Value, principal

While the principal value of the book is for the professional chemist or student of chemistry, it should also be of value to many people not especially educated as chemists. Workers in the natural sciences—physicists, mineralogists, biologists, pharmacists, engineers, patent attorneys, and librarians—are often called upon to solve problems dealing with the properties of chemical products or materials of construction. Eor such needs this compilation supplies helpful information and will serve not only as an economical substitute for the costly accumulation of a large library of monographs on specialized subjects, but also as a means of conserving the time required to search for... 

Gel polymers such as AG are used as part or all of the polychloroprene component ia an adhesive. Its principal value is ia providing relatively high... 

In this section, well-known properties of second-order positive-definite symmetric tensors and functions involving them will be cited without proof. The principal values and principal vectors (m = 1, 2, 3) of a symmetric second-order tensor A are given by... 

Associated with /4 is a quadric surface, which for positive-definite symmetric A is an ellipsoid, whose intercepts with the principal directions are the principal values. When expanded, (A.72) becomes... 

Note that 0" < A< 60". The invariants A , and form a cylindrical coordinate system relative to the principal coordinates, with axial coordinate / A equally inclined to the principal coordinate axes, with radial coordinate /3t, and with angular coordinate The plane A" = 0 is called the II plane. Because the principal values can be ordered arbitrarily, the representation of A through its invariants in n plane coordinates has six-fold symmetry. 

This monomer has been used as the basis of a laminating resin and as a reactive diluent in polyester laminating resins, but at the present time its principal value is in moulding compositions. It is possible to heat the monomer under carefully controlled conditions to give a soluble and stable partial polymer in the form of a white powder. The powder may then be blended with fillers, peroxide catalysts and other ingredients in the same manner as the polyester alkyds to form a moulding powder. Similar materials may be obtained from diallyl isophthalate. 

With the new coordinate system only the three diagonal components axx, ayy, and olzz referred to as principal values of a are nonzero. The halfaxes of the ellipsoid are a 2, ay]/2, and aj1/2. If the polarizability ellipsoid... 

Table 1 Isotropic and Principal Value Shieldings, and Intensity Measures for the Benzene Molecule. See text.

The ESR spectrum of C6H6 " trapped in CFCI3 at 15 K is shown in Figure la and agrees with that reported previously [18]. The principal values of the hyperfine coupling were obtained from previous ESR and ENDOR measurements [17, 18]. The best agreement with experiment was obtained with the axes oriented as in Table 4. In the latter study, the simulated ENDOR spectra were insensitive to the orientation of the tensor axes, however, and the assignment was made on the basis of molecular orbital calculations [9]. The tensor data are reproduced here for convenience (see Table 4). 

Radical IV can be considered as a unique phosphorus radical species. Reduction of the parent macrocycle with sodium naphtalenide in THF at room temperature gave a purple solution. The FPR spectrum displayed a signal in a 1 2 1 pattern, with flp(2P)=0.38 mT. DFT calculations on radical IV models indicated a P-P distance of 2.763 A (P - P is3.256 A in the crystal structure of the parent compound and the average value of a single P-P bond is 2.2 A). According to the authors, the small coupling constant arises from the facts that the principal values of the hyperfine tensor are of opposite sign and that the a P P one electron bond results from overlap of two 3p orbitals [88]. 

Principal Values of 29Si Chemical Shift Tensors for Disilenes (ppm)... 

The g-tensor principal values of radical cations were shown to be sensitive to the presence or absence of dimer- and multimer-stacked structures (Petrenko et al. 2005). If face-to-face dimer structures occur (see Scheme 9.7), then a large change occurs in the gyy component compared to the monomer structure. DFT calculations confirm this behavior and permitted an interpretation of the EPR measurements of the principal g-tensor components of the chlorophyll dimers with stacked structures like the P 00 special dimer pair cation radical and the P700 special dimer pair triplet radical in photosystem I. Thus dimers that occur for radical cations can be deduced by monitoring the gyy component. 

Petrenko, A., K. Redding et al. (2005). The influence of the structure of the radical cation dimer pair of aromatic molecules on the principal values of a g-tensor DFT predictions. Chem. Phys. Lett. 406 327-331. 

The physical interpretation of the anisotropic principal values is based on the classical magnetic dipole interaction between the electron and nuclear spin angular momenta, and depends on the electron-nuclear distance, rn. Assuming that both spins can be described as point dipoles, the interaction energy is given by Equation (8), where 6 is the angle between the external magnetic field and the direction of rn. 

A sinusoidal plot of grf>2 vs.

crystal plane gives another set of Ks that depend on other combinations of the gy, eventually enough data are obtained to determine the six independent values of gy (g is a symmetric matrix so that gy = gy,). The g-matrix is then diagonalized to obtain the principal values and the transformation matrix, elements of which are the direction cosines of the g-matrix principal axes relative to the crystal axes. An analogous treatment of the effective hyperfine coupling constants leads to the principal values of the A2-matrix and the orientation of its principal axes in the crystal coordinate system. 

These equations have three solutions (i) 9 = 0 (ii) 9 = 90°, q> = 0 and (iii) 9 = (p = 90°. Since 9 and cp are in the g-matrix axis system, observable features are expected for those fields corresponding to orientations along the principal axes of the g-matrix. This being the case, the principal values of the g-matrix are obtained from a straightforward application of eqn (4.10). 

Since the g-matrix has only three principal values and there are almost always many potentially interacting molecular orbitals, there is rarely sufficient information to interpret a g-matrix with complete confidence. When a well-resolved and reliably assigned optical spectrum is available, the energy differences, E0—Em, are known and can be used in eqn (4.11) to estimate the contribution of the corresponding MOs to the g-matrix. Extended Hiickel MO (EHMO) calculations can be useful (but do not trust EHMO energies ), but one is most commonly reduced to arguments designed to show that the observed g-matrix is consistent with the interpretation placed on the hyperfine matrix. 

In general, the g- and nuclear hyperline coupling matrices, g and A can be written in diagonal form with three principal values, i.e., gx, gy, g. and A,x, Aiy, Aiz. In textbooks on ESR6a 30,33 35 it is usually assumed that the same set of principal axes diagonalizes all the relevant matrices. While this is sometimes true, there are many instances where the principal axes are non-coincident.36... 

Since the energy differences, AEx2 y2 and AEXZ are expected to be comparable, the parameter Q is probably not far from unity. For Q — 1, eqn (4.47) has a particularly simple form, tan 2/1 = I 2a/b so that, for small b/a, we expect /i a 0 and /i 45°, which is entirely consistent with experiment. The axial hyperfine matrix is in agreement with experiment, and the principal values of the g-matrix can also be rationalized with reasonable values of Q and b/a. A small rotation of dyz about the y-axis might reflect the small displacements of the phosphorus atoms from the idealized octahedral positions. 

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