Rotational and Vibrational Spectroscopy
Introduction:
You are familiar with spectroscopy when you look at a rainbow because you are seeing a continuous spectrum of visible light. The spectrum contains discrete wavelengths that correspond to discrete colors. However, spectroscopy extends into longer and shorter wavelengths that are not visible to your eye. Examples of wavelengths that are not visible include radio waves, which are significantly larger than visible wavelengths. The other side of the spectrum includes shorter wavelengths which correspond to energy within atoms. This energy is described through three subcategories of spectroscopy: emission, scattering, and absorption. Emission spectroscopy uses the intensity and wavelength of emitted light from a sample to find the identity of elements and their quantity within the sample. Scattering (Raman) spectroscopy provides information regarding the energy transitions of the molecules in the sample. Absorption spectroscopy measures the energy that a system takes to reach the energy required to eject an electron (ionization energy).
Figure 1: Spectrum of Wavelengths

The following equation shows the relationship between wavelengths and frequency, where c is the speed of light (2.99792458 x 10^{8} m/s)
c=λv
The above equation shows that wavelengths (λ), measured in meters, and frequency (v), measured in s^{1}, are inversely related. Therefore we can infer that with longer wavelengths, we will have a relatively lower frequency. Frequency is directly proportional to energy; this is seen in the following equation.
E=hv
Where E (J) stands for energy, h is Planck’s constant (6.626069 x 10^{34} Js). Therefore, we can infer that when frequency increases so does energy. In addition, it is evident that when wavelengths increase, energy will decrease.
Background Information:
Rotational: Rotational energies are described in terms of molecular moments of inertia, where the molecule rotates around a center of gravity, usually the nuclei. Quantum mechanics denotes that rotational energy is quantized, so only certain energy levels are permitted. In addition to the energy of rotational molecules being quantized, the angular momentum of the molecule is also fixed to certain values. Consequently, the rotational absorption spectrum depicts a series of equallyspaced lines. The information gained from rotational spectroscopy, which is measured in microwave radiation, is used to deduce the molecular geometry of the compounds in the sample. The rigid rotor model provides an explanation about rotating systems and how the energy is quantized. For more information regarding the rigid motor model, refer to The Hamiltonian.
Figure 2: Rotation of molecule along x,y,z axes 
Vibrational: Vibrational spectroscopy analyzes the motion of diatomic molecules that oscillate along the bond length, using the harmonic oscillator and the Morse potential energy functions. This oscillation can be likened to a spring connecting the two molecules, with the frequency of the oscillation depending upon the strength of the bond between the molecules. The reason for this vibration is that atoms that are close to one another tend to attract one another resulting in a decrease in the distance between atoms; however, when atoms are too close, the repulsion force between the electrons pushes the atoms away from one another. This produces the vibration as the atoms oscillate between these two forces. [input picture to the left] the following pictures are only for polyatomic molecules (3 or more atoms)
Figure 3: Different vibrational possibilities for polyatomic compounds

Harmonic/Anharmonic: Harmonic motion is motion in which a moving particle/wave oscillates at a constant frequency within a fixed amount of space. In a harmonic system, the amount of energy needed for a transition approaches infinity as the distance from the nucleus increases. A harmonic system follows a parabolic motion because the energy is conserved. At the maximum of the curve, the potential energy (V(x)) is greatest, but at the minimum, the potential energy is 0 (kinetic energy (KE(x)) is at its maximum). The following equation shows this relationship:
V(x) = 1/2kx^{2}
In the above equation, V(x) is the potential energy, k is Hooke's constant which depends on the system, and x is the distance (m). In contrast, Anharmonic motion refers to a particle that does not osciallate at a constant frequency, although being subject fixed amount of space. In the case of anharmonic motion, the harmonic potential is no longer the only factor that describes the potential energy of the system. For example, there could be an additional constant or variable that is added to the original potential equation V(x) given above. Therefore, the potential energy curve does not follow a parabolic curve. The following potential energy curve shows the curve for anharmonic motion:
Graph 1: Anharmonic Potential Energy Curve

The following equation shows the relationship between the potential energy of potential energy:
The importance of this above equation is not based upon understanding the variables but to realize the difference that exists between the anharmonic and harmonic equations. In the case of harmonic systems, there is a selection rule in place that states transitions can only be made between adjacent energy levels, such as between energy levels 4 and 5. In anharmonic systems, however, this rule is relaxed, and anharmonic systems can transition between non adjacent energy levels, such as energy levels 2 and 4. This rule is relaxed due to the interactions with the surrounding atoms that are normally absent from a harmonic system. The major importance of anharmonic motion is that it reflects real life systems, unlike harmonic oscillations. As a real life example, Anharmonic motion contributes to why water can appear blue. Water is able absorb enough energy to transfer from the infrared to visible light by absorbing the wavelengths that correspond to red light. In turn, blue light is reflected.
Video 1: Visual representation of harmonic motion

Since the systems can be modeled as wavefunctions, we can find the state of the system by solving the Schrodinger equation. In order to do so, we make use of one of two Hamiltonian operators, which are listed below.
Harmonic Oscillator Hamiltonian
Rigid Rotor Hamiltonian
For more information regarding the use of the Hamiltonian, click The Hamiltonian.
Vibrational vs. Rotational:
Vibrational

Rotational

Vibration & Rotational

Measures infrared (IR)

Measures microwaves

Molecular properties can be analyzed with a spectrum

Often used to identify molecules on a surface and bonding orientation

Used to deduce molecular geometry


Can be used with molecules in the solid or liquid state.

