Orthogonality bound modes

Equations (27.1) and (27.2) are hybrid quantized expressions in which the bound modes orthogonal to the reaction coordinate are treated quantum mechanically, that is, the partition functions (T, s) and 0 T) are computed quantum mechanically... 

Treating bound modes quantum mechanically, the adiabatic separation between s and u is equivalent to assuming that quantum states in bound modes orthogonal to s do not change throughout the reaction (as s progresses from reactants to products). The reaction dynamics is then described by motion on a one-mathematical-dimensional vibrationally and rotationally adiabatic potential... 

Figure 1. Harmonic frequencies for bound Figure 2. Potential energy along the minimum modes orthogonal to the reaction path as a energy path and relative ground-state adiabatic function of the reaction coordinate for the potential energy [eq. (2)] as a function of...

In order to determine the amplitude Oj of a bound mode in the expansion of Eq. (11-2), we require an orthogonality condition. This condition is derived... 

In Section 31-3 we show that each bound mode is orthogonal to the radiation field in Eq. (11-2). Hence Eq. (31-15) gives... 

We recall from Section 11-4 that each mode is orthogonal to the radiation field and all other bound modes. Assuming that the fiber is nonabsorbing, we take the cross product of Eq. (20-la) with h, Eq. (20-lb) with ej and integrate over the infinite cross-section A. We deduce from Eq. (11-13) that... 

As we now have orthogonality relations and normalization expressions for leaky modes, results which were derived for bound modes in earlier chapters can simply be extended to apply to leaky modes. These include the perturbation expressions of Chapter 18, the modal amplitudes due to illumination in Chapter 20, and the excitation and scattering effects of current sources in Chapters 21 to 23. We give an example of leaky-mode excitation by a source in Section 24—23. 

In Section 11-4, we showed that each bound mode is orthogonal to every other bound mode and to all radiation modes. The orthogonality properties of radiation modes are derived in Section 31-3. On a nonabsorbing waveguide, the th and kth forward-propagating radiation modes obey... 

The first application of the conjugated form of the reciprocity theorem demonstrates orthogonality of bound, radiation and leaky modes of nonabsorbing waveguides. Consider two modes propagating in the forward direction with propagation constants Pj nd The subscripts jand k may refer to two different bound modes, but, in the case... 

We showed how to determine the radiation modes of weakly guiding waveguides in Sections 25-9 and 25-10, starting with the transverse electric field e, which is constructed from solutions of the scalar wave equation. However, unlike bound modes, the corresponding magnetic field h, of Eq. (25-23b) does not satisfy the scalar wave equation. This means that the orthogonality and normalization of the radiation modes differ in form from that of the bound modes in Table 13-2, page 292, as we now show. 

The radiation field of the scalar wave equation can be represented by the continuum of scalar radiation modes discussed above, or by a discrete summation of scalar leaky modes and a space wave. This is clear by analogy with the discussion of vector radiation and leaky modes for weakly guiding waveguides in Chapters 25 and 26. Scalar leaky modes have solutions P of Eq. (33-1) below their cutoff values when P becomes complex. Many of the properties of bound modes derived in this chapter also apply to leaky modes. For example, the orthogonality condition of Eq. (33-5a) applies to leaky modes, provided only that the cross-sectional area A. is replaced by the complex area A of Section 24-15 to ensure that the line integral of Eq. (33-4) vanishes. 

The second kind of transition state resonance, as illustrated in Figure 23.20b, is known as the vibrational threshold resonance. This type of resonance corresponds to the energetic threshold for a quantized dynamical bottleneck in the transition-state region. This quasi-bound state can be characterized by two vibrational quantum numbers (for a three-atom system), corresponding to the modes of motion orthogonal to the unbound reaction coordinate. This kind of resonance has been found experimentally in... 

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