Pure rotation spectra

The selection rules governing allowed changes in the rotational quantum number J (and K in the case of the symmetric top) depend on whether changes are taking place in other quantized molecular properties at the same time. They are therefore different for a pure rotational transition, for a vibrational transition with associated rotational changes, or for an electronic transition with associated vibrational and rotational changes. These selection rules are all based on symmetry, but here we simply present the results, rather than attempt to use symmetry to derive them. 

The basic requirement for the observation of a pure rotation spectrum is that there should be an oscillating dipole associated with a molecular rotation. To a lirst approximation this is the same as saying that the molecule must have a permanent dipole. Only molecules of point groups or C (including the low-symmetry cases Cl and Cs) can have permanent dipole moments, so molecules belonging to these point groups will all have rotational spectra. 

Rotational spectrum of a rigid non-symmetric linear molecule. 

The selection rules for pure rotational transitions of symmetric tops are A7 = 1, AAi = 0 for direct absorption or emission, and A/= 1 or 2, AAi = 0 for the Raman effect. We obtain simple spectra in both cases, with a single series of lines (A/= +1) in absorption and two series (A7 = +1 and +2) in the Raman effect. Neglecting centrifugal distortion, these series have constant spacings of 2B or AB, and lines for all values of K coincide. If there is centrifugal distortion, separate lines can be observed for the different K values that are possible for each value of J, with frequencies (for A7 = 1) given by 

The pure rotation spectrum of an asymmetric top is very complex, and cannot be reduced to a formula giving line positions. Instead, it has to be dealt with by calculation of the appropriate upper and lower state energies (Section 7.2.2). The basic selection rule, A7 = 0, 1, applies to absorption/emission spectra, and there are other selection rules. These depend on the symmetry of the inertial ellipsoid, which is always Dan, but the orientations of the dipole moment components depend on the symmetry of the molecule itself. For the rotational Raman effect A7= 2 transitions are allowed as well. The selection rules for pure rotational spectra are described in more detail in the on-line supplement for Chapter 7. 

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Microwave spectra (giving pure rotational spectra) are especially usefiil for the detection of interstellar molecular ions (in some cases the microwave spectrum has first been observed in interstellar spectra ). 

This vibrational cooling is sufficient to stabilize complexes that are weakly bound by van der Waals or hydrogen-bonding forces. The pure rotational spectra and structure of species such as... 

Quadrupole coupling constants for molecules are usually determined from the hyperfine structure of pure rotational spectra or from electric-beam and magnetic-beam resonance spectroscopies. Nuclear magnetic resonance, electron spin resonance and Mossbauer spectroscopies are also routes to the property. There is a large amount of experimental data for and halogen-substituted molecules. Less data is available for deuterium because the nuclear quadrupole is small. 

Z. f. Physik, vol. 34, p. 227 (1925), from pure rotation spectra with half quantum numbers. 

For a polar molecule (/xo 0) the first term on the far right is nonzero only if the initial and final vibrational states are the same, viz. v = u. This case applies to the pure rotational spectra of gaseous molecules, as observed in the microwave region. The second term in Eq. (98) applies to vibrational transitions. The matrix elements of interest are (t> jc u), which are given by... 

Pure rotational spectra only appear for molecules with permanent dipole moments and vibrational spectra require a change of dipole during the motion. However, electronic spectra are observed for all molecules, and changes in the electron distribution in a molecule are always accompanied by dipole changes. As a result even homonuclear molecules (H2 or N2) which have no rotation or vibration spectra, do give electronic spectra with vibrational and rotational structure from which rotational constants and bond vibration frequencies may be derived. 

Kratzer and Loomis as well as Haas (1921) also discussed the isotope effect on the rotational energy levels of a diatomic molecule resulting from the isotope effect on the moment of inertia, which for a diatomic molecule, again depends on the reduced mass. They noted that isotope effects should be seen in pure rotational spectra, as well as in vibrational spectra with rotational fine structure, and in electronic spectra with fine structure. They pointed out the lack of experimental data then available for making comparison. 

Low energy photons in the far IR can only modify the term ERol. This leads to pure rotational spectra that can be easily studied for small diatomic gases. However, in the mid IR, photons have sufficient energy to modify Vib and Fr,. This leads to vibrational-rotational spectra (Fig. 10.6). Each vibrational transition is accompanied by tens of individual rotational transitions. The molecule becomes an oscillating rotor for which energy VR approximately corresponds to the following values, where 7 (./ = 0, 1, 2, 3,...) and V (V = 0, 1, 2) are the rotational and vibrational quantum numbers, respectively. 

Prolate symmetric top, 199, 211 Propane, dipole moment of, 225 Proper axis of symmetry, 53 Proper rotation, 395-396 Proton, 178 Pseudovector, 434 Pulse laser, 137,139 Purcell, E. M., 328, 360 Purely electronic energy, 57 Pure-rotation spectra, 165... 

