MOS Final Report

1
MOS: Molecular Spectroscopy
Leland Breedlove, Andrew Hartford, Roman Hodson, and Kandyss Najjar
Abstract
This set of experiments uses Fourier-Transform infrared spectroscopy (FTIR) to determine
the molecular characteristics of various molecules. The data from these experiments provides good
insight into the rovibrational levels of carbon monoxide, the effectiveness of the greenhouse gases
NO2 and CH4, and the X-State and B-State of molecular iodine. The obtained results from the
carbon monoxide experiment are close to the literature values, and provide evidence that carbon
monoxide acts more like a harmonic oscillator than an anharmonic oscillator because its
anharmonicity constant is small compared to the other obtained constants. In addition, while the
obtained values are all less than the literature values, the global warming potentials of the
greenhouse gases NO2 and CH4 indicate that NO2 is a more effective greenhouse gas than CH4, as
expected from theory, due to NO2’s time horizon in the atmosphere. Lastly, the results from the
absorption and emission of molecular iodine provide molecular constants for the X-State and B-
State. These values are close to the literature values, excluding the anharmonicity constants due to
extrapolation error. While the calculated equilibrium bond length for the X-State is less than the
literature value, the results show that the X-State has a smaller equilibrium bond length than the
B-State, which is expected from theory as the equilibrium bond length increases with increasing
vibrational energy. In essence, all three experiments provide the expected trends from the theory.
Introduction

2
This set of experiments is concerned with the determination of structural features of certain
molecules, as well as global warming potentials of various greenhouse gases, through molecular
spectroscopy. Molecular spectroscopy studies the response of molecular structure to
electromagnetic radiation in the form of absorption and emission as well as any energy level
transitions that occur during these processes.1 In addition, it depends on nuclear and electronic
configurations as well as molecular behavior to distinguish molecules.
Central to molecular spectroscopy are rules pertaining to the energy of movement of nuclei
and electrons, as well as their respective frequencies.2 These rules adhere to the quantum
mechanical basis of energy quantization. For all particles, the kinetic and potential energy is
dependent on their motion. In the x, y, z space realm, the number of degrees of freedom associated
with n particles is 3n. When examining a molecule, the reference is its center of mass outlined by
Equation 1
𝑟0 =
1
𝑀
∑ 𝑚 𝑖𝑖 𝑟𝑖 (1)
where M is the total mass of the system, mi is the mass of a particle, ri is the distance of the particle
from the center, and ro is the center of mass. For the nucleus, the vibrational, rotational, and
translational aspects of motion are all carried out with respect to this center of mass. Electronic
motion is spatially arranged with respect to molecular orbitals as electrons are significantly smaller
than nuclei resulting in a fixed configuration about them. Energy quantization notes the discrete
energy levels associated with different wave functions in the form v=0, v=1, v=2, etc. for
vibrational motion, J = 0, J = 1, J = 2, etc. for rotational motion, and S0, S1, S2 etc. for electronic
motion as shown in Figure 1.

3
Figure 1. Energy Levels of Electronic, Vibration, and Rotational Energy 2
The strength of transition between two energy levels is dependent on the dipole moment of a
molecule dependent on Equation 2
𝑃𝑖→𝑓| ∫ 𝜓𝑓𝑖𝑛𝑎𝑙
∗
µ𝜓𝑖𝑛𝑡𝑖𝑎𝑙 𝑑𝑟|2
(2)
where µ=∑ 𝑞𝑖𝑖 𝑟𝑖
where the 𝜓 terms are the wavefunctions of the particles, q is the charge of the particles, and r is
the length of the bond. The transitions that occur between these states are governed by selection
rules that determine whether a particular absorption transition is permitted. When an electron is
excited rotationally between energy levels such as v=0 and v=1, the excited state values must
follow that ΔJ = ±1.2 For vibrations, the selection rule states that Δv=±1.2
Rovibrational spectroscopy consists of analyzing the coupled rotational and vibrational
aspects of molecules using infrared radiation in the form of light. Infrared radiation (IR) has

4
enough energy to cause molecules to rotate and vibrate with rotations and vibrations represented
in Equations 3 and 4
F(J) = BJ(J+1) (3)
G(v) = (v + ½)νe (4)
where F(J) represents the rotational energy and G(v) represents the vibrational energy.3 In addition,
J and v represent the quantum numbers of the rotational and vibrational states respectively, νe
represents the frequency constant in wavenumbers (cm-1), and B represents the rotational constant
in wavenumbers. The equations for constants νe and B are represented in Equations 5 and 6, with
the moment of inertia represented in Equation 7
𝑣𝑒 =
1
2𝜋𝑐
√
𝑘
𝜇
(5)
𝐵 =
ℎ
8𝜋2 𝑐𝐼
(6)
𝐼 = 𝜇𝑅 𝑒
2
(7)
where c is the speed of light ( m s-1), k is the force constant (N m-1), 𝜇 is the reduced mass (kg), h
is Planck’s constant (J s), I is the moment of inertia (kg m2), and Re is the equilibrium bond length
(m).3 The force constant is proportional to the strength of the covalent bond, as it shows how stiff
the bond is.4 Stiffer bonds are more difficult to stretch and compress, and therefore require a greater
amount of energy to do so. As a result, stiffer bonds vibrate faster and absorb at higher
wavenumbers.4 The equilibrium bond length is the internuclear distance when the internuclear
potential energy is at a minimum, as shown by the Lennard-Jones potential in Figure 2. It is the
thermal motion of the molecule that causes the iodine atoms to move around this equilibrium
position.5

5
Figure 2. Lennard-Jones Potential Diagram 5
The negative derivative of potential energy is force, as shown in Equation 21.
−𝑑𝑈
𝑑𝑟
= 𝐹( 𝑟) (21)
Therefore, on the Lennard-Jones potential diagram, the area to the left of the minimum is the
repulsive force the atoms feel, and the area to the right of the minimum is the attractive force the
atoms feel.5 The equilibrium bond distance is at the minimum of the curve, where the repulsive
and attractive forces cancel.5 Therefore, a small equilibrium distance corresponds to a larger force
constant. In addition, following Equations 5 and 7, the moment of inertia is directly proportional
to the size of the force constant. The terms I and Re combine to make the reduced mass term, shown
in Equation 22
𝜇 =
𝑚1 𝑚2
𝑚1+ 𝑚2
(22)

6
Because 𝜇 is directly proportional to the force constant, diatomic molecules with larger masses
will therefore have larger force constants. We determined the molecular constants for carbon
monoxide in this experiment using rovibrational spectroscopy.
While Equations 3 and 4 provide a good model for rotations and vibrations alone, when
coupled they create interferences which need to be assessed. During a vibrational state transition,
the molecule experiences a force which causes the average bond length to increase.3 This increase
in bond length affects the rotational constant B, and therefore needs the terms Be and αe to account
for this bond length increase, as represented in Equation 8.
𝐵 𝑣 = 𝐵 𝑒 − 𝛼 𝑒 (𝑣 +
1
2
) (8)
In addition, Equation 8 implies that rotations are not based on a rigid rotor, so as the value of J
increases, the centrifugal distortion will cause the bond length to increase as well. This increase in
bond length due to centrifugal distortion, represented by the constant De, and is provided by
Equation 9.
𝐹( 𝐽) = 𝐵 𝑣 𝐽( 𝐽 + 1) − 𝐷 𝑒 𝐽2( 𝐽 + 1)2
(9)
So far, the vibrational transitions have been based on the harmonic oscillator model, as shown in
Equation 3 and in Figure 3.

