Tensor determinant

Patriciu, A., Chirikjian, G.S., Pappu, R.V. Analysis of the conformational dependence of mass-metric tensor determinants in serial polymers with constraints. J. Chem. Phys. 2004,121,12708-20. 

Vaara et al.24 have published a very important and exhaustive review on spin spin coupling tensors determined either experimentally or by quantum chemical calculations. Among several other topics, comparisons of anisotropic contributions calculated with different approaches are given in this review. 

For typical solutes where the mass tensor determinant at each iq position can be considered as essentially independent of the other coordinates, we may then write... 

The measured crystal optical activity, in general, can be either of molecular origin or due to the chiral helical arrangement of chiral or achiral molecules in the crystal, or both. The two factors are difficult to separate. Kobayashi defined a chirality factor r = (pc — ps)/pc = 1 — pslpc, where pc is the rotatory power per molecule of a randomly oriented crystal aggregate derived from the gyration tensors determined by HAUP, and ps that in solution [51]. It is a measure of the 4 crystal lattice structural contribution to the optical activity and represents the severity of the crystal lattice structural contribution to the optical activity, and represents the severity of the restriction of the freedom of molecular orientation by forming a crystal lattice. Quartz is a typical example of r = 1, as it does not contain chiral molecules or ions and its optical activity vanishes in random orientation (ps = 0). 

Two types of terms appear in the stress tensor a in eqn (5.28), terms of the form and The divergence of this tensor determines the force... 

The second-order expansion given in eqn (6.96) recovers all of the physical quantities needed to describe a quantum system and determine its properties the charge density p and its gradient vector field Vp define atoms and determine many of their properties in a stationary state the current density determines the system s magnetic properties and the change in p in a time-dependent system and, finally, the stress tensor determines the local and average mechanical properties of the system. Thus, one does not need all the... 

The chemical shifts and principal values of the shift tensors determined for adenosine and guanosine in the solid state by using the FIREMAT experiment are summarized in Table 14. 

The radicals arising by X-ray irradiation of a single crystal of acridine are formed by the addition of a hydrogen atom to the heteroatom. Spin densities have been deduced from the proton hyperfine tensors determined by the ENDOR technique. The unpaired electron is... 

With Box = 0 and Boy = 0, the terms dominating the resonance frequency are given in the last row. Thus only three elements of the gradient tensor determine the resonance frequency in first order. The others, though nonzero, by (2.1.13) and (2.1.14), can be neglected. The three relevant terms are often concatenated to form the gradient vector G,... 

Fig. 3.2.3 Relationships between coordinate systems for the description of molecular order. The orientation of the laboratory coordinate system in the principal axes system of the coupling tensor determines the angular dependence of the resonance frequency. The orientations of the preferential sample direction in a molecule-fixed coordinate frame determines the orientational distribution function.

Fig. 7. The 13C powder lineshape calculated for the Cp ring hopping between chemically equivalent sites in [A1C14] [AlCp 2]+ using the 13C-shielding tensor determined from ab initio electronic structure calculations.12 The figure illustrates how the lineshape varies with the ring hopping rate k. The orientation of the principal axes of the 13C-shielding tensor determined from the ab initio calculations is shown in the inset.

Kehrer et al.(S) Orientation of EFG tensor determined Vzz oriented ca. along shortest H- -Br hydrogen bond... 

In a homogeneous bulk phase the force across any unit area is equal in all directions, except in the surface phase s. Here this force depends on the direction and is called pressure which is assumed to be a tensor (in the bulk phase the pressure is a scalar). The normal and tangential components of the pressure tensor determine the interfacial tension y. According to Bakker (1928) we obtain... 

Schmitz C, John M et al (2006) Efficient chi-tensor determination and NH assignment of paramagnetic proteins. J Biomol NMR 35(2) 79-87... 

Single crystal nmr spectra have been obtained for the resonance in K2Pt(CN)4Bro.3(H20)3, and the chemical shift tensor determined (565). The principal values (relative to the carboxyl carbon in acetic acid) have been reported as uu = 261 lOppm (parallel to the CN bond), <722 = — 48 10 ppm (perpendicular to the CN bond and Pt-Pt chain) and <733 = —10 5 ppm (parallel to the Pt-Pt chain). The primary effect on the chemical shift tensor of the cyanide bonding to the platinum and of the solid state structure is the nonzero asymmetry produced around the CN bond (565). 

The distance r between the atom, which the electron is centered on, and the nucleus is obtained by identifying bmax with the numerically largest principal value of the dipolar coupling tensor determined experimentally. The formula applies to the case of a localized unpaired electron. The method can be elaborated to cases... 

The calculated eigenvector for the maximal principal value of a specific lithium ion is compared with the maximal values and vectors for all Li hfc tensors determined experimentally. The Li hfc tensors are assigned to specific Li" " ions in the crystal structure by this comparison [29]. 

Formulae for the yz-, and zx-planes are obtained as in Section 3.3.2 by cyclic permutations of the X, y, and z indices. Thus, to obtain the tensor gA g the separation A between adjacent hyperfine lines and the. g-factor are both measured at each angle 0 in the three planes. The tensor is then obtained by an algebraic procedure [12] using the g-tensor determined from the same set of measurements. The analysis is simpler for isotropic g, see Exercise E3.9. 

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