INDEX vector coupling

In internal reflection, at angles of incidence larger than the critical angle, electromagnetic radiation is totally reflected (attenuated total reflectance, ATR. see Section 16.2.2.4 and Fig. 5). This special ca.se is very important in analysis for two approaches. First, simple transportation of radiation within the fiber (or a waveguide). Second, in total reflection, an evane.scent field appears in which the electrical field vector decays exponentially in the optically less dense medium. Every change within the medium with lower refractive index influences the field vector coupled to the field in the optically denser medium. Therefore, the totally reflected radiation contains information about effects on the other side of the phase boundary (the medium with lower refractive index) [20], [144]. Various principles to interogate this effect are known and used in evanescent field sensors. 

With the brief discussion of index, it is now possible to identify and compare some aspects of the high-index behavior of the constant-pressure and the compressible stagnation-flow equations. To understand the structure of the DAE system, it is first necessary to identify all variables that are not time differentiated (i.e., the x vector). In the constant-pressure formulation, neither the axial velocity u nor the pressure curvature A has time derivatives. By introducing the axial momentum equation, the compressible formulation introduces du/dt. To be of value in reducing the index, however, the momentum equation must be coupled to the other equations. The coupling is accomplished through pressure, which is included as a dependent variable. The variable A is not time differentiated in either formulation. 

First, we consider the case of two tlu electrons, later we will extend the results to more particles. Since the one electron spin-orbit coupling is negligible [13], we will work in the LS (Russel-Saunders) molecular approximation. Two electron basis ket vectors are denoted by a single index I ... 

This Hamiltonian is similar to the usual electron-phonon Hamiltonian, but the vibrations are like localized phonons and q is an index labeling them, not the wave-vector. We include both diagonal coupling, which describes a change of the electrostatic energy with the distance between atoms, and the off-diagonal coupling, which describes the dependence of the matrix elements tap over the distance between atoms. 

Equation (32) gives the phase of the vector in question for a single nucleus at the time of exchange. In the case of weakly coupled spin systems, this relationship remains valid and just has to be amended with index j (the value of tojP is either oofP or o>j depending on the parity of r). 

Here B is an optical constant, or is the total polarizability of the particle, and n is the number of components in each particle. The indexes i and j refer to components of the same particle. If the assumption of independent particles was not made, then the indexes could refer to components of any two particles, and the autocorrelation expression could not be written as a simple sum of contributions from individual particles. The spatial vector r(r) refers to the center of mass of the particle. R(r). In the case of a nonspherical particle (arbitrary shape), Eq. (I0) would describe the coupled motion of the center of mass and the relative arrangement of the components of the particle. For spherical particles, translational and rotational motion arc uncoupled and we have a simplified expression for the electric field time correlation function ... 

The construction of the blocks of the coupling matrix C is performed most straightforwardly in the CSF basis as discussed in Section III. The full list of transition density matrices D" and d" is quite large and, even for CSF expansions of only a few hundred in lengths, cannot be held in memory. There are two approaches that have been proposed for the construction of the coupling matrix C, both of which involve an orderly computation of the transition density matrices. The first such approach employs the index-driven unitary group approach as discussed in Section III. With this approach the vectors of transition density matrices, for a fixed set of orbital indices, are computed for all of the CSFs simultaneously. For example, for the orbital indices p, q, r and s, three vectors are constructed simultaneously from the coupling coefficients with elements denoted as dj ", dj , and These vectors are then used to update the elements of the four blocks of the matrix C. 

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