Vibrational 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... 

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

Vibrational Energy Levels A diatomic molecule has a single set of vibrational energy levels resulting from the vibration of the two atoms around the center of mass of the molecule. A vibrating molecule is usually approximated by a harmonic oscillatord for which... 

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... 

The Anharmonic Oscillator Model. The harmonic oscillator model for diatomic molecules predicts that the vibrational energy levels of a molecule will be equally spaced. If this were true, an overtone band would appear at a frequency (or wavenumber) exactly twice the fundamental. What actually occurs is the appearance of an overtone band at a frequency slightly lower than twice the fundamental and we must therefore modify the simple equations for a harmonic oscillator to take this observation into account. 

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. 

We have seen in Section 1.3.6 how the vibrational energy levels of a diatomic molecule, treated in the harmonic oscillator approximation, are given by... 

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. 

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. 

Figure 9.12. Vibrational energy levels of a diatomic molecule...

More accurate characterization of the energy levels—beyond the Morse potential— might include additional anharmonic contribution of the form (v- -1) (Ogyg. Vibrational tran.sitions of diatomic molecules occur in the infrared, in the broad range of 50—15, 000 cm A molecule will absorb or emit radiation only if it has a nonzero dipole moment. Thus heteronuclear diatomic molecules such as HCl are infrared (IR) active, while homonuclear diatomics such as H2 and CI2 are not. 

Figure 9.3 Potential energy (E) of diatomic molecule vibration. For a small displacement vibration (Ar), the vibration motion can be treated as harmonic. Quantum theory divides the potential energy of vibration into several levels as v0, v, v2, v3,. ..

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... 

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,... 

Bond dissociation enthalpies for many diatomic neutral molecules including, for example, metal hydrides, have been obtained by the Birge-Sponer extrapolation method.This method is based on the analysis of the vibrational energy levels in a molecule obtained from spectroscopic studies. 

We expect, therefore, that a reasonable approximation to the vibrational energy levels of a diatomic molecule would be the harmonic-oscillator vibrational energy levels Eqs. (4.45) and (4.23) give... 

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),... 

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