Inertial moment tensors

The original rs-method is based on equations presented by Kraitchman [4], He had noticed that the numerically dominant part of the inertial moment tensor for an isotopomer in which (exactly) one atom has been substituted, can be replaced by the known inertial moments of the parent molecule, and that the remainder of the tensor then depends only on the Cartesian coordinates of the substituted atom. Equating the roots of the secular equation for this hybrid tensor to the experimental inertial moments of the isotopomer, Kraitchman obtained his famous equations which, quite in contrast to any r0-type method, always aim at the determination of the Cartesian coordinates of just this one substituted atom. The rest of the molecule is irrelevant, it does not influence the result and need not be known. Costain has checked, for several small molecules, the invariance of -determined bond lengths when employing different pairs of parent molecule and isotopomer [7]. He found it superior to the comparable r0-data and recommended to prefer -structures for... 

In mechanics, the inertial properties of a rotating rigid body are fully described by its inertial moment tensor I. We can simplify the subsequent equations if we employ in place of I the closely related planar moment tensor P, apparently first used by Kraitchman [4], At any stage of the calculations, however, an equivalent equation could be given which involves I instead of P. The principal planar moments P (g = x, y, z) are the three eigenvalues of the planar moment tensor P and the principal inertial moments Ig the eigenvalues of the inertial moment tensor I. Pg, Ig, and the rotational constants Bg = f/Ig are equivalent inertial parameters of the problem investigated (/conversion factor). 

The inertial moment tensor I and the planar moment tensor P are related by,... 

Kuz min et al. (15) pointed out a standard result of classical mechanics If a configuration of particles has a plane of symmetry, then this plane is perpendicular to a principal axis (19). A principal axis is defined to be an eigenvector of the inertial tensor. Furthermore, if the configuration of particles possesses any axis of symmetry, then this axis is also a principal axis, and the plane perpendicular to this axis is a principal plane corresponding to a degenerate principal moment of inertia (19). 

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... 

The Q, a, and 31 tensors are all defined in the principal inertial axes systems. Qzz is the scalar quadrupole moment of the nucleus [defined by the convention in Eq. (11)] and Q is the field-gradient tensor at the nucleus described again in the principal inertial axes systems. All other terms have been defined previously. 

I is the moment of inertia tensor if the x, y, z axes are chosen to be the principal inertial axes of the molecule a, b, c), I is then diagonal with principal components ha, hb, he For a linear molecule (including diatomics), ha = 0 and hb = he- In the inertial axis system equation (8.76) becomes simply... 

To obtain a simple form of the balance of moment of momentum, we confine its formulation to inertial frame with angular moment (3.88) having point y fixed here (although we use here the inertial frame fixed with distant stars, resulting formulations are valid in any inertial frame as will be shown at the end of this section). Again, the main reason for that is the nonobjectivity of x, y, v in (3.88), cf. (3.25), (3.38) generalization of this balance in the arbitrary frame will be discussed below but we note that the main local result—symmetry of stress tensor (3.93) below—is valid in the arbitrary frame. 

Balance of moment of momentum (3.93) expressed through the symmetry of a stress tensor (at least for mechanically nonpolar materials, cf. Rem. 17) is valid in any frame, even noninertial. Finally we can see that because (3.48) is valid for transformations between any inertial frames, the balances of angular moment related to fixed y (3.89)-(3.91) are valid in any inertial frame and not only in those fixed with distant stars. 

Elements of the g tensor for the rotational magnetic moment in units of the nuclear magneton and referred to the principal inertial axes from the linear and quadratic rotational Zeeman effect [1, 11] ... 

In references (Santamaria Holek, 2005 2009 2001), the Smoluchowski equation was obtained by calculating the evolution equations for the first moments of the distribution function. These equations constitute the hydrodynamic level of description and can be obtained through the Fokker-Planck equation. The time evolution of the moments include relaxation equations for the diffusion current and the pressure tensor, whose form permits to elucidate the existence of inertial (short-time) and diffusion (long-time) regimes. As already mentioned, in the diffusion regime the mesoscopic description is carried out by means of a Smoluchowski equation and the equations for the moments coincide with the differential equations of nonequilibrium thermodynamics. 

Molecules in the tetrahedral, octahedral, and icosahedral point groups behave like spherical electron distributions in this respect the induced dipole moment is independent of the molecule s orientation in the electric field. Most molecules, however, are more easily polarized along one axis than another. The polarizability in this case is actually represented by a matrix called the polarizability tensor, with elements that describe the polarizability along the molecule s principal inertial axes. 

Then if Y, is the /th column of the scaled coordinate matrix Y, we have BY, = X, Y, for i = 1,..., 3. It follows that these columns are proportional to eigenvectors of the scaled estimated Gram matrix B, while the moments of inertia Xi, X2, X.3 are the corresponding eigenvalues. Since the eigenvectors have unit norm, the diagonal form of the inertial tensor implies that the constant of proportionality is VXJ. 

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