Rotational Constant

In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigid rotor is the ''linear rotor'' requiring only two angles to describe, for example of a diatomic molecule. More general molecules are 3-dimensional, such as water (asymmetric rotor), ammonia (symmetric rotor), or methane (spherical rotor).

Linear rotor

The linear rigid rotor model consists of two point masses located at fixed distances from their center of mass. The fixed distance between the two masses and the values of the masses are the only characteristics of the rigid model. However, for many actual diatomics this model is too restrictive since distances are usually not completely fixed. Corrections on the rigid model can be made to compensate for small variations in the distance. Even in such a case the rigid rotor model is a useful point of departure (zeroth-order model).

Classical linear rigid rotor

The classical linear rotor consists of two point masses $m_1$ and $m_2$ (with reduced mass $\mu = \frac$) at a distance $R$ of each other. The rotor is rigid if $R$ is independent of time. The kinematics of a linear rigid rotor is usually described by means of spherical polar coordinates, which form a coordinate system of R³. In the physics convention the coordinates are the co-latitude (zenith) angle $\theta \,$, the longitudinal (azimuth) angle $\varphi\,$ and the distance $R$. The angles specify the orientation of the rotor in space. The kinetic energy $T$ of the linear rigid rotor is given by
:
2T = \mu R^2 \left dot^2 + (\dot\varphi\,\sin\theta)^2\right=
\mu R^2 \begin\dot & \dot\end
\begin
1 & 0 \\
0 & \sin^2\theta \\
\end
\begin\dot \\ \dot\end
=
\mu \begin\dot & \dot\end
\begin
h_\theta^2 & 0 \\
0 & h_\varphi^2 \\
\end
\begin\dot \\ \dot\end,

where $h_\theta = R\,$ and $h_\varphi= R\sin\theta\,$ are scale (or Lamé) factors.

Scale factors are of importance for quantum mechanical applications since they enter the Laplacian expressed in curvilinear coordinates. In the case at hand (constant $R$)
:
\nabla^2 = \frac\left
\frac \frac \frac
+ \frac \frac \frac
\right=
\frac\left \frac\frac
\sin\theta\frac
+ \frac\frac
\right

The classical Hamiltonian function of the linear rigid rotor is
:
H = \frac\left ^2_ + \frac\right

Quantum mechanical linear rigid rotor

The linear 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:

:$I = \mu R^2$

where $\mu$ is the reduced mass of the molecule and $R$ is the distance between the two atoms.

According to quantum mechanics, the energy levels of a system can be determined by solving the Schrödinger equation:

:$\hat H \Psi = E \Psi$

where $\Psi$ is the wave function and $\hat H$ is the energy ( Hamiltonian) operator. For the rigid rotor in a field-free space, the energy operator corresponds to the kinetic energy of the system:

:$\hat H = - \frac \nabla^2$

where $\hbar$ is reduced Planck constant and $\nabla^2$ is the Laplacian. The Laplacian is given above in terms of spherical polar coordinates. The energy operator written in terms of these coordinates is:

:\hat H =- \frac \left \left ( \sin \theta \right) + \right/math>

This operator appears also in the Schrödinger equation of the hydrogen atom after the radial part is separated off. The eigenvalue equation becomes
:$\hat H Y_\ell^m (\theta, \varphi) = \frac \ell(\ell+1) Y_\ell^m (\theta, \varphi).$
The symbol $Y_\ell^m (\theta, \varphi)$ represents a set of functions known as the spherical harmonics. Note that the energy does not depend on $m \,$. The energy
:$E_\ell = \ell \left (\ell+1\right)$
is $2\ell+1$-fold degenerate: the functions with fixed $\ell\,$ and $m=-\ell,-\ell+1,\dots,\ell$ have the same energy.

