Diatomic molecules vibrational energy

We have seen in Section 6.1.3.2 that, for diatomic molecules, vibrational energy levels, other than those with v = 1, in the ground electronic state are very often obtained not from... 

The vibrational selection rule for the harmonic oscillator, Au = 1, applies to polyatomic molecules just as it did to diatomic molecules. Vibrational energy can, therefore, change in units of hcoi/ln. Transitions in which one of the three normal modes of energy changes by Au = - -1 (for example Ui = 0 1, U2 = U3 = 0 or 1 = 1) 2 = 3, i>3 = 2 3) result from absorption of a photon having one of three fundamental frequencies of the molecule. In the actual case, anharmonicities also allow transitions with Au, = 2, 3,... so that, for example, weak absorption also occurs at 2coi, 3(Ui, etc. and at coi + coj, 2vibrational transitions often play major roles in planetary spectroscopy. 

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... 

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,. ..

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...

The only variable in the vibrational energy equation E = v i)/zv is the quantum number t . If the quantum number changes by +1 then the diatomic molecule gains energy by an amount AE equal to the above energy equation subtracted from the same equation where i + 1 is substituted for v. 

Make a plot of Cy as a function of temperature for an ideal gas composed of diatomic molecules vibrating as a harmonic oscillator according to the energy level expression of E (kj mohi) = 23.0 (n + 1/2). 

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. 

Although a diatomic molecule can produce only one vibration, this number increases with the number of atoms making up the molecule. For a molecule of N atoms, 3N-6 vibrations are possible. That corresponds to 3N degrees of freedom from which are subtracted 3 translational movements and 3 rotational movements for the overall molecule for which the energy is not quantified and corresponds to thermal energy. In reality, this number is most often reduced because of symmetry. Additionally, for a vibration to be active in the infrared, it must be accompanied by a variation in the molecule s dipole moment. 

These results do not agree with experimental results. At room temperature, while the translational motion of diatomic molecules may be treated classically, the rotation and vibration have quantum attributes. In addition, quantum mechanically one should also consider the electronic degrees of freedom. However, typical electronic excitation energies are very large compared to k T (they are of the order of a few electronvolts, and 1 eV corresponds to 10 000 K). Such internal degrees of freedom are considered frozen, and an electronic cloud in a diatomic molecule is assumed to be in its ground state f with degeneracy g. The two nuclei A and... 

As an illustrative example, consider the vibrational energy relaxation of the cyanide ion in water [45], The mechanisms for relaxation are particularly difficult to assess when the solute is strongly coupled to the solvent, and the solvent itself is an associating liquid. Therefore, precise experimental measurements are extremely usefiil. By using a diatomic solute molecule, this system is free from complications due to coupling... 

To compare the relative populations of vibrational levels, the intensities of vibrational transitions out of these levels are compared. Figure B2.3.10 displays typical potential energy curves of the ground and an excited electronic state of a diatomic molecule. The intensity of a (v, v ) vibrational transition can be written as... 

Diatomic molecules have only one vibrational mode, but VER mechanisms are paradoxically quite complex (see examples C3.5.6.1 and C3.5.6.2). Consequently there is an enonnous variability in VER lifetimes, which may range from 56 s (liquid N2 [18]) to 1 ps (e.g. XeF in Ar [25]), and a high level of sensitivity to environment. A remarkable feature of simpler systems is spontaneous concentration and localization of vibrational energy due to anhannonicity. Collisional up-pumping processes such as... 

Since depends on nuclear coordinates, because of the term, so do and but, in the Bom-Oppenheimer approximation proposed in 1927, it is assumed that vibrating nuclei move so slowly compared with electrons that J/ and involve the nuclear coordinates as parameters only. The result for a diatomic molecule is that a curve (such as that in Figure 1.13, p. 24) of potential energy against intemuclear distance r (or the displacement from equilibrium) can be drawn for a particular electronic state in which and are constant. 

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... 

Just as the electrical behaviour of a real diatomic molecule is not accurately harmonic, neither is its mechanical behaviour. The potential function, vibrational energy levels and wave functions shown in Figure f.i3 were derived by assuming that vibrational motion obeys Hooke s law, as expressed by Equation (1.63), but this assumption is reasonable only... 

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... 

In a diatomic molecule one of the main effects of mechanical anharmonicity, the only type that concerns us in detail, is to cause the vibrational energy levels to close up smoothly with increasing v, as shown in Figure 6.4. The separation of the levels becomes zero at the limit of dissociation. 

In the lowest approximation the molecular vibrations may be described as those of a harmonic oscillator. These can be derived by expanding the energy as a function of the nuclear coordinates in a Taylor series around the equilibrium geometry. For a diatomic molecule this is the intemuclear distance R. 

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... 

If, now, we continue warming the substance sufficiently, we will reach a point at which the kinetic energies in vibration, rotation, and translation become comparable to chemical bond energies. Then molecules begin to disintegrate. This is the reason that only the very simplest molecules—diatomic molecules—are found in the Sun. There the temperature is so high (6000°K at the surface) that more complex molecules cannot survive. 

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... 

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