Excitation probability, formula

The probability for a transition to the first excited state of rotational vibrations to occur per unit time is readily estimated by formula (A2.25) if the harmonic... 

Application of the F-D theorem produced [122] several significant results. Apart from the Nyquist formula these include the correct formulation of Brownian motion, electric dipole and acoustic radiation resistance, and a rationalization of spontaneous transition probabilities for an isolated excited atom. 

Calculation of the pre-exponential factor in eqn. (7) is connected with the analysis of electron motion in parabolic coordinates. The first time such calculations were conducted was by Lanczos [12]. The formulae he obtained were cumbersome and we shall not give them here. The simple formula for the probability of ionization of a slightly excited atom is given in ref. 13 as... 

Another clue to why this branching ratio changes in this counterintuitive way with laser intensity is to note that the three-photon signal is peaked near v = 15, while the two-photon signal is peaked near v = 7. This implies that high vibrational excitation of the ion enhances the curve crossing necessary to produce the two-photon signal. This is exactly the trend observed in the Landau-Zener formula calculations performed by Zavriyev et al. [50], In their calculations on H2 the probability to cross the... 

The TOF spectrum of the Cl or Br atom is measured as a function of wavelength, with the polarization of the laser selected so that only TTq is excited. The peaks that are observed in the spectrum can be identified with atoms in either the 3/2 Dr pl/2 state> so that the experimental data can be used to determine the probability, P, for the diabatic crossing. The Landau-Zener formula for this probability is given by... 

Following from formula (4.54), the transfer of energy on excitation of molecules has a noticeable probability even in the case where the impact parameter is much greater than their size d. Since the intermolecular spacings in a condensed medium are of order of d, a charged particle interacts with many of its molecules. The polarization of these molecules weakens the field of the particle, which, in its turn, weakens the interaction of the particle with the molecules located far from the track. This results in that the actual ionization losses are smaller than the value we would get by simply summing the losses in collisions with individual molecules given by formula (5.1). This polarization (density) effect was first pointed out by Swann,205 while the principles of calculation of ionization losses in a dense medium were developed by Fermi.206... 

A nontrivial result with respect to the well-known formula of Migdal (1941) for the probability of the -decay-induced excitation of an atom is the justification in the above derivation for taking the electron wave functions at the equilibrium nuclear configuration of the parent molecule, as well as the... 

Formula for the Probability of Electron Excitation in the MO LCAO Approximation... 

As seen previously, the chemical reactions studied most often are the exchange ones. Those requiring several potential energy surfaces of excited states (diabatic reactions) are worth special mention, since they most certainly define a domain of application with a future for classical trajectories. An electron jump from one surface to another requires either to be given a statistical probability of occurence by the Landau Zener formula (or one of its improved versions " ) or to be described by means of complex-valued classical trajectories as a direct and gradual passage in the complex-valued extension of the potential surfaces (generalization of the classical S-matrix ). 

We do not want to discuss the singlet excited states here, because they require at least a two-determinantal wave function for their description and most probably also an MCSCF (Multiconfiguration SCF) treatment including all single excitations is necessary. The derivation of the necessary formulae has been already completed [95] and the numerical application of them is in progress in our Laboratory. 

Table 3 Energy shifts of K- and L-shell electrons in hydrogen-like due to various collective excitations. Upper half The contributions fixim low-lying nuclear states are calculated using experimental energies and transition probabilities [69]. Lower half The contributions from giant resonance states. Excitation energies and corresponding reduced electric transition strengths are again estimated based on empirical formulae. Notations are the same as in Table 2.

In this relation, Vj is the vibrational quantum number of a non-stable negative iort, and Fq, (in s ) are probabilities of transitions between vibrational states. The cross section of the resonant vibrational excitation process (2-148) can be found in the quasi-steady-state approximation using the Breit-Wigner formula ... 

The effective activation energy in this case is E /a. Relation (2-205) describes rate coefficients after averaging over the Maxwelhan distribution ftmction for translational degrees of freedom. Probabihties for reactiorrs of excited molecules without averaging can be fotmd based on the Le Roy formula (Le Roy, 1969), which gives the probability Pv(Ey, Et) of elementary reaction as a ftmction of vibrational and translational energies ( v and Et, respectively) ... 

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