Products of inertia

For a nonlinear molecule the rotational energy levels are a function of three principal moments of inertia /A, /B and /c- These are moments of inertia around three mutually orthogonal axes that have their origin (or intersection) at the center of mass of the molecule. They are oriented so that the products of inertia are zero. The relationship between the three moments of inertia, and hence the energy levels, depends upon the geometry of the molecules. 

Different notations are sometimes used for moments and products of inertia, e.g. if ... 

The corresponding cross-terms are called products of inertia and are defined as ... 

Principal moments of inertia are the moments of inertia corresponding to that particular and unique orientation of the axes for which one of the three moments has a maximum value, another a minimum value, and the third is either equal to one or the other or is intermediate in value between the other two. The corresponding axes are called principal axes of a molecule (or principal inertia axes). Moreover, the products of inertia all reduce to zero and the corresponding inertia matrix is diagonal. Conventionally, principal moments of inertia are labeled as ... 

The b coordinates of all three non-hydrogen atoms were imaginary when the Kraitchman equations were used to locate the position of the nuclei in isothiocyanic acid, HNCS [218]. To determine a reasonable structure for this planar molecule, the authors used the first moment condition for the b coordinates and the product of inertia condition. The second moment condition was used also but with the substitution moment of inertia 4 instead of the ground state moment /q. The authors argued that this procedure gives a more reliable structure instead of a hybrid, particularly since Watson... 

Similarly, a product of inertia relation may be used, as for example,... 

By contrast it may be shown that the solution of Eq. (18) is invariant to linear combination of the components of X. Therefore, determination of Cartesian coordinates or determination of internal coordinates should lead to the same result. It should be pointed out, however, that the principal axis conditions may be taken into account in different ways in Cartesian coordinate and internal coordinate calculations. We include the center-of-mass and product-of-inertia relations along with the moments when we fit Cartesian coordinates. On the other hand, internal coordinates are independent of the origin of the coordinate system. Thus, holding some Cartesian coordinates constant while others are varied is not the same as holding some internal coordinates constant while others are varied. This is true even if the same atomic positions are involved. 

Values of the nontrivial first moments and product of inertia for the various structures of 2-chloropropane are also given at the bottom of Table 2. The values are essentially zero for all but the substitution structures. The reasons for this difference were given above. The first moments for the rt structure are somewhat larger than is typical for rs structures, but the product of inertia is about average size. 

We recommend that experimental uncertainties in the coordinates be assessed by Eq. (10) and the vibration/rotation contributions be assessed by the Costain rule [Eq. (14)1 or by first moment or product of inertia relations. Then, either the procedure introduced by Tobiason and Schwendeman20 should be used to propagate the uncertainties into distances or angles, or the two contributions should be added together and used with Eq. (18) to generate distance and angle uncertainties. 

C SUB P5IXY CALC, products OF INERTIA ABOUT X-AXIS AND... 

All the singly substituted isotopic species are needed to obtain a complete stracture. However, it is sometimes impossible, e.g. when the molecule contains atoms having only one stable nuclide such as P, or difficult, e.g. for a chemical reason, to make conplete isotopic substitutions. In these cases, one is forced to use a first-moment equation or a condition that the cross-products of inertia be zero. In some cases even a part or all of the three moments of inertia of the parent species are used. The stracture thus obtained is in reahty a Itybrid of the and ro structures. Nevertheless, even in these cases the stracture is usually called r. ... 

Here I x, lyy, hz are the instantaneous moments of inertia with respect to the moving x, y, z axes hy, lyz, and Izt are the products of inertia. These quantities are not constants but functions of the positions of the... 

This means that for calculating the product of inertia moments, we need to consider the pyramidal cordiguration of the molecules. 

The other products of inertia, ly, I y, hz, and Izx, are defined analogously. Only three of the products of inertia have distinct values, because lyz = Izy and so on. For calculating the moments of inertia and the products of inertia, we neglect the masses of the electrons and include only the nuclei in the sums since the masses of the electrons are small compared to those of the nuclei. 

Calculate the three principal moments of inertia for the water molecule, assuming a bond length of 96 pm and a bond angle of 104.5°. You must first find the location of the center of mass in the molecule. Assume its isotopes are and H. Pick a product of inertia and show that it vanishes. 

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