Bound states wavefunction

Bound states are readily included in the line shape formalism either as initial or final state, or both. In Eq. 6.61 the plane wave expression(s) are then replaced by the dimer bound state wavefunction(s) and the integration(s) over ky and/or kjj2 are replaced by a summation over the n bound state levels with total angular momentum J n or J . The kinetic energy is then also replaced by the appropriate eigen energy. In this way the bound-free spectral component is expressed as [358]... 

As r — oo the wavefunctions are oscillatory sine and cosine functions, as shown by Eqs. (2.14). For small r the wavefunctions of the continuum are functionally identical to the bound wavefunctions, differing only in their normalization. Since continuum waves extend to r = oo they cannot be normalized in the same way as a bound state wavefunction. We shall normalize the continuum waves per unit... 

We note that this wavefunction differs from the bound state wavefunctions by a factor of n 3/2. The factor of n 3/2 is, in essence, the factor obtained by converting... 

The scattering/and g wavefunctions we have used are normalized per unit energy, and now we consider how to normalize the wavefunctions based on them in different energy regions. First we consider the bound states. We require a bound state wavefunction to satisfy... 

This expression is valid for any state, as long as ( n. The factor v%2 accounts for the fact that v2t) is normalized per unit energy. The overlap integral (vb v2tB) between two bound state wavefunctions would not have the factor v22 and would be equal to one for v2 = vh and zero for v2 different from vb by an integer, as expected. We can write the cross section as... 

The nuclear wavefunctions are continuum, i.e., scattering, wavefunctions which asymptotically behave like free waves rather than decaying to zero like the bound-state wavefunctions, scattering wavefunctions fulfil distinct boundary conditions in the limit R — oo. 

In Section 2.1 we derived the expression for the transition rate kfi (2.22) by expanding the time-dependent wavefunction P(t) in terms of orthogonal and complete stationary wavefunctions Fa [see Equation (2.9)]. For bound-free transitions we proceed in the same way with the exception that the expansion functions for the nuclear part of the total wavefunction are continuum rather than bound-state wavefunctions. The definition and construction of the continuum basis belongs to the field of scattering theory (Wu and Ohmura 1962 Taylor 1972). In the following we present a short summary specialized to the linear triatomic molecule. 

In addition to the direct methods, in which one calculates first the continuum wavefunctions and subsequently the overlap integrals with the bound-state wavefunction, there are also indirect methods, which encompass the separate computation of the continuum wavefunctions the artificial channel method (Shapiro 1972 Shapiro and Bersohn 1982 Balint-Kurti and Shapiro 1985) and the driven equations method (Band, Freed, and Kouri 1981 Heather and Light 1983a,b). Kulander and Light (1980) applied another method, in which the overlap of the bound-state wavefunction with the continuum wavefunction is directly propagated. The desired photodissociation amplitudes are finally obtained by applying the correct boundary conditions for R —> oo. 

As in the one-dimensional case, the general form of the spectrum is of Gaussian-type and can be regarded as a reflection of the two-dimensional bound-state wavefunction at the upper-state PES. It is noteworthy that ... 

Fig. 6.4. Schematic illustration of the multi-dimensional reflection principle in the adiabatic limit. The left-hand side shows the vibrationally adiabatic potential curves en(R). The independent part of the bound-state wavefunction in the ground electronic state is denoted by

The general shape of the cross sections as functions of E, n, and j resembles the initial distribution function in the electronic ground state. If the parent molecule is initially in its lowest vibrational state, the partial cross section behaves qualitatively like a three-dimensional bell-shape function of E,n, and j. If the photodissociation starts from an excited vibrational state, a(E, n,j) will exhibit undulations which reflect the nodal structures of the corresponding bound-state wavefunction [more of this in Chapter 13 see also Shapiro (1981) and Child and Shapiro (1983)]. 

In analogy to Equation (4.3) we expand the time-dependent wavepacket created by the delta-pulse in the excited electronic state in terms of the bound-state wavefunctions 4,1/. Using (4.5) we obtain ... 

