Transition frequencies or wavenumbers

In practice, what is measured experimentally is not energy but frequency, in the millimetre wave and microwave regions, or wavenumber, in the far infrared. Therefore we convert the energy levels of Equation (5.10) to what are known as term values F J) having dimensions of either frequency, by dividing by h, or wavenumber, by dividing by he, giving 

The use of the symbols F J) and B for quantities which may have dimensions of frequency or wavenumber is unfortunate, but the symbolism is used so commonly that there seems little prospect of change. In Equations (5.11) and (5.12) the quantity B is known as the rotational constant. Its determination by spectroscopic means results in determination of intemuclear distances and represents a very powerful structural technique. 

The transition intensity is proportional to the square of the transition moment, which is given by 

The molecule must have a permanent dipole moment (pi 0). 

Rule 1 shows that transitions are allowed in heteronuclear diatomic molecules such as CO, 

AMj = 0, 1, a rule which is important only if the molecule is in an electric or magnetic field (see Equation 1.61). 

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The ubiquitous use of the word Tine to describe an experimentally observed transition goes back to the early days of observations of visible spectra with spectroscopes in which the lines observed in, say, the spectmm of a sodium flame are images, formed at various wavelengths, of the entrance slit. Although, nowadays, observations tend to be in the form of a plot of some measure of the intensity of the transition against wavelength, frequency or wavenumber, we still refer to peaks in such a spectmm as lines. 

The energy of a particular transition is, therefore, proportional to the frequency or wavenumber (v = 1 /A) and inversely proportional to the wavelength (equation 1.2). 

In regions of the spectrum where a tunable laser is available it may be possible to use it to obtain an absorption spectrum in the same way as a tunable klystron or backward wave oscillator is used in microwave or millimetre wave spectroscopy (see Section 3.4.1). Absorbance (Equation 2.16) is measured as a function of frequency or wavenumber. This technique can be used with a diode laser to produce an infrared absorption spectrum. When electronic transitions are being studied, greater sensitivity is usually achieved by monitoring secondary processes which follow, and are directly related to, the absorption which has occurred. Such processes include fluorescence, dissociation, or predissociation, and, following the absorption of one or more additional photons, ionization. The spectrum resulting from monitoring these processes usually resembles the absorption spectrum very closely. 

The use of infra-red or ultraviolet spectroscopy to examine the molecular groups present in a chemical compound is familiar to any chemist. One of the main uses of this technique is to apply a range of electromagnetic frequencies to a sample and thus identify the frequency at which a process occurs. This can be characteristic of, say, the stretch of a carbonyl group or an electronic transition in a metal complex. The frequency, wavelength or wavenumber at which an absorption occurs is of most interest to an analytical chemist. In order to use this information quantitatively, for example to establish the concentration of a molecule present in a sample, the Beer-Lambert law is used ... 

Included in Table III is the comparison of the transition frequencies calculated from the energies obtained in our calculations with the experimental transition frequencies of Dabrowski [125]. To convert theoretical frequencies into wavenumbers, we used the factor of 1 hartree = 219474.63137 cm . For all the frequencies our results are either within or very close to the experimental error bracket of 0.1 cm . We hope that the advances in high-resolution spectroscopy will inspire remeasurements of the vibrational spectrum of H2 with the accuracy lower than 0.1 cm. With such high-precision results, it would be possible to verify whether the larger differences between the calculated and the experimental frequencies for higher excitation levels, which now appear, are due to the relativistic and radiative effects. 

Calculate the energy associated with transitions with the following frequencies, wavelength or wavenumber. What type of molecular transition is associated with each transition (a) V = 3 X IO Hz (b) 2 = 254 nm (254X lO m) (c) 2 — l.O cm ... 

Infrared Absorption is a single-photon process. Here, also, kiR = K 0 applies. Thus, infrared absorption detects only phonons at the F point of the first BZ. In this case, we have oo = L2, where ho) is the quantum energy of the infrared radiation. The frequencies or the wavenumbers of the optical phonons in molecular crystals are of the order of 3 THz or 100 cm" thus the wavelengths of infrared absorption are of the order of 100 /xm. Infrared spectroscopy of phonons in molecular crystals is therefore in fact far-infrared spectroscopy. The symmetry selection rules are complementary to those for Raman scattering for vibrations with u and g states w g transitions are allowed and g g transitions are forbidden. 

Infrared (IR) spectra were recorded on a Perkin-Elmer 281 IR spectrophotometer. The observed frequencies are expressed in wavenumbers (cm ) using the 1601 cm line of a polystyrene film as a standard. Spectra of oils and liquids were obtained neat as a smear on sodium chloride or potassium chloride plates, and those of solids were performed by using Nujol mulls or KBr pellets. Vibrational transition frequencies are reported in wavenumbers (cm, with the intensity of the bands being assigned the following classifications week (w), medium (m), shoulder (sh), strong (s), broad (br). 

So far as rule 2 is concerned, since AJ is conventionally taken to refer to J -J", where J is the quantum number of the upper state and J" that of the lower state of the transition, AJ = — 1 has no physical meaning (although it emerges from the quantum mechanics). It is commonly, but incorrectly, thought that AJ = +1 and AJ = — 1 refer to absorption and emission, respectively in fact AJ = +1 applies to both. Transition wavenumbers or frequencies are given by... 

Raman spectra of adsorbed species, when obtainable, are of great importance because of the very different intensity distributions among the observable modes (e.g., the skeletal breathing frequency of benzene) compared with those observed by infrared spectroscopy and because Raman spectra of species on oxide-supported metals have a much wider metal oxide-transparent wavenumber range than infrared spectra. Such unenhanced spectra remain extremely weak for species on single-crystal surfaces, but renewed efforts should be made with finely divided catalysts, possibly involving pulsed-laser operation to minimize adsorbate decomposition. Renewed efforts should be made to obtain SER and normal Raman spectra characterizing adsorption on surfaces of the transition metals such as Ni, Pd, or Pt, by use of controlled particle sizes or UV excitation, respectively. 

Fig. 5.2 shows the dependence of n on v for T = 300 K, again with both frequency (Hz) and wavenumber (cm-1) used as abscissae. It is useful to recall that at 300 K, kTIh 6 x 1012 Hz or kT/hc 200 cm-1. This point is apparent in Fig. 5.2, since n = 1 at A-1 = 200 cm-1. Since the vacuum fluctuations, which lead to spontaneous emission, are given by n = 1/2, black body radiation at frequencies greater than kT/hn, where n 1, does not lead to significant effects. For an atom in its ground state with transitions at 104 cm-1, black body induced transitions are unimportant, since n 1. However, for a Rydberg state with transitions at 10 cm-1, where n 10, the black body induced transition rates can... 

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