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Encyclopedia > Rigid rotor

The rigid rotor is a mechanical model that is used to explain rotating systems. The model consists of two point masses located at fixed distances from their center of mass. The fixed distance between the two masses is an important characteristic of the model. However, for many actual systems this distance is not completely fixed. Fortunately, corrections can be made to compensate for small variations in the distance and therefore the rigid rotor model can still be used to produce fairly accurate results.

1 Rigid Rotor and Quantum Mechanics
2 Selection Rules for the Rigid Rotor
3 Non-Rigid Rotor
4 References
5 See also

Rigid Rotor and Quantum Mechanics

The rigid rotor model can be used in quantum mechanics to predict the rotational energy of a diatomic molecule. The rotational energy depends on the moment of inertia for the system, $I$ . In the center of mass reference frame, the moment of inertia is equal to: Fig. ... Diatomic molecules are molecules formed of exactly two atoms, of the same or different chemical elements. ... Moment of inertia quantifies the rotational inertia of an object, i. ... It has been suggested that this article or section be merged with Center of gravity. ... Moment of inertia quantifies the rotational inertia of an object, i. ...

$I = μ R 2$

where $μ$ is the reduced mass of the molecule and $R$ is the distance between the two atoms. Reduced mass is a concept that allows one to solve the two-body problem of mechanics as if it were a one body problem. ...

According to quantum mechanics, the energy ( $E$ ) of a system can be determined using the Schrödinger equation: Fig. ... In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, describes the time-dependence of quantum mechanical systems. ...

$hat H Y = E Y$

where $Y$ is the wave function (sometimes the letter $ψ$ is used instead) and $hat H$ is the energy (Hamiltonian) operator. For the rigid rotor, the energy operator corresponds to the kinetic energy of the system: In the most restricted usage in quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued square integrable function ψ defined over a portion of space normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared... The Hamiltonian, denoted H, has two distinct but closely related meanings. ... Kinetic energy is energy that a body has as a result of its speed. ...

$hat H = - frac{hbar^2}{2I} nabla^2$

where $hbar$ is Planck's constant divided by $2π$ and $nabla^2$ is a symbol known as the Laplacian. The Laplacian is the second derivative with respect to the spatial coordinates. For the rigid rotor model, the energy operator is often written in terms of spherical coordinates: A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...

$hat H =- frac{hbar^2}{2I} left [ {1 over sin theta} {partial over partial theta} left ( sin theta {partial over partial theta} right ) + {1 over {sin^2 theta}} {partial^2 over partial phi^2} right]$

The wave functions for the rigid rotator, Y $(θ,φ)$ , can be separated a product of two functions. One of these functions depends only on $θ$ and the other depends only on $φ$ . Multiplied together, these functions result in the complete wave function for the rigid rotor model: In the most restricted usage in quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued square integrable function ψ defined over a portion of space normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared...

$Y_{l,m}(theta,phi) = left [ sqrt{ {(2l+1) over 2} {( l - left |m right |)! over (l + left |m right |)!} } P_l^left |m right | (cos theta) right ] left [ sqrt {{1 over 2pi}} exp (imphi) right ]$

where $sqrt{ {(2l+1) over 2} {( l - left |m right |)! over (l + left |m right |)!} }$ is a normalization constant that depends on the quantum numbers $l$ and $m$ . The symbol $P_l^left |m right | (cos theta)$ represents a set of functions known as the spherical harmonics, and which also depend on the $l$ and $m$ quantum numbers. A quantum number is any one of a set of numbers used to specify the full quantum state of any system in quantum mechanics. ... In mathematics, the spherical harmonics are an orthogonal set of solutions to Laplaces equation represented in a system of spherical coordinates. ...

Although the wave functions for the rigid rotor may appear complicated, the equation for the energy of the system is much more compact: In the most restricted usage in quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued square integrable function ψ defined over a portion of space normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared...

$E_l = {hbar^2 over 2I} l left (l+1right )$

In units of wave numbers, a unit that is often used for rotational-vibrational spectroscopy, this equation is: Wavenumber in most physical sciences is a wave property inversely related to wavelength, having units of inverse length. ...

$bar E_l = bar {B}l left (l+1right )$

where $bar B$ is known as the rotational constant and:

$bar B = {h over 8pi^2 cI}$

A typical rotational spectrum consists of a series of peaks that correspond to transitions between levels with different values of the angular momentum quantum number ( $l$ ). Consequently, rotational peaks appear at energies corresponding to an integer multiple of ${2bar B}$ . A quantum number is any one of a set of numbers used to specify the full quantum state of any system in quantum mechanics. ... Rotational spectroscopy studies the absorption of electromagnetic radiation (typically in the microwave region of the electromagnetic spectrum) by molecules. ...

Selection Rules for the Rigid Rotor

Typically, rotational transitions can only be observed when the angular momentum quantum number changes by 1 ( $Delta l = pm 1$ ). This selection rule arises from the first-order perturbation theory treatment of rigid rotor using the time-dependent Schrödinger equation. According to this treatment, rotational transitions can only be observed when the $z$ -component of the dipole transition moment: A quantum number is any one of a set of numbers used to specify the full quantum state of any system in quantum mechanics. ...

$left ( mu_z right )_{21} = int psi_2^*mu_zpsi_1, dtau$

is non-zero. This means that the molecule must have a permanent dipole moment in order to have a spectroscopically observable rotational transition. When this integral is evaluated using the wave function for the rigid rotor and the properties of the spherical harmonics the result is:

$left ( mu_z right )_{l,m;l^',m^'} = mu int_0^{2pi} dphi int_0^pi dtheta sin theta cos theta Y_{l^1}^{m^1} left ( theta , phi right )^* Y_l^m left ( theta , phi right )$

Using the normalization constant for wave function and the orthogonality of the spherical harmonics, it is possible to determine which values of $l$ , $m$ , $l'$ , and $m'$ will result in nonzero values for the dipole transition moment integral. This constraint results in the observed selection rules for the rigid rotor:

$Δ m = 0$

$Delta l = pm 1$

Non-Rigid Rotor

The rigid rotor is commonly used to describe the rotational energy of diatomic molecules but it not a completely accurately description of such molecules. This is because molecular bonds (and therefore the interatomic distance $R$ ) is not completely fixed; the bond between the atoms stretches out as molecule rotates faster (higher values of the rotational quantum number $l$ ). This effect can be accounted for by introducing a correction factor known as the centrifugal distortion constant ( $D$ ): A quantum number is any one of a set of numbers used to specify the full quantum state of any system in quantum mechanics. ...

$bar E_l = {E_l over hc} = bar {B}l left (l+1right ) - bar {D}l^2 left (l+1right )^2$

where

$bar D = {4 bar {B}^3 over bar {v}^2}$

$bar v$ is the fundamental vibrational frequency of the bond. This frequency is related to the reduced mass and the force constant (bond strength) of the molecule according to

$bar v = {1over 2pi c} left ({k over mu } right )^{1 over 2}$

The non-rigid rotor is an acceptably accurate model for diatomic molecules but is still somewhat imperfect. This is because, although the model does account for bond stretching due to rotation, it ignores any bond stretching due to vibrational energy in the bond.

References

McQuarrie, Donald A (1983). Quantum Chemistry, Mill Valley, Calif. : University Science Books. ISBN 093570213X.

Contents

Rigid Rotor and Quantum Mechanics GA_googleFillSlot("encyclopedia_square");

Selection Rules for the Rigid Rotor

Non-Rigid Rotor

References

See also

Rigid Rotor and Quantum Mechanics