Tables transition probability

Table 8.8 A list of all (independent) elementary 1-dimensional k = 2,r = V rules supporting a QCA-II quantum dynