Selection rules for rotational spectroscopy

Which of the foUowing molecules wih absorb or emit light due to rotational energy transitions  

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Linear molecules also have the z component of their total angular momentum quantized, and if the total angular momentum is changing by 1 unit, then that unit may be in the z-axis. But then again, it may not, so the M quantum number might either stay the same or change by 1 unit. Therefore, the selection rule for Mj is written as 

We have already seen that symmetric top molecules have a second quantum number, K, used to define the quantized figure-axis component of the angular momentum. Interaction of the light field with the molecule s dipole, which must lie on the figure axis, is such that the electromagnetic field cannot change the molecule s rotation with respect to the figure axis. Therefore, the quantum number K is used to define the selection rule 

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Finally, for the determination of selection rules for rotational spectroscopy it is necessary to find the wavefimcdons for this problem. This subject will be left for further development as given in numerous texts on molecular spectroscopy. 

Although the most fundamental selection rule for rotational spectroscopy is that the molecule should have a nonvanishing permanent dipole moment, we note that molecules without a permanent dipole moment can have perturbation-allowed rotational spectrum [22]. For spherical tops, for example, centrifugal distortion effects can produce a small permanent dipole moment that allows the observation of the rotational spectrum [1, 37]. 

Both infrared and Raman spectra are concerned with measuring molecular vibration and rotational energy changes. However, the selection rules for Raman spectroscopy are very different from those of infrared - a change of polarisability... 

Fortunately, molecular symmetry can help us, for 2.1, 2. Ill and 2.IV have different rotational axes and mirror planes. We will see in due course that this can affect the selection rules for vibrational spectroscopy techniques, which in turn affect how many peaks we would expect to observe in the spectra of the various forms. Symmetry also changes the numbers of equivalent atoms we have in a molecule, and this influences the number of peaks we would observe in, for example, a NMR spectrum. 

The selection rules for Raman transitions are different from those of absorption or emission, and this makes it possible to observe transitions that are forbidden in emission or absorption spectroscopy. The Raman selection rules for rotational and vibrational transitions are ... 

The selection rules for rotational transitions in linear polyatomic molecules are also the same as for diatomic molecules. The transition AJ is equal to 1 in infrared spectroscopy and -nl in purely rotational spectroscopy (i.e. microwave spectroscopy) but only if the molecule has a non-zero dipole moment (see Section 6.7). A rotational transition for H-C=C-C1 will be observed whereas for H-C=C-H no rotational transition will be observed due to its zero dipole moment. 

This spectrum is called a Raman spectrum and corresponds to the vibrational or rotational changes in the molecule. The selection rules for Raman activity are different from those for i.r. activity and the two types of spectroscopy are complementary in the study of molecular structure. Modern Raman spectrometers use lasers for excitation. In the resonance Raman effect excitation at a frequency corresponding to electronic absorption causes great enhancement of the Raman spectrum. 

The methyl iodide molecule is studied using microwave (pure rotational) spectroscopy. The following integral governs the rotational selection rules for transitions labeled J, M, K... 

Beyond such electronic symmetry analysis, it is also possible to derive vibrational and rotational selection rules for electronic transitions that are El allowed. As was done in the vibrational spectroscopy case, it is conventional to expand i j (R) in a power series about the equilibrium geometry of the initial electronic state (since this geometry is more characteristic of the molecular structure prior to photon absorption) ... 

Spectroscopy is concerned with the observation of transitions between stationary states of a system, with the accompanying absorption or emission of electromagnetic radiation. In this section we consider the theory of transition probabilities, using time-dependent perturbation theory, and the selection rules for transitions, particularly those relevant for rotational spectroscopy. 

As in the case of absorption and fluorescence emission spectroscopy, selection rules apply for the Raman transitions between rotational energy levels. However, since two photons are involved in the process, each of angular momentum Lphoton = 1. angular momentum conservation requires that the difference between the initial and final rotational levels must be two. As the selection rule for pure Raman spectra, one finds... 

Molecules that don t have a permanent dipole moment are rotating, of course, but they do not follow the gross selection rule for pure rotational spectra. Their rotations cannot be observed directly using microwave spectroscopy. 

As with rotational spectroscopy, there are several ways of stating selection rules for spectral transitions involving vibrational states of molecules. There is a gross selection rule, which generalizes the appearance of absorptions or emissions involving vibrational energy levels. There is also a more specific, quantum-number-based selection rule for allowed transitions. Finally, there is a selection rule that can be based on group-theoretical concerns, which were not considered for rotations. 

