Diatomic molecule anharmonic

Selection Rules. The harmonic-oscillator, rigid-rotor selection mles are Av = 1 and AJ = 1 that is, infrared emission or absorption can occur only when these allowed transitions take place. For an anharmonic diatomic molecule, the A7 = 1 selection mle is still valid, but weak transitions corresponding to An = 2, 3, etc. (overtones) can now be observed. Since we are interested in the most intense absorption band (the fundamental ), we are concerned with transitions from various J levels of the vibrational ground... 

The absorption bands in the spectrum can thus be interpreted as if they originated from an anharmonic diatomic molecule. This is the reason why NIR spectra are often said to become simpler at higher energy. Experimentally, it is found that the inversion from normal to local mode character... 

As an introduction, the chapter begins with the anharmonic diatomic molecule. Then we study the thermal properties (free energy, equation of state, thermal expansion and specific heat) of the classical anharnx)nic linear chain. Two important concepts are introduced the Gvuneisen pavametev and the quasiharmonic approximation. In this approximation, the temperature dependence of the force constants and phonon frequencies is only due to the... 

Figure 6.4 Potential energy curve and energy levels for a diatomic molecule behaving as an anharmonic oscillator compared with those for a harmonic oscillator (dashed curve)...

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. 

For diatomic molecules, B0 is the rotational constant to use with equation (10.125), while Be applies to equation (10.124). They are related by Bq = Be 2 - The moment of inertia 70(kg-m2) is related to 50(cm ) through the relationship /0 = h/ 8 x 10 27r22 oc), with h and c expressed in SI units. For polyatomic molecules, /a, /b, and Iq are the moments of inertia to use with Table 10.4 where the rigid rotator approximation is assumed. For diatomic molecules, /0 is used with Table 10.4 to calculate values to which we add the anharmonicity and nonrigid rotator corrections. 

For diatomic molecules, lj0 is the vibrational constant to use with equation (10.125) for calculating anharmonicity and nonrigid rotator corrections, while J)e and tDe-Ve... 

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

Table A4.5 summarizes the equations for calculating anharmonicity and nonrigid rotator corrections for diatomic molecules. These corrections are to be added to the thermodynamic properties calculated from the equations given in Table A4.1 (which assume harmonic oscillator and rigid rotator approximations).

The following equations are used to calculate the anharmonicity and nonrigid rotator corrections to the thermodynamic properties of diatomic molecules. 

Knaap E. W. Vibrational dephasing of diatomic molecules in liquids role of anharmonicity of the diatom, Chem. Phys. Lett. 58, 221-4 (1978). 

The energy of the diatomic molecule, as given by Bq. (79) does not take into account foe anharmonicity of the vibration. Tte.qjE fect of fop cubic and qoartic terms in Eq. (73) can be evaluated by application of the theory of perturbation (see Chapter 12). 

The interatomic potential function for the diatomic molecule was described in Section 6 5. In the Taylpr-series development of this function (6-72)3 cubic and higher terms were neglected in the harmonic approximation. It is now of interest to evaluate the importance of these so-called anharmonic terms with the aid of the perturbation theory outlined above. If cubic and quartic... 

Fig. 3.1 Born-Oppenheimer vibrational potentials for a diatomic molecule corresponding to the CH fragment. The experimentally realistic anharmonic potential (solid line) is accurately described by the Morse function Vmorse = De[l — exp(a(r — r0)]2 with De = 397kJ/mol, a = 2A and ro = 1.086 A (A = Angstrom = 10 10m). Near the origin the BO potential is adequately approximated by the harmonic oscillator (Hooke s Law) function (dashed line), Vharm osc = f(r — ro)2/2. The harmonic oscillator force constant f = 2a2De...

To obtain the allowed energy levels, Ev, for a real diatomic molecule, known as an anharmonic oscillator, one substitutes the potential energy function describing the curve in Fig. 3.2c into the Schrodinger equation the allowed energy levels are... 

FIGURE 3.2 (a) Vibration of diatomic molecule, HC1, (b) potential energy of an ideal harmonic oscillator, and (c) an anharmonic oscillator described by the Morse function. 

The vibrational levels (6.54) are only approximate, because we neglected the anharmonic cubic, quartic,... terms in the potential energy (6.7). For a diatomic molecule, anharmonicity adds the correction - hvexe v + )2 to... 

---