Partial photodissociation cross sections

The partial cross section gives the probability of absorbing light and producing a particular final product quantum state. The total photodissociation cross section is clearly given by the sum over all partial photodissociation cross sections ... 

In the following we will call the a u,n,j) partial photodissociation cross sections.t They are the cross sections for absorbing a photon with frequency u and producing the diatomic fragment in a particular vibrational-rotational state (n,j). Partial dissociation cross sections for several photolysis frequencies constitute the main body of experimental data and the comparison with theoretical results is based mainly on them. Summation over all product channels (n,j) yields the total photodissociation cross section or absorption cross section ... 

With the excited-state wavefunction given by (3.38) the partial photodissociation cross sections approximately reduce to... 

The time-dependent formalism would be rather limited if it yielded only the total cross section. However, that is not the case all partial photodissociation cross sections a(Ef,n) can be also extracted from the time-dependent wavepacket. We assume that for large times the wavepacket has completely left the interaction zone and moves entirely in the asymptotic region where the interaction potential Vi(R,r) is zero. Then, the asymptotic conditions (2.59) for the stationary continuum wavefunctions can be inserted into (4.3) yielding... 

In the time-independent approach one has to calculate all partial cross sections before the total cross section can be evaluated. The partial photodissociation cross sections contain all the desired information and the total cross section can be considered as a less interesting by-product. In the time-dependent approach, on the other hand, one usually first calculates the absorption spectrum by means of the Fourier transformation of the autocorrelation function. The final state distributions for any energy are, in principle, contained in the wavepacket and can be extracted if desired. The time-independent theory favors the state-resolved partial cross sections whereas the time-dependent theory emphasizes the spectrum, i.e., the total absorption cross section. If the spectrum is the main observable, the time-dependent technique is certainly the method of choice. 

The partial photodissociation cross sections for absorbing a photon with energy Ephoton and producing the fragment in a particular vibrational-rotational state (n, j) are similarly defined as... 

The partial photodissociation cross sections are calculated by a sum over N trajectories with initial conditions TQtk which are randomly selected from a uniform distribution in the multi-dimensional phase-space,... 

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

We consider the photofragmentation of a triatomic molecule, ABC — A + BC(j), within the model outlined in Section 3.2. The vibrational coordinate of BC is fixed and the total angular momentum is zero. According to (5.23), the classical approximation of the partial photodissociation cross section for producing BC in rotational state j is given by... 

Fig. 7.6. (a) Energy dependence of a Lorentzian line-shape function with width KT centered at the resonance energy (Ei + 6E). (b) Partial photodissociation cross sections a(E,0) as given by (7.23). All of them have the same width hT the values at the maximum scale like the partial decay rates Tp. 

So far we have considered only the decay of the initial quasi-bound states and the influence on the total spectrum. Recalling that the total rate is the sum of all partial rates we can recast (7.22) as a sum of partial photodissociation cross sections, each being defined by... 

All partial photodissociation cross sections resonance energies Ei + 6E and have the same width AEp yHM irrespective of the final fragment state as illustrated in Figure 7.6(b). t... 

A much clearer picture evolves when one decomposes the total spectrum into the partial photodissociation cross sections a(, n,j) for absorbing a photon with wavelength A and producing NO in a particular vibrational-rotational state with quantum numbers (n,j). Experimentally this is accomplished by measuring so-called photofragment yield spectra. The idea is, in principle, simple the NO product is probed by laser-induced fluorescence (LIF). However, instead of scanning the wavelength Alif of the probe laser (in order to determine the final rotational state distribution) one fixes Alif to a particular transition NO(2n, nj) —>... 

Fig. 7.14. Photofragment yield spectra for the photodissociation of C1NO through the T electronic state. The lower part depicts the total absorption spectrum crt<)t(A) measured at room temperature. The three separate spectra in the upper part correspond to the (unnormalized) partial photodissociation cross sections cr(, n,j) for producing NO in a particular vibrational state n as indicated. The rotational state varies between 1.5 and 4.5 in these three cases. The additional spectrum in the lower part is the n = 0 cross section originating from excitation in the So —> Si electronic band. Recall that the sum of all partial cross sections yields the total spectrum. Adapted from Qian, Ogai, Iwata, and Reisler (1990).

The partial photodissociation cross sections for producing OH in different vibrational states n also depend — like the absorption spectrum — sensitively on the initial vibrational level of the parent molecule. Let us... 

Fig. 13.5. Partial photodissociation cross sections a(E,n) following the photodissociation of the 40 ) and the 31 ) vibrational states of H20(X) as functions of the energy in the A state. The quantum number n specifies the vibrational state of the OH product n = 0 (solid line), n = l (dashed line), and n = 2 (dotted line). E = 0 corresponds to three ground-state atoms. The vertical lines mark the total energies in the excited state corresponding to the two photolysis wavelengths A2 = 239.5 and 218.5 nm in the experiment of Vander Wal, Scott, and Crim (1991). Reproduced from Weide, Hennig, and Schinke (1989).

In order to evaluate partial photodissociation cross sections (vibrational and rotational product distributions) i.e. ABC + hv A + BC n,K) the wave function can be projected onto the different rovibrational eigenstates of the molecular fragment BC at fixed distance R between the two fragments. The chosen R should be on the asymptote of the potential energy surface where the two fragments do not interact. Balint-Kurti et at [87] have shown that the partial cross section is given by... 

In both cases, la and lb, the total photodissociation cross section is completely determined by the short-time dynamics in the Franck-Condon region. In contrast, the partial cross sections, which determine the vibrational, rotational, and electronic-state distributions of the products, involves longer time dynamics. To obtain all of the relevant information about the reaction, the wavepacket evolution must be followed out into the product region of the potential energy surface and projected onto the various different vibrational and rotational states of the fragments. The partial cross section for scattering... 

The partial photodissociation cross section to a particular final state in the golden rule approximation is proportional to the following quantity... 

The partial photodissociation cross section for absorbing a photon with frequency co and producing the fragments in a particular quantum state a is given by ... 

Partial photodissociation cross sections can be obtained by propagating the wave packet to long times, until 4>ex(0 moves freely in the exit channels, and projecting it on the asymptotic states, i.e., " ... 

Let us assume that the upper state is degenerate with substates F, all corresponding to the same total energy Ef. The photon excites each of these states simultaneously because the resonance condition ujfi ui holds for all of them. The absorption cross section is consequently composed of several partial absorption cross sections cr(u), [3) each being defined as in (2.27) with Ff) replaced by F ). We will come back to this in Section 2.5 when discussing photodissociation. 

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