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Figure 3. Simple Harmonic Oscillator Model 6
However, the harmonic oscillator model is only useful for low quantum numbers, as this
model does not account for bond dissociation or repulsive effects. In addition, the simple harmonic
oscillator forbids vibrational transitions which do not follow a change in vibrational level of Δv =
±1. However, such transitions can occur when enough energy is presented in the system, such as
the first overtone which corresponds to a molecule’s being excited from the ground vibrational
state to the second excited vibrational state.6 Therefore, another model known as the anharmonic
oscillator (shown in Figure 4) is used which accounts for these deviations from the simple model.

8
Figure 4. The Anharmonic Oscillator 6
The anharmonic oscillator model shows the average bond length to change with increasing
quantum numbers, as well as that the vibrational energy levels are no longer equally spaced for a
molecule.6 The anharmonic oscillator demonstrates an increasing average bond length for
increasing quantum numbers. In addition, the anharmonic oscillator shows a decreasing width of
spacing of energy levels at higher excitation, as the curve provides less constraint than the
harmonic oscillator parabola.6 This is an effective model to use in rovibrational spectroscopy, as
it provides another constant, the anharmonicity constant (χe) shown in Equation 10, which accounts
for the deviations in bond length due to increasing vibrational levels.
𝐺( 𝑣) = (𝑣 +
1
2
) 𝑣𝑒 + 𝑥 𝑒 𝑣𝑒(𝑣 +
1
2
)2
(10)
Rovibrational spectroscopy characterizes the structure of molecules by their rotational
energy levels corresponding to specific vibrational levels.7 We will identify rovibrational
characteristics of carbon monoxide through use of a FTIR spectrometer to develop an IR spectrum.
An FTIR spectrometer is of use in both inorganic and organic chemistry realms as it is capable of
determining structural characteristics from IR exposure. A major part of the spectrometer is the

9
Michelson interferometer which handles both the radiation exposure of a sample and the Fourier
Transformation required to develop a spectrum. From the source, a beam of light is split and
reflected off a motorized mirror, subsequently recombining to run through a sample. A detector
obtains the interferogram and Fourier transforms it into a spectrum. A block diagram of a
Michelson interferometer in Figure 5 outlines the major components.
Figure 5. Block Diagram of a Michelson Interferometer 8
We will use the rovibrational spectra to determine the fundamental transition and first
overtone of carbon monoxide between the ranges of 1950-2275 cm-1 and 4100-4400 cm-1,
respectively.9 A fundamental transition corresponds to Δv = +1, whereas the first overtone
corresponds to Δv = +2 for the CO molecule. Overtones correspond to Δv = ±𝑛 transitions, but
the probability of overtone transitions decreases as n increases.6 The anharmonic model shows the
overtones to be usually less than a multiple of the fundamental frequency.6 While the first overtone
corresponds to a higher energy, it is expected that its intensity will be less than that of the
fundamental.
We will determine the band center frequency (v0) to calculate rovibrational characteristics
like the anharmonicity constant (𝜒 𝑒) and equilibrium frequency (𝜐𝑒) by plotting m vs wavelength
of the fundamental and overtone spectra as well as using Equation 11

10
𝜈0 = 𝜈𝑒 − 2𝜈𝑒 𝜒 𝑒 (11)
We will also quantify the band force constant k by assessing the transition of the ground
state to first excited state as if it were a harmonic oscillator and using Equation 12 where ħ is
Planck’s constant divided by 2π, µ is the reduced mass, ω is the angular frequency, and k is the
band force constant, shown in Equation 12.
ħ𝜔 = ħ√
𝑘
µ
(12)
The internuclear distance, or the bond length between atoms is when the systematic
potential energy is at its lowest level. The bond length is assumed to be identical for both the
ground and first excited energy state, and therefore we will use the transition frequency difference
for this calculation. Diatomic molecules such as CO and HCl have a center frequency spectrum
shown in Figure 6.
Figure 6. Center Frequency Spectrum of HCl 10

11
Using the internuclear distance, we calculated the moment of inertia for a diatomic
molecule using Equation 7. This formula corresponds to a singular point mass; for a defined space
containing multiple point masses, the moment of inertia is the summation of these terms.
We also determined the global warming potentials (GWP) of various greenhouse gases
including N2O and CH4 using IR absorbance. GWP represents the amount of heat trapped by
greenhouse gases when they are exposed to IR radiation emitted from the earth’s surface. The
GWP for a molecule is determined with respect to quantity, strength, and location of IR absorption
bands of the molecule with respect to the earth’s emitted IR radiation. GWP has been of interest
among researchers and political activists alike as it is a way of quantifying the adverse effects and
levels of harm these gases have on climate change. For example, the 1997 Kyoto Climate
Conference aimed to reduce emissions of six common greenhouse gases determined to have high
GWP’s to levels around 5.2% below 1990 levels by 2012.9 Radiation forcing capacity is the
summation of the IR spectrum and the emission of blackbody radiation from earth. It is equivalent
to the GWP in proportion to the time of residence the gases have in the atmosphere. In order to
obtain IR spectra, we will fill the gas cell of the FTIR spectrometer with samples at pressures
compatible with Beer’s Law (60 Torr for both N2O and CH4). Once obtained, we integrated the
spectra at 10 cm-1 intervals between 500-1500 cm-1 per the Pinnock et. Al model.11 The radiation
forcing capacity of a sample can be determined relative to a reference gas. The reference gas is
normally CO2.The radiation forcing capacity is given by Equation 13 where RFA is the radiation
forcing capacity per 1 kg increase of sample, A(t) is time decay of the sample pulse sample, and
RFR and R(t) are that of the reference.
GWP =
𝑅𝐹 𝐴
∗∫ 𝐴( 𝑡) 𝑑𝑡
𝑇𝐻
0
𝑅𝐹 𝑅 ∗∫ 𝑅( 𝑡) 𝑑𝑡
𝑇𝐻
0
(13)

