Effective moments of inertia

The effective moment of inertia / and the friction coefficient / could easily be estimated. The force constant k associated with the relative motion of the lobes was determined from an empirical energy function. To do so, the molecule was opened in a step-wise fashion by manipulating the hinge region and each resulting structure was energy minimized. Then, the interaction energy between the two domains was measured, and plotted versus 0. 

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. 

We see from (4.104) that, although the vibrational quantum number is not changing, the frequency of a pure-rotational transition depends on the vibrational quantum number of the molecule undergoing the transition. (Recall that vibration changes the effective moment of inertia, and thus affects the rotational energies.) For a collection of diatomic molecules at temperature T, the relative populations of the energy levels are given by the Boltzmann distribution law the ratio of the number of molecules with vibrational quantum number v to the number with vibrational quantum number zero is... 

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 manifold of energy levels for small J then has the appearance shown in Fig. 7. In higher approximation it would be necessary to take account of centrifugal distortion which causes the effective moments of inertia to increase slightly with J. For highly asymmetric tops the evaluation of the energy levels is difficult but approximate values can be interpolated from published tables (Erlandsson, 1956 King ef al., 1943, 1949). 

I is the effective moment of inertia of a dipole (we consider here a linear molecule), determined by the relation (149). The spectral function L(z), calculated for thermal equilibrium, is linearly related to the spectrum C° of the dipolar autocorrelation function (ACF) C°(f) (VIG, p. 137 GT, p. 152) as... 

It should be noted that relation (2.51) is valid within the sudden approximation. However, the relaxation of heavy particle impurities typically involves motion that is slow compared with vibrations of the host lattice (i.e., the tunneling takes place in the adiabatic limit). The net effect of the adiabatic approximation is to renormalize the effective moment of inertia of the particle. This approach was used, for example, to describe vacancy diffusion in light metals. The evolution of the rate constant from Arrhenius behavior to the low-temperature plateau was described within the framework of one-dimensional tunneling of a... 

Effective moment of inertia of active internal rotations. 

Table 7.9 Reactant System Characteristic Frequencies in Selected Debye Solvents Estimated Using Equation (7.10.12) Together with the Solvent Molecule s Effective Moment of Inertia and the Reciprocal of the Longitudinal Relaxation Time...

The effective moment of inertia in the vibrational state v defined as... 

The experimental parameters actually determined are the effective rotational constants B from which the effective moments of inertia are obtained by the relation ... 

For a diatomic molecule the effective moment of inertia has a well-defined physical significance since the effective rotational constant is inversely proportional to the average value of the square of the bond length. Thus the effective moment of inertia for a diatomic molecule is given by ... 

The bond length determined from the effective moment of inertia is therefore ... 

It should be emphasized that the physical significance attributable to the effective moment of inertia of a diatomic molecule does not extend to polyatomic molecules where the vibrational effects are more complicated. This point is sometimes overlooked in the literature. 

For polyatomic molecules the physical significance of the effective moment of inertia is complicated by the presence of Coriolis terms. Thus one has the relation ... 

Note that the term quadratic in the normal coordinates now contains first derivatives as well as second due to the effect of inverting the components of the moment of inertia tensor. To get an expression for the effective moments of inertia, Eq. (13) must first be vibrationally averaged and reinverted to give ... 

However, there are two common operationally defined types of structure that are determined from effective moments of inertia. The more common, the so-called effective or r0 structure, is somewhat loosely defined. In practice, any structural parameter that requires for its determination fitting one or more of the second moment relations is designated as r0. r0 structures are not uniquely defined since, for any over-determined system, the value of structural parameters obtained depends somewhat on the manner in which the data are treated and the values are isotopically dependent. This problem is examined in more detail by Schwendeman (this volume). 

As discussed by Laurie in an earlier paper, the observed rotational constants in any one vibrational state can be converted into effective moments of inertia through the relation... 

For a planar molecule, the effective moments of inertia show a small positive inertia defect... 

Consider a ribbon-like structure with a continuous angular deviation that describes the average twist between monomer units (Figure 5.3). In a continuous limit, the average effective moment of inertia that governs the radius of curvature of the ribbon-like stnicture is given by [24] ... 

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