Inertial tensor, principal

If the considered molecule cannot be assimilated to a sphere, one has to take into account a rotational diffusion tensor, the principal axes of which coincide, to a first approximation, with the principal axes of the molecular inertial tensor. In that case, three different rotational diffusion coefficients are needed.14 They will be denoted as Dx, Dy, Dz and describe the reorientation about the principal axes of the rotational diffusion tensor. They lead to unwieldy expressions even for auto-correlation spectral densities, which can be somewhat simplified if the considered interaction can be approximated by a tensor of axial symmetry, allowing us to define two polar angles 6 and

symmetry axis of the considered interaction) in the (X, Y, Z) molecular frame (see Figure 5). As the tensor associated with dipolar interactions is necessarily of axial symmetry (the relaxation vector being... 

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). 

Let us first examine a few special cases that cover most common point groups. A linear molecule, such as HCN (point group Coov) or acetylene (Dxl), will lie along one principal axis, say the z axis, so that the first eigenvalue of the inertial tensor vanishes and the other is doubly degenerate alternatively, by the second case in Eq. 3 x, = v, = 0 for all i, and thus % = 0. 

The q matrix is the negative of the electric-field gradient. Like the inertial tensor and the polarizability tensor, q is symmetric (since the order of partial differentiation is immaterial), and we can make an orthogonal transformation to a new set of axes a, ft, y such that q is diagonal, with diagonal elements qaa, q, q. Note, however, that the origin for q is at the nucleus in question and the axes for which q is diagonal need bear no relation to the principal axes of inertia (unless the nucleus happens to lie on a symmetry element). 

For each of the following, state whether the principal axes of the inertial tensor have the same orientation in the molecule as the principal axes of the polarizability tensor, (a) H20 (b) HDO (c) D20. 

When the principal inertial axes system (PAS) is used as the coordinate system, the inertial tensor of the whole molecule is diagonal, and thus... 

In low-symmetry molecules, diagonal and off-diagonal matrix elements of the electronic dipolar coupling tensor may contribute to ( h[)0 ) J ssl b ). Therefore, they are specified mostly in terms of their Cartesian components. If symmetry is C2V or higher, the off-diagonal matrix elements of the tensor operator in Eq. [163] vanish (i.e., the principal axes diagonalizing the SCC tensor coincide with the inertial axes). For triplet and higher multiplicity states, one then obtains... 

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). 

With/ running over x, y and z all inertial products are constrained to zero, and consequently the molecular axes are principal axes of the instantaneous tensor of inertia. 

The spin-rotation constants can be measured at zero field according to Eq. (26), where we write the Si tensor in terms of the molecular-fixed principal inertial values by using the direction cosines... 

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... 

Components of the Q tensor along the principal inertial axes a, b, c (b bisects the bond angle, c 1 molecular plane) from the first- and second-order Zeeman effect of several pure rotational transitions [26, 27,32,37] areQaa = -1.6 1.4, Qbb= + 2.1 1.1, and Qcc = -0.5 1.9. 

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] ... 

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. 

For asymmetric top molecules, the principal axis system of the inertia tensor and the field gradient tensor do not coincide in general. In the case of a completely non-symmetric position of the quadmpolar nucleus in the molecule, none of the components Xgg- of the field gradient tensor eqnals zero. If a nucleus lies on a plane which contains the principal inertial axes g and g and which is a symmetry plane of the molecule, then the off-diagonal elements Zgg and Xg g" vanish. 

In many cases the coupling tensor cannot be determined completely. In order to discuss bond properties, one frequently assumes that the bond direction coincides with one of the principal axes of the quadmpole coupling tensor (bond axis system), yielding information if the position of the principal inertial axes is known. 

Z -[MHz] components of the nuclear quadrupole coupling tensor in the principal inertial axes system... 

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