Several other forms of nonlinear spectroscopy have been developed that are not strictly based on Raman-active vibrations or rotations. Degenerate four-wave mixing (DFWM) is a technique where all input and output frequencies are identical. Because it does not involve the generation of light at new frequencies, it can rely on non-local mechanisms other than the local electronic polarizability (e.g. electrostric-tion). The selection rules for DFWM are closely related to those of one-photon techniques (e.g. absorption). DFWM using infrared beams is therefore used to probe infrared absorbing transitions instead of Raman-active transitions. 

The selection rules for atomic spectra are determined in the same way as it was done previously for vibration-rotation spectroscopy (see Section 6.7). For a hydrogen atom, the dipole moment operator of the incoming photon, H, is operated on and integrated over all space between the initial state n, /, m, with the final state n , m/ . Allowed transitions occur when the integral... 

The dipole and polarization selection rules of microwave and infrared spectroscopy place a restriction on the utility of these techniques in the study of molecular structure. However, there are complementary techniques that can be used to obtain rotational and vibrational spectrum for many other molecules as well. The most useful is Raman spectroscopy. 

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.)... 

In many cases, the infrared and Raman rotation-vibration spectra contribute complementary structure data, particularly for highly symmetric molecules. Due to the significantly different selection rules a greater line density is observed for Raman due to a larger selection of allowed changes in the rotational energy compared to infrared gas spectra. Raman spectroscopy is, on these grounds, also a valuable supplement to infrared studies. 

Rotational features of almost aU H-bonded complexes in the gaseous phase appear in the microwave region, with wavenumbers less than 10 cm They correspond to transitions between pure rotational levels, pure meaning that vibrations remain unchanged, or no vibrational transition accompanies such rotational transitions. Rotational features, however, also appear in the IR spectra of these H-bonded complexes. IR bands correspond to transitions between various vibrational levels of a molecule. When this molecule is isolated, as in the gas phase, these transitions are always accompanied by transitions between rotational levels that obey the same selection rules as pure rotational transitions detected in microwave spectroscopy. The information conveyed by these rotational features in IR spectra are therefore most similar to those conveyed by microwave spectra, even if the mechanism at the origin of their appearance is different. Their interests lie in the use of an IR spectrometer, a common instrument in many laboratories, instead of a microwave spectrometer, which is a much more specialized instrament. However, the resolution of usual IR spectrometers are lower than that of microwave spectrometers that use Fabry-Perot cavities. This IR technique has been used in the case of simple H-bonded dimers with relatively small moments of inertia, such as, for instance, F-H- -N C-H (3). Such complexes are far from simple to manipulate, but provide particularly simple IR spectra with a limited number of bands that do not show any overlap. 

Pure rotational transitions of symmetrical diatomic molecules like dihydrogen are forbidden in infrared spectroscopy by the dipole selection rule but are active in Raman spectroscopy because they are anisotropically polarisable. They are in principle observable in INS although the scattering is weak except for dihydrogen. These rotational transitions offer the prospect of probing the local environment of the dihydrogen molecule, as we shall see in this chapter. 

The symmetry of an isolated atom is that of the full rotation group R+ (3), whose irreducible representations (IRs) are D where j is an integer or half an odd integer. An application of the fundamental matrix element theorem [22] tells that the matrix element (5.1) is non-zero only if the IR DW of Wi is included in the direct product x of the IRs of ra and < f. The components of the electric dipole transform like the components of a polar vector, under the IR l)(V) of R+(3). Thus, when the initial and final atomic states are characterized by angular momenta Ji and J2, respectively, the electric dipole matrix element (5.1) is non-zero only if D(Jl) is contained in Dx D(j 2 ) = D(J2+1) + T)(J2) + )(J2-i) for j2 > 1 This condition is met for = J2 + 1, J2, or J2 — 1. However, it can be seen that a transition between two states with the same value of J is allowed only for J 0 as DW x D= D( D(°) is the unit IR of R+(3)). For a hydrogen-like centre, when an atomic state is defined by an orbital quantum number , this can be reduced to the Laporte selection rule A = 1. This is of course formal, as it will be shown that an impurity state is the weighted sum of different atomic-like states with different values of but with the same parity P = ( —1) These states are represented by an atomic spectroscopy notation, with lower case letters for the values of (0, 1, 2, 3, 4, 5, etc. correspond to s, p, d, f, g, h, etc.). The impurity states with P = 1 and -1 are called even- and odd-parity states, respectively. For the one-valley EM donor states, this quasi-atomic selection rule determines that the parity-allowed transitions from Is states are towards np (n > 2), n/ (n > 4), nh (n > 6), or nj (n > 8) states. For the acceptor states in cubic semiconductors, the even- and odd-parity states labelled by the double IRs T of Oh or Td are indexed by + or respectively, and the parity-allowed transition take place between Ti+ and... 

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