-CH4 and due to a high effective value of the rotational quantum
number for the monomer in the complex (j*). In methane dimer, this parameter has
been calculated for j = 1 and found to have a value of 0.9533 which is also higher than
the values of Ar-CH4 , Kr- CH4 and closer to the value of Ne-CH4 complex. This
indicates that the monomer in methane dimer is rotating more freely than in other
methane complexes. For example, the Coriolis constant in Ar-CH4 complex causes
stronger mixing between the K-levels in the system indicating a relatively stronger
intermolecular potential compared to the potential of methane dimer complex. This
can be inferred from the two values of (j*) for both complexes. The term α is the
splitting parameter between K = 0 and K = ± 1 levels before the Coriolis coupling
takes place. In a free rotor limit the splitting approach -2B`, i.e. α = -B`, this shows
that α +B` measures the effect of the anisotropy of the intermolecular potential (233)
.
Therefore, the smaller values of α + B` indicates a very weak anisotropic potential
102
like in which reflect the case Ne-CH4 and paraH2- CH4 complexes where the
monomer is close to free rotor limit.
7.4 Conclusion
The data analysis concludes that the geometry of the methane dimer complex is not
the equilibrium structure concluded by the ab initio theoretical studies. The analysis
also shows that the motion of the methane monomer in the dimer complex is close to
free rotor limit. Finally, the work on the other spectral regions of the triply degenerate
bending mode ν4 of methane monomer that have been measured in this work [P (1), Q
(1), R (1) and R (2)] is still in progress which in addition to analyzed data of the R (0)
spectral region in this thesis will help in the ongoing calculations of the intermolecular
potential of methane dimer complex.
103
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