The Rotation and Vibration of Diatomic Molecules

When the coupling of the angular momenta, exclusive of nuclear spin, is described by Hund s case (b), three ways of including the nuclear spin may be considered. In the first, known as case the nuclear spin / is coupled to N, forming an intermediate F which is then coupled with S to form/ . The corresponding basis kets take the form r, A N, A, I, Fi F, S, F),hut they are unlikely to be used because the coupling of 5 to iV is invariably much stronger than that between / and N. 

In the second scheme to be considered, labelled case (bjg.s), S and / are coupled to form an intermediate G, which is then coupled with N to form F. This scheme is appropriate when the hyperfine interaction between S and / is strong compared with spin-rotation coupling, and we will meet it elsewhere, most notably in the Hj ion. The basis kets are written in the form ri. A S, I, G G, N, F) and the hyperfine matrix elements is this basis are calculated in later chapters. The most natural extension of Hund s case (b), known as case (bpj), is that in which/, theresultantofiVand5 coupling, is coupled with /to form F. The corresponding basis kets are r]. A, N, S, J /, /, F) and we will often meet matrix elements calculated in this basis. 

In chapter 2 we showed how the wave equation of a vibrating rotator was derived tinough a series of coordinate transformations. We discussed the solutions of this wave equation in section 2.8, and the particular problem of representing the potential in which tile nuclei move. We outlined the relatively simple solutions obtained for a harmonic oscillator, the corrections which are introduced to take accoimt of anharmonicity, and derived an expression for the rovibrational energies. Our treatment was relatively brief, so we now return to this subject in rather more detail. 

It is instructive to start with the simplest possible model of a rotating diatomic molecule, the so-called dumbbell model, as Uluslrated in figine 6.19. The two atoms, of masses mi and m2, are regarded as point-hke, and are fastened a distance R apart 

Substitution for R and Ri in equation (6.153) gives an important result for the moment 

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Figure 1. Translation, rotation, and vibration of a diatomic molecule. Every molecule has three translational degrees of freedom corresponding to motion of the center of mass of the molecule in the three Cartesian directions (left side). Diatomic and linear molecules also have two rotational degrees of freedom, about rotational axes perpendicular to the bond (center). Non-linear molecules have three rotational degrees of freedom. Vibrations involve no net momentum or angular momentum, instead corresponding to distortions of the internal structure of the molecule (right side). Diatomic molecules have one vibration, polyatomic linear molecules have 3V-5 vibrations, and nonlinear molecules have 3V-6 vibrations. Equilibrium stable isotope fractionations are driven mainly by the effects of isotopic substitution on vibrational frequencies.

This results from the mathematical fact that an exponential of a sum is a product of the exponentials of each of the terms in the sum, e.g., 6 + " = e e .) We should keep in mind that Equations 11.41 and 11.42 correspond to an approximation. For instance, vibrational-rotational coupling prevents strict separation of the vibrational and rotational parts of a molecular Hamiltonian. For now, we will use Equation 11.42 to obtain q via obtaining each of the elements in the product. We shall also rely on model problems presented in Chapters 7 and 8 as idealizations of rotation and vibration of a diatomic molecule. 

Isotope Effect. When an isotopic substitution is made in a diatomic molecule, the equilibrium bond length and the force constant k are unchanged, since they depend only on the behavior of the bonding electrons. However, the reduced mass ft does change, and this will affect the rotation and vibration of the molecule. In the case of rotation, the isotope effect can be easily stated. From the definitions of B and I, we see that... 

There are excellent coincidences between vibration rotation transitions of the NO fundamental in the X n and some strong CO-laser lines near 5.2 jim. If we shine a few watts of this co-laser light into an absorption cell containing NO and Ar, those y- and p-bands occur at IR-laser power densities of less than 1 kW/cm. This means that there must be a way for the energy that is put into the fundamental vibration of the diatomic molecule to get up the ladder of vibrational states to the level of electronic excitation or to the dissociation limit [3,2/3,3]. For a diatomic molecule, particularly at these low power densities, multiphoton excitation is not possible. 

In this paper we investigate, in relation to the previous paper, the influence of the electron spin on the stationary states of a diatomic molecule in the 5-state. Since the rotation and vibration of the nuclei perturb the electronic motion, as it would take place for fixed nuclei, only a little, we first compute the influence of the electron spin for fixed nuclei. The x-axis lies through the nuclei and the position of the fcth electron is given by Zfc. The number of electrons is n, the resulting... 

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. 

It should be noted that, in addition to possessing electronic and vibrational energy, a diatomic molecule may possess kinetic energy associated with the rotation and translation of the molecule as a whole. These two additional terms will be of little concern to us in most of this discussion and will be taken into account only when special need arises. 

Aside from vibration and rotation constants, an important piece of information available from electronic spectra is the dissociation energies of the states involved. In electronic absorption spectroscopy, most of the diatomic molecules will originate from the c"=0 level of the ground electronic state. The vibrational structure of the transition to a given excited electronic state will consist of a series of bands (called a progression) representing changes of 0—>0, 0—>1, 0- 2,..., 0— t nax, where... 

The microwave experiment studies rotational structure at a given vibrational level. The spectra are analyzed in terms of rotational models of various symmetries. The vibration of a diatomic molecule is, for instance, approximated by a Morse potential and the rotational frequencies are related to a molecular moment of inertia. For a rigid classical diatomic molecule the moment of inertia I = nr2 and the equilibrium bond length may be calculated from the known reduced mass and the measured moment, assuming zero centrifugal distortion. 

In order to determine the vibrational and rotational states of molecules, we shall consider the case of the diatomic molecule. To a first approximation we shall assume that the vibrations are harmonic (see Chapter i equations 1.12 et seq). We have... 

As shown in Chapter 17, the rotations and vibrations of diatomic or polyatomic molecules make additional contributions to the energy. In a monatomic gas, these other contributions are not present thus, changes in the total internal energy AU measured in thermodynamics can be equated to changes in the translational kinetic energy of the atoms. If n moles of a monatomic gas is taken from a temperature Ti to a temperature Tx, the internal energy change is... 

Crystallographic studies provide two grades of intemuclear distances R, between nuclei at special positions (frequently with a precision 0.005 to 0.002 A) determined by geometric coefficients times the unit cell parameters, and between two nuclei which are both (or at least one) on general positions (typically 10 times less precise). It should be noted that R values with 4 or 5 decimals 48) usually are derived from micro-wave rotational spectra, especially of gaseous diatomic molecules, giving the time average of R 2. The thermal vibrations at room temperature frequently have an amplitude of 0.05 to 0.1 A which would only be decreased by a factor around 2 or 3 at the absolute zero. 

The rotational and vibrational factors in the partition function of the diatomic molecule A are given by equations (16.24) and (16.30), respectively. These are independent of the pressure (or volume) and require no adjustment or correction to the standard state. 

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