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 NO 2 and CH 4 , 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 NO 2 and CH 4 indicate that NO 2 is a more effective greenhouse gas than CH 4 , as expected from theory, due to NO 2 ’s time horizon in the atmosphere. Lastly, the results from the 1
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Transcript
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.
1
Introduction
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=1M ∑
i
mi 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
2
v=0, v=1, v=2, etc. for vibrational motion, J = 0, J = 1, J = 2, etc. for rotational motion, and S 0,
S1, S2 etc. for electronic motion as shown in Figure 1.
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∨∫ψ final¿ µψ intial dr ¿2 (2)
where µ=∑i
qi 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
3
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
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
2 πc √ kμ
❑
(5 )
B= h
8 π2cI(6)
I=μ Re2(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
4
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
Figure 2. Lennard-Jones Potential Diagram 5
The negative derivative of potential energy is force, as shown in Equation 21.
−dUdr
=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
5
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=m1m2
m1+m2 (22)
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.
Bv=Be−αe (v+ 12 )(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 )=Bv J (J +1 )−D e J 2 ( 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.
6
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.
7
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+ 12 ) ve+xe ve (v+
12)
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
8
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 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.
9
We will determine the band center frequency (v0) to calculate rovibrational characteristics
like the anharmonicity constant ( χe ¿ and equilibrium frequency ¿) by plotting m vs wavelength
of the fundamental and overtone spectra as well as using Equation 11
ν0=ν e−2 νe χe (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√ kµ
(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.
10
Figure 6. Center Frequency Spectrum of HCl 10
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.
11
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∗∫0
TH
A (t ) dt
RFR∗∫0
TH
R (t ) dt
(13)
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∗( 1000MW A )∗∫
0
TH
e−t / τ A
dt
RFR∗( 1000MW R )∗∫
0
TH
e−t / τR
dt
(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.
12
A = -log(II o
)
(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
NN o
=e−EkB 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.
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.
13
Figure 7. I2 Absorbance Spectrum at 40oC 12
Figure 8. B-state and X-state of I212
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
14
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.
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
15
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, 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.
16
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.
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,
17
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 = ∑v '=0
∞
∑v=0
∞
Sv' , v
2 (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
18
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 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
19
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.
2000 2050 2100 2150 2200 22500
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
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
2627
252426
28
27
2928
Wavelength (cm-1)
Abso
rban
ce (A
U)
P-Branch R-BranchP-Branch R-Branch
J values
20
Figure 12. CO Fundamental Absorbance Spectra
4125 4175 4225 4275 43250.08
0.1
0.12
0.14
0.16
0.18
0.2
24-24
23
-23
22
-222120
-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
89
7
10
9
11
810
1213
14
1211
15
13
16
1415
1718
1920
1618
21
1719
2223
202221
24252324
Wavelength (cm-1)
Abso
rban
ce (A
U)
P-Branch R-Branch
J-values
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.
R} e = sqrt {{6.626 x {10} ^ {-34} {kg {m} ^ {2}} over {{s} ^ {2}} s} over {8 {π} ^ {2} left (2.99 x {10} ^ {10} {cm} over {s} right ) left (126.9045 {g} over {mol} x {mol} over {6.022 x {10} ^ {23}} x {kg} over {1000 g} right ) 0.05822451 {cm} ^ {-1`}}} ¿