**Table of Contents:**

I. Introduction

II. Rotational Motion

III. Rigid Rotor Model

IV. Setting Up the Schrodinger Equation

V. Differences between Classical Mechanics and Quantum Mechanics

VI. Rotational Transitions

VII. Rotational Spectroscopy

VIII. Scientific Applications

IX. References

** **

**I. Introduction**

Rotational motion is important in the everyday world because without it, many fundamental chemical activities in our daily lives cannot happen, and we would also be unable to study the absorption of electromagnetic radiation by molecules. All linear diatomic molecules are subject to rotation, and they are modeled by the rigid rotor model. In the following wiki, we will explain the concept of rotational motion, set up the Schrodinger Equation for the rigid rotor model, describe the differences between classical and quantum mechanics for rotation, and talk about rotational transitions and spectroscopy. In the end, we will describe some exciting scientific applications of rotation!

**II. Rotational Motion**

Rotational spectroscopy is the study of the absorption of electromagnetic radiation by molecules, which is only practical in the gas phase where rotational motions are quantized. The rotational energy levels of molecules are quantized, and this quantization is related to the moment of inertia, I. For a linear diatomic molecule, there is only one moment of inertia, which takes place perpendicular to the axis of symmetry. ^{2}

The rotational motion of a diatomic molecule is best illustrated using a rigid rotor model. The model consists of a rigid body rotating about a fixed axis. The atoms follow concentric, circular orbits in a plane perpendicular to the rotational axis. In the absence of an external force, this rotational axis passes through the center of mass of the two atoms.

Equations are required to describe the rotational dynamics of molecules. Since only the angle θ changes with time, the rate of rotation ω can be calculated in terms of θ, the value of which has SI units of radians per second.

ω = d θ/dt

The instantaneous velocity is:

v = ω r

The angular momentum of molecules, used to describe the overall state of a physical system, is given by:

L = I ω

where I is the moment of inertia (m_{1}r_{1 }+ m_{2}r_{2}) in units of kg•m^{2}. The angular momentum has units of kg•m^{2}/s.^{2}

The kinetic energy of a rotating mass m is described by the expression

KE = ½ I ω^{2}

Work goes into changing the rotational kinetic energy of an object;

W = ΔKE = τ r

where τ is torque and r is the angular distance.^{1}

**III. Rigid Rotor Model**

The linear rigid rotor model consists of two point masses located at fixed distances from their center of mass. A diatomic molecule is an example of this. The only characteristics of the rigid model consist of the aforementioned fixed distance and the values of the masses. However, actual diatomic molecules’ distances are not always completely fixed, and thus corrections on this model must be made.

Above is a picture of a rigid rotor molecule. Angular momentum is pointing up, as shown by the green arrow.^{3}

The simplest model of rotational motion that can be described quantum mechanically is a rigid rotor consisting of two masses held a fixed distance apart about its center of mass. This rotation is **assumed** to be free of any outside potential energy. There is a direct correspondence between classical mechanical equations of rotational motion and angular momentum in quantum systems.

A rigid rotor rotating with angular velocity w and with moment of inertia I = µ R^{2} has angular momentum |L| = I ω and an energy of E = L^{2}/2I. In order to convert these classical equations to quantum equations we need to replace momentum with its quantum operator and solve Schrödinger's equation.^{4}

**IV. Setting Up the Schrodinger Equation**

Wave functions for the rigid rotor model:

Hamiltonian Operator is:

is the Laplacian operator, which is just the differential operator, and can be represented in either cartesian coordinates or spherical coordinates. Here, we will just show it in spherical coordinates:

There are two assumptions that we need to make at this time:

- The distance between the two masses is fixed (even though it isn’t, we will just assume this to make life easier), so the terms in the Laplacian containing the derivative of r would be zero.
- The orientation of the masses is described by just θ and Ф and in the absence of electric or magnetic fields the energy is independent of orientation. The potential energy portion of the Hamiltonian is thus zero.

**Schrodinger Equation** is now:

After solving the Schrodinger equation, which we will not show here, you can get energy to be simplified to: E=BJ(J+1)

When a molecule is irradiated with photons of light, it can absorb the radiation and undergo an energy transition. Energy of transition must be equivalent to the energy of the photon of light absorbed E=hv. From our equation E=BJ(J+1), the energy difference for a diatomic molecule between rotational levels J to J+1 is…

B(J+1)(J+2)-BJ(J+1)=2B(J+1), where J=0,1,2,3

E_{rot,J} = hBJ(J+1), where B = h/(8π^{2}I) and J = 0, 1, 2, ... corresponds to the rotational quantum number. Because ∆J = ±1 is one of the selection rules for rotational transitions, the rotational kinetic energy levels for transitions are limited to increments of 2Bh.

The rotation of a diatomic molecule can be described very well using this model, which treats the molecules fixed by a certain bond length as a spinning dumbbell.^{5}

**V. Differences between Classical Mechanics and Quantum Mechanics**

**Classical Mechanics:**

Hamiltonian solution to the rigid rotor is H = K.E., since H=K.E. + U. Potential energy, U, is zero because the distance between particles does not change in our model.

Since H = K.E., we can say, K.E. = 1/2mv^{2}.

After some rearrangement, we come up with the equation K.E. = 0.5Iω^{2} for classical mechanics.

**Quantum Mechanics:**

We must utilize the internal Hamiltonian and the Schrodinger Equation to solve for energy. We get E=J(J+1)Bh. The transition energy between two energy levels is a multiple of 2. The energy is quantized here. We can see the differences between classical mechanics and quantum mechanics upon comparing the different energies and how we approach and solve the equations.^{6}

**VI. Rotational Transitions**

In quantum mechanics, a rotational transition is an abrupt change in angular momentum. The rotations of diatomic molecules can occur only when the molecules have a permanent dipole moment and when there is a difference between the molecule's center of charge and center of mass. Rotational transitions occur when the electric field E created by incident electromagnetic waves causes a molecule to spin by exerting a torque on the molecule.

