Rotational Energy Levels of Diatomic Molecules

I 2.1 Rotational Energy Levels of Diatomic Molecules, K I 2.2 Vibrational Energy Levels of Diatomic Molecules, 10 I 2.3 Electronic Stales of Diatomic Molecules, 11 I 2.4 Coupling of Rotation and Electronic Motion in Diatomic Molecules Hund s Coupling Cases, 12 1-3 Quantum States of Polyatomic Molecules, 14... 

The simplest vibration-rotation spectra to interpret are those of diatomic molecules. The rotational energy levels of diatomic molecules are characterized by a single rotational quantum number, J. If the molecule is assumed to be a rigid rotor (i.e., its bond length remains constant no matter how rapidly the molecule rotates), the rotational energy is given by... 

Diatomic Molecules (Spin Neglected), 258. Symmetry Properties of the Wave Functions, 261. Selection Rules for Optical Transitions in Diatomic Molecules, 262. The Influence of Nuclear Spin, 265. The Vibrational and Rotational Energy Levels of Diatomic Molecules, 268. The Vibrational Spectra of Polyatomic Molecules, 273. 

Although the interpretation of rotational spectra of diatomic molecules is relatively simple, such spectra lie in the far infrared, a region that at present is not as easily accessible to study as are the near infrared, visible, cr ultraviolet. Consequently, most information about rotational energy levels has actually been obtained, not from pure rotation spectra, but from rotation-vibration spectra. Molecules without dipole moments have no rotation spectra, and nonpolar diatomic molecules lack rotation-vibration spectra as well, Thus, II2, N2, 02, and the molecular halogens have no characteristic infrared spectra. Information about the vibrational and rotational energy levels of these molecules must be obtained from the fine structure of their electronic spectra or from Raman spectra. 

Written to be the definitive text on the rotational spectroscopy of diatomic molecules, this book develops the theory behind the energy levels of diatomic molecules and then summarises the many experimental methods used to study the spectra of these molecules in the gaseous state. 

Table 0.1 Rotational and vibrational energy levels of diatomic molecules...

A diatomic molecule has two axes around which it can physically rotate see Figure 2. These axes are equivalent, and correspond to a single moment of inertia L I determines the spacing of rotational energy levels of the molecule, and is used to define the rotational constant B, where B = h/(8ai I) and h is Planck s constant. According to quantum mechanics, rotational energy levels can only take on certain discrete values, i.e. they are quantized, and we label them with a quantum number called J. The energies of the rotational levels are ... 

EBK) semiclassical quantization condition given by Eq. (2.72). In contrast to the RKR method for diatomics, a direct method has not been developed for determining potential energy surfaces from experimental anharmonic vibrational/rotational energy levels of polyatomic molecules. Methods which have been used are based on an analytic representation of the potential energy surface (Bowman and Gazdy, 1991). At low levels of excitation the surface may be represented as a sum of quadratic, cubic, and quartic normal mode coordinates (or internal coordinate) terms, that is,... 

Because molecules are not rigid, the rotational energy levels for diatomic molecules differ slightly from rigid-rotor levels. From (6.52) and (6.55), the two-particle rigid-rotor levels are = BhJ J -l-1). Because of the anharmonicity of molecular vibration (Fig. 4.6), the average internuclear distance increases with increasing vibrational quantum number v, so as v increases, the moment of inertia I increases and the rotational constant B decreases. To allow for the dependence of B on v, one replaces B in E by The mean rotational constant B for vibrational level v is - Ug v + 1/2),... 

The rotational energy levels of (a) a heavy diatomic molecule and (b) a light diatomic molecule. Note that the energy levels are closer together for the heavy diatomic molecule. Microwaves arc absorbed when transitions take place between neighboring energy levels. 

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. 

