2.9 Rigid Rotator (***) When we eventually study the structure and spectra of molecules, it will be a welcome surprise to find that the rotation of most diatomic molecules may be described quantum mechanically by the rigid rotator, a particularly simple system. What properties of the molecule can be physically observed? Rigid rotator and non-rigid rotator approximations. Rigid-Rotor model of diatomic molecule Measured spectra Physical characteristics of molecule Line spacing =2B B I r e Accurately! The rotational constant depends on the distance ($$R$$) and the masses of the atoms (via the reduced mass) of the nuclei in the diatomic molecule. The rotational energies for rigid molecules can be found with the aid of the Shrodinger equation. To second order in the relevant quantum numbers, the rotation can be described by the expression . 4. Rigid rotors can be classified by means of their inertia moments, see classification of rigid … &= \frac{\hbar^2}{2I}[2 + 3J_i + J_i^2 -J_i^2 - J_i]\\ The effect of isotopic substitution. Rotational Raman spectra. The selection rules for the rotational transitions are derived from the transition moment integral by using the spherical harmonic functions and the appropriate dipole moment operator, $$\hat {\mu}$$. Moment of Inertia and bond lengths of diatomic and linear triatomic molecule. Example: CO B = 1.92118 cm-1 → r CO = 1.128227 Å 10-6 Å = 10-16 m Ic h 8 2 2 r e Intensities of spectral lines 12 2. the bond lengths are fixed and the molecule cannot vibrate. Classification of molecules iii. The rotational motion of a diatomic molecule can adequately be discussed by use of a rigid-rotor model. K. P. Huber and G. Herzberg, Molecu-lar Spectra and Molecular Structure, Vol. The simplest of all the linear molecules like : H-Cl or O-C-S (Carbon Oxysulphide) as shown in the figure below:- 9. Hint: draw and compare Lewis structures for components of air and for water. Equation $$\ref{5.9.8}$$ predicts a pattern of exactly equally spaced lines. Demonstrate how to use the 3D regid rotor to describe a rotating diatomic molecules; Demonstate how microwave spectroscopy can get used to characterize rotating diatomic molecules ; Interprete a simple microwave spectrum for a diatomic molecule; To develop a description of the rotational states, we will consider the molecule to be a rigid object, i.e. Molecules are not rigid rotors – their bonds stretch during rotation As a result, the moment of inertia I change with J. How do we describe the orientation of a rotating diatomic molecule in space? The rotation of a rigid object in space is very simple to visualize. Isotope effect vii. Example: CO B = 1.92118 cm-1 → r CO = 1.128227 Å 10-6 Å = 10-16 m Ic h 8 2 2 r e These rotations are said to be orthogonal because one can not describe a rotation about one axis in terms of rotations about the other axes just as one can not describe a translation along the x-axis in terms of translations along the y- and z-axes. Linear Molecules. In this section we examine the rotational states for a diatomic molecule by comparing the classical interpretation of the angular momentum vector with the probabilistic interpretation of the angular momentum wavefunctions. In these cases the energies can be modeled in a manner parallel to the classical description of the rotational kinetic energy of a rigid object. This decrease shows that the molecule is not really a rigid rotor. Vibrational satellites . 05.20.-y. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Linear molecules behave in the same way as diatomic molecules when it comes to rotations. &= 2B(J_i + 1) \end{align*}\], Now we do a standard dimensional analysis, \[ \begin{align*} B &= \frac{\hbar^2}{2I} \equiv \left[\frac{kg m^2}{s^2}\right] = [J]\\ with $$J_i$$ and $$J_f$$ representing the rotational quantum numbers of the initial (lower) and final (upper) levels involved in the absorption transition. The rotational constant for 79 Br 19 F is 0.35717cm-1. We want to answer the following types of questions. Quantum theory and mechanism of Raman scattering. Rotation along the axis A and B changes the dipole moment and thus induces the transition. Missed the LibreFest? The Non-Rigid Rotor When greater accuracy is desired, the departure of the molecular rotational spectrum from that of the rigid rotor model can be described in terms of centrifugal distortion and the vibration-rotation interaction. Of \ ( \ref { 5.9.8 } \ ): rotation of a diatomic molecule regularly. 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