Introducing the ''rotational constant'' $B$, we write,
:$E_\ell = B\; \ell \left (\ell+1\right)\quad \textrm\quad B \equiv \frac.$
In the units of reciprocal length the rotational constant is,
:$\bar B \equiv \frac = \frac = \frac,$
with ''c'' the speed of light. If cgs units are used for $h$, $c$, and $I$, $\bar B$ is expressed in cm⁻¹, or wave numbers, which is a unit that is often used for rotational-vibrational spectroscopy. The rotational constant $\bar B(R)$ depends on the distance $R$. Often one writes $B_e = \bar B(R_e)$ where $R_e$ is the equilibrium value of $R$ (the value for which the interaction energy of the atoms in the rotor has a minimum).

A typical rotational absorption spectrum consists of a series of peaks that correspond to transitions between levels with different values of the angular momentum quantum number ($\ell$) such that $\Delta l = +1$, due to the selection rules (see below). Consequently, rotational peaks appear at energies with differences corresponding to an integer multiple of $2\bar B$.

Selection rules

Rotational transitions of a molecule occur when the molecule absorbs a photon particle of a quantized electromagnetic (em) field Depending on the energy of the photon (i.e., the wavelength of the em field) this transition may be seen as a sideband of a vibrational and/or
electronic transition. Pure rotational transitions, in which the vibronic (= vibrational plus electronic) wave function does not change, occur in the microwave region of the electromagnetic spectrum.

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 a first-order perturbation theory approximation of the time-dependent Schrödinger equation. According to this treatment, rotational transitions can only be observed when one or more components of the dipole operator have a non-vanishing transition moment. If $z$ is the direction of the electric field component of the incoming electromagnetic wave, the transition moment is,
:$\langle \psi_2 , \mu_z , \psi_1\rangle = \left ( \mu_z \right )_ = \int \psi_2^*\mu_z\psi_1\, \mathrm\tau .$

A transition occurs if this integral is non-zero. By separating the rotational part of the molecular wavefunction from the vibronic part, one can show that this means that the molecule must have a permanent dipole moment. After integration over the vibronic coordinates the following rotational part of the transition moment remains,

:$\left ( \mu_z \right )_ = \mu \int_0^ \mathrm\phi \int_0^\pi Y_^ \left ( \theta , \phi \right )^* \cos \theta\,Y_l^m\, \left ( \theta , \phi \right )\; \mathrm\cos\theta .$
Here $\mu \cos\theta \,$ is the ''z'' component of the permanent dipole moment. The moment $\mu$ is the vibronically averaged component of the dipole operator. Only the component of the permanent dipole along the axis of a heteronuclear molecule is non-vanishing.
By the use of the orthogonality of the spherical harmonics $Y_l^m\, \left ( \theta , \phi \right )$ 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:

:$\Delta m = 0 \quad\hbox\quad \Delta l = \pm 1$

Non-rigid linear rotor

The rigid rotor is commonly used to describe the rotational energy of diatomic molecules but it is not a completely accurate description of such molecules. This is because molecular bonds (and therefore the interatomic distance $R$) are not completely fixed; the bond between the atoms stretches out as the 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 $\bar$ (bars on top of various quantities indicate that these quantities are expressed in cm⁻¹):

:$\bar E_l = = \bar l \left (l+1\right ) - \bar l^2 \left (l+1\right )^2$

where

:$\bar D =$

:$\bar$ is the fundamental vibrational frequency of the bond (in cm⁻¹). This frequency is related to the reduced mass and the force constant (bond strength) of the molecule according to

:$\bar = \sqrt$

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 (anharmonicity in the potential).

Arbitrarily shaped rigid rotor

An arbitrarily shaped rigid rotor is a rigid body of arbitrary shape with its center of mass fixed (or in uniform rectilinear motion) in field-free space R³, so that its energy consists only of rotational kinetic energy (and possibly constant translational energy that can be ignored). A rigid body can be (partially) characterized by the three eigenvalues of its moment of inertia tensor, which are real nonnegative values known as ''principal moments of inertia''.
In