Thus, the stationary wavefunction becomes on resonance essentially a bound-state wavefunction with an amplitude which is proportional to the corresponding lifetime. The larger the survival time in the well region the larger is the magnitude of the stationary wavefunction. 

The calculation of photodissociation cross sections requires the overlap of the continuum wavefunctions with the bound-state wavefunction multiplied by the transition dipole function. Employing for both wave-functions the expansion in terms of the Qj p and utilizing (11.13) leads to radial integrals of the form... 

Fig. 12.3. Schematic illustration of the vibrational-translational coupling term Vi(R), the bound-state wavefunction Xi° (R)> and the continuum wavefunction Xo0)(R)- The overlap of these three functions determines the dissociation rate T according to Equation (12.5).

The intermolecular term has the same general form as the absorption cross section in the case of direct photodissociation, namely the overlap of a set of continuum wavefunctions with outgoing free waves in channel j, a bound-state wavefunction, and a coupling term. For absorption cross sections, the coupling between the two electronic states is given by the transition dipole moment function fi (R,r, 7) whereas in the present case the coupling between the different vibrational states n and n is provided by V (R, 7) = dVi(R, r, 7)/dr evaluated at the equilibrium separation r = re. 

Multimodal reflection structures exist only (or most prominently) if the bound-state wavefunction has one or several nodes along the dissociation path or, expressed in different words, if the parent molecule is excited in the direction of the dissociation path. 

Using the two-laser excitation scheme illustrated in Figure 11.7 Vander Wal, Scott, and Crim (1991) measured part of the absorption spectrum for excitation of the 40 ) state. Figure 1.9 shows the comparison with the result of a two-dimensional calculation. The corresponding bound-state wavefunction, depicted in Figure 13.4, qualitatively resembles the 130 ) wavefunction with one additional node along the reaction path. Accordingly, the absorption spectrum has three instead of two pronounced... 

Because of the appreciable excitation along the H-0 bond in the parent molecule, the bound-state wavefunction ko4 extends much further into the H+OD than into the D+OH channel. The consequence is that it overlaps h+od at relatively low energies where the overlap with the other wavefunction, I,d+oh is essentially zero. The result is a relatively small but finite cross section for the production of OD while the cross section for OH is practically zero. 

Fig. 13.7. Contour plot of the A-state PES for a bending angle ae = 104°. Energy normalization is such that E — 0 corresponds to H+O+H. Superimposed are contours of luAX Oil2 where hax is the X —> A transition dipole function and 04 is the bound-state wavefunction of HOD with four quanta of excitation in the O-H bond. The filled circle indicates the barrier and the two especially marked contours represent the energies for the two photolysis wavelengths A2 = 239.5 and 218.5 nm used in the experiment. Adapted from Vander Wal et al. (1991).

Calculation of all bound-state wavefunctions of(Ef) in the electronic ground state. 

Calculation of continuum wavefunctions 4>i(.E, n) in the excited electronic state for all possible final states n and their overlap with all bound-state wavefunctions for a sufficiently narrow grid of energies. 

This property is actually not surprising as the series limit is approached, the bound state wavefunction acquires more and more nodes, and tends to the oscillatory function of the continuum. The position of the nodes is related to the phase in the continuum, and we may expect that the two are connected, since the wavefunction at very high n must change smoothly into the free electron s wavefunction just above the series limit. In QDT, as for H, the wavefunction for r > ro preserves this... 

Recapitulation is an important property because it provides us with an immediate interpretation of the physical meaning of the quantum defect p for large enough n, the bound state wavefunctions possess an oscillatory inner part which defines a phase, and is nearly independent of energy if p is nearly constant in energy. A change in the value of p corresponds to a shift in the radial position of all the nodes. As one tends to the series limit, the oscillatory part grows. Continuum functions, of course, become... 

The time-dependent wave packet propagation can be employed to obtain bound state energies and bound state wavefunctions without the need to diagonalize the Hamiltonian matrix. The application of the method to bound state calculation is quite straightforward. If the Hamiltonian supports bound states 4> with eigenenergies Etn one can expand any given initial wave packet in this eigenbasis set,... 

Once the eigenenergy is obtained, it is straightforward to obtain the bound state wavefunction by performing a Fourier transform... 

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