L tensor elements

The L are the familiar L matrix elements the coefficients L" and L u, etc., will be called the second and third derivative elements of the L tensor, following the notation introduced by Hoy et al,12 In equation (44) an unrestricted summation is to be understood over all indices repeated as a superscript and a subscript in the terms on the right-hand side this convention will be followed in all the later equations of this section. The use of subscript/ superscript notation for the indices on the L matrix and L tensor elements, which is used throughout the equations of this section, simplifies the rather complex algebra involved in the non-linear co-ordinate transformations. Equation (44) may be compared with equation (39) for the co-ordinates Ri, which contains only linear terms. 

The first derivative L tensor elements, are determined as the elements of the L matrix from the preliminary harmonic calculation. The second and third derivative L tensor elements have been determined in one of two ways in the literature of anharmonic calculations. The first method involves setting up an... 

The second method of determining the L tensor is that proposed by Hoy et al.,1 in which closed analytical formulae are obtained for the second and third derivative L tensor elements in terms of the first derivative elements, by direct differentiation of Hi with respect to Qr. The resulting formulae are given in equations (22)—(36) of ref. 12, and they will not be repeated here. These formulae, which may be directly programmed, greatly simplify the determination of the L tensor. The existence of these formulae emphasize that the L tensor may be entirely determined from a knowledge of the L matrix. 

In this case the second derivative L tensor element L22 is particularly important. Thus on substituting for 8r in the stretching potential V = fr8r2,we obtain large terms in Q i, QrQ2, and Qi, showing that 4>u, and may all be expected to show large contributions from frr. Table 12 confirms this. 

A simplified method exists" for setting up the required nonlinear coordinate transformations from curvilinear internal coordinates to simple normal coordinates. The transformation coefficients are called the L tensor elements, and the transformation equations can simply be written as... 

One convenient set of equations to calculate the elements of the L tensor has been given for the five basic types of internal coordinates (stretch, bend, linear bend, out-of-plane, and torsion). These equations can also be generalized. For example, calculation of the L tensor elements for the bond stretching coordinate can be accomplished through equations almost identical to those given in equation (37). 

If the origin is located at a center of symmetry, for each atom at r with vibration tensor U , there will be an equivalent atom at — r with the same vibration tensor. When the observational equations for these two atoms are added, the terms involving elements of S disappear since they are linear in the components of r. The other terms, involving elements of the T and L tensors, are simply doubled, like the U" components. 

Figure 11. 15N NMR MAS NMR spectra and tensor results for [15N4]-ring labeled metalloporphyrins. A, 8.45 Tesla 15N MAS NMR spectrum of Fe(tetraphenylporphyrinate)(PhNO)( 1 -methylimidazole), vr = 2.6 kHz. B, graph showing correlation between experimental and DFT computed I5N tensor elements for Fe(TPP)(PhNO)( 1 -Melm) ( ) and Fe(TPP)(CO)(l-MeIm) (O). The mean experimental and theoretical shieldings over the four non-equivalent sites in each molecule are shown since the solid state shifts are not specifically assigned. Slope = 0.847, R2 = 0.995.

To obtain the quadrupole Hamiltonian of a spin in a magnetic field the Hamiltonian needs to be transformed from the PAS to the LAB frame, keeping only those terms that commute with L. This is called truncation of a Hamiltonian and is only valid when Hq << Hz (the high field approximation). To perform the transformation it is much more convenient if second-rank irreducible spherical tensors are used. The Cartesian and spherical tensor elements (T) can be related (see Schmidt-Rohr and Spiess 1994 and Eq. 8, in Man 2000), with two of the more common elements being... 

Since second-order nonlinear optical materials are anisotropic, their optical properties are described by tensors as discussed previously in Sect. 2.1.2. For a nonlinear optical process, the -th order nonlinear polarization is due to n interacting electric held vectors and is described by an (n -I-1) rank tensor composed of 3"+ tensor elements. In nonlinear optics, several fields with different frequencies l can be present simultaneously so that the electric field and the polarization are represented by... 

Table 6. Isotropic 15N shielding (triso) and 15N shielding tensor element (022) of solid polypeptides [Leu, X] containing 15N-labelled L-leucine residue in the a-helix...

Figure 29 shows the plots of criso and shielding tensor elements are displaced over a wide range of L-leucine content (20 100%). Therefore, the 022 value contains some information about neighbouring amino acid sequence effects and any higher ordered interactions, as well as conformation-dependent interactions. 

Simulations [69, 70] of the spinning sideband patterns of P spectra obtained at 80.9 MHz with a low MAS speed (1-2 kHz) (not shown here) were carried out on all the polysiloxane-immobilized phosphine samples [71] to provide values of the principal elements of the chemical shift tensor, which often are valuable in elucidating chemical structure. Two model compounds, diphenyl-isopropylphosphine oxide and l,6-bis(diphenylphosphino)hexane, were used to provide CSA tensor elements that would be approximately representative for the polysiloxane-immobilized phosphine oxide and phosphine moieties, respectively. The experimental P spectra were then simulated on the basis of the principle tensor elements obtained on the two model compounds 117.1, 86.8 and -93.7 ppm for diphenyl-isopropylphosphine oxide and 7.3, -29.8 and -40.9 ppm for l,6-bis(diphenylphosphino)hexane. The simulations showed that the P chemical shift anisotropies of both phosphine and phosphine oxide moieties are qualitatively similar within each of these two categories for all the samples examined, implying very similar... 

In Eq. (I.l) Ja, Jt>, and Je stand for the a, b, and c-components of the overall angular momentum measured in units of h. They are referred to the principal axis system of the molecular moment of inertia tensor. Because of the mutual compensation of positive and negative contributions, the absolute values of the g-tensor elements are usually smaller than 1 (typically on the order of 0.01 to 0.1 as may be checked in Table AI of the Appendix). In many cases the off-diagonal elements of the g-tensor in Eq. (I.l) will be zero because of molecular symmetry. Formaldehyde or 1,2-difluorobenzene may serve as examples. 

Now consider the case where 7)/ is a tensor in the laboratory-fixed coordinate frame. Then the spherical components in Eq. (7.C.13) are also in the laboratory frame, and we denote this by writing these elements as Tf (L). The elements Tf (L) that appear in Eq. (7.C.13) can be related through Eq. (7.C.8) to the spherical components in the molecule or body-fixed coordinate system... 

Similar arguments apply to the point groups Di and C2 . For groups of lower symmetry, where the axes of M do not coincide with the molecular axes, fewer tensor elements vanish by symmetry. Thus for a resonance transition between g Ag) and in C2/, the scattering tensor of a bg mode... 

Only theoretical values of the quadrupole moment are available. Ab initio calculations gave the tensor elements xx = 0.9701, 0yy=-0.3670, and -0.6030 (y L molecular plane, z = C2 axis with the positive direction from H atoms to N) [4]. 

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