Rotational constants inertia

The only tenn in this expression that we have not already seen is a, the vibration-rotation coupling constant. It accounts for the fact that as the molecule vibrates, its bond length changes which in turn changes the moment of inertia. Equation B1.2.2 can be simplified by combming the vibration-rotation constant with the rotational constant, yielding a vibrational-level-dependent rotational constant. 

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... 

The moment of inertia 1 determines the rotational constant 0 = h /IT, which is the parameter that controls the strength of quantum effects. The other parameter of the model, which is the quadrupolar coupling constant K, can be conveniently taken as the energy and temperature scale. We can thus reduce all quantities related to energies by K, and define, e.g., the dimensionless temperature = k T/K, energy = E/K, and rotational... 

Table 10.1 Moments of inertia and rotational constants of some common molecules.

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. 

Tables 10.1, 10.2, and 10.3e summarize moments of inertia (rotational constants), fundamental vibrational frequencies (vibrational constants), and differences in energy between electronic energy levels for a number of common molecules or atoms/The values given in these tables can be used to calculate the rotational, vibrational, and electronic energy levels. They will be useful as we calculate the thermodynamic properties of the ideal gas.

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). 

Rotational constants G = A, B or C are inversely proportional to principal moments of inertia Ia through the expressions G = h/Sn2Ia, where a refers to one of the three principal inertia axis directions a, b or c. The Ia are related to the coordinates of the atoms i in the principal axis system via the... 

In Equation 12.8 Be is the rotational constant, Be = h/(8jt2I), (I is the moment of inertia), coe is the vibrational frequency, 27T(oe = (k/ix)1, (k the vibrational force constant and x the reduced mass), re the equilibrium bond length (isotope independent to reasonable approximation), and ae is the vibration-rotation interaction constant... 

Five isotopomers of Sia were studied in Ref (20), and are labeled as follows Si- Si- Si (I) Si- Si- Si (II) Si- Si- Si (III) Si- "Si- Si (IV) Si- Si- °Si (V). Rotational constants for each (both corrected and uncorrected for vibration-rotation interaction) can be found towards the bottom of Table I. Structures obtained by various refinement procedures are collected in Table II. Two distinct fitting procedures were used. In the first, the structures were refined against all three rotational constants A, B and C while only A and C were used in the second procedure. Since truly planar nuclear configurations have only two independent moments of inertia (A = / - 4 - 7. = 0), use of B (or C) involves a redundancy if the other is included. In practice, however, vibration-rotation effects spoil the exact proportionality between rotational constants and reciprocal moments of inertia and values of A calculated from effective moments of inertia determined from the Aq, Bq and Co constants do not vanish. Hence refining effective (ro) structures against all three is not without merit. Ao is called the inertial defect and amounts to ca. 0.4 amu for all five isotopomers. After correcting by the calculated vibration-rotation interactions, the inertial defect is reduced by an order of magnitude in all cases. 

For water, the equilibrium OH bond distance is 0.957 A and the equilibrium bond angle is 104.5°. (a) Locate the principal axes and then calculate the equilibrium moments of inertia, (b) Calculate the rotational constants Ae, Be, and Ce. (c) Calculate the frequencies corresponding to transitions between the 7 = 0 level and each of the three 7 = 1 levels ignore the effects of zero-point vibrations, (d) Which of the transitions in part (c) are actually allowed for H20 (e) Give the frequencies of the allowed 7 = 1 <2-branch transitions. 

The fifth term in (4.67) represents an interaction between vibration and rotation, and ae is called a vibration-rotation coupling constant. [Do not confuse ae with a in (4.26).] As the vibrational quantum number increases, the average internuclear distance increases, because of the anharmonicity of the potential-energy curve (Fig. 4.4). This increases the effective moment of inertia, and therefore decreases the rotational energy. We can define a mean rotational constant Bv for states with vibrational quantum number v by... 

The primary significance of microwave spectroscopy for chemistry is in determination of molecular structure. Assignment of microwave spectral lines to transitions between specific rotational levels allows determination of the rotational constants A0, B0, and C0, and the corresponding moments of inertia. The moments of inertia are dependent on the molecular bond distances and bond angles. 

For working problems, the following constant allows one to convert moments of inertia to rotational constants ... 

Analysis of the rotational fine structure of IR bands yields the moments of inertia 7°, 7°, and 7 . From these, the molecular structure can be fitted. (It may be necessary to assign spectra of isotopically substituted species in order to have sufficient data for a structural determination.) Such structures are subject to the usual errors due to zero-point vibrations. Values of moments of inertia determined from IR work are less accurate than those obtained from microwave work. However, the pure-rotation spectra of many polyatomic molecules cannot be observed because the molecules have no permanent electric dipole moment in contrast, all polyatomic molecules have IR-active vibration-rotation bands, from which the rotational constants and structure can be determined. For example, the structure of the nonpolar molecule ethylene, CH2=CH2, was determined from IR study of the normal species and of CD2=CD2 to be8... 

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 ... 

Example of a Rotational Constant The ground-state rotational band of 152Gd is shown in Figure 6.11. Use the energy separation between the 2+ and 0+ levels to estimate the rotational constant in keV, the moment of inertia in amu-fm2, and then compare your result to that obtained to the rigid-body result with a deformation parameter of 3 = 0.2. Finally, evaluate the irrotational flow moment of inertia for this nucleus. 

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