Planar moment tensors

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 rotational motion of the rigid set of mass points about any axis through its center of mass in the absence of exterior forces is known as the free rotation of the rigid body. The planar moment tensor for this motion, with the position vectors ra referred to an arbitrary basis system, can be compactly written as a dyadic (T denotes transposition) [8,32],... 

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

In a practical case, many of the Ama(s) will vanish (for an atom that has never been substituted, all Ama(s) vanish). The planar moment tensor of the parent, with reference to its own PAS (cf. Eq. 11),... 

Kraitchman s basic idea was to introduce into this first (diagonal) tensor term of Eq. 35 the three experimental principal planar moments of the parent PgWexp(l) as obtained from the MRR spectrum and treat them as independent experimental information. This is the essential distinguishing feature between any retype and any rQ-type method and all differences between the two types of treatment may be traced back to this fact. The first term of Eq. 35 is now written as P p(l). It >s then convenient to replace the notation for the planar tensor P[11(s) (Eq. 35) by I V) to distinguish this new function (Eq. 36) from Eq. 35. Note that Eq. 36, in contrast to Eq. 35, depends explicitly on the positions of only those atoms that have actually been substituted in the isotopomer 5 ... 

For a planar shear fault (with normal v in the X3 direction and displacement discontinuity [u] in the Xi direction, say, so that vj = V2 = 0 and [ 2] = [M3] = 0) in a homogeneous isotropic medium (Cya = kSyShnS/ + IfidikSji), Eq. 8 gives a moment tensor with two nonzero components Mi3 = M31 = /jAu, where u is the average slip. This corresponds to a pair of force couples, one with forces in the Xi direction and moment arm in the Xj direction, and the other with these directions interchanged. 

EFISHG yields projections of the /3 tensor on the direction of the molecular dipole moment (z-axis). Hence a specific linear combination of elements is obtained and not a unique -value that is sufficient to characterize the molecular second-order NLO response. This is a serious limitation of the technique some components of /3 may be large but will not show up in the experimental results because their projection on the direction of the molecular ground-state dipole is zero. However, the use of polarized incident light with polarization directions parallel and perpendicular to the externally applied electric field allows the extraction of further information on the /3 tensor. For planar molecules conjugated in the yz plane, components with contributions of the X direction may be safely ignored. Two linear combinations, /3 and of tensorial elements may then be determined (Wortmann et al., 1993), (123) and (124) ... 

Figure 2.7-6 A Assignment of the Cartesian coordinate axes and the symmetry operations of a planar molecule of point group C2,.. B Character table, 1 symbol of the point group after Schoen-flies 2 international notation of the point group 3 symmetry species (irreducible representations) 4 symmetry operations 5 characters of the symmetry operations in the symmetry species +1 means symmetric, -1 antisymmetric 6 x, y, z assignment of the normal coordinates of the translations, direction of the change of the dipole moment by the infrared active vibrations, R, Ry, and R stand for rotations about the axes specified in the subscript 7 x, xy,. .. assign the Raman active species by the change of the components of the tensor of polarizability, aw, (Xxy,. ...

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... 

As an example consider a planar, near oblate symmetric top molecule. If small amplitude vibrations are assumed, we may expect that the ju-tensor elements are of the order of the reciprocal principal moments of inertia. However, for a planar configuration, withIc-Ia+Ib, it is easily seen that the generally smallest element, ncc, may reach extreme values when evaluated in a PAS using Eq. (2.63). A numerical example illustrates this ... 

Fig. 12.4. The direction of the induced dipole moment may differ from the direction of the electric field applied (due to the tensor character of the polarizability and hyperpolarizabilities). Example the vinyl molecule in a planar conformation. Assume the following Cartesian coordinate system x (horizontal in the Figure plane), y tvertical in the Figure plane and (peipendicular to the Figure plane), and the external electric field E = (0, y, 0). Tlie compmient x rf the induced dipole mcxnent is equal to [within the accuracy of linear terms, Eq. (12.19)] = M.v — ft0.v yv y. I ind.y ftind, Due to the...

For the case of a two-dimensional analysis, when motion is assumed planar, the moment of inertia in Eq. (5.18) takes on a single value. In the case of a three-dimensional analysis, I becomes a 3 X 3 inertia tensor. The main diagonal of the inertia tensor is constant and the off-diagonal elements vanish when the principle axis of inertia is aligned with the axes of the ACS. The diagonal matrix in Eq. (5.38) reflects this alignment and is the form used in Eq. (5.18) for a three-dimensional analysis in which the moments are expressed in the ACS of the segment. 

The local pressure tensor pfi) suffers from ambiguities similar to those of die energy nshy sinoe it can be expressed in terms of the viiial of the intermolecular potential (S 4.3), but again there are the constraints that any consistent definition must lead to invariant values for observaUe quantifies. In a fluid with a gradient only in the z-direction mechanical stability requires that Pn(z) is a constant, and equal to p and p. The zeroth-moment of the difference n (z)-p,first moment is apparently not, and have discussed in 4.8 the diflBculties that follow if this moment is identified with the planar limit of the surface of tension that enters into the thermodynamic discussion of the spherical drop. 

It is worth mentioning that Osman et al. [74] found theoretically that, in planar dyes, such as anthraquinone, the order tensor deviates strongly from cylindrical symmetry. The direction of the optical transition moment does not, in general, coincide with one of the principal axes of the order tensor. Based on their calculations they found that only those anthraquinone dyes that have a small angle between the transition moment and the molecular axis show a good dichroic ratio [74]. 

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