12
In order to determine GWP in terms of mass as opposed to per molecule as in Equation 13,
we used Equation 14 as shown below where τ is the atmospheric lifetime and MW is molecular
mass.
GWP =
𝑅𝐹 𝐴
∗(
1000
𝑀𝑊 𝐴
)∗∫ 𝑒−𝑡/𝜏 𝐴
𝑑𝑡
𝑇𝐻
0
𝑅𝐹 𝑅 ∗(
1000
𝑀𝑊 𝑅
)∗∫ 𝑒−𝑡/τ 𝑅
𝑑𝑡
𝑇𝐻
0
(14)
Absorbance spectroscopy is another aspect of this experiment, which works by measuring
the transmittance of light after it passes through the analyte. This transmittance relates to the energy
level transition from ground to an excited state. Transmittance is related to absorbance by Equation
15, where I0 is the initial intensity of light and I is the transmitted intensity.
A = -log(
𝐼
𝐼𝑜
) (15)
Absorbance uses a broad spectrum of visible light to raise the electrons to a range of higher
vibronic energy levels.14 Vibronic modes describe the simultaneous vibrational and electronic
transitions of a molecule.14 The broad spectrum allows for observation of multiple excited states.
An important aspect of absorbance is the population of the ground and excited states. The
population of these excited states is further described by the Boltzmann distribution in Equation
16
𝑁
𝑁𝑜
= 𝑒
−𝐸
𝑘 𝐵 𝑇 (16)
where N is the population of the excited state, No is the population of the ground state, E is the
energy (J), kB is the Boltzmann constant (J K-1), and T is temperature (K). The distribution states
that at higher temperatures, the populations of the ground and excited states become more equal.

13
The absorbance spectrum shows the vibrational level of the B-State. The B-state describes
the potential energy of the excited mode, which is a low-lying bound excited state.12 The other
state observed is the X-state, which describes the potential energy of the ground state.12 In this
experiment, we observed the B-state of molecular iodine through the use of its absorbance
spectrum. An example of an absorbance spectrum for iodine is provided in Figure 7, with the B-
state and X-states shown in Figure 8.
Figure 7. I2 Absorbance Spectrum at 40oC 12
Figure 8. B-state and X-state of I2
12

14
As indicated in Figure 7, the absorbance spectrum consists of cold and hot bands. A cold
band is a transition from the lowest vibrational level of the ground electronic state to a certain
vibrational level in the B-state.12 On the other hand, a hot band is a vibrational transition between
two excited states.12 By taking the absorption spectrum and plotting wavenumber vs. v’ + ½, we
determined the spectroscopic constants for the B-state from a fourth order polynomial fit. The
spectroscopic constants are Te – G”(0), ve, vex’e, vey’e, E*, and D’e.12 The constant Te – G”(0)
corresponds to the energy offset between the two potential wells, where T’e is the spacing between
the bottoms of the two potential wells, and G”(0) is the vibrational energy in the ground state.12 In
addition, ve represents the fundamental vibrational number of the B-State, and x’e and y’e are
anharmonicity constants.12 The other constants, E* and D’e, correspond to the energy it takes to
move molecular iodine from the lowest vibrational energy level of the X-state to the dissociation
limit of the B-state, and the well depth of the B-state, respectively.12
Emission is similar to absorbance except the molecule is subjected a single wavelength of
light, in this case 514.5 nm. This selected wavelength excites the molecule to a singular excited
state. From this state, the molecule then relaxes back to various vibronic levels in the X-state.12
These relaxations are measured and reveal the nature of the ground states. An example of an I2
emission spectrum is shown in Figure 9.

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Figure 9. I2 Absorbance Spectrum 12
The peaks of the emission spectrum are referred to as bandheads. The bandhead represents
the highest energy point in the spectrum reached by the R branch.13 For molecular iodine, the
bandhead and band origin are close together. The band origin can be assumed equal to the peak
bandhead because at room temperature the rotational levels of I2 are largely populated.12 This large
population means that the band maximum is at lower energy than the band origin.13 The numbers
above of the bandheads correspond to the vibrational level which the molecular iodine relaxes to
after it is excited. As shown in Figure 9, the larger numbers correspond to a higher wavelength,
which means that the vibrational energy levels at these numbers correspond to a higher energy.12
By taking the emission spectrum and plotting Δv vs. v” + ½, we obtained the spectroscopic
constants for the X-state. The value of Δv is obtained by subtracting the wavenumber
corresponding to the v” value from the wavenumber of the laser (19429.7694 cm-1).12 The
spectroscopic constants of interest for molecular iodine are G”(0), v”e, vex”e, vey”e, D”e, and
E(I*).12 As stated earlier, G”(0) corresponds to the vibrational energy in the ground state.12 Also,
similar to the B-state, the v”e, vex”e, and vey”e constants refer to the fundamental vibrational
number of the X-state, and the two anharmonicity constants of the X-state, respectively.12 Lastly,

16
the D”e and E(I*) constants correspond to the well depth of the X-state and the excitation energy
corresponding to the lowest 2P1/2 ← 2P3/2 atomic transition of iodine, respectively.12
We examined the emission and absorbance properties of I2 for its ground state and excited
state when exposed to an argon laser. When our I2 sample is exposed to the argon laser at a short
wavelength, the electrons will temporarily excite vibrationally and electronically before relaxing
back to the ground state. The emission in this experiment occured in the form of fluorescence, as
depicted in Figure 10.
Figure 10. Types of Emission 14
The emission levels for the spectra will be developed in relation to wavelength of the laser. The
expected peak of the absorption spectrum of iodine is at a peak of 500-600 cm-1. According to
Stokes Law, the emission peak is typically lower in intensity than the absorbance peak as the
general trend follows for a loss of vibrational energy when going from excited to ground state.14
This process is known as the Stokes Shift depicted in Figure 11.

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Figure 11. Excitation and Emission spectrum of a Common Fluorochrome 14
The Franck-Condon Principle is a means of describing the intensity of a vibronic transition
within a molecule. It states that when a molecule undergoes an electronic transition, there is not a
major change in its nuclear configuration due to the inability of the nucleus to react vibrationally
before the transition ends, due to the massive size of the nucleus compared to the electrons.15 The
Born-Oppenheimer approximation accounts for this inability and quantifies the vibrational and
rotational motion separately as shown in Equation 17, where each value of E corresponds to the
energy of the type of transition.1
E = Eelec + Evib + Erot (17)
The cause of the vibrational state of the nucleus is the Coulombic forces that arise after the
transition. Integrating the wavefunctions for the ground and excited states determines their
overlap. Squaring the overlap terms provides the Franck-Condon factor as shown in Equation 18.
FC = ∑ ∑ 𝑆 𝑣′
,𝑣
∞
𝑣=0
∞
𝑣′
=0
2
(18)