The Vibration and Rotation of Molecules.—The nature of the vibrational motion and the values of the vibrational energy levels of a molecule are determined by the electronic energy function, such as that shown in Figure VII-1. The simplest discussion of the vibrational motion of a diatomic molecule is based upon the approximation of the energy curve in the neighborhood of its minimum by a parabola that is, it is assumed that the force between the atoms of the molecule is proportional to the displacement of the internuclear distance from its equilibrium value r.. This corresponds to the approximate potential function... 

It is well known from the Bom-Oppenheimer separation [1] that the pattern of energy levels for a typical diatomic molecule consists first of widely separated electronic states (A eiec 20000 cm-1). Each of these states then supports a set of more closely spaced vibrational levels (AEvib 1000 cm-1). Each of these vibrational levels in turn is spanned by closely spaced rotational levels ( A Emt 1 cm-1) and, in the case of open shell molecules, by fine and hyperfine states (A Efs 100 cm-1 and AEhts 0.01 cm-1). The objective is to construct an effective Hamiltonian which is capable of describing the detailed energy levels of the molecule in a single vibrational level of a particular electronic state. It is usual to derive this Hamiltonian in two stages because of the different nature of the electronic and nuclear coordinates. In the first step, which we describe in the present section, we derive a Hamiltonian which acts on all the vibrational states of a single electronic state. The operators thus remain explicitly dependent on the vibrational coordinate R (the intemuclear separation). In the second step, described in section 7.55, we remove the effects of terms in this intermediate Hamiltonian which couple different vibrational levels. The result is an effective Hamiltonian for each vibronic state. 

The vibration-rotation energy levels of a diatomic molecule are given in traditional notation (Herzberg, 1950) by the expression... 

Dubost H 1984 Spectroscopy of vibrational and rotational levels of diatomic molecules in rare-gas crystals Inert Gases. Potentials, Dynamics, and Energy Transfer in Doped Crystals (Springer Ser. Chem. Phys. 34) ed M L Klein (Berlin Springer) pp 145-256... 

The electronic spectra of isotopically varied diatomic molecules reflect the effects of changes in the vibrational and rotational energy levels of both the ground and the excited electronic states. This subject is of great historical importance since the isotopes 0, " 0, and were all identified in... 

In addition to electronic energy states, molecules posses both rotational and vibrational energy levels. Assuming a fixed distance between two atoms (rigid rotor approximation), the Schrddinger equation yields for the allowed rotational energy levels of a diatomic molecule... 

If electrons and other particles act like waves, they should obey a wave equation. In 1926, Erwin Schrodinger published a series of four articles containing a wave equation for de BrogUe waves, which we now call the Schrodinger equation. The first three articles presented the time-independent version of the wave equation and applied it to the hydrogen atom, rotation and vibration of diatomic molecules, and the effect of an external electric field on energy levels. The time-dependent version of the equation was reported in the fourth article. ... 

For a linear polyatomic molecule like acetylene or cyanogen, the rotational energy levels are the same as those of diatomic molecules in Eq. (22.2-18). Equation (25.4-13) can be used for the rotational partition function with the appropriate symmetry number and moment of inertia. The rotational energy levels of nonlinear polyatomic molecules are more complicated than those of diatomic molecules. The derivation of the rotational partition function for nonlinear molecules is complicated, and we merely cite the result ... 

The RRHO approximation and analysis of the infrared spectrum of formulates a picture of the vibration-rotation energy levels of a diatomic molecule. The energy difference between vibrational energy levels is large with respect to the rotational energy levels. A vibrational state v will have an infinite manifold of J rotational states. This is depicted in Figure 6-4. 

As for diatomic molecules, there are stacks of rotational energy levels associated with all vibrational levels of a polyatomic molecule. The resulting term values S are given by the sum of the rotational and vibrational term values... 

The name dissociation energy is given to the work required to break up a diatomic molecule which is in its lowest rotation-vibrational state, and to leave the two particles (either atoms or ions) at rest in a vacuum. This quantity, which will be denoted by D , corresponds to the length of the arrow in Fig. 7 or Fig. 8a, where the length is the vertical distance between the lowest level of the molecule and the horizontal line which... 

---