Principal axis frame

Fundamental constants (Cx), spatial tensors in the principal axis frame ((fi3 m,)F), and spin tensors (Tjm) for chemical shielding (a), J coupling (J), dipole-dipole (IS), and quadrupolar coupling (Q) nuclear spin interactions (for more detailed definition of symbols refer to [50])... 

In the principal axis frame of the alignment tensor, A, the dipolar coupling between two nuclei, P and Q, as a function of the polar coordinates, 6 and tp, is given by... 

The nuclear spin interaction tensor is most readily expressed in its principal axis frame where only the M = 0, 2 terms are nonzero (and only the M = 0 term is nonzero for axial symmetry). It can then be expressed in the laboratory frame via... 

Figure 2 Schematic view of the rotor and shielding tensor placed in a magnetic field. The angle y is the angle between B0 and the principal z-axis of the shielding tensor b is the angle between the z-axis of the shielding tensor principal axis frame and the spinning axis.

The differences in the diagonal elements, in the principal frame, represents the differences in length of the principal axes. The average radius of the ellipsoid is arbitrary. In the principal axis frame of the ellipsoid, there are only two independent components the cylindrical Vjz component, and a rhombic component. 

In PAHC the leftmost matrix R on the right-hand side of Eq. (15) identifies a body frame (the principal-axis frame). The Jacobi vectors referred to this body frame, p, and p2, are expressed as [cf. Eq. (2)]... 

In Eq. (19), the angle 0 specifies the orientation of the principal-axis frame of the three-atom system and has nothing to do with the shape of the molecule. The continuous change in 0 causes the ordinary rotation of the system without changing the system shape. On the other hand, the continuous change in cp in... 

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... 

The Wigner rotations describe the coordinate transformations from the principal axis frame (P ) in which the tensor describing the interaction X is diagonal, via a molecule-fixed frame (C) and the rotor-fixed frame (R) to the laboratory frame (L) as illustrated in an ORTEP representation in Fig. 1. 

The Euler angles represent transformations between the principal axis frame (P) and the peptide plane (E) coordinate system, see Figure 2a. 

Here, Aq is the constant inherent to the nucleus, and are the polar angles that define the direction of Bq in the principal axis frame for the field gradient tensor, and 17 is the anisotropic parameter. For example. Fig. 3.11 shows the principal axis frame XpypZp (Cp) for the phenylene deuterium in the ortho position. As seen in this figure, the Zp axis is parallel to the direction of the C—bond, the yp axis is perpendicular to the phenylene plane and the Xp... 

Fig. 3.11. Coordinate transformations among different frames Cp, principal axis frame Cm, molecular frame Cr, reference frame and Ci, laboratory frame.

In Equation (3.51), Tr(cr /) is expressed by Tr(tr,y) in the principal axis frame, because the trace does not change by the unitary transformation. On the other hand, A b should be described by A20 in the principal axis frame through appropriate coordinate transformations depending on the molecular motion allowable to the system. 

Here, the matrices appearing on the righthand side of this equation are the transformation matrices between the two respective coordinates as shown in Fig. 3.14. Using this R A o in Equation (3.52) is described by the corresponding A20 in the principal axis frame as follows ... 

Fig. 3.14. Schematic representation of the coordinate transformation from the principal axis frame (Cp) through the molecular frame (Cm) and the reference frame (Cr) to the laboratory frame (Ci ).

Figure 64. ORTEP stereo drawings for thermal ellipsoids of Leu-29 in BPTI in the dynamics (upper) and stereochemical (lower) principal-axis frames.

The matrices x° and A° are the equilibrium values of these matrices. It should be noted that in an Eckart frame A° = 0. Moreover, if the Eckart frame corresponds to the principal axis frame in the equilibrium configuration, then Hr takes the familiar form of a Hamiltonian for rigid-body motion ... 

The elastic deformation resistance of rubbers and their temperature dependence must be viewed on a thermodynamic basis. Moreover, the mechanistic response of rubbers is also best understood in a principal-axis frame of reference. 

The g- and superhyperfine tensors, in general, are eharacterized on the basis of symmetry, and their relative orientation is fixed in the moleeular referenee frame (usually defined by molecular symmetry axes). The Hamiltonian that describes the electronic Zeeman interaction in a principal axis frame (no terms of the form gij, where i J) is... 

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