Diatomic molecules rotational spectra

As mentioned in section B. 1., only one rotational constant, (B + C), per molecule is available from the observed a-type spectrum (AK = 0, AJ = 1), and the results are given in Table 1. (B + C)/2 is essentially an effective diatomic molecule rotational constant, and therefore depends most strongly on the distance between the monomers of the complex. The rotational constants in Table 1 were best fitted with a model in which O. .. H—O distances were 2.67 A and O. .. H—N distances were 2.71 A. These O. .. H—O distances are somewhat shorter than the 2.73 A distance in formic acid dimer and the 2.76A distances in acetic acid dimer68. However, they are... 

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

The spectrum calculated in the secular non-adiabatic approximation reproduces some special peculiarities of the spectra observed. In following papers (see Table 7.1) for diatomic molecules the dependence of the resolved spectra components on the rotational quantum number was described. As an example, the experimental dependence %(j) = Tj+1. /r is shown in Fig. 7.5. 

The rotational microwave spectrum of a diatomic molecule has absorption lines (expressed as reciprocal wavenumbers cm ) at 20, 40, 60, 80 and 100 cm . Calculate the rotational partition function at 100 K from its fundamental definition, using kT/h= 69.5 cm" at 100 K. 

Within this "rigid rotor" model, the absorption spectrum of a rigid diatomic molecule should display a series of peaks, each of which corresponds to a specific J ==> J+l transition. The energies at which these peaks occur should grow linearally with J. An example of such a progression of rotational lines is shown in the figure below. 

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. 

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. 

As we saw in Chapter 1, the importance of numbers in chemistry derives from the fact that experimental measurement of a particular chemical or physical property will always yield a numerical value to which we attach some significance. This might involve direct measurement of an intrinsic property of an atom or molecule, such as ionization energy or conductivity, but, more frequently, we find it necessary to use theory to relate the measured property to other properties of the system. For example, the rotational constant, B, for the diatomic molecule CO can be obtained directly from a measurement of the separation of adjacent rotational lines in the infrared spectrum. Theory provides the link between the measured rotational constant and the moment of inertia, I, of the molecule by the formula ... 

The distinction between a truly continuous absorption spectrum and a banded absorption spectrum for diatomic molecules may be made by instruments of relatively low resolving power. Even though individual rotational lines are not resolved, a discrete spectrum will have sharp band heads and the appearance will in no way resemble the appearance of a continuum. 

Measurement and assignment of the rotational spectrum of a diatomic or other linear molecule result in a value of the rotational constant. In general, this will be B0. which relates... 

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... 

The conclusion is that if the spectrum can be analysed in terms of equations (3)—(7), then the force constants can be determined. The bond length re can be determined from the equilibrium rotational constant Bc then the quadratic force constant /3 can be determined either from the harmonic wavenumber centrifugal distortion constant De then the cubic force constant /3 can be determined from aB and finally the quartic force constant /4 can be determined from x. It is necessary to determine the force constants in this order since in each case we depend upon already knowing the preceding constants of lower order. The values of re,f2,f3, and /4 calculated in this way for a number of diatomic molecules are shown in Table 2. 

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. 

Spectral studies of rotational energy levels have proved most profitable for linear molecules having dipole moments, particularly diatomic molecules (for example, CO, NO, and the hydrogen halides). The moment of inertia of a linear molecule may be readily obtained from its rotation spectrum and for diatomic molecules, interatomic distances may he calculated directly from moments of inertia (Exercise 14d). For a mole-... 

In practice values of B are also often quoted in cm-1. For the simple rigid rotor the rotational quantum number J takes integral values, J = 0, 1, 2, etc. The rotational energy levels therefore have energies 0, 2B, 6B, 12B, etc. Elsewhere in this book we will describe the theory of electric dipole transition probabilities and will show that for a diatomic molecule possessing a permanent electric dipole moment, transitions between the rotational levels obey the simple selection rule A J = 1. The rotational spectrum of the simple rigid rotor therefore consists of a series of equidistant absorption lines with frequencies 2B, 4B, 6B, etc. 

There can be no question that the most important species with a 3 E ground state is molecular oxygen and, not surprisingly, it was one of the first molecules to be studied in detail when microwave and millimetre-wave techniques were first developed. It was also one of the first molecules to be studied by microwave magnetic resonance, notably by Beringer and Castle [118]. In this section we concentrate on the field-free rotational spectrum, but note at the outset that this is an atypical system O2 is a homonuclear diatomic molecule in its predominant isotopomer, 160160, and as such does not possess an electric dipole moment. Spectroscopic transitions must necessarily be magnetic dipole only. 

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