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5. Molecular rotation 5.1 Moments of inertia The key molecular parameter needed to describe molecular rotations is the moment of inertia, I, of the molecule defined as I mi ri 2 i Definition of moment of inertia. In this molecule there are three identical atoms attached to the B atom and three different but mutually identical atoms attached to the C atom. In this example, the center of mass lies on an axis passing through the B and C atom, and the perpendicular distances are measured from the axis. 5.2 Rotational properties The rotational properties of any molecule can be expressed in terms of the moment of inertia about three perpendicular axes set in the molecule. For linear molecules the moment of inertia around the internuclear axis is zero. (why??) An asymmetric rotor has three different moments of inertia; all three rotation axes coincide at the centre of mass of the molecule. 1 5.3 Rigid rotors We shall suppose initially that molecules are rigid rotors (i.e., do not distort under the stress of rotation). There are four types then: have one moment of inertia equal to zero (CO2, HCl, HCCH) (diatomics: I=mR2) have three equal moments of inertia (CH4, SF6) have two equal moments of inertia (NH3, CH3Cl) have three different moments of inertia (H2O, CH3OH) 5.4 Rotational energy levels The rotational energy levels of a rigid rotor may be obtained by solving the appropriate SE. However, there is a ‘short cut’ to the exact expressions: The classical energy of a body rotating about an axis a is E a 1 2 I a a 2 A body free to rotate about three axes has an energy E 1 1 1 2 2 2 I a a I bb I cc 2 2 2 Angular momentum about the axis a is 2 2 J a I aa , therefore 2 J J J E a b c 2I a 2I b 2I c 2 5.5 Spherical rotors When all three moments of inertia are equal to some value, I, (CH4, SF6), the classical expression for energy is J Jb Jc J2 E a 2I 2I 2 2 2 where J is the magnitude of the angular momentum. The quantum expression is obtained by therefore J 2 J ( J 1) 2 J=0, 1, 2, … 2 E J J ( J 1) 2I J=0, 1, 2, … 5.6 The rotational constant B Definition of the rotational constant B B 4cI EJ J ( J 1)hcB J=0, 1, 2, … rotational term (a wavenumber): F ( J ) BJ ( J 1) The rotational energy levels of a linear or spherical rotor. Note that the energy separation between neighboring levels increases as J increases. 3 5.7 Degeneracies The angular momentum of the molecule has a component on an external, laboratory-fixed axis, M J which is quantized: M J 0,1,..., J The significance of the quantum number MJ. (a) When MJ is close to its maximum value, J, most of the molecular rotation is around the laboratory z-axis. (b) an intermediate value of MJ. (c) When MJ=0 the molecule has no angular momentum about the z-axis. 5.8 Centrifugal distortion The effect of rotation on a molecule. The centrifugal force arising from rotation distorts the molecule, opening out bond angles and stretching bonds slightly. The effect is to increase the moment of inertia of the molecules and hence to decrease its rotational constant. The effect is taken into account empirically: F ( J ) BJ ( J 1) DJ J 2 ( J 1) 2 DJ: centrifugal distortion constant 4 5.9 Rotational transitions Typical values of B for ‘small’ molecules: 0.1 – 10 cm -1. Therefore, rotational transitions lie in the microwave region of the spectrum! A molecule must have a permanent electric dipole moment (it must be polar) for the observation of a pure rotational spectrum. Diatomic homonuclear and symmetric linear molecules as well as spherical rotors are normally inactive. 5.9.1 Rotational selection rules For a linear molecule the transition moment vanishes unless: J 1 M J 0,1 When a photon is absorbed by a molecule, the angular momentum of the combined system is conserved. If the molecule is rotating in the same sense as the spin of the incoming photon, then J increases by 1 (absorption). 5 5.10 The appearance of rotational spectra When selection rules are applied for a rigid symmetric or linear rotor allowed wavenumbers for J+1 J absorptions are: ( J 1 J ) 2 B( J 1) J=0, 1, 2, … The rotational energy levels of a linear rotor, the transitions allowed by the selection rule DJ=±1, and a typical pure rotational absorption spectrum. Measurement of n’s gives B and hence bond lengths in case of diatomic molecules. 5.11 The intensities of spectral lines The Boltzmann distribution implies that the population decays exponentially with J: N J Ng J e EJ kT and if the degeneracy gJ of level J is 2J+1 (linear rotor): N J (2 J 1)e hcBJ ( J 1) kT For a ‘typical’ molecule (OCS, B=0.2 cm -1) at room temperature, Jmax is 22. 6 5.12 Rotational Raman spectra An electric field applied to a molecule results in its distortion, and the distorted molecule acquires a contribution to its dipole moment (even if it is nonpolar initially). The polarizability may be different when the field is applied (a) parallel or (b) perpendicular to the molecular axis (or, in general, in different directions relative to the molecule); if that is so, then the molecule has an anisotropic polarizability. 5.12.1 Rotational Raman spectra-selection rules The distortions induced in a molecule by an applied electric field returns to its initial value after a rotation of only 180o (that is, twice a revolution). This is the origin of the DJ=±2 selection rule in rotational Raman spectroscopy. 7 5.12.1 Rotational Raman spectra-selection rules All linear molecules and diatomics (whether homonuclear or heteronuclear) have anisotropic polarizabilities, and so are rotationally Raman active. Note that spherical rotors (e.g. CH4, SF6) are rotationally Raman inactive as well as microwave inactive! Rotational Raman selection rules are: Linear rotors: J 0,2 5.12.2 Rotational Raman spectra – energy levels The rotational energy levels of a linear rotor and the transitions allowed by the DJ=±2 Raman selection rules. The form of a typical rotational Raman spectrum is also shown. The Rayleigh line is much stronger than depicted in the figure, it is shown as a weaker line to improve visualization of the Raman lines. 8