Projected tensors determinants

Note that the form of the projection tensor P depends on the form chosen for the hard components of Z v Specifically, values of the mixed soft-hard components of P, , which are not specified by the definition of a generalized projection tensor given in Section VIII, are determined in this context by the values chosen for the mixed components of Z v, which specify correlations between hard and soft components of the random forces that are not specified by Eq. (2.295) for Z v... 

In this section, we develop some useful relationships involving the determinants and inverses of projected tensors. Let S ap be the Riemannian representation of an arbitrary symmetric covariant tensor with a Cartesian representation S v We may write the Riemannian representation in block matrix form, using the indices a,b to denote blocks in which a or p mns over the soft coordinates and i,j to represent hard coordinates, as... 

We first prove a theorem given by Fixman relating the determinants of projected tensors Sab and T ->, which is stated in Eq. (2.28). The proof given here follows that given for the mass matrix in Ref. 35. Define a 3N x 3N matrix... 

Derivatives of the determinants S and f of the generic projected tensors Sah and Tij defined in Eqs. (2.20) and (2.24) may be expressed compactly in terms of the reciprocal vectors that are generated by applying Eqs. (2.207) and (2.208) to the corresponding Cartesian tensors S v and respectively. Using Eq. (A.14) to differentiate In 5 with respect to a soft variable gives... 

The affine tensor is unambiguously determined by the projective tensor Gag and gij fixes a Riemannian metric with element of arc... 

The characterisation of the angular dependence of the interaction of two dipole tensors A1 A2 and B B2 is therefore straightforward, namely it depends on the projection angle of the two bonds between A1 and A2 and between B1 and B2. The orientation and magnitude of the chemical shift anisotropy (CSA) tensor, which also can cause cross-correlated relaxation, is not know a priori and therefore needs to be determined experimentally or... 

Fixman has shown [2] that, for any covariant symmetric tensor S ap defined in the full space, with an inverse = (5 ) in the full space, the determinants S and f of the projections of S and T onto the soft and hard subspaces, respectively, are related by... 

The dual axial vector in 4-space is constructed geometrically from the integral over a hypersurface, or manifold, a rank 3-tensor in 4-space antisymmetric in all three indices [101]. In three-dimensional space, the volume of the parallelepiped spanned by three vectors is equal to the determinant of the third rank formed from the components of the vectors. In four dimensions, the projections can be defined analogously of the volume of the parallelepiped (i.e., areas of the hypersurface) spanned by three vector elements < dl, dx and dx". They are given by the determinant... 

We now turn to the 6-z plane, after having determined seven of the nine strain-rate elements of the strain-rate tensor. Figure 2.8 illustrates the two-dimensional projection a differential element on a 6-z surface for some r value. 

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. 

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) ... 

The most traditional experimental determination of p is the electric field-induced second harmonic (EEISH) method, which requires the molecules to be aligned in solution by an electric field, by means of their static dipole moment (po). The EEISH signal is therefore proportional to po and to p <>c (projection of p on po), which is assumed to be equal to p in most cases. The bulk NLO properties are frequently evaluated as the efficiency of a powdered sample in second-harmonic generation (SHG), or as the d components of the x tensor. 

Figure 5 shows the calculated optical absorption spectrum, obtained from an analysis of the dipole matrix elements of Fe8Br62+. The two projections correspond to two most different polarization axes which have been determined by diagonalizing the bare polarizability tensor. The calculated and experimental electronic structures were found to be relatively good agreement. Despite this fact the anisotropy parameters calculated for this molecule seem to overestimate the experimental results by about a factor of two. 

Two points, however, should be taken into account. First, natural crystals can show significant variability that depends upon the growth conditions and locality (e.g., solid solutions and incorporation of impurities). It is necessary to measure the bulk crystal structure of such samples before it is possible to determine the surface structure using the CTR approach for such samples. Second, the CTR intensities can depend on the type of form factors (e.g., neutral or ionic form factors) used in the bulk structure analysis. At minimum, the calculated bulk Bragg reflectivities must reproduce the observed values precisely internal consistency requires that we use the same atomic form factors that were used in the determination of the bulk crystal structure. Similarly, the bulk vibrational amplitudes derived from the original bulk crystal structure analysis must be used. In many cases, vibrational amplitudes are anisotropic and are therefore described by a tensor. The appropriate projection of the vibrations for each scattering condition, Q, needs to be included in the expression for Fuc-... 

The spin projection factors 7/3 and —4/3 relate the tensors of the coupled system to the local tensors and reflect the orientation of the local spins relative to the system spin, S = Sa -f Sb- Since gjb > 2.0 (y = x, y, z) the (-4/3) factor of gb leads to g values below g = 2.00. " For applied magnetic fields B > 0.05 T the electronic Zeeman interaction in Equation (5) is at least 20 times larger than the hyperfine interactions. Consequently, the expectation value of the electronic spin, , is determined by the electronic Zeeman term, allowing us to replace the spin operator S... 

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