Diatomic molecules vibration-rotation spectra

We previously found the selection rule A7 = 1 for a 2 diatomic-molecule vibration-rotation or pure-rotation transition. The rule (4.138) forbids A/ = 1 for homonuclear diatomics this gives us no new information as far as vibration-rotation spectra are concerned, since the absence of a dipole moment insures the absence of a vibration-rotation or pure-rotation spectrum, anyway. 

Diatomic molecules provide a simple introduction to the relation between force constants in the potential energy function, and the observed vibration-rotation spectrum. The essential theory was worked out by Dunham20 as long ago as 1932 however, Dunham used a different notation to that presented here, which is chosen to parallel the notation for polyatomic molecules used in later sections. He also developed the theory to a higher order than that presented here. For a diatomic molecule the energy levels are observed empirically to be well represented by a convergent power-series expansion in the vibrational quantum number v and the rotational quantum number J, the term... 

Vibrationally excited diatomic molecules will only emit if they are polar, and most of the available results are for reactions which produce diatomic hydrides. Because of their unusually small reduced mass, these molecules have high frequency and very anharmonic vibrations, and their rotational levels are widely spaced. Consequently, their spectra can be resolved more easily than those of nonhydrides, where there are many more individual lines in the vibration-rotation spectrum. Furthermore, the molecular dynamics of these reactions are particularly interesting because of the special kinematic features that arise when an H atom is involved in a reactive collision and because these... 

Figure 3.9 Energy levels and transitions giving rise to vibration-rotation spectra in a diatomic molecule. The upper state and lower state quantum numbers are /, J ) and v", J"), respectively. The schematic spectrum at bottom shows line intensities weighted by rotational state populations, which are proportional to 2J" + 1) exp[—hcBJ J" + 1 )/kT]. The rotational constants B - and B. are assumed to be equal, resulting in equally spaced rotational lines. This assumption is clearly not valid in the HCI vibration-rotation spectrum in Fig. 3.3.

Figure 6.14 Rotational fine structure in a parallel vibrational band for a prolate symmetric top in which (A B ) - (A" - B") is small >1" = 5.28cm" A = 5.26 cm" , and B" = B = 0.307 cm V The origin positions are closely spaced, and the spectrum resembles the vibration—rotation spectrum of a diatomic molecule. Horizontal energy scale is in cm V...

Fig. 1.22. The vibrational-rotational energy levels and the vibrational-rotational spectrum of a diatomic molecule. To show the relations, the spectra are positioned so that the zero wavenumber point is on the energy level from which the transition originates. The vertical coordinate is shown in cm" or Elhc. ...

Thus, the vibration-rotation spectrum of a rigid diatomic molecule consists of a series of equally spaced lines above and below vq that correspond to A7 = +1 and AJ = — 1, respectively. The series of lines below vq AJ = — 1) is known as the P branch of the band, while the lines above vq AJ = +1) are known as the R branch. Because AJ 0, there is no absorption line at Vq. ... 

Write a computer program to calculate the relative intensities of the spectral fines in the fundamental band of the vibration-rotation spectrum of a diatomic molecule, assuming that the absorbance is displayed in the spectrum. Set the maximum absorbance of the first line of the P branch equal to 1. Assume the Boltzmann probability distribution and assume that the transition dipole moments for all transitions are equal. Use your program to calculate the relative intensities for the first 15 lines in each branch of the HCl spectrum... 

Laser excitation is not only a means for providing the necessary energy for chemistry to take place. We begin the discussion of the selectivity achievable by the excitation and the specificity of the detection for the special case of diatomic molecules. Then the spectrum is simpler in that there is only one vibrational mode as well as only one rotational constant. But once the molecule is electronically excited we will find it necessary to allow for the breakdown of the Born-Oppenheimer separation. 

In low-resolution infrared spectroscopy of diatomic molecules, the rotational fine structure is lost, and some feature in a spectrum assigned to be a fundamental transition is but a single peak. The frequency of that peak is taken to be the vibrational frequency in the absence of any more precise experiments, and that is the extent of the information obtained. This low-resolution information corresponds mostly with a nonrotating picture or else a rotationally averaged picture of the molecule s dynamics to the extent that we can analyze the data, we need only consider pure vibration. To understand the internal dynamics of polyatomic molecules, it is helpful to start with a low-resolution analysis. This means neglecting rotation, or presuming the molecules to be nonrotating. 

Figure 6.7 Rotational transitions accompanying a vibrational transition in (a) an infrared spectrum and (b) a Raman spectrum of a diatomic molecule...

Absorption and Emission Spectra of Small Molecules. In diatomic molecules the number of vibrational and rotational levels is small, so that their energy spacing remains relatively large. Their absorption spectra are therefore line spectra which correspond to transitions to stable , associative excited states, but if a dissociative excited state is reached then the absorption spectrum becomes a continuum since such states have no vibrational levels. 

The vibration-rotation hamiltonian of a polyatomic molecule is more complicated than that of a diatomic molecule, both because of the increased number of co-ordinates, and because of the presence of Coriolis terms which are absent from the diatomic hamiltonian. These differences lead to many more terms in the formulae for a and x values obtained from the contact transformation, and they also lead to various kinds of vibrational and rotational resonance situations in which two or more vibrational levels are separated by so small an energy that interaction terms in the hamiltonian between these levels cannot easily be handled by perturbation theory. It is then necessary to obtain an effective hamiltonian over these two or more vibrational levels, and to use special techniques to relate the coefficients in this hamiltonian to the observed spectrum. 

Schematic vibrational-rotational infrared spectrum for a diatomic molecule.

The allowed changes in the rotational quantum number J are AJ = 1 for parallel (2 ) transitions and A7= 0, 1 for perpendicular (II ) transitions. Parallel transitions such as for acetylene thus have P i J= 1) and R(AJ = +1) branches with a characteristic minimum between them, as shown for diatomic molecules such as HCl in Fig. 37-3 and for the HCN mode in Fig. 2. However, perpendicular transitions such as Vs for acetylene and V2 for HCN (Fig. 2) have a strong central Q branch (AJ = 0) along with P and R branches. This characteristic PQR-Yersus-PR band shape is quite obvious in the spectrum and is a useful aid in assigning the symmetries of the vibrational levels involved in the infrared transitions of a hnear molecule. 

As an example of the rotation-vibration band of a diatomic molecule, the nitrogen-broadened spectrum of C 0 is shown in Fig. 4.3-2. An additional band appears, less intense, but shifted, which is attributed to the isotope. Fig. 4.3-3 displays an... 

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