Channels INDEX

The above equations have to be complemented with rules for dealing with the singularities emerging when PHP has eigenvalues at energy E, which is the case primarily when the Wp) states are continuum states. In such a case, we denote /p) = c,E 1) and ) = Ic.ii"), with c standing for all quantum numbers related to a continuum channel (the channel index). The notation E = E - iO, serves to remind of the incoming boundary conditions used for example, for the c,E ) states one has... 

All the phenomena described above are absent in a 2D-junction when the effects of transverse mode quantization are neglected [7]. We have considered the limiting case of a single (transverse) channel because this is the case when the effects induced by a dispersion asymmetry in the electron spectrum are most pronounced. The anomalous supercurrent Eq. (7) is a sign alternating function of the transverse channel index since for neighboring channels the spin projections of chiral states are opposite [4]. Besides, the absolute value of the dispersion asymmetry parameter decreases with transverse-channel number j. So, for a multichannel junction the effects related to a dispersion asymmetry phenomenon will be strongly suppressed and they completely disappear in the pure 2D case. 

Formulation using i-coupled dipole matrix elements The scattering amplitudes Sy can also be related to dipole matrix elements Dy which are defined on the basis of j/7-coupling and classified by the channel index y — For these dipole matrix elements one has [Hua80]... 

The effects of competition can be estimated by extending the statistical method introduced above for handling kinetic shifts [46]. This is accomplished by introducing a branching ratio for multiple channels indexed using j (Eq. 7) ... 

The sytem is observed before and after the collision in time-dependent channel states ,(t)). The channel index i stands not only for the channel quantum numbers n, j, m, v but also for the relative momentum kj. The entrance channel is denoted by i = 0. The Schrodinger equation of motion for the channel i is, in atomic units. 

It is convenient at this stage to redefine the notation for energy. We drop the channel index from the total energy E, since it is the same in all channels. This is expressed for example by the energy-conserving delta function in (6.39). We now use a subscripted energy variable to refer to the kinetic energy of an electron, for example in channel 0 or i. The density of... 

The degree of accuracy of the adiabatic approximation can be inferred from Eq. (4.27). The nonadiabatic terms are proportional to the adiabatic channel index n and to h2/m. For fixed E, the electronic quantum index is proportional to Emtl2/h, hence the nonadiabatic terms are proportional to Eh/mtl2. 

The coupled-channels method may be developed within the language of wave-mechanics, or more formally (and more compactly) by means of operator equations. The common feature of both approaches is that the total scattering state is expanded in internal states of reactants and products. The nature of the colliding particles and the quantum numbers of the interna] states define the reaction channel index c = a, b,. We begin with the wave-mechanical approach, some of whose features have been presented in the section on statistical theories. For the total wavefunction [pa of reactants in channel a, with relative wave vector ka, we can write... 

It is more economical to use hyperspherical coordinate systems " for HLH systems. For collinear configurations, these coordinates are also plane polar coordinates, but the turning center is located at the origin. These coordinates have had a wide application to collinear re-actions,especially those of the HLH variety. The hyperspherical radius p is independent of the arrangement channel index... 

Following Garrett and Truhlar, we define the angle y (not to be confused with the arrangement channel index) as the bond angle between... 

For systems that behave statistically, it is straightforward to include the effects of competition in the threshold analysis. This is achieved by modifying Equation (8) to include multiple channels, indexed by j. 

The power of this variational result is that it applies as written also for reactive scattering provided one expands the definition of the channel label also to include an arrangement label. Thus for an atom-diatom reaction, A+BC( , j, m)-> AB( , j, m )-fC, the initial channel index i denotes the collection (a, u, j, m), labelling the arrangement a and quantum state (i , j,m) of that arrangement the final channel label in this case is / = (c, /, m ). 

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