Dissociation wavefunction partial

For fixed total energy E, Equation (2.59) defines one possible set of Nopen degenerate solutions I/(.R, r E, n),n = 0,1,2,..., nmax of the full Schrodinger equation. As proven in formal scattering theory they are orthogonal and complete, i.e., they fulfil relations similar to (2.54) and (2.55). Therefore, the (R,r E,n) form an orthogonal basis in the continuum part of the Hilbert space of the nuclear Hamiltonian H(R, r) and any continuum wavefunction can be expanded in terms of them. Since each wavefunction (R, r E, n) describes dissociation into a specific product channel, we call them partial dissociation wavefunctions. 

After having defined the partial dissociation wavefunctions l>(R,r E,n) as basis in the continuum, the derivation of the absorption rates and absorption cross sections proceeds in the same way as outlined for bound-bound transitions in Sections 2.1 and 2.2. In analogy to (2.9), the total time-dependent molecular wavefunction T(t) including electronic (q) and nuclear [Q = (R, r)] degrees of freedom is expanded within the Born-Oppenheimer approximation as... 

The individual partial cross sections are even more structured than the total cross sections and a simple explanation of the energy dependences is probably impossible, except at the low-energy tail of the spectrum. For total energies below the barrier of the A-state PES (E = —2.644 eV in this normalization), the dissociative wavefunction is mainly confined to the two H + OH channels with little amplitude in the intermediate region. It therefore overlaps only that part of the ground-state wavefunction which extends well into the two exit channels. There, however, the 40-) and the 31-) states behave completely differently (see Figure 13.4),... 

The y>Ee(R) are the radial free-state wavefunctions (see Chapter 5 for details). The free state energies E are positive and the bound state energies E(v,S) are negative v and ( are vibrational and rotational dimer quantum numbers t is also the angular momentum quantum number of the fth partial wave. The g( are nuclear weights. We will occasionally refer to a third partition sum, that of pre-dissociating (sometimes called metastable ) dimer states,... 

The partial wavefunctions I> Ey n) do not depend on the initial state they are exclusively governed by the dynamics in the excited electronic state. The definition in (2.70), on the the other hand, involves the overlap with the initial wavefunction and therefore tot(Ef) depends on the particular state of the parent molecule in the electronic ground state. tot has no practical advantage because its construction requires the dissociation amplitudes t(Ef,n) which already contain all the desired information. 

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