We now consider the pure-rotation spectra of gaseous diatomic molecules. As noted in the previous section, a diatomic molecule with an electric dipole moment can undergo a pure-rotational radiative transition with AJ = +1 or — 1, corresponding to absorption or emission. Rotational spectra are studied as absorption spectra. For a transition with the initial and final rotational quantum numbers J and J+1, respectively, and with no change in electronic or vibrational levels, we find from (4.67)... 

Analysis of the rotational fine structure of IR bands yields the moments of inertia 7°, 7°, and 7 . From these, the molecular structure can be fitted. (It may be necessary to assign spectra of isotopically substituted species in order to have sufficient data for a structural determination.) Such structures are subject to the usual errors due to zero-point vibrations. Values of moments of inertia determined from IR work are less accurate than those obtained from microwave work. However, the pure-rotation spectra of many polyatomic molecules cannot be observed because the molecules have no permanent electric dipole moment in contrast, all polyatomic molecules have IR-active vibration-rotation bands, from which the rotational constants and structure can be determined. For example, the structure of the nonpolar molecule ethylene, CH2=CH2, was determined from IR study of the normal species and of CD2=CD2 to be8... 

Recall that homonuclear diatomic molecules have no vibration-rotation or pure-rotation spectra due to the vanishing of the permanent electric dipole moment. For electronic transitions, the transition-moment integral (7.4) does not involve the dipole moment d hence electric-dipole electronic transitions are allowed for homonuclear diatomic molecules, subject to the above selection rules, of course. [The electric dipole moment d is given by (1.289), and should be distinguished from the electric dipole-moment operator d, which is given by (1.286).] Analysis of the vibrational and rotational structure of an electronic transition in a homonuclear diatomic molecule allows the determination of the vibrational and rotational constants of the electronic states involved, which is information that cannot be provided by IR or microwave spectroscopy. (Raman spectroscopy can also furnish information on the constants of the ground electronic state of a homonuclear diatomic molecule.)... 

Most atmospheric visible and DV absorption and emission involves energy transitions of the outer electron shell of the atoms and molecules involved. The infrared spectrum of radiation from these atmospheric constituents is dominated by energy mechanisms associated with the vibration of molecules. The mid-infrared region is rich with molecular fundamental vibration-rotation bands. Many of the overtones of these bands occur in the near infrared. Pure rotation spectra are more often seen in the far infrared. Most polyatomic species found in the atmosphere exhibit strong vibration-rotation bands in the 1 - 25 yin region of the spectrum, which is the region of interest in this paper. The richness of the region for gas analysis... 

Absorption of microwave radiation to excite molecular rotation is allowed only if the molecule has a permanent dipole moment. This restriction is less severe than it may sound, however, because centrifugal distortion can disturb the molecular symmetry enough to allow weak absorption, especially in transitions between the higher rotational states which may appear in the far IR (c. 100cm-1). Microwave spectroscopy can provide a wealth of other molecular data, mostly of interest to physical chemists rather than inorganic chemists. Because of the ways in which molecular rotation is affected by vibration, it is possible to obtain vibrational frequencies from pure rotational spectra, often more accurately than is possible by direct vibrational spectroscopy. 

Although the interpretation of rotational spectra of diatomic molecules is relatively simple, such spectra lie in the far infrared, a region that at present is not as easily accessible to study as are the near infrared, visible, cr ultraviolet. Consequently, most information about rotational energy levels has actually been obtained, not from pure rotation spectra, but from rotation-vibration spectra. Molecules without dipole moments have no rotation spectra, and nonpolar diatomic molecules lack rotation-vibration spectra as well, Thus, II2, N2, 02, and the molecular halogens have no characteristic infrared spectra. Information about the vibrational and rotational energy levels of these molecules must be obtained from the fine structure of their electronic spectra or from Raman spectra. 

Information Derived From Pure Rotational Spectra.77... 

It will be clear that pure rotational spectra of more complex orbital and spin states, most of which arise in molecules containing transition metal atoms, are still relatively sparse. This will almost certainly change as experimental techniques develop further a further stimulus is the growing recognition of the importance of these molecules in interstellar and circumstellar space. 

Again molecules with centre of symmetry have no dipole moment and thus do not show any pure rotational spectra. 

Most fundamental rotation-vibration bands are located in the mid-infrared region from 4000 - 400 cm". A few vibrational bands appear in the far infrared where purely rotational spectra of light molecules with two or three atoms are also observed. This is in contrast to heavier polyatomic molecules the study of their rotational spectra is the domain of the microwave spectroscopist who employs different equipment, particularly, monochromatic tunable radiation sources. Rotational constants determined from IR-work are therefore usually less accurate than those obtained by microwave spectroscopy. 

Rusk and Gordy have investigated the pure rotational spectra of KBr in the 1.5 to 5.0 mm range of the microwave region... 

Typical energy-level schemes for dipolarly unstable systems with indications of the allowed rotational, tunneling, and tunneling-rotational transitions in two limit cases, when the rotational frequency is larger than the tunneling one and when the inverse inequality takes place, are presented in Figs. 1 and 2. The expected pure rotational spectra for three concrete sets of parameter values are shown in Figs. 3 to 5. If A = 0 (or... 

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