Principal axis components

For a symmetric top, symmetry requires the dipole moment to lie along the symmetry axis, so that two of the three principal-axis components of d must vanish. In deriving the symmetric-top wave functions in Section 5.5, we assumed that the c axis was the symmetry axis hence to use the eigenfunctions (5.68) to find the selection rules, we must take da = db — 0, dcJ= 0. For a symmetric top, we thus must evaluate only the three integrals IXOc, lYoc anc Azof The three relevant direction cosines are given in (6.64) and Problem 5.15 they are independent of x- Since the integral... 

This moment of inertia is essential for the analysis of rotational spectra of molecules. For anisotropic solids or for molecules, the moment of inertia I is a second-rank tensor, with three principal-axis components I, I2, and f3. This moment of inertia is important when the body rotates with angular frequency co radians per second (co FIz), or with v revolutions per second. 

In organic radicals in solution, the y-factor anisotropy cannot be detected one needs oriented samples. In crystals of free radicals, this anisotropy is easily measured—for example, in crystals of sodium formate (Na+ HCOO-) the principal-axis components are gxx = 2.0032, gyy = 1.9975, and gzz = 2.0014. If there is some spin-orbit interaction in an organic molecule (e.g., if a compound contains S or Cl), then y-values as high as 2.0080 are encountered. In disordered powders with narrow EPR lineshapes, the y-factor anisotropy can produce considerable distortion in the overall signal, due to averaging of the y-tensor. 

Figure 2. Illustration of magnetic ineqnivalence for crystallographically eqnivalent sites of low point symmetry, nsing the chemical shift anisotropy (CSA). Oval represents a cross-section of the snrface for the CSA (Eqn. 1), with axes for the principal axis components corresponding to 5n and 833 (822 is normal to the page).

The nuclear shielding tensor is usually assumed to be symmetric although this may not be strictly valid [144]. Information concerning the principal axis components may be obtained from NMR studies of solids or other systems where the molecular motion is not isotropic. In a common NMR experiment on an isotropic liquid only an averaged shielding value, will influence the NMR resonance frequency,... 

The spectral features corresponding to all principal axis components of nuclear hyperfine and dipolar coupling interactions are clearly resolved and... 

Structural Studies Based on -Factor Measurements. - Accurate measurements of spin-label magnetic parameters provide important data about the structure of nitroxide radicals. These data are also required for the analysis of the molecular dynamics of spin probes. The high resolution of HF EPR makes possible accurate determination of the principal axis components of the -matrix and y4-tensor from powder pattern spectra eliminating the need to prepare and study single crystals. 

In linear, spherical and synnnetric tops the components of a along and perpendicular to the principal axis of synnnetry are often denoted by a and respectively. In such cases, the anisotropy is simply Aa = tty -If the applied field is oscillating at a frequency w, then the dipole polarizability is frequency dependent as well a(co). The zero frequency limit of the dynamic polarizability a(oi) is the static polarizability described above. 

The electric field gradient is again a tensor interaction that, in its principal axis system (PAS), is described by the tluee components F Kand V, where indicates that the axes are not necessarily coincident with the laboratory axes defined by the magnetic field. Although the tensor is completely defined by these components it is conventional to recast these into the electric field gradient eq = the largest component,... 

The principal axis of the cone represents the component of the dipole under the influence of the thermal agitation. The component of the dipole in the cone results from the field that oscillates in its polarization plane. In this way, in the absence of Brownian motion the dipole follows a conical orbit. In fact the direction of the cone changes continuously (because of the Brownian movement) faster than the oscillation of the electric field this leads to chaotic motion. Hence the structuring effect of electric field is always negligible, because of the value of the electric field strength, and even more so for lossy media. 

The coupling tensor Rlm in the laboratory frame is time dependent due to the motions of spin-bearing molecules. It can be expressed in terms of the rotational transformation of the corresponding irreducible components pln in the principal axis system (PAS) to the laboratory frame by... 

Note that, by this definition, f is inversely proportional to the rate of increase of the /th component of 50,/ in (6.259). Likewise, in directions of rapid decrease, the maximum half-length of a principal axis is limited to two. Using the definition of the difference vector 50o = 0o - 0m1. the initial EOA for the tabulated point 09] is defined169 by... 

The ZFS is assumed to be cylindrically symmetric (only the /q component is different from zero) and of constant magnitude. The static part of the Hzfs is obtained by averaging the Wigner rotation matrix Dq q[ pm(0] over the anisotropic distribution function, Pip pj. The principal axis of the static ZFS is, in addition, assumed to coincide with the dipole-dipole IS) axis. Eq. (48) becomes equivalent to Eq. (42), with the /q component scaled by Z)q q[ 2pm(0] The transient part of the Hzfs can be expressed in several ways, the simplest being 92) ... 