18
Although the summation terms range to infinity, the finite nature of overlap creates limitation
attributed to the finite number of absorption states of a molecule.17
Experimental
The first week of experimentation consisted of obtaining the rovibrational spectrum of
carbon monoxide (CO) using the FTIR spectrophotometer. Any time we were not using the gas
sample cell, we kept it in a desiccator because of its moisture sensitivity. Using the gas manifold,
we evacuated the gas sample cell using the vacuum, using the digital gauge to monitor the pressure
of any gases which could have been left inside it. We then placed the evacuated cell in the FTIR
spectrophotometer and collected a background spectrum. After collecting the background
spectrum, we then used the gas manifold to fill the gas sample cell with 100 mmHg of CO. We
collected CO spectra at resolutions of 4, 2, 1, 0.5, and 0.25 cm-1, collecting a new background after
each run. After collecting an adequate spectra showing the fundamental and first overtones, we
then shut down the program we used to obtain the spectra, and evacuated the gas sample cell,
storing it in the desiccator.
During the second week of experimentation we used the gas manifold to fill the gas sample
cell with N2O and CH4, using the FTIR spectrophotometer to determine their GWPs. In order to
prevent N2O from entering the vacuum pump, which could cause an explosion, we made a gas trap
to collect the N2O and any other contaminant gases. This process consisted of filling the trap dewar
with liquid nitrogen and placing it under the gas trap. After setting up the gas trap, we then checked
for leaks in the gas manifold, using the digital pressure gauge. After confirming the absence of
leaks, we then filled the gas sample cell with 60.1 mmHg N2O and took a spectrum of it using the
FTIR spectrophotometer, after taking the appropriate background. After collecting an adequate

19
N2O spectrum, we evacuated the gas sample cell, filled it with 60.0 mmHg CH4, and repeated the
spectrum collecting steps. After collecting both spectra, we concluded the procedure by evacuating
the gas sample cell and placing it back in the dessicator.
The third week’s procedure consisted of taking the absorption and emission spectra of
molecular iodine. After adjusting the sample holders in an appropriate manner for absorption, we
took a reference background, adjusting the integration time to an appropriate value. After taking
the reference background, we then collected a dark background, and then took an absorbance
spectrum of the molecular iodine sample using a halogen light source, adjusting the signal to noise
ratio by increasing the number of scans until we obtained an appropriate spectrum. We then set up
the detector perpendicular to the laser beam in order to prepare for the collection of the emission
spectrum. After properly aligning the laser, we then covered the sample with a dark cloth to prevent
fluorescence, and then we took a dark background of the sample. After taking the dark background,
we then placed a filter in front of the path of the laser to maximize the region of interest as well as
to minimize the laser signal. We then turned on the laser and collected the emission spectrum,
adjusting the signal to noise ratio by increasing the number of scans. After collecting an adequate
spectrum, we ended the experiment by closing the shutter, turning off the laser, and placing the
iodine samples back in their containers.
Results and Discussion
The first week of experimentation consisted of obtaining the rovibrational spectrum of CO,
from which we obtained spectroscopic constants. We used the FTIR to determine the fundamental
and first overtones of CO, as shown in Figures 12 and 13.

20
Figure 12. CO Fundamental Absorbance Spectra
2728
-28
26
-27
25
-26
2423
-25
-24
2221
-23
20
-22
1918
-21
-20
1716
-19
15
-18
1113
-17
14 12
-16
-15
-14
-13
8
-12
10
-11
9
-10
7
-9
-8
6
-7
5
-6
-5
4 3
-4
-3
2 1
-2
-1 1
0
2
1
3 4
2
5
43
6
5
7
6
8
7
9
8
10
9
11
10
12
11
13
12
14
13
15
14
16
17
1516
18
19
17
20
18
21
19
22
22
23
20
24
21
25
23
26
27
25
2426
28
27
29
28
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2000 2050 2100 2150 2200 2250
Absorbance(AU)
Wavelength (cm-1)
P-Branch R-BranchP-Branch R-Branch
J values

21
Figure 13. CO First Overtone Spectra
As shown in Figures 12 and 13, the fundamental and first overtones are located in the
literature value ranges, from 1950-2275 cm-1 for the fundamental and 4100-4400 cm-1 for the first
overtone. This data makes sense as the first overtone corresponds to the second excited vibrational
state of the molecule, which is at a higher energy than the fundamental. In addition, we located the
P and R branches on these spectra, which allowed us to find the m values (located above the peaks
in Figures 12 and 13) corresponding to each J value. The R branch corresponds to ΔJ = +1, and
therefore has positive m values starting at 0.18 On the other hand, the P branch corresponds to ΔJ
= -1, and therefore has negative m values.18 However, its m values cannot start at 0 because the
value of J’ (the excited rotational state) cannot be -1.18 We then plotted the wavelength versus the
m values for the fundamental and first overtones, obtaining cubic, quadratic, and linear fits, shown
in Figures 14 and 15.
24
-24
23
-23
22
-22
2120
-21
-20
19
-19
18
-18
1716
-17
1514
-16
12
-15
13
-14
-13
-12
11
-11
10
-10
9 8
-9
-8
7
-7
6 4
-6
5
-5
3
-4
2
-3
1
-2
-1
0
1
2
1
3
2
4
3
5 6
4 5
7
6
8
9
7
10
9
11
8 10
12
13
14
12
11
15
13
16
14
15
17
18
19
20
1618
21
17
19
22
23
2022
21
24
25
23
24
0.08
0.1
0.12
0.14
0.16
0.18
0.2
4125 4175 4225 4275 4325
Absorbance(AU)
Wavelength (cm-1)
P-Branch R-Branch
J-values

22
Figure 14. CO Fundamental Absorbance
Figure 15. CO First Overtone
y = -6E-05m3 - 0.0143m2 + 3.7663m + 2142.8
R² = 0.9997
y = -0.0143m2 + 3.7376m + 2142.8
R² = 0.9997
y = 3.7234m + 2138.7
R² = 0.9964
2000
2050
2100
2150
2200
2250
-30 -20 -10 0 10 20 30
Wavelength(cm-1)
m values
Cubic
Quadratic
Linear
y = 3.7673m + 4252.2
R² = 0.9862
y = -0.0349m2 + 3.8018m + 4259.6
R² = 1
y = -1E-05m3 - 0.0349m2 + 3.8058m + 4259.6
R² = 1
4100
4150
4200
4250
4300
4350
-30 -20 -10 0 10 20 30
Wavelength(cm-1)
m values
Linear
Quadratic
Cubic

23
As shown in Figures 14 and 15, the graphs both have R2 values close or equal 1, which
shows the reliability of the obtained functions. We used the values obtained from the cubic
functions, and Equation 19 (shown below) to determine the spectroscopic and molecular constants,
shown in Tables 1-3. The calculations for these constants are provided in the appendix section.
ν(m) = νo + (2Be - 2αe)m – αem2 – 4Dem3 (19)
Table 1. Fundamental and First Overtone Wavenumbers
Fundamental (cm-1) First Overtone (cm-1)
2142.9 4259.6
Table 2. Rovibrational Spectra Constants
Equilibrium
Frequency
(cm-1)
αe (cm-1) Be (cm-1) De (cm-1) χe (cm-1)
Experimental
Value
2168.8 0.0143 1.90 1.5 x 10-5 0.00599
Literature
Value19 2169.8 0.0175 1.9313 6.2 x 10-6 0.00612
Percent error 0.0461% 18.3% 1.62% 142% 2.12%
Table 3. Molecular Constants
Moment of Inertia
(kg m2)
Equilibrium bond
(Å)
Force Constant
(N/m)
Experimental Value 1.47 x 10-46 1.14 1903
Literature Value19 1.4490 x 10-46 1.1281 1902
Percent error 1.45% 1.05% 0.0526%
As shown in Table 2, the experimental values are close to the literature values for the larger
spectroscopic constants, such as the equilibrium frequency. On the other hand, for smaller
constants, such as De and αe, the percent errors are large. This large percent error means that these
constants are not as well defined as the larger values. In addition, the larger values contribute more

