Electronic transition probabilities

There are two basic physical phenomena which govern atomic collisions in the keV range. First, repulsive interatomic interactions, described by the laws of classical mechanics, control the scattering and recoiling trajectories. Second, electronic transition probabilities, described by the laws of quantum mechanics, control the ion-surface charge exchange process. 

In this chapter, we discussed the significance of the GP effect in chemical reactions, that is, the influence of the upper electronic state(s) on the reactive and nonreactive transition probabilities of the ground adiabatic state. In order to include this effect, the ordinary BO equations are extended either by using a HLH phase or by deriving them from first principles. Considering the HLH phase due to the presence of a conical intersection between the ground and the first excited state, the general fomi of the vector potential, hence the effective... 

Stueckelberg derived a similar fomiula, but assumed that the energy gap is quadratic. As a result, electronic coherence effects enter the picture, and the transition probability oscillates (known as Stueckelberg oscillations) as the particle passes through the non-adiabatic region (see [204] for details). 

The one exception to this is the INDO/S method, which is also called ZINDO. This method was designed to describe electronic transitions, particularly those involving transition metal atoms. ZINDO is used to describe electronic excited-state energies and often transition probabilities as well. 

In photoluminescence one measures physical and chemical properties of materials by using photons to induce excited electronic states in the material system and analyzing the optical emission as these states relax. Typically, light is directed onto the sample for excitation, and the emitted luminescence is collected by a lens and passed through an optical spectrometer onto a photodetector. The spectral distribution and time dependence of the emission are related to electronic transition probabilities within the sample, and can be used to provide qualitative and, sometimes, quantitative information about chemical composition, structure (bonding, disorder, interfaces, quantum wells), impurities, kinetic processes, and energy transfer. 

The fact that detailed balance provides only half the number of constraints to fix the unknown coefficients in the transition probabilities is not really surprising considering that, if it would fix them all, then the static (lattice gas) Hamiltonian would dictate the kind of kinetics possible in the system. Again, this cannot be so because this Hamiltonian does not include the energy exchange dynamics between adsorbate and substrate. As a result, any functional relation between the A and D coefficients in (44) must be postulated ad hoc (or calculated from a microscopic Hamiltonian that accounts for couphng of the adsorbate to the lattice or electronic degrees of freedom of the substrate). Several scenarios have been discussed in the literature [57]. 

All ab initio applications of multiple scattering theory in dilute substitutional alloys rely on the one-to-one correspondence configuration. This holds both for the calculation of transition probabilities [7], represented by Eq. (7), and the electronic structure [8], represented by the Green s function equation [9]... 

Electron Correlation The theory should have a limiting case of exact total energies, electron binding energies and corresponding transition probabilities. 

Figure 24. Electron-transfer rate versus electronic coupling strength. The temperature is T = 500 K. Solid line with circle-present results from Eq. (126) with the transition probability averaged over the seam surface. Solid line with square-present results with the transition probability taken at the minimum energy crossing point (MECP). Dashed line-Bixon-Jortner theory Ref. [109]. Dotted line-Marcus s high temperature theory. Taken from Ref. [28].

The elementary act of an electrochemical redox reaction is the transition of an electron from the electrode to the electrolyte or conversely. Snch transitions obey the Franck-Condon principle, which says that the electron transition probability is highest when the energies of the electron in the initial and final states are identical. 

At a small value of this parameter, the system can perform multiple passages through the transitional configuration before the electron transfer occurs. The exact expression for the transition probability taking into account these multiple passages has the form... 

The occupation of energy level 8 depends on its position with respect to the Fermi level and should be taken into account in calculation of the transition probability. The number of electrons within a given energy interval de is equal to p(8)/(8)d8. 

Matsuda and Hata [287] have argued that the species that are detectable using OES only form a very small part (<0.1%) of the total amount of species present in typical silane deposition conditions. From the emission intensities of Si and SiH the number density of these excited states was estimated to be between 10 and 10 cm", on the basis of their optical transition probabilities. These values are much lower than radical densities. lO " cm . Hence, these species are not considered to partake in the deposition. However, a clear correlation between the emission intensity of Si and SiH and the deposition rate has been observed [288]. From this it can be concluded that the emission intensity of Si and SiH is proportional to the concentration of deposition precursors. As the Si and SiH excited species are generated via a one-electron impact process, the deposition precursors are also generated via that process [123]. Hence, for the characterization of deposition, discharge information from OES experiments can be used when these common generation mechanisms exist [286]. 

Fig. 9.24 Theoretical calculations of nuclear forward scattering for the relaxation rates as indicated for a system with electron spin S = 1/2, hyperfine parameters A y jg fi = 50 T, and AF.q = 2 mm s in an external field of 75 mT applied perpendicular to k and O . The transition probabilities co in ((9.8a) and (9.8b)) are expressed in units of mm s , with 1 mm corresponding to 7.3 10 s. (Taken Ifom [30])...

The electron is excited from a filled initial state f below the Fermi level F to an empty final state f above F. Momentum conservation will be provided by a lattice vector or in some cases by a surface vector. The transition probability is mainly determined by the optical excitation matrix element containing the joint density of states. 

Robinson and Frosch<84,133> have developed a theory in which the molecular environment is considered to provide many energy levels which can be in near resonance with the excited molecules. The environment can also serve as a perturbation, coupling with the electronic system of the excited molecule and providing a means of energy dissipation. This perturbation can mix the excited states through spin-orbit interaction. Their expression for the intercombinational radiationless transition probability is... 

Note that in the reference model all the interactions of the electron with the medium polarization VeP are included in Eqs. (8) determining the electron states. The dependence of A and B on the polarization and intramolecular vibrations was entirely neglected in most calculations of the transition probability [the approximation of constant electron density (ACED)]. This approximation, together with Eqs. (4)-(7), resulted in the parabolic shape of the diabatic PES Ut and Uf. The latter differed only by the shift... 

This new approach enables us to consider all the physical effects due to the interaction of the electron with the medium polarization and local vibrations and to take them into account in the calculation of the transition probability. These physical effects are as follows ... 

Polarization fluctuations of a certain type were considered in the configuration model presented above. In principle, fluctuations of a more complicated form may be considered in the same way. A more general approach was suggested in Refs. 23 and 24, where Eq. (16) for the transition probability has been written in a mixed representation using the Feynman path integrals for the nuclear subsystem and the functional integrals over the electron wave functions of the initial and final states t) and t) for the electron ... 

Based on the results obtained in the investigation of the effects of modulation of the electron density by the nuclear vibrations, a lability principle in chemical kinetics and catalysis (electrocatalysis) has been formulated in Ref. 26. This principle is formulated as follows the greater the lability of the electron, transferable atoms or atomic groups with respect to the action of external fields, local vibrations, or fluctuations of the medium polarization, the higher, as a rule, is the transition probability, all other conditions being unchanged. Note that the concept lability is more general than... 

Note that the results obtained are in accordance with the lability principle. The smaller U is, the more labile are the electrons in the adatom and the stronger is the distortion of the shape of the free energy surfaces, leading to a decrease of the activation free energy and to an increase of the transition probability. 

The modulation of the charge of the adsorbed atom by the vibrations of heavy particles leads to a number of additional effects. In particular, it changes the electron and vibrational wave functions and the electrostatic energy of the adatom. These effects may also influence the transition probability and its dependence on the electrode potential. 

---