Only can be used in the gas phase

Can be used with molecules in the gas phase

Vibrational systems are described using Morse or harmonic potentials

Rotational systems are described by the rigid rotor model.

Both described by the Schrodinger Equation

Table 1: Comparison between Rotational and Vibrational Spectroscopy

Further Analysis:
How does a microwave work?
Microwaves emit frequency at around 10^{11}10^{9} Hz, wavelengths at 10^{3}10^{1 }m, and energy at 10^{2.5}10^{5.5} J. The energy that is released is absorbed by the water molecules in the food, which causes them to rotate and generate heat. The energy from the microwaves is released when the food steams. If metal is placed in the microwave, there is no available water molecules, so the metal will heat up very rapidly and possibly generate arcs, which is why metal is forbidden from microwaves.
How are vibrational spectra affected by anharmonicity? What is an example of an
anharmonic system?
As mentioned earlier, the potential energy equation for anharmonic systems include additional terms that influence the energy value. Therefore the vibrational spectra (IR) of absorption intensity between energy states of anharmonic waves is not constant. This is different than harmonic waves, because the difference in energy between states is constant. Anharmonicity is a realistic system, while a harmonicity is a theoretical system. Another effect of anharmonicity on a vibrational spectra is that an electron is able to transfer between nonadjacent energy levels (e.g. 1 > 4). This is reflected in the spectra because the representation of the anharmonic spectra would display larger absorption peaks On the other hand, harmonic spectra would display less absorption since harmonicity is limited by a selection rule that confines energy transitions to only adjacent energy levels (e.g. 2 > 3).
An example of an anharmonic system includes any atom with a changing displacement of its electron cloud.
What is the Morse potential and why does it differ from the harmonic potential? How are
the energy levels different?
The Morse potential is the potential energy for anharmonic models to explain a single vibrational stretch, the variation of bond lengths, vibrational excitations, and zeropoint energy (which is the energy of the ground state). Harmonic potential is also used to understand zeropoint energy (albeit for a harmonic system), frequencybondorder relationships (vibrational frequency is inversely proportional to bond order), and quantization of consistent wave patterns.
Practice Problems:
1. Which of the following correctly describes the relationship between wavelength and frequency?
a.) Wavelength and frequency are directly proportional.
b.) Wavelength and frequency are not related.
c.) Wavelength and frequency are inversely proportional.
d.) Wavelength is the square root of frequency.
2. Which of the following statements is true?
I: Anharmonic oscillators have constant changes in transitions of energy levels.
II: The amount of energy required for a transition in a harmonic oscillator approaches infinity.
III: An anharmonic oscillator is able to display larger absorptions in his spectra.
a.) I
b.) III
c.) II and III
d.) I and III
e.) I, II, and III
3. The harmonic potential is used to understand...
a.) zero point energy
b.) frequencybondorder
c.) variation of bond lengths
d.) a and b
e.) all of the above
4. Fill in the blanks of the following statement.
Spectroscopy includes three subcategories of ____, _____, and ______.
5. Is the following statement true or false?
A wavelength of 5.0 x 10^{8}m corresponds to xrays on the electromagnetic spectrum.
answers can be found at the bottom of the page
References:
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Brauer, Brina, R. Benny Gerber, Martin Kabelac, Pavel Hobza, Joost M. Bakker, Ali G. Abo Riziq, and Mattanjah S. De Vries. "Vibrational Spectroscopy of the GC Base Pair: Experiment, Harmonic and Anharmonic Calculations, and the Nature of Anharmonic Couplings." J. Phys. Chem. A. 109.31 (2005): 6974984. Print.
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"CHAPTER 13: Molecular Spectroscopy I: Rotational and Vibrational Spectroscopy." N.p., n.d. Web. 17 Sept. 2012. <http://iweb.tntech.edu/snorthrup/chem3510/Chapter%2013.pdf>.
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"Lecture 16: Rotational Spectroscopy." N.p., 4 Nov. 2010. Web. 17 Sept. 2012. <http://depts.washington.edu/chemcrs/bulkdisk/chem455A_aut10/notes_Lecture%2016.pdf>.
Lim, Kieren F. "The Effect of Anharmonicity on Diatomic Vibration: A Spreadsheet Simulation." Journal of Chemical Education 82.8 (2005): 1263264. Print.
"Morse Potential." Wikipedia. Wikimedia Foundation, 27 Sept. 2012. Web. 01 Oct. 2012. <http://en.wikipedia.org/wiki/Morse_potential>.
PelletierPhysics. "Simple Harmonic Motion." YouTube. YouTube, 06 Aug. 2007. Web. 22 Sept. 2012. <http://www.youtube.com/watch?v=eeYRkW8V7Vg>.
Randall D. Knight. Physics for scientists and engineers with modern physics. A strategic approach. Pearson Education. 2004. pg. 1302.
"Vibrational Spectroscopy." 5.61 Physical Chemistry. N.p., n.d. Web. 17 Sept. 2012.
Answers:
1. c 2. b 3. d 4. absorption, emission, and scattering. 5. false; 5.0 x 10^{8}m corresponds to ultraviolet. Xrays exist at wavelengths between 1.0 x 10^{10 }m and 1.0 x 10^{8} m.^{ }
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