24
to the rovibrational frequency than the smaller values. For instance, as shown in Table 2, the
rotational constants Be and αe contribute more to the rovibrational transitions than the
anharmonicity contant χe. A smaller anharmonicity constant means that the molecule acts more like
the ideal harmonic oscillator. In addition, the centrifugal distortion constant, De, is roughly 1000
times smaller than the anharmonicity constant. This previous statement means that the centrifugal
distortion caused by rotation contributes the least amount to the rovibrational states. The
experimental molecular constants shown in Table 3 are all close to those found in the literature.
These close values show that FTIR is an effective method of determining spectroscopic and
molecular constants.
To calculate the global warming potentials (GWPs) during this experiment, we took the
absorbance spectrum of CH4 and N2O,which provided the frequencies where CH4 and N2O absorb
Earth’s blackbody radiation. The absorbance spectra obtained in the lab as well as where CH4 and
N2O absorb Earth’s blackbody radiation are provided in Figures 16-19.
-0.05
0.15
0.35
0.55
0.75
0.95
495.00 695.00 895.00 1095.00 1295.00 1495.00
Absorbance(AU)
Wavenumber (cm-1)

25
Figure 16. CH4 Absorbance Spectrum
Figure 17. Absorbance Spectrum of Methane and Blackbody Radiation of Earth
Figure 18. N2O Absorbance Spectrum
-1
0
1
2
3
4
500 600 700 800 900 1000 1100 1200 1300 1400 1500
s*1018orf
Frequency (cm-1)
Blackbody Radiation of Earth
GHG Spectrum
-0.10
0.10
0.30
0.50
0.70
0.90
495.00 695.00 895.00 1095.00 1295.00 1495.00
Absorbance(AU)
Wavenumber (cm-1)

26
Figure 19. Absorbance Spectrum of N2O and Blackbody Radiation of Earth
As shown in Figure 16, CH4 has a peak near 1300 cm-1, which means it absorbs at that
wavenumber value. This data correlates with Figure 17 as the CH4 peak has overlap with the
blackbody radiation of earth near 1300 cm-1. In addition, as shown in Figure 18, N2O has
absorbance peaks near 600 cm-1, 1200 cm-1, and 1300 cm-1. However, as shown in Figure 19, N2O
only has overlap with the blackbody radiation of earth near 1300 cm-1. The lack of peaks in Figure
19 is due to the pressure of N2O used in the experiment. At higher pressures, the GHG and
blackbody spectra will overlap.
We used the data collected from the absorbance spectra to calculate the GWPs of CH4 and
N2O. GWP is a measurement of the ability for a gas to trap heat in the atmosphere. The calculated
GWP values are provided in Table 4.
Table 4. Greenhouse Gas (GHG) GWP Values
GHG
Lifetime
(Years)
Time Horizon
(Years)
Calculated GWP Literature GWP
Percent
Difference (%)
N2O 120
20 73.3 93 21.1
100 69.3 88 21.3
-1
0
1
2
3
4
500 600 700 800 900 1000 1100 1200 1300 1400 1500
s*1018orf
Frequency (cm-1)
Blackbody Radiation of Earth
GHG Spectrum

27
500 60.9 77 20.9
CH4 15
20 33.3 37 10.0
100 11.6 13 10.9
500 5.9 6 2.46
Looking at Table 4, the GWPs for N2O are larger than those of CH4, which means that N2O
traps more heat in the atmosphere and therefore is a more effective GHG than CH4. This data
correlates well with the literature values, as the literature GWPs for N2O are greater than those of
CH4. Nitrous oxide has larger GWPs because it has a larger atmospheric lifetime than CH4, and
therefore decays less rapidly than CH4. However, as shown in Table 4, the calculated GWPs are
all less than the literature values, even though the pressures of the gases were 60.0 Torr for CH4
and 60.1 Torr for N2O, which correspond to the linear range of Beer’s Law. Smaller GWPs mean
that a GHG traps less heat in the atmosphere than gases with larger GWPs. These smaller GWPs
for N2O and CH4 portray that they are not as effective as holding in heat as the literature states,
which cannot be trusted. The smaller calculated GWPs could be due to some possible reasons. One
reason for this difference could be difficulties retaining a vacuum during the filling of the IR cell
as well as possible contamination. However, the effects of contamination were minimal due to the
precautions of the experimental set-up. In addition, we could have found more accurate data by
using smaller cm-1 intervals instead of using 10 cm-1 intervals, as the literature values were taken
at 2 cm-1 intervals.9 Also, the windows of the IR cell were different than that used in the literature,
which would affect the transmission limits. In addition, the literature does not state exactly what
pressures were used during experimentation, which leads to uncertainty in its values. Variances in
pressures also have the ability to change the GWP by a significant amount. For example when
using the GWP model, pressure values of 65 Torr and 60.1 Torr for N2O differ in values by 10%.
However, the literature also states that a major source of uncertainty in the GWP is the

28
determination of the atmospheric lifetime of the GHG.9 Elrod et. al states that GHGs with longer
lifetimes are more accurately modeled by the Pinnock et. al model because the gases are more well
mixed globally.9 However, for our data the gas with the larger atmospheric lifetime, N2O, was less
accurately represented by this model. The biggest source of error is that the model is an exponential
decay.9 In reality CO2 follows three different rates of decay, which means that this simplified
model does not accurately portray the GWP for both CH4 and N2O as well. Using a model which
accurately portrays the decay of the molecules would provide more accurate results. In essence,
while the GWP model shows N2O to be the more effective GHG than CH4, which agrees with the
literature, the effectiveness of N2O and CH4 as GHGs are underestimated, as their GWPs are less
than the literature values.
During the third week of experimentation we determined the B-State and X-State constants
for molecular iodine using absorption and emission, respectively. Figures 20 and 21 provide the
absorbance spectrum for molecular iodine and the bandhead energy versus v’ + ½, respectively,
with Table 5 providing the B-State spectroscopic constants.
Figure 20. I2 Absorbance Spectrum

