Rotational Spectra

In classical mechanics, the energy of an particle which rotates around an axis is given by $E = \frac{1}{2} I\omega^2$. Here $\omega$ is the angular velocity in radian per second and $I = m r^2$ is the moment of inertia, where $m$ is the particle mass and $r$ is the distances between the particle and the axis. For an arbitrary object the rotation energy can be written in the following form:

$$E = \frac{1}{2}I_A\omega^2_A + \frac{1}{2}I_B\omega^2_B + \frac{1}{2}I_C\omega^2_C,$$ (30)

where $A,B,C$ are so-called principal axes of rotation.

Using $J_i = I_i \omega_i$, with $i=A,B,C$, where $J_i$ is angular momentum, we get

$$E = \frac{1}{2}\frac{J_A^2}{I_A} + \frac{1}{2}\frac{J_B^2}{I_B} + \frac{1}{2}\frac{J_C^2}{I_C}$$ (31)

In general, the moment of inertia $I$ of a polyatomic molecule over the axis $A$ is defined by

$$I_A = \sum_i m_ir_i^2$$ (32)

where $m_i$ is the mass of an $i$-th nucleous and $r_i$ is its distance from the axis of rotation $A$.

To begin with, we assume that the shape of the molecule does not depend on rotation, i.e. that the the bond lengths are not affected by centrifugal forces. This molecule is called rigid rotor in contrast to an elastic rotor. We will consider the following cases, see Table 1:

Table 1:

Spherical top	$I_A = I_B = I_C = I$	e.g. $CH_4$, $CCl_4$, $SF_6$
	$I_A = I_B = I_{\bot}$ and $I_C = I_{\Vert}$
Symmetric top	$I_{\bot} < I_{\Vert}$ oblate	e.g. benzene
	$I_{\bot} > I_{\Vert}$ prolate	e.g. methyl iodide $CH_3I$
Linear Rotor 1	$I_A = 0$, $I_B = I_C$	all diatomic molecules and
		e.g. $CO_2$, $N_2O$, $C_2H_2$
Asymmetric top	$I_A \ne I_B \ne I_C$	water $H_2O$
	$I_C \geq I_B \geq I_A$	nitrogen dioxide $NO_2$
		formaldehyde $H_2CO$
		methanol $CH_3OH$

Auf diesem Webangebot gilt die Datenschutzerklärung der TU Braunschweig mit Ausnahme der Abschnitte VI, VII und VIII.