Photodissociation amplitude

Inserting (3.38) into the expression for the partial photodissociation amplitudes yields... 

The sudden approximation is easy to implement. One solves the onedimensional Schrodinger equation (3.43) for several fixed orientation angles 7, evaluates the 7-dependent amplitudes (3.47), and determines the partial photodissociation amplitudes (3.46) by integration over 7. Because of the spherical harmonic Yjo(x, 0) on the right-hand side of (3.46), the integrand oscillates rapidly as a function of 7 if the rotational... 

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. 

Theoretically, the calculation of photodissociation cross sections for excited vibrational states proceeds in exactly the same way as for the dissociation of the lowest level. The basic quantities are the photodissociation amplitudes (2.68) with the initial wavefunction being... 

The matrix elements (4>i(.E,n) pio f of(Ef)) are just the partial photodissociation amplitudes (2.68) which are required in the calculation of dissociation cross sections for vibrationally excited parent molecules. The actual calculation proceeds in the following way ... 

The photodissociation probability into the state characterized by n at energy E, Pn(E i), is given by the square of Aa(E i), the photodissociation amplitude for observing the free state exp(—iEt/1i) E, n 0) in the long-time limit. That is,... 

Equation (2.83) has been often used [28, 29] to describe photodissociation in the following way At t = 0, the light pulse creates the wave packet Ae Et), which subsequently evolves under the action of expStrictly speaking, for Eq. (2.83) to hold, and hence for this qualitative interpretation to be valid, e(c%,) must vary more slowly with energy than any of the other variables in Eq. (2.28). In particular, it must vary more slowly than the energy (or frequency) dependence of the photodissociation amplitudes (E, n- deff[E,). 

Die collection of final channel quantum numbers n in the photodissociation amplitude E,n,q Aeg Ei,Ji,Mj) must now include the scattering angles == 0k) and, in the case of IBr, the quantum number of primary interest... 

Since the state E, n", N", t) contains the effect of the full Hamiltonian at time fq then the photodissociation amplitude A(E, n, N, t i, A)) into the final state will energy E, internal quantum numbers n and radiation field described by N, starting in the initial state ] , initial state and the incoming fully interacting state. That is,... 

Transient, or time-resolved, techniques measure tire response of a substance after a rapid perturbation. A swift kick can be provided by any means tliat suddenly moves tire system away from equilibrium—a change in reactant concentration, for instance, or tire photodissociation of a chemical bond. Kinetic properties such as rate constants and amplitudes of chemical reactions or transfonnations of physical state taking place in a material are tlien detennined by measuring tire time course of relaxation to some, possibly new, equilibrium state. Detennining how tire kinetic rate constants vary witli temperature can further yield infonnation about tire tliennodynamic properties (activation entlialpies and entropies) of transition states, tire exceedingly ephemeral species tliat he between reactants, intennediates and products in a chemical reaction. 

Figure 3, Wavepacket dynamics of the photodissociation of NOCl, shown as snapshots of the density (wavepacket amplitude squared) at various times, The coordinates, in au, are described in Figure b, and the wavepacket is initially the ground-state vibronic wave function vertically excited onto the 5i state. Increasing corresponds to chlorine dissociation. The density has been integrated over the angular coordinate. The 5i PES is ploted for the geometry, 9 = 127, the ground-state equilibrium value,...

Therefore, we can use this somewhat artificial wavepacket, when it has been determined by direct integration of the time-dependent Schrodinger equation, to extract the transition amplitudes and hence the photodissociation cross sections for all energies. 

The transition from direct to indirect photodissociation proceeds continuously (see Figure 7.21) and therefore there are examples which simultaneously show characteristics of direct as well as indirect processes the main part of the wavepacket (or the majority of trajectories, if we think in terms of classical mechanics) dissociates rapidly while only a minor portion returns to its origin. The autocorrelation function exhibits the main peak at t = 0 and, in addition, one or two recurrences with comparatively small amplitudes. The corresponding absorption spectrum consists of a broad background with superimposed undulations, so-called diffuse structures. The broad background indicates direct dissociation whereas the structures reflect some kind of short-time trapping. 

The above results deal with photodissociation. A similar formulation can be applied to an inelastic scattering event, for example, the scattering of A + BC from an initial state E, m 0) into a final state E, n 0) of A + BC. This problem can be phrased as our asking for the amplitude of making a transition from a free state E, m 0) at t —> —oo to free state E, n 0 at t — oo. Since E, m+) is known to have evolved from E, m 0) and E, n ) is known to evolve to state E, n 0), the answer is given by the expansion coefficients of an outgoing state in terms of the incoming states. That is,... 

Figure 3.8 Contour plot of the yield of Bi- ( Piy2) (percentage of Br as product) in the. photodissociation of IBr from an initial bound state in X Sq with v — 0, Jj = 1, M,- = 0. I Results arise from simultaneous (wl,co3) excitation (co3 = 3cO ) with to, = 6657.5 cm-1. Abscissa is labeled by the amplitude parameter s = x2/(l +x2) and ordinate by the relative phase parameter d3 — 3 0], equivalent to 3 — 3 j of the text. (Taken from Fig. 4, Ref. [63],) ...

Suppressing for the moment all channel indices m (which can be readily included), save for the final direction k, we square the amplitude to obtain Pq(E, It), the probability of photodissociation into channel q at angles k = (Bh, k),... 

Finally, we make a few additional remarks. First, note that a pure number state is a3j= state whose phase 0k is evenly distributed between 0 and 2n. This is a consequence of the commutation relation [3] between Nk and e,0 <. Nevertheless, dipole mafKi w elements calculated between number states are (as all quantum mechanical amplitudes) well-defined complex numbers, and as such they have well-defined phajje j S Thus, the phases of the dipole matrix elements in conjunction with the mode ph f i f/)k [Eq. (12.15)] yield well-defined matter + radiation phases that determine the outcome of the photodissociation process. As in the weak-field domain, if only gJ one incident radiation mode exists then the phase cancels out in the rate expres4<3 [Eq. (12.35)], provided that the RWA [Eqs. (12.44) and (12.45)] is adoptedf However, in complete analogy with the treatment of weak-field control, if we irradh ate the material system with two or more radiation modes then the relative pb between them may have a pronounced effect on the fully interacting state, phase control is possible. 

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