Single transition approximation

While the uncoupled Hartree-Fock method and the single transition approximation have the merit of computational simplicity, they suffer, however, in particular from the usually unsatisfactory description of electronically excited states with a single-determinant wavefunction. 

The reaction kinetics approximation is mechanistically correct for systems where the reaction step at pore surfaces or other fluid-solid interfaces is controlling. This may occur in the case of chemisorption on porous catalysts and in affinity adsorbents that involve veiy slow binding steps. In these cases, the mass-transfer parameter k is replaced by a second-order reaction rate constant k. The driving force is written for a constant separation fac tor isotherm (column 4 in Table 16-12). When diffusion steps control the process, it is still possible to describe the system hy its apparent second-order kinetic behavior, since it usually provides a good approximation to a more complex exact form for single transition systems (see Fixed Bed Transitions ). 

Electron correlations show up in two ways in the measured cross sections. If the initial target state is well described by the independent particle Hartree-Fock approximation, the experimental orbital (6) is the Hartree-Fock orbital. Correlations in the ion can then lead to many transitions for ionisation from this orbital, rather than the expected single transition, the intensities of the lines being proportional to the spectroscopic factors S K... 

The single exponential approximation works especially well for observables that are less sensitive to the location of initial distribution, such as transition probabilities and correlation functions. 

Fig. 17. A comparision of the temperature dependence of the line-shape function (G) of the transition probability for the multimode case (solid line) as against a single mode approximation (dashed line). Here the phonon frequency spectrum (A) is assumed to be of Gaussian form, A a>) = 2 2) 1,2exp [—(to — cu0)2/2

Specifically, the eleven profiles include the obvious single transitions, i.e., the rotovibrational transitions in just one of the two colliding H2 molecules these are the Si(0), Si(l), and Qi (1) transitions in one of the two interacting molecules. Double transitions in both collisional partners are also taking place, such as the simultaneous transitions gi(l)+So(0) (which occur near the Si(0) transition frequency) and Q (1) + So(l) (near Si(l)), Fig. 3. 32. Intensities of all these lines are known from theory (classical multipole approximation, Chapter 6) their superposition reproduces the measurement closely, Fig. 3.33. 

T,(R) have to be extracted from experimental information on the individual transitions, in the same way as that described for a single transition in the previous sections. Such detailed studies have not been made. So far only cross section ratios have been discussed, and it has been tacitly assumed that these ratios are good approximations to the electronic branching ratios. From the classical formulas (II. 10) and (II.11) it is evident that this assumption is correct under the following two conditions (1) the true branching ratios at(R) must be nearly constant within the relevant R range, so that we may write... 

If it happens that a single transition overwhelmingly dominates the spectral series then the f-value for this transition will be approximately given by the sum rule, equation (16). Such a situation may be expected to prevail when the ( and (/ +1) wavefunctions overlap very strongly and are radially separated from the core, e.g., in s—>p transitions. Transitions to be compared in the case ofthe alkali elements are [22]... 

We shall first briefly describe the phase-integral approximation referred to in item (i). Then we collect connection formulas pertaining to a single transition point [first-order zero or first-order pole of Q2(z) and to a real potential barrier, which can be derived by... 

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