Above is a picture of a diatomic molecule in an electric field. The torque acts in a direction parallel to the electric field, causing rotational transitions.^{1}

The electric field of light can exert a torque only on molecules with a dipole moment, causing the molecule to either rotate more quickly (excitation of rotational energy) or more slowly (de-excitation of rotational energy).^{7} The requirement for a molecule to have a permanent dipole is known as a selection rule. Other selection rules for rotational transitions are ∆J = ±1 and ∆M_{J} = 0, where J is the angular momentum quantum number, and M_{J} is the angular momentum spin quantum number.

As seen by solving the Schrodinger equation for a rigid rotor molecule, rotational energy, or E_{rot,J} = hBJ(J+1), where B = h/(8π^{2}I) and J = 0, 1, 2, ... corresponds to the rotational quantum number. Because ∆J = ±1 is one of the selection rules for rotational transitions, the rotational kinetic energy levels for transitions are limited to increments of 2Bh.

Above is a picture of the allowed energy levels for rotational transitions.^{8}

The rotational kinetic energy of the rigid rotor can be expressed in terms of the angular momentum, so angular momentum is also quantized.^{4} When a molecule experiences a loss in angular momentum, it transitions to a lower rotational energy state. Similarly, when a molecule experiences a gain in angular momentum, it transitions to a higher rotational energy state. In addition to angular momentum, the spin angular momentum of a molecule is also quantized. The allowed values of spin angular momentum are given by S = ħ, where s = n/2, with n being any non-negative integer. During rotational transitions, the spin angular momentum of a molecule is conserved, as shown by the selection rule ∆M_{J} = 0.

**VII. Rotational Spectroscopy**

Rotational spectroscopy studies the absorption and emission of electromagnetic radiation by molecules when they experience rotational transitions. Rotational spectroscopy is commonly used to determine the spectra of linear diatomic molecules, such as the one shown below. This occurs typically in the microwave region of the electromagnetic spectrum. Molecules with a permanent net dipole moment will exhibit pure rotational spectrums. Spectroscopy shows the rotational-vibrational spectrum of different molecules, which occur in the microwave and infrared regions.

Above is a picture of the rotational-vibrational spectrum of HCl.^{6}

Homonuclear diatomic molecules, such as dioxygen (O_{2}) and dihydrogen (H_{2}) do not exhibit pure rotational spectrum because they do not have a dipole moment. However, electromagnetic radiation can be used to induce dipole moments in these molecules by displacing the electron densities of the atoms and making the charge distributions in the molecules asymmetric. When this occurs, homonuclear diatomic molecules will exhibit a rotational spectrum.

**VIII. Scientific Applications**

Microwave spectroscopy is commonly used in physical chemistry to determine the structure of small molecules with high precision. It can be used to determine such characteristics as the bond lengths, angular velocities, and angular momentums of molecules. Other techniques for determining molecular structure, such as X-ray crystallography, do not work as well, especially for gases, and are not as precise. These techniques are more useful for determining the structures of large molecules, for which microwave spectroscopy is not useful.

Microwave spectroscopy is one of the main methods by which the components of the universe are determined. It is especially useful for detecting molecules in the interstellar medium (ISM), or the matter that exists in the space between the star systems in a galaxy. For example, it was discovered that long-chain carbon molecules exist in the ISM. Following this discovery, the collaborative experiments of Harry Kroto, Rick Smalley, and Robert Curl, led to the discovery of C_{60}, or buckminsterfullerene, for which they were awarded the 1996 Nobel prize in chemistry.^{9}

**IX. References**

1. "Rotational Dynamics." *Wiki*. Web. 30 Oct. 2011. <http://www.ch.ic.ac.uk/local/physical/mi_4.html>.

2. "Rotational Spectroscopy Molecule Molecules Inertia Spectra." *Business, Economy, Market Research, Finance, Income Tax Informations*. Web. 30 Oct. 2011. <http://www.economicexpert.com/a/Rotational:spectroscopy.html>.

3. "Rigid Rotor." *Middlebury College: Tertiary Web Server*. Web. 30 Oct. 2011. <http://cat.middlebury.edu/~chem/chemistry/class/physical/quantum/help/rigid_rotor/rigid_rotor.html>.

4. "Energy Calculation for Rigid Rotor Molecules." Web. 30 Oct. 2011. <http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/rotqm.html>.

5. Sherrill, David. "The Rigid Rotor." 15 Aug. 2006. Web. 30 Oct. 2011. <http://vergil.chemistry.gatech.edu/notes/quantrev/node24.html>.

6. Huh, Ian. "Rigid Rotor." *ChemWiki*. Web. 30 Oct. 2011. <http://chemwiki.ucdavis.edu/Physical_Chemistry/Quantum_Mechanics/Quantum_Theory/Free_Particles/Rigid_Rotor>.

7. "Rotational_spectroscopy." *Chemeurope.com – The Chemistry Information Portal from Laboratory to Process*. Web. 30 Oct. 2011. <http://www.chemeurope.com/en/encyclopedia/Rotational_spectroscopy.html>.

8. "Diatomic Molecules." *Under Construction*. Web. 30 Oct. 2011. <http://electron6.phys.utk.edu/qm2/modules/m1-3/molecules.htm>.

9. "Buckminsterfullerene." *Royal Society of Chemistry | Advancing the Chemical Sciences*. Web. 08 Nov. 2011. <http://www.rsc.org/chemsoc/timeline/pages/1985.html>.

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