29
Figure 21. I2 Bandhead Energy Versus v’ + ½.
Table 5. B-State Spectroscopic Constants
Spectroscopic
Constants
Experimental Values
(cm-1)
Literature Values12
(cm-1)
Percent Error (%)
T’e – G”(0) 15690 15661.99 0.501
v’e 119.23 125.67 5.12
vex’e 0.4381 0.7504 41.6
vey’e -0.0076 -0.00414 83.6
E* 19658.58 20043.2 1.92
De 3968.578 4381.2 9.42
The bandheads in Figure 20 show the vibronic transitions from the ground state to varying
excited states. These bandheads are where the unresolved vibrational-electronic lines are the
strongest.15 We then plotted the bandhead energy versus v’ + ½ (Figure 21) which provided us
with a cubic fit, which we then used to calculate the spectroscopic constants for the B-State, shown
y = -0.0076x3 - 0.4381x2 + 119.23x + 15690
R² = 0.9999
16500
17000
17500
18000
18500
19000
19500
20000
0 20 40 60 80 100
Wavenumbers(cm-1)
v' + 1/2

30
in Table 5. Looking at Table 5, the spectroscopic constants are relatively close to the literature
values, besides the anharmonicity constants, vex’e and vey’e. The error in this part of the data
analysis is due to the extrapolation of the cubic fit in order to find the maximum of the function,
which corresponds to De. In addition, even though the R2 is practically equal to 1, there is always
estimation error associated with extrapolation, which can lead to erroneous results, particularly in
the case of the anharmonicity constants.12
In addition to finding the B-State spectroscopic constants, we found the X-State
spectroscopic constants by taking the emission spectrum of molecular iodine. Figures 22 and 23
show aspects of the emission spectrum of molecular iodine, whereas Figure 24 shows the bandhead
energy versus v” + ½ and Table 6 provides the spectroscopic constants of the X-State.
Figure 22. I2 Emission Spectrum
1
2
3 4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25 26
27
28
29
30
31
32
34
36
0
5000
10000
15000
20000
25000
500 550 600 650 700 750 800
Intensity
Wavelength (nm)

31
Figure 23. I2 Emission Spectrum Doublet at v” = 0 Used to Find B”
Figure 24. I2 Bandhead Energy vs v” + ½
Table 6. X-State Spectroscopic Constants
508.54 nm 509.14 nm
-20
0
20
40
60
80
100
120
140
508 508.5 509 509.5 510
Intensity
Wavelength (nm)
y = -0.0012x4 + 0.0754x3 - 2.1404x2 + 224.93x - 138.15
R² = 1
0
1000
2000
3000
4000
5000
6000
7000
8000
0 10 20 30 40 50 60 70 80
Wavenumber(cm-1)
v" + 1/2

32
Spectroscopic
constants
Experimental values
(cm-1)
Literature Values12
(cm-1)
Percent Error (%)
G”(0) 138.15 107.11 29.0
v”e 224.93 214.53 4.85
vex”e 2.14 0.6130 249
vey”e -0.5177 0.0754 73303.88
D”o 11731.229 12440.2 5.70
E(I*) 7927.4 7602.98 4.27
As shown in Figures 22 and 23, the bandheads provide the vibronic transitions from the
excited state (v’ = 43) to varying ground states, where larger values of v” correspond to higher
energy vibronic levels. We plotted the bandhead energies versus v” + ½ to find the spectroscopic
constants of the X-State, as shown in Table 6. Like the B-State spectroscopic constants, the X-
State spectroscopic constants are close to the literature values, except for the anharmonicity
constants. Extrapolation of the fourth order fit is the cause of this error, even though the R2 value
is equal to 1. Looking at Figure 22, the largest Franck-Condon factor is at v” = 5 because it has the
greatest intensity, and therefore corresponds to the greatest vibrational-electronic overlap.15
We then used these spectroscopic constants to find the Morse Potentials for both the X-
State and B-State. In order to generate these potentials from the spectroscopic constants, we used
Equations 20 and 21
𝐸 = 𝑇𝑒 + 𝐷 𝑒(1− 𝑒(𝛽(𝑅−𝑅 𝑒)
)2
(20)
𝛽 = 𝜈𝑒 𝜋√
2𝜇𝑐
ℎ𝐷 𝑒
(21)
where R is bond length (Å), Re is equilibrium bond length (Å), μ is the reduced mass of molecular
iodine (g), νe is the equilibrium frequency (cm-1), h is Planck’s constant (J s), and c is the speed of
light (m/s). After resolving the bandheads corresponding to the v” = 0 transition (Figure 23), we
calculated the value of Re using Equation 6. We plotted these equations for both the X-State and

33
B-State, which provided the Morse Potentials, shown in Figure 25, with the spectroscopic
constants for the Morse Potentials provided in Tables 7 and 8.
Figure 25. X-State and B-State Morse Potential Curves
Table 7. Morse Potential Spectroscopic Constants For X-State
Spectroscopic constants Experimental Values Literature Values12 Percent Error
(%)
D”
e Used Lit. Value 12547.3 cm-1 n/a
R”e 151 pm 266.64 pm 43.4
Te 15828.15 cm-1 15769.1 cm-1 0.374
v”e 224.93 cm-1 214.53 cm-1 4.85
Table 8. Morse Potential Spectroscopic Constants For B-State
Spectroscopic constants Experimental Values Literature Values12 Percent Error
(%)
D’
e 3968.578 cm-1 4381.2 cm-1 9.42
R’e Used Lit. Value 3.0267 Å n/a
Te 15828.15 cm-1 15769.1 cm-1 0.374
v’e 84.603 cm-1 125.67 cm-1 32.7
0
5000
10000
15000
20000
25000
30000
35000
40000
0 1 2 3 4 5 6
Energy(cm-1)
Bond Length (Å)
X-State
B-State
T'e
E(I*)
D"e
E*
D'e
D"o

34
In determining the Morse Potentials, we used the literature value of D”e because
extrapolation of our curve in Figure 24 provided a negative value. The Morse Potentials in Figure
25 cannot entirely be trusted even though they follow the theoretical shape of Morse Potentials, as
the value for R”e is smaller than the literature value. However, this value is still smaller than R’e,
which follows the theory, because bond lengths at higher vibrational levels will increase due to the
higher energy stretching the length of the bond. Other than the small value of R”e, the other
constants are relatively close to the literature values. In comparison with Figure 8, the equilibrium
bond length of the X-State should be larger. Our data shows smaller overlap between the two
potential wells. According to the Franck-Condon principle, the only vibronic transitions occur
within the overlap swathe which contains the potential wells of the X-State and B-State. A larger
swathe allows for more transitions to occur. Therefore, our data shows less transitions than are
actually possible. After plotting these Morse Potentials, we then plotted the Morse Potentials using
the program FCIntensity, by using our spectroscopic constants. Figures 26 and 27 show the Morse
Potentials and Franck-Condon factor intensities, respectively, for the X-State and B-State.

