Rotational constants, diatomic molecules

Measurement and assignment of the rotational spectmm of a diatomic or other linear molecule result in a value of the rotational constant. In general, this will be Bq, which relates... 

As in diatomic molecules the structure of greatest importance is the equilibrium structure, but one rotational constant can give, at most, only one structural parameter. In a non-linear but planar molecule the out-of-plane principal moment of inertia 4 is related to the other two by... 

As is the case for diatomic molecules, rotational fine structure of electronic spectra of polyatomic molecules is very similar, in principle, to that of their infrared vibrational spectra. For linear, symmetric rotor, spherical rotor and asymmetric rotor molecules the selection mles are the same as those discussed in Sections 6.2.4.1 to 6.2.4.4. The major difference, in practice, is that, as for diatomics, there is likely to be a much larger change of geometry, and therefore of rotational constants, from one electronic state to another than from one vibrational state to another. 

Much of the additional material is taken up by what 1 have called Worked examples . These are sample problems, which are mostly calculations, with answers given in some detail. There are seventeen of them scattered throughout the book in positions in the text appropriate to the theory which is required. 1 believe that these will be very useful in demonstrating to the reader how problems should be tackled. In the calculations, 1 have paid particular attention to the number of significant figures retained and to the correct use of units. 1 have stressed the importance of putting in the units in a calculation. In a typical example, for the calculation of the rotational constant B for a diatomic molecule from the equation... 

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... 

The quantity I (= /jiR ) is the moment of inertia for the diatomic molecule with the intemuclear distance fixed at Re and Be is known as the rotational constant (see Section 5.4). 

The molecular constants o , B, Xe, D, and ae for any diatomic molecule may be determined with great accuracy from an analysis of the molecule s vibrational and rotational spectra." Thus, it is not necessary in practice to solve the electronic Schrodinger equation (10.28b) to obtain the ground-state energy o(R). 

In treating the vibrational and rotational motion of a diatomic molecule having reduced mass (i, equilibrium bond length re and harmonic force constant k, we are faced with the following radial Schrodinger equation ... 

The principles discussed for diatomic molecules generally apply to polyatomic molecules, but their spectra are much more complex. For example, instead of considering rotation only about an axis perpendicular to the internuclear axis and passing through the center of mass, for nonlinear molecules, one must think of rotation about three mutually perpendicular axes as shown in Fig. 3-lb Hence we have three rotational constants A, B, and C with respect to these three principal axes. 

We saw that homonuclear diatomic molecules exhibit no pure-rotation or vibration-rotation spectra, because they have zero electric dipole moment for all internuclear separations. The Raman effect depends on the polarizability and not the electric dipole moment homonuclear diatomic molecules do have a nonzero polarizability which varies with varying internuclear separation. Hence they exhibit pure-rotation and vibration-rotation Raman spectra. Raman spectroscopy provides information on the vibrational and rotational constants of homonuclear diatomic molecules. 

Analogous to (4.31) for the diatomic molecule, the rotational constants of a polyatomic molecule are defined as... 

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... 

Table 4.1 lists rotational and vibrational constants for some diatomic-molecule states. The most extensive compilation of such data is Bourcier. 

The vibration-rotation energy expression (4.67) contains the rotational constant Be, which depends on the equilibrium intemuclear separation Re hence investigation of vibrational and rotational spectra of diatomic molecules allows their structure to be determined. 

Recall that homonuclear diatomic molecules have no vibration-rotation or pure-rotation spectra due to the vanishing of the permanent electric dipole moment. For electronic transitions, the transition-moment integral (7.4) does not involve the dipole moment d hence electric-dipole electronic transitions are allowed for homonuclear diatomic molecules, subject to the above selection rules, of course. [The electric dipole moment d is given by (1.289), and should be distinguished from the electric dipole-moment operator d, which is given by (1.286).] Analysis of the vibrational and rotational structure of an electronic transition in a homonuclear diatomic molecule allows the determination of the vibrational and rotational constants of the electronic states involved, which is information that cannot be provided by IR or microwave spectroscopy. (Raman spectroscopy can also furnish information on the constants of the ground electronic state of a homonuclear diatomic molecule.)... 

As we saw in Chapter 1, the importance of numbers in chemistry derives from the fact that experimental measurement of a particular chemical or physical property will always yield a numerical value to which we attach some significance. This might involve direct measurement of an intrinsic property of an atom or molecule, such as ionization energy or conductivity, but, more frequently, we find it necessary to use theory to relate the measured property to other properties of the system. For example, the rotational constant, B, for the diatomic molecule CO can be obtained directly from a measurement of the separation of adjacent rotational lines in the infrared spectrum. Theory provides the link between the measured rotational constant and the moment of inertia, I, of the molecule by the formula ... 

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