Spin tensor

In the language of Section A.4, s and d are indifferent but s is not, involving extra terms in Q. In order to render the stress rate indifferent, the extra terms must be cancelled out. This may be done using the spin tensor w defined in (A.l Ij), following the steps leading to (A.68). The result is... 

Consequently, the stretching tensor and the convected rate of spatial strain are indifferent, but the spin tensor is not, involving the rate of rotation of the coordinate frame. From (A.24) and (A.26)... 

In order to eliminate the last two terms on the right, it may be noted that the spin tensor in (A.612) gives rise to terms in Q. Postmultiplying (A.612) by the stress leads to... 

When the length scale approaches molecular dimensions, the inner spinning" of molecules will contribute to the lubrication performance. It should be borne in mind that it is not considered in the conventional theory of lubrication. The continuum fluid theories with microstructure were studied in the early 1960s by Stokes [22]. Two concepts were introduced couple stress and microstructure. The notion of couple stress stems from the assumption that the mechanical interaction between two parts of one body is composed of a force distribution and a moment distribution. And the microstructure is a kinematic one. The velocity field is no longer sufficient to determine the kinematic parameters the spin tensor and vorticity will appear. One simplified model of polar fluids is the micropolar theory, which assumes that the fluid particles are rigid and randomly ordered in viscous media. Thus, the viscous action, the effect of couple stress, and... 

Fundamental constants (Cx), spatial tensors in the principal axis frame ((fi3 m,)F), and spin tensors (Tjm) for chemical shielding (a), J coupling (J), dipole-dipole (IS), and quadrupolar coupling (Q) nuclear spin interactions (for more detailed definition of symbols refer to [50])... 

It is also possible to employ highly correlated reference states as an alternative to methods that employ Hartree-Fock orbitals. Multiconfigu-rational, spin-tensor, electron propagator theory adopts multiconfigura-tional, self-consistent-field reference states [37], Perturbative corrections to these reference states have been introduced recently [38],... 

As in Eq. (64), the electron spin spectral densities could be evaluated by expanding the electron spin tensor operators in a Liouville space basis set of the static Hamiltonian. The outer-sphere electron spin spectral densities are more complicated to evaluate than their inner-sphere counterparts, since they involve integration over the variable u, in analogy with Eqs. (68) and (69). The main simplifying assumption employed for the electron spin system is that the electron spin relaxation processes can be described by the Redfield theory in the same manner as for the inner-sphere counterpart (95). A comparison between the predictions of the analytical approach presented above, and other models of the outer-sphere relaxation, the Hwang and Freed model (HF) (138), its modification including electron spin... 

The evaluation of a spin operator times an occupation number vector is faciliated by noting that the core is a singlet spin tensor, since ata ajjj is a singlet spin operator (Eq. 5.25). The action of Sz on I na np> becomes... 

The eigenvalue of Qz is N — 2 for a state of gN. We can now consider that the identical components of g1 and g together form a quasi-spin tensor of rank 1/ 2, whose array of ranks we can now indicate by writing G(l - - -]. The e, operators can be broken down into parts that have well-defined quasi-spin ranks however, it turns out that e2 is a quasi-spin scalar, which can be used to explain some similar matrix elements of e2 in g 2 and g 4 [10]. 

The spatial velocity gradient a = grad va can be decomposed into symmetric and skew-symmetric parts as la = sym a + skw 1 = dQ + wQ., where da and wa are the deformation rate and the spin tensors, respectively. 

So far we know the selection rules for spin-orbit coupling. Further, given a reduced matrix element (RME), we are able to calculate the matrix elements (MEs) of all multiplet components by means of the WET. What remains to be done is thus to compute RMEs. Technical procedures how this can be achieved for Cl wave functions are presented in the later section on Computational Aspects. Regarding symmetry, often a complication arises in this step Cl wave functions are usually determined only for a single spin component, mostly Ms = S. The Ms quantum numbers determine the component of the spin tensor operator for which the spin matrix element (S selection rules dictated by the spatial part of the ME. 