35
Figure 26. Morse Potential From FCIntensity Program
Figure 27. FC Intensities From FCIntensity Program
Looking at Figure 26, the Morse Potentials are not accurate as the equilibrium bond length
for the X-State is larger than that of the B-State, because higher vibrational energies stretch the
equilibrium bond length. In addition, the FC Intensities graph (Figure 27) from the FCIntensity
program show that the largest Franck-Condon factor is located near 800 nm. However, we
determined the largest Franck-Condon factor from the emission spectrum at v” = 5, corresponding
to a wavelength of 544 nm. The results from our calculations are more reliable than those of the
FCIntensity program because the equilibrium bond length of the X-State is less than that of the B-
State.
Conclusion
The results from the CO experiment show the accuracy of the FTIR, as the experimental
values are close to the literature values. The data shows the Be and αe constants contribute most to
the rovibrational frequency, with the smaller constants such as χe contributing the least. In addition

36
there is also an increase in percent error for the smaller constants because the smaller constants are
not as well defined as the larger ones. In addition, the findings show that CO acts more like a
harmonic oscillator as its anharmonicity constant is small compared to the other constants.
The GHG data shows N2O to be a more effective greenhouse gas than CH4, because N2O
has a larger GWP than CH4. A larger GWP means that a greenhouse gas is more efficient at
trapping heat within the atmosphere. While all the experimental GWP values are less than the
literature values, the results show the expected result that N2O is a more efficient greenhouse gas
than CH4, which is due to N2O’s large time horizon in the atmosphere. The deviations from
literature values are mainly due to the inaccuracy of the Pinnock et. al model, which uses a simple
exponential decay model. In reality, molecules can have different decay models, such as CO2
which uses three different rates of decay.9
The results from the third week provide insight into the different vibronic levels of the X-
State and B-State, as well as provide a model of the Morse Potentials for each state. The constants
obtained from the absorption and emission data are all relatively close to the literature values,
besides the anharmonicity constants, due to error stemming from extrapolation. However, the most
important aspect of this data analysis is the Morse Potentials. The experimental Morse Potentials
show the correct anharmonic oscillator curves, shown in Figure 8. However, the experimental
Morse Potentials show the R”e to be less than the literature value, which ultimately leads to less
overlap swathe between the X-State and B-State. This smaller swathe leads to a smaller Franck-
Condon factor than expected from the theory.

37
References:
1. Molecular Spectroscopy. Seton Hall University Chemistry Department.
https://hplc.chem.shu.edu/NEW/Undergrad/Molec_Spectr/molec.spectr.general.html (accessed
Mar 8, 2015).
2. MIT. (n.d.). Principles of Molecular Spectroscopy. Retrieved March 23, 2015, from
http://web.mit.edu/ 5.33/www/lec/spec4.pdf.
3. McQuarrie, Donald A.; Simon, John D. Physical Chemistry: a Molecular Approach;
University Science Books: United States of America, 1997; 495-506.
4. MSU. (n.d.). The Nature of Vibrational Spectroscopy. Retrieved March 27, 2015, from
https://www2.chemistry.msu.edu/faculty/reusch/virttxtjml/Spectrpy/InfraRed/irspec1.htm.
5. UC Davis. (n.d.). Lennard-Jones Potential. Retrieved March 27, 2015, from
http://chemwiki.ucdavis.edu/Physical_Chemistry/Intermolecular_Forces/Lennard-
Jones_Potential.
6. University of Liverpool. (n.d.). Vibrational Spectroscopy. Retrieved March 13, 2015, from
http://osxs.ch.liv.ac.uk/java/spectrovibcd1-CE-final.html.
7. Tipler, Paul A. and Llewellyn, Ralph A., Modern Physics, 3rd Ed., W.H. Freeman, 1999.
8. University of California at Davis Chemistry Department. FTIR Block Diagram [Image].
9. Elrod, M. J. J. Chem. Ed. 1999, 76, 1702-05.
10. Georgia State University. Center frequency spectrum of HCl [Image].
11. Pinnock, S.; Hurley, M. D.; Shine, K. P.; Wallington, T. J.; Smyth, T. J. J. Geophys. Res.
Atmos. 1995, 100, 23227–23238.
12. Williamson, J. C. (2007). Teaching the Rovibronic Spectroscopy of Molecular Iodine.
Journal of Chemical Education, 84(8), 1355-1359.

38
13. University of Colorado. (n.d.). Band Spectra and Dissociation Energies. Retrieved March 27,
2015, from http://chem.colorado.edu/chem4581_91/images/stories/BS.pdf.
14. Microscopy Resource Center. Jablonski Energy Diagram; Excitation and Emission Spectrum
[Image].
15. Franck-Condon Principle. University of California at Davis Chemistry Department.
http://chemwiki.ucdavis.edu/Physical_Chemistry/Spectroscopy/Electronic_Spectroscopy/Franck-
Condon_Principle (accessed Mar 8, 2015)
16. Properties of Iodine Molecules. University of Hannover.
https://hplc.chem.shu.edu/NEW/Undergrad/Molec_Spectr/molec.spectr.general.html (accessed
Mar 8, 2015).
17. Dunbrack, R.L. J. Chem Ed. 1986, 63, 953-55.
18. Konstanz. (n.d.). Electronic Spectroscopy of Molecules I - Absorption Spectroscopy.
Retrieved March 27, 2015, from http://www.uni-konstanz.de/FuF/Bio/folding/3-Electronic
%20Spectroscopy%20r.pdf.
19. Mina-Camilde, N., & Manzanares, C. (1996). Molecular Constants of Carbon Monoxide at v
= 0, 1, 2, and 3. Journal of Physical Chemistry, 73(8), 804-807.

39
Appendix: Sample Calculations
1. Moment of inertia
𝐵 𝑒 =
ℎ
8𝜋2 𝑐𝐼
𝐼 =
ℎ
8𝜋2 𝑐𝐵 𝑒
𝐼 =
6.62606957𝑥10−34
𝐽𝑠
(8𝜋229979245800
𝑐𝑚
𝑠
)1.90𝑐𝑚−1
𝐼 = 1.47𝑥10−46
𝑘𝑔 𝑚2
2. Equilibrium Bond Length
𝐼 = 𝜇 𝑟2
𝑟 = √
𝐼
𝜇
𝑟 =
√
(1.47𝑥10−46 𝑘𝑔 𝑚2)(1000
𝑔
𝑘𝑔
)(6.02214129𝑥1023 𝑚𝑜𝑙−1)
(12.0107
𝑔
𝑚𝑜𝑙
)(15.9994
𝑔
𝑚𝑜𝑙
)
12.0107
𝑔
𝑚𝑜𝑙
+ 15.9994
𝑔
𝑚𝑜𝑙
𝑟 = 1.14 𝐴̇