For the nitrogen hyperfine tensors, there is no satisfactory empirical scheme for estimating the various contributions, so that Table II compares the total observed tensor to the DSW result. The tensors are given in their principal axis system, with perpendicular to the plane of the heme and along the Cu-N bond. The small values (0.1 - 0.2 MHz) found for A O in the nonrelativistic limit are not a consequence of orbital motion (which must vanish in this limit) but are the result of inaccuracies in the decomposition of the total tensor into its components, as described above. 

The only orthotropic particles for which comprehensive experimental results are available are square bars, rectangular parallelepipeds with one pair of square faces. Symmetry then shows that the two principal resistances corresponding to translation with square faces parallel to the direction of motion are equal. These resistances will be denoted by c 2, while the resistance for translation normal to the square faces will be called cy. Consider such a particle in arbitrary translation at velocity U. Figure 4.11 shows a section of the particle parallel to the square faces (72 is the component of U in this plane, and the angle between U2 and principal axis 2 is 0. From Eq. (4-5), the drag components are as shown in Fig. 4.11. Hence the drag component parallel to U2 is... 

For the P-phase of PVDF the fc-axis is a principal axis of the polarizability tensor of the repeat unit in the absence of an applied field, only the component of parallel to the (>-axis is nonzero. Equation (3) may thus be expressed in scalar form, where Ap is also directed along the b-axis. 

The antisymmetric tensor is generally not observable in NMR experiments and is therefore ignored. The symmetric tensor is now diagonalized by a suitable coordinate transformation to orient into the principal axis system (PAS). After diagonalization there are still six independent parameters, the three principal components of the tensor and three Euler angles that specify the PAS in the molecular frame. 

The principal-axis dipole-moment components da, db, and dc depend on the configuration of the nuclei relative to one another, but are independent of the spatial orientation of the molecule they thus depend on the vibrational coordinates, but not on the Eulerian angles. We have... 

The state of stress in a flowing liquid is assumed to be describable in the same way as in a solid, viz. by means of a stress-ellipsoid. As is well-known, the axes of this ellipsoid coincide with directions perpendicular to special material planes on which no shear stresses act. From this characterization it follows that e.g. the direction perpendicular to the shearing planes cannot coincide with one of the axes of the stress-ellipsoid. A laboratory coordinate system is chosen, as shown in Fig. 1.1. The x- (or 1-) direction is chosen parallel with the stream lines, the y- (or 2-) direction perpendicular to the shearing planes. The third direction (z- or 3-direction) completes a right-handed Cartesian coordinate system. Only this third (or neutral) direction coincides with one of the principal axes of stress, as in a plane perpendicular to this axis no shear stress is applied. Although the other two principal axes do not coincide with the x- and y-directions, they must lie in the same plane which is sometimes called the plane of flow, or the 1—2 plane. As a consequence, the transformation of tensor components from the principal axes to the axes of the laboratory system becomes a simple two-dimensional one. When the first principal axis is... 

Figure 9(b) shows the data points plotted in the scattering diagram on PC 1 and PC 2 by the principal component analysis. The first principal axis reflects bitterness and sweetness. The second principal axis reflects sourness and umami. Amino acids are classified clearly into five groups by the taste sensor. 

The dependence of the principal components of the nuclear magnetic resonance (NMR) chemical shift tensor of non-hydrogen nuclei in model dipeptides is investigated. It is observed that the principal axis system of the chemical shift tensors of the carbonyl carbon and the amide nitrogen are intimately linked to the amide plane. On the other hand, there is no clear relationship between the alpha carbon chemical shift tensor and the molecular framework. However, the projection of this tensor on the C-H vector reveals interesting trends that one may use in peptide secondary structure determination. Effects of hydrogen bonding on the chemical shift tensor will also be discussed. The dependence of the chemical shift on ionic distance has also been studied in Rb halides and mixed halides. Lastly, the presence of motion can have dramatic effects on the observed NMR chemical shift tensor as illustrated by a nitrosyl meso-tetraphenyl porphinato cobalt (III) complex. 

Not all 9 components of a are typically reported when describing the chemical shielding tensor. Instead, the 3 principal components (or eigen values) in a principal axis system (PAS) are reported. The CSA tensor can also be described by three additional parameters 1) the isotropic value (or trace), aiso, of the shielding tensor and is defined as... 

Low-symmetry LF operators are time-even one-electron operators that are non-totally symmetric in orbit space. They thus have quasi-spin K = 1, implying that the only allowed matrix elements are between 2P and 2D (Cf. Eq. 28). Interestingly in complexes with a trigonal or tetragonal symmetry axis a further selection rule based on the angular momentum theory of the shell is retained. Indeed in such complexes two -orbitals will remain degenerate. This indicates that the intra-t2g part of the LF hamiltonian has pseudo-cylindrical D h symmetry. As a result the 2S+1L terms are resolved into pseudo-cylindrical 2S+1 A levels (/l = 0,1,..., L ). It is convenient to orient the z axis of quantization along the principal axis of revolution. In this way each A level comprises the ML = A components of the L manifold. In a pseudo-cylindrical field only levels with equal A are allowed to interact, in accordance with the pseudo-cylindrical selection rule ... 

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