Given a molecule that possesses C2p symmetry, let us try to figure out how to calculate ( Ai ffsol Bi) from wave functions with Ms = 1. The coupling of an Ai and a B state requires a spatial angular momentum operator of B2 symmetry. From Table 11, we read that this is just the x component of It. A direct computation of (3A2, Ms = 1 t x spin-orbit Hamiltonian with x symmetry and So correspondingly for the zero-component of the spin tensor. This is the only nonzero matrix element for the given wave functions. 

We wish to express each of these terms in terms of cartesian components. First we note that the components of the second-rank spin tensor are defined by... 

The first-rank spherical spin tensor components in (8.472) may now be rewritten in cartesian form using the definitions... 

The derivation above is not the only way to obtain the required result, but it is straightforward, if somewhat tedious. The reduced matrix element of the fourth-rank spin tensor, T4, S. S, S), which can arise in the analysis of higher spin states, is obtained by further use of the recursion relationship given by Edmonds [80], See also the general expression given in equation (5.134) of chapter 5. 

The first two terms are Zeeman terms and the third represents the hyperfine interaction of the electron and nuclear spins, / b and are the Bohr and nuclear magnetons respectively, S is a fictitious effective spin (S = 2 for a simple Kramers doublet), and / is the nuclear spin tensor. The hyperfine tensor is further split into Fermi contact, dipolar, and orbital components according to ... 

The magnetic spin dipole-dipole interaction is the most important source of nuclear spin relaxation for spin half (/ = ) nuclei. Apart from the relative orientations of the spins, the dipole-dipole interaction also depends on the length and orientation of the vector between the spins. Formally, it can be expressed as a tensor product of the 1st rank spin tensors, and and a 2nd rank... 

This contains an TCP of the TpaL tensor, which is derived from the electron spin and dipole-dipole interaction tensor(See equation (11)). Hence, the first question we confront is whether those tensors are correlated or not. In case they are not the total TCP can be decomposed into a product of auto correlations for the the electron spin and dipole-dipole interaction tensor, respectively. In case they are, however, it is necessary to consider the whole TCP and the electron spin has to be correlated with the dipole-dipole interaction tensor. The time dependence in the electron spin tensor can be obtained by integrating the time dependent Schrbdinger equation for the electron spin under the electron spin Hamiltonian. The electron spin is just like the nuclear spin precessing around the external magnetic field and influenced by molecular dynamics. 

The quadrupole tensor is proportional to the 2nd rank spin tensor of a single spin, and an expression for the relaxation time in the extreme narrowing limit is easily derived (See Appendix A.2). 

The elements V2,-m describe the spatial interaction in a reference axis system of choice and are connected to Qz.m by a rotation as shown in Eq. (4). The spin tensor elements 72, are given in terms of the Cartesian angular momentum operators by t... 

Z < M 5>[Cs, + tfSf] which contains the singlet-spin tensor operator. 

Another reason for carrying out a spin-tensor decomposition of the effective interaction, is to get an assessment of the features which have to be incorporated into effective interactions for heavier... 

Fig. 13. Spin-tensor decompositions of the effective interaction for the central component. The upper part shows the results obtained with the bare G-matrix for the Bonn A, B and C potentials. The lower figure exhibits the results derived from the LS effective interaction discussed in the text. The numbering of the matrix elements follows the Table 6.

Fig. 14. Spin-tensor decomposition of the effective interaction for the tensor component. Notations as in the previous...

Therefore, it is ensured that the Eulerian quantity can be caluclated by using the Lagrangien quantity. From the velocity gradient tensor, two new tensors, rate of deformation tensor, D, and spin tensor, W, can be defined ... 

The velocity gradient L is decomposed into its symmetric part D, called the stretch tensor or rate-of-deformation tensor, and its anti-symmetric part W, called the spin tensor ... 

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