40
3. Vibrational Force Constant
𝑣𝑒 =
1
2𝜋𝑐
√
𝑘
𝜇
𝑘 = 𝜇(2𝜋𝑐𝑣𝑒)2
𝜇 =
(12.0107
𝑔
𝑚𝑜𝑙
)(15.9994
𝑔
𝑚𝑜𝑙
)
12.0107
𝑔
𝑚𝑜𝑙
+ 15.9994
𝑔
𝑚𝑜𝑙
= 6.86052081
𝑔
𝑚𝑜𝑙
𝑘 = (𝜇)(
𝑚𝑜𝑙
6.022𝑥1023
)(2∗ 2168.8𝑐𝑚−1
∗ 𝜋 ∗ 29979245800
𝑐𝑚
𝑠
)2
𝑘 = 1903
𝑁
𝑚
NOTE: Values were taken from cubic fits from the fundamental and first overtone graphs.
4. Calculating αe
𝛼 𝑒 𝑚2
= (0.0143𝑐𝑚−1
)𝑚2
𝛼 𝑒 = 0.0143 𝑐𝑚−1
5. Calculating Be
(2𝐵 𝑒 − 2𝛼 𝑒) 𝑚 = (3.7663𝑐𝑚−1
)𝑚
𝛼 𝑒 = 0.0143 𝑐𝑚−1
𝐵 𝑒 =
3.7663𝑐𝑚−1
+ 2(0.0143 𝑐𝑚−1)
2
𝐵 𝑒 = 1.90 𝑐𝑚−1

41
6. Calculating De
4𝐷 𝑒 𝑚3
= (6 𝑥 10−5
𝑐𝑚−1
)𝑚3
𝐷 𝑒 = 1.5 𝑥 10−5
𝑐𝑚−1
7. Calculating equilibrium frequency νe
𝜐𝑜 = 𝜐𝑒 − 2𝜐𝑒 𝜒 𝑒
𝜒 𝑒 =
𝜐𝑒 − 𝜐𝑜
2𝜐𝑒
𝜐𝑜 𝑜𝑣𝑒𝑟𝑡𝑜𝑛𝑒 = 2𝜐𝑒 − 6𝜐𝑒
2𝜐𝑒
𝜐𝑜 𝑜𝑣𝑒𝑟𝑡𝑜𝑛𝑒 = 4259.6 𝑐𝑚−1
𝜐𝑜 = 2142.8 𝑐𝑚−1
𝜐𝑒 = 2168.8 𝑐𝑚−1
8. Calculating 𝜒 𝑒
𝜒 𝑒 =
2𝜐𝑒
𝜐𝑒 = 2168.8 𝑐𝑚−1
𝜐𝑜 = 2142.8 𝑐𝑚−1
𝜒 𝑒 = 0.00599 𝑐𝑚−1
9. Percent error of moment of inertia
%𝑒𝑟𝑟𝑜𝑟 =
𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑣𝑎𝑙𝑢𝑒(𝑎𝑐𝑐𝑒𝑝𝑡𝑒𝑑 − 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙)
𝑎𝑐𝑐𝑒𝑝𝑡𝑒𝑑
𝑥 100%
% 𝑒𝑟𝑟𝑜𝑟 =
𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑣𝑎𝑙𝑢𝑒( 1.47 𝑥 10−46
𝑘𝑔 𝑚2
− 1.449 𝑥 10−46
𝑘𝑔 𝑚2)
1.449 𝑥 10−46 𝑘𝑔 𝑚2
𝑥 100%
% 𝑒𝑟𝑟𝑜𝑟 = 1.45%

42
10. Calculating D”o
𝐷"
𝑜 = 19429.7694 𝑐𝑚−1
− 𝑝𝑒𝑎𝑘 𝑒𝑛𝑒𝑟𝑔𝑦
𝑝𝑒𝑎𝑘 𝑒𝑛𝑒𝑟𝑔𝑦 = 7698.54 𝑐𝑚−1
𝐷"
𝑜 = 11731.2 𝑐𝑚−1
11. Calculating E(I*)
𝐸( 𝐼∗) = 𝐸∗
− 𝐷"
𝑜
𝐸∗
= 19658.58 𝑐𝑚−1 ( 𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑑 𝑓𝑟𝑜𝑚 𝑒𝑥𝑡𝑟𝑎𝑝𝑜𝑙𝑎𝑡𝑖𝑜𝑛 𝑡𝑜 𝑚𝑎𝑥𝑖𝑚𝑢𝑚)
𝐸( 𝐼∗) = 19658.58 𝑐𝑚−1
− 11731.2 𝑐𝑚−1
𝐸( 𝐼∗) = 7927.4 𝑐𝑚−1
12. Calculating spectroscopic constant R”e
𝑅"
𝑒 = √
ℎ
8 𝜋2 𝑐 𝜇 𝐵"
𝐵"
=
(19664 .13655 −19640.96391 ) 𝑐𝑚−1
398
= 0.05822451 cm-1
𝑅"
𝑒
= √
6.626 𝑥 10−34 𝑘𝑔 𝑚2
𝑠2 𝑠
8 𝜋2 (2.99 𝑥 1010 𝑐𝑚
𝑠
) (126.9045
𝑔
𝑚𝑜𝑙
𝑥
𝑚𝑜𝑙
6.022 𝑥 1023 𝑥
𝑘𝑔
1000 𝑔
)0.05822451 𝑐𝑚−1`
𝑅"
𝑒 = 1.51 𝑥 10−10
𝑚 = 1.51 𝑎𝑛𝑔𝑠𝑡𝑟𝑜𝑚𝑠
13. Calculating β (for X-State)
𝛽 = 𝜈𝑒 𝜋√
2𝜇𝑐
ℎ𝐷 𝑒

43
𝛽 = 224.93 𝑐𝑚−1
𝜋√
2 (126.9045
𝑔
𝑚𝑜𝑙
)(3 𝑥 108 𝑚
𝑠
)
𝑘𝑔
1000 𝑔
𝑚𝑜𝑙
6.022 𝑥 1023
(6.626 𝑥 10−34 𝑘𝑔 𝑚2
𝑠2 𝑠)(12547.3
1
𝑐𝑚
)(
100 𝑐𝑚
𝑚
)
𝛽 = 1.94 𝑥 107
𝑐𝑚−1
14. Calculating Te
𝑇𝑒 − 𝐺"(0) = 15690 𝑐𝑚−1
𝐸∗
= 19658.58 𝑐𝑚−1 ( 𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑑 𝑓𝑟𝑜𝑚 𝑒𝑥𝑡𝑟𝑎𝑝𝑜𝑙𝑎𝑡𝑖𝑜𝑛 𝑡𝑜 𝑚𝑎𝑥𝑖𝑚𝑢𝑚)
𝐺"(0) = 138.15 𝑐𝑚−1( 𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑑 𝑓𝑟𝑜𝑚 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛 𝑑𝑎𝑡𝑎)
𝐷 𝑒 = 39678.578 𝑐𝑚−1
(𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑑 𝑓𝑟𝑜𝑚 𝐸∗
− [𝑇𝑒 − 𝐺"(0) ])
𝑇𝑒 = −(𝐷 𝑒 − 𝐺"(0) − 𝐸∗
)
𝑇𝑒 = 15828.15 𝑐𝑚−1

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