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.

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: $I\; =\; \backslash \backslash mu\; R^2$ where $\backslash \backslash mu$ is the reduced mass of the molecule and $R$ is the distance between the two atoms. According to quantum mechanics, the energy ($E$) of a system can be determined using the Schrödinger equation: $\backslash \backslash hat\; H\; Y\; =\; E\; Y$ where $Y$ is the wave function (sometimes the letter $\backslash \backslash psi$ is used instead) and $\backslash \backslash hat\; H$ is the energy (Hamiltonian) operator. For the rigid rotor, the energy operator corresponds to the kinetic energy of the system: $\backslash \backslash hat\; H\; =\; -\; \backslash \backslash frac\{\backslash \backslash hbar^2\}\{2I\}\; \backslash \; abla^2$ where $\backslash \backslash hbar$ is Plank's constant divided by $2\backslash \backslash pi$ and $\backslash \; abla^2$ 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: $\backslash \backslash hat\; H\; =-\; \backslash \backslash frac\{\backslash \backslash hbar^2\}\{2I\}\; \backslash \backslash left\; [\; \{1\; \backslash \backslash over\; \backslash \backslash sin\; \backslash \backslash theta\}\; \{\backslash \backslash partial\; \backslash \backslash over\; \backslash \backslash partial\; \backslash \backslash theta\}\; \backslash \backslash left\; (\; \backslash \backslash sin\; \backslash \backslash theta\; \{\backslash \backslash partial\; \backslash \backslash over\; \backslash \backslash partial\; \backslash \backslash theta\}\; \backslash ight\; )\; +\; \{1\; \backslash \backslash over\; \{\backslash \backslash sin^2\; \backslash \backslash theta\}\}\; \{\backslash \backslash partial^2\; \backslash \backslash over\; \backslash \backslash partial\; \backslash \backslash phi^2\}\; \backslash ight]$ The wave functions for the rigid rotator, Y$(\backslash \backslash theta,\; \backslash \backslash phi)$, can be separated a product of two functions. One of these functions depends only on $\backslash \backslash theta$ and the other depends only on $\backslash \backslash phi$. Multiplied together, these functions result in the complete wave function for the rigid rotor model: $Y\_\{l,m\}(\backslash \backslash theta,\backslash \backslash phi)\; =\; \backslash \backslash left\; [\; \backslash \backslash sqrt\{\; \{(2l+1)\; \backslash \backslash over\; 2\}\; \{(\; l\; -\; \backslash \backslash left\; |m\; \backslash ight\; |)!\; \backslash \backslash over\; (l\; +\; \backslash \backslash left\; |m\; \backslash ight\; |)!\}\; \}\; P\_l^\backslash \backslash left\; |m\; \backslash ight\; |\; (\backslash \backslash cos\; \backslash \backslash theta)\; \backslash ight\; ]\; \backslash \backslash left\; [\; \backslash \backslash sqrt\; \backslash \backslash exp\; (im\backslash \backslash phi)\; \backslash ight\; ]$ where $\backslash \backslash sqrt\{\; \{(2l+1)\; \backslash \backslash over\; 2\}\; \{(\; l\; -\; \backslash \backslash left\; |m\; \backslash ight\; |)!\; \backslash \backslash over\; (l\; +\; \backslash \backslash left\; |m\; \backslash ight\; |)!\}\; \}$ is a normalization constant that depends on the quantum numbers $l$ and $m$. The symbol $P\_l^\backslash \backslash left\; |m\; \backslash ight\; |\; (\backslash \backslash cos\; \backslash \backslash theta)$ represents a set of functions known as the spherical harmonics, and which also depend on the $l$ and $m$ quantum numbers. Although the wave functions for the rigid rotor may appear complicated, the equation for the energy of the system is much more compact: $E\_l\; =\; \{\backslash \backslash hbar^2\; \backslash \backslash over\; 2I\}\; l\; \backslash \backslash left\; (l+1\backslash ight\; )$ In units of wave numbers, a unit that is often used for rotational-vibrational spectroscopy, this equation is: $\backslash \backslash bar\; E\_l\; =\; \backslash \backslash bar\; \{B\}l\; \backslash \backslash left\; (l+1\backslash ight\; )$ where $\backslash \backslash bar\; B$ is known as the rotational constant and: $\backslash \backslash bar\; B\; =\; \{h\; \backslash \backslash over\; 8\backslash \backslash pi^2\; cI\}$ 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 $\{2\backslash \backslash bar\; B\}$.

Selection Rules for the Rigid Rotor

Typically, rotational transitions can only be observed when the angular momentum quantum number changes by 1 ($\backslash \backslash Delta\; l\; =\; \backslash \backslash pm\; 1$). 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: $\backslash \backslash left\; (\; \backslash \backslash mu\_z\; \backslash ight\; )\_\{21\}\; =\; \backslash \backslash int\; \backslash \backslash psi\_2^$

• \\mu_z\\psi_1, d\\tau

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: $\backslash \backslash left\; (\; \backslash \backslash mu\_z\; \backslash ight\; )\_\{l,m;l^\text{'},m^\text{'}\}\; =\; \backslash \backslash mu\; \backslash \backslash int\_0^\{2\backslash \backslash pi\}\; d\backslash \backslash phi\; \backslash \backslash int\_0^\backslash \backslash pi\; d\backslash \backslash theta\; \backslash \backslash sin\; \backslash \backslash theta\; \backslash \backslash cos\; \backslash \backslash theta\; Y\_\{l^1\}^\{m^1\}\; \backslash \backslash left\; (\; \backslash \backslash theta\; ,\; \backslash \backslash phi\; \backslash ight\; )^$

• Y_l^m \\left ( \\theta , \\phi \ight )

Using the normalization constant for wave function and the orthogonality of the spherical harmonics, it is possible to determine which values of $l$, $m$, $l\text{'}$, and $m\text{'}$ will result in nonzero values for the dipole transition moment integral. This constraint results in the observed selection rules for the rigid rotor: $\backslash \backslash Delta\; m\; =\; 0$ $\backslash \backslash Delta\; l\; =\; \backslash \backslash pm\; 1$

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 ($D$): $\backslash \backslash bar\; E\_l\; =\; \{E\_l\; \backslash \backslash over\; hc\}\; =\; \backslash \backslash bar\; \{B\}l\; \backslash \backslash left\; (l+1\backslash ight\; )\; -\; \backslash \backslash bar\; \{D\}l^2\; \backslash \backslash left\; (l+1\backslash ight\; )^2$ where $\backslash \backslash bar\; D\; =\; \{4\; \backslash \backslash bar\; \{B\}^3\; \backslash \backslash over\; \backslash \backslash bar\; \{v\}^2\}$ $\backslash \backslash bar\; v$ 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 $\backslash \backslash bar\; v\; =\; \{1\backslash \backslash over\; 2\backslash \backslash pi\; c\}\; \backslash \backslash left\; (\{k\; \backslash \backslash over\; \backslash \backslash mu\; \}\; \backslash ight\; )^\{1\; \backslash \backslash over\; 2\}$ 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: Quantum Chemistry, McQuarrie, Donald A.

See also

*Rotational spectroscopy *Vibrational spectroscopy *Infrared spectroscopy *Spectroscopy

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