Dissociation wavefunction

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... 

For illustration purposes we find it useful to define the total dissociation wavefunction... 

However, the total dissociation wavefunction is useful in order to visualize the overall dissociation path in the upper electronic state as illustrated in Figure 2.3(a) for the two-dimensional model system. The variation of the center of the wavefunction with r intriguingly illustrates the substantial vibrational excitation of the product in this case. As we will demonstrate in Chapter 5, I tot closely resembles a swarm of classical trajectories launched in the vicinity of the ground-state equilibrium. Furthermore, we will prove in Chapter 4 that the total dissociation function is the Fourier transform of the evolving wavepacket in the time-dependent formulation of photodissociation. The evolving wavepacket, the swarm of classical trajectories, and the total dissociation wavefunction all lead to the same general picture of the dissociation process. 

Fig. 3.2. Two-dimensional potential energy surface V(R, 7) (dashed contours) for the photodissociation of C1CN, calculated by Waite and Dunlap (1986) the energies are given in eV. The closed contours represent the total dissociation wavefunction tot R,l E) defined in analogy to (2.70) in Section 2.5 for the vibrational problem. The energy in the excited state is Ef = 2.133 eV. The heavy arrow illustrates a classical trajectory starting at the maximum of the wavefunction and having the same total energy as in the quantum mechanical calculation. The remarkable coincidence of the trajectory with the center of the wavefunction elucidates Ehrenfest s theorem (Cohen-Tannoudji, Diu, and Laloe 1977 ch.III). Reprinted from Schinke (1990).

Fig. 4.2. Evolution of a two-dimensional wavepacket (R, r t). 4>(i) is shown at several instants. The model system is the same as the one in Figure 2.3. According to (4.11), the wavepacket is related to the time-independent total dissociation wavefunction depicted in Figure 2.3(a), by a Fourier trans-...

The wavepacket in Figure 4.2 follows essentially the same route as the time-independent total dissociation wavefunction tot Ef), defined in Equation (2.70), which is shown for a particular energy in Figure 2.3(a). This coincidence does not come as a surprise, however. If we multiply f(t) with elEft/h and integrate over t we obtain... 

The time-independent total dissociation wavefunction tot Ef) is the Fourier transform of the time-dependent wavepacket y (t). 

Fig. 10.16. Final rotational state distributions of NO following the dissociation of C1NO through the T state. The quantum numbers n and k specify the initial vibrational and bending excitation of the ClNO(Ti) complex. The undulations for the excited bending states reflect the nodal structures of the dissociation wavefunction at the transition state. The open and the filled circles indicate different P and Q branches. The corresponding absorption spectrum is depicted in Figure 7.14. Adapted from Qian, Ogai, Iwata, and Reisler (1990).

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),... 

If the total energy increases and eventually exceeds the barrier, the two dissociation wavefunctions begin to spread over both channels. This causes the overlap of with d+OH to increase while at the same... 

The reason the CAS process works is that the Bc2 CAS wavefunction has the flexibility to dissociate into the product of two CAS Be wavefunctions ... 

This turns out to be common behaviour for HF wavefunctions wherever strong bonds are made or broken, the HF wavefunction will tend to give incorrect dissociation products. 

The proposed scenario is mainly based on the molecular approach, which considers conjugated polymer films as an ensemble of short (molecular) segments. The main point in the model is that the nature of the electronic state is molecular, i.e. described by localized wavefunctions and discrete energy levels. In spite of the success of this model, in which disorder plays a fundamental role, the description of the basic intrachain properties remains unsatisfactory. The nature of the lowest excited state in m-LPPP is still elusive. Extrinsic dissociation mechanisms (such as charge transfer at accepting impurities) are not clearly distinguished from intrinsic ones, and the question of intrachain versus interchain charge separation is not yet answered. 

From a theoretical perspective, the object that is initially created in the excited state is a coherent superposition of all the wavefunctions encompassed by the broad frequency spread of the laser. Because the laser pulse is so short in comparison with the characteristic nuclear dynamical time scales of the motion, each excited wavefunction is prepared with a definite phase relation with respect to all the others in the superposition. It is this initial coherence and its rate of dissipation which determine all spectroscopic and collisional properties of the molecule as it evolves over a femtosecond time scale. For IBr, the nascent superposition state, or wavepacket, spreads and executes either periodic vibrational motion as it oscillates between the inner and outer turning points of the bound potential, or dissociates to form separated atoms, as indicated by the trajectories shown in Figure 1.3. 

Table 6.9 Comparison of experimental C-H bond dissociation energies at 0 K (kJ/mol) with those calculated with wavefunction-based electronic structure methods.

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