Spherical components

The scalar spherical component A and the tensor deviator A of A are defined by... 

Recrystallization. The recrystallization of a solid may result in the production of a higher temperature lattice modification, which permits increased freedom of motion of one or more lattice constituents, e.g. a non-spherical component may thereby be allowed to rotate. Such reorganizations are properly regarded as premelting phenomena and have been discussed by Ubbelohde [3]. The mechanisms of phase transitions have been reviewed by Nagel and O Keeffe [21] (see also Hannay [22]). 

Results similar to those for the nitrogen compounds discussed here have been obtained by analysis of C, N, and O atoms in a number of nucleotides and nucleosides (Pearlman and Kim 1985). Finally, a test of the kappa refinement using the theoretical densities of 28 diatomic molecules proved it to be quite successful in reproducing the theoretical radial distribution of the spherical component of the atomic density (Brown and Spackman 1991). 

To preserve the shell structure of the spherical component of the valence density, the radial function of the bonded atom may be described by the isolated atom radial dependence, modified by the k expansion-contraction parameter. 

In the derivation of the traceless quadrupole moments from the electrostatic moments, the spherical components are subtracted. Thus, the quadrupole moments can be derived from the second moments, but the opposite is not the case. Spackman (1992) notes that the subtraction introduces an ambiguity in the comparison of quadrupole moments from theory and experiment. The spherical component subtracted is not that of the promolecule, but is based on the distribution itself. It is therefore generally not the same in the two densities being compared. On the other hand, the moments as defined by Eq. (7.1) are based on the total density without the intrusion of a reference state. 

The EFG tensor elements can be obtained by differentiation of the operator in expression (8.5) for Ea to each of the three directions / . This procedure removes the spherical component, which does not affect the electric field, and yields the traceless result... 

Including the spherical component, the electric field gradient can be interpreted as the second-moment tensor of the distribution p(r)/ r — r 5. 

The operators for the potential, the electric field, and the electric field gradient have the same symmetry, respectively, as those for the atomic charge, the dipole moment, and the quadrupole moment discussed in chapter 7. In analogy with the moments, only the spherical components on the density give a central contribution to the electrostatic potential, while the dipolar components are the sole central contributors to the electric field, and only quadrupolar components contribute to the electric field gradient in its traceless definition. 

The integral over energy must also include a sum over the discrete states if such states exist. The dipole moment may be written as fi(R) = n(R) R, or, in spherical components,... 

The induced dipole moment p is expressed in the form of Eq. 4.18 in spherical components [314, 317]. As was seen in [43], we obtain the spectral function as a multiple sum of incoherent components,... 

The H 1 form a basis set of functions of the orientations of three molecules which transform as a vector. The i/-th spherical component of the dipole moment n may be expanded in terms of this basis, according to... 

Tower Burst. If the energy of the detonation is sufficient to vaporize the entire tower mass, the particle population is like that described for the land surface burst. If, however, the entire tower is not vaporized, the particle population will consist of three identifiable components— the crystalline and glass components of the surface detonation plus a metal sphere population which arises from melted (not vaporized) tower materials resolidifying as spheres. Such spheres are metallic rather than metal oxide and exhibit the density and magnetic properties of the tower material. The size range of the spherical component is from a few microns to perhaps a few hundred microns diameter. If we indicate by... 

In order to utilize the mathematical apparatus of the angular momentum theory, we have to write all the operators in. /-representation, i.e., to express them in terms of the quantities which transform themselves like the eigenfunctions of operators j2 and jz. For example, the explicit form of the spherical components of the spin operator in. /-representation, according to [28], is as follows ... 

The angular momentum operator is a first-rank tensor whose Cartesian and spherical components transform into each other as... 

Irreducible Cartesian components Spherical components Compound operator made of vectors... 

The relation between the spherical components AJ0( ) of a general tensor A of rank 2 and the cartesian components A, ( ) are given in Appendix 4. Equations (3.36) will form the basis for derivation of selection rules for rotation-internal motion transitions of SRMs presented in the next section. They also may serve for derivation of the transformation properties of the electric and magnetic dipole moment operators referred to the laboratory system (VH G... 

Here Q is the quadrupole moment of the molecule, which has the value of 0.49 a.u. for H2 at the equilibrium internuclear separation, and / 2(cos 0) is the second-degree Legendre function, 9 being the angle between R and r. The polarization potential, also with spherical and non-spherical components, has the asymptotic form... 

The same dendrimer discussed in question 14.1 is formed in a yield of 98.5 % for each unit reaction (the addition of each spherical component). What will be the yield of pure dendrimer at each generation Do you think it would be feasible to construct a dendrimer along these lines, which reaches enough generations to isolate the dendrimer core from the external medium entirely ... 

Here Wgn are the coefficients of the expansion of the electrostatic free energy, which can be obtained from the free energy Wfl(li), according to Equation (2.269). T2n are the irreducible spherical components of the (second rank) surface tensor, which describe the anisometry of the molecular shape, and can be calculated in the form of integrals over the molecular surface [25]. Given the nematic potential the distribution function... 

While powerful contemporary techniques permit the use of molecular cavities of complex shape [3], it is instructive to note a few cases based on idealized representations of solute cavity and charge density. Cavities are typically constructed in terms of spherical components. Marcus popularized two-sphere models, [5,38] which can be used to model CS, CR, or CSh processes (see Section 3.5.2), where the two spheres are associated with the D and A sites, and initial and final charge densities are represented by point charges (qD and qA) at the sphere origins. If a single electron is transferred, Ap corresponds to A = 1 in units of electronic charge (e), and Aif is given by [5,38]... 

Here MJ1 denotes the operator of the with spherical component of the multipole moment of order l, and the reaction field moments are given by,... 

We will present the effective Hamiltonian terms which describe the interactions considered, sometimes using cartesian methods but mainly using spherical tensor methods for describing the components. These subjects are discussed extensively in chapters 5 and 7, and at this stage we merely quote important results without justification. We will use the symbol T to denote a spherical tensor, with the particular operator involved shown in brackets. The rank of the tensor is indicated as a post-superscript, and the component as a post-subscript. For example, the electron spin vector A is a first-rank tensor, T1 (A), and its three spherical components are related to cartesian components in the following way ... 

The spherical components of the new first-rank tensor in (1.57) are defined, in the molecule-fixed axes system, by... 

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