Rotational inertia tensor

Here Wj are components of angular velocity relative to coordinate axes, are components of rotational inertia tensor. The dimension of Sly is cube of length, therefore the tensor is interpreted as equivalent volume. Note, that the relation (8.14) can be represented similar to (8.5), namely, as... 

For molecules that are non-linear and whose rotational wavefunctions are given in terms of the spherical or symmetric top functions D l,m,K, the dipole moment Pave can have components along any or all three of the molecule s internal coordinates (e.g., the three molecule-fixed coordinates that describe the orientation of the principal axes of the moment of inertia tensor). For a spherical top molecule, Pavel vanishes, so El transitions do not occur. 

In practical applications of the expressions for the rotational g tensor, equations (2), (4), (5), or (6), the nuclear masses in the nmment of inertia tensor I are generally approximated with atomic masses and is approximated with the center of atomic masses. This introduces a correction term to the moment of inertia tensor which is actually closely related to the nuclear contribution to the rotational g tensor [3,11,38]. Going to second order of perturbation theory for the electronic contributions one would obtain a further correction term to the moment of inertia tensor which is similar to the electronic contribution to the rotational g tensor [3,4]. 

All together one would obtain an effeetive moment of inertia tensor which includes the rotational g tensor again. This correction is normally ignored for polyatomic molecules, but allows to estimate the rotational g factor of diatomic molecules from field-free rotation-vibration spectra [5,10,11]. 

Identity element, 387-388 Identity operation, 54, 395 Improper axis of symmetry, 53 Improper rotation, 396 Index of refraction, 132 INDO method, 71, 75-76 and ESR coupling constants, 380 and force constants, 245 and ionization potentials, 318 and NMR coupling constants, 360 Induced dipole moment, 187 Inertial defect, 224-225 Inertia tensor, 201... 

Eq. (3.21) discussed in Section 3.3.2 is only valid if the motion of the molecules under study has no preferential orientation, i.e. is not anisotropic. Strictly speaking, this applies only for approximately spherical bodies such as adamantane. Even an ellipsoidal molecule like trans-decalin performs anisotropic motion in solution it will preferentially undergo rotation and translation such that it displaces as few as possible of the other molecules present. This anisotropic rotation during translation is described by the three diagonal components Rlt R2, and R3 of the rotational diffusion tensor. If the principal axes of this tensor coincide with those of the moment of inertia - as can frequently be assumed in practice - then Rl, R2, and R3 indicate the speed at which the molecule rotates about its three principal axes. 

The first quantitative estimate of the rotational diffusion tensor for simple molecules was accomplished by Grant et al. [163], By solving the Woessner equations, they were able to show e.g. for trans-decalin that the molecule rotates preferentially like a propeller, i.e. about the axis perpendicular to the plane of the molecule. The values given as a measure of the rotational frequencies do not correlate with the moments of inertia, but instead with the ellipticities of the molecule as defined [163]. They are accessible from the ratios of the interatomic distances perpendicular to the axes of rotation, and can be adopted as a measure of the number of solvent molecules that have to be displaced on rotation about each of the three axes. 

For less symmetric molecules one has to resort to computer programs [164] to solve the Woessner equations. The orientation of the rotational diffusion tensor is usually defined by assuming that its principal axes coincide with those of the moment of inertia tensor. This assumption is probably a good approximation for molecules of low polarity containing no heavy atoms, since under these conditions the moment of inertia tensor roughly represents the shape of the molecule. 

This study is the first where semiquantitative use of relaxation data was made for conformational questions. A similar computer program was written and applied to the Tl data of several small peptides and cyclic amino acids (Somorjai and Deslauriers, 1976). The results, however, are questionable since in all these calculations it is generally assumed that the principal axis of the rotation diffusion tensor coincides with the principal axis of the moment of inertia tensor. Only very restricted types of molecules can be expected to obey this assumption. There should be no large dipole moments nor large or polar substituents present. Furthermore, the molecule should have a rather rigid backbone, and only relaxation times of backbone carbon atoms can be used in this type of calculation. 

The calculated Euler angles (a = 50°, /3 = 60°, and y = 40°), which determine the relative orientation between the principal-axis system of the rotational diffusion tensor and that of the moment of inertia tensor, indicate a significant shift between the two tensors. This result is expected because of the fact that molecule 31 contains a number of polar groups and hydrogen-bonding centers, leading to strong intermolecular interactions. 

FIG. 12.—Orientation of the principal-axis system of inertia tensor (x. y. z1) and that of the rotational diffusion tensor (x,y,z) for compound 31. The principal diffusion axis x is perpendicular to the plane of the drawing. [Reproduced with permission from P. Dais, Carbohydrate Res., 263 (1994) 13-24, and Elsevier Science B.V.]... 

Somewhat similar conclusions apply to the rotational magnetic moment g tensor for a diatomic molecule. The component of the moment of inertia tensor along the intemuclear axis is zero, and the two perpendicular components are, of course, equal. Consequently the rotational magnetic moment Zeeman interaction can be represented by the simple term... 

We define the order of the singular values as a > a2 > 31. The planar and collinear configurations give a3 0 and a2 a3 = 0, respectively. Furthermore, we let the sign of a3 specify the permutational isomers of the cluster [14]. That is, if (det Ws) = psl (ps2 x ps3) > 0, which is the case for isomer (A) in Fig. 12, fl3 >0. Otherwise, a3 < 0. Eigenvectors ea(a = 1,2,3) coincide with the principal axes of instantaneous moment of inertia tensor of the four-body system. We thereby refer to the principal-axis frame as a body frame. On the other hand, the triplet of axes (u1,u2,u3) or an SO(3) matrix U constitutes an internal frame. Rotation of the internal frame in a three-dimensional space, which is the democratic rotation in the four-body system, is parameterized by three... 

Fig. III.10. The 7 = 2 rotational levels of an asymmetric top molecule as a function of the rotational constant B. A =25.48366 GHz and C = 14.09795 GHz have been fixed to their values for ethyleneoxide. As soon as the moment of inertia tensor becomes asymmetric, the if-degeneracy of the limiting prolate (left) and oblate (right) symmetric tops is lifted. The actual B value for ethyleneoxide is marked by a dagger. Both conventions of labelling the rotational levels, the /r designation and the Jk-K designation are shown...

Fig. IV. 1. Coordinate systems used in the derivation of the classical Hamiltonian. The difference between the nuclear center of mass (n.c.m.) and the molecular center of mass (m.c.m.) which is typically on the order of 10 to 10- A is vastly exagerated for illustration. C r, By, (bz) are the basis vectors of the space fixed coordinate system. Ba, et, (Bc) are the rotating basis vectors of the principle moment of inertia tensor of the rigid nuclear frame...

Neither of the above results include the effects of the instantaneous distortion of the molecule, as the p, u are taken to be constants. In both these calculations we neglect the coordinate dependence of the inverse moment of inertia tensor in Eq. (5) and evaluate p, in the equilibrium configuration. This definition was the one of several definitions considered by Frederick et al. (84) that leads to the smallest standard deviation. If we were to include the coordinate dependence of p in our definition of the rotational energy operators, then both A v and AEA would be substantially larger. 

This constitutive property is termed rotational inertia (or moment of inertia) because, historically, it has been thought to oppose the start of the rotation of an object (which exhibits some resistance or inertia). The hat (circumflex) over the rotational inertia symbol means that it is not a scalar but an operator. Effectively, in the most general case, the inductive relation is not linear and the rotational inertia is a tensor. If the relativistic model for translational mechanics is relatively amenable, this is not the case in rotation during a translation because of the variation of radius with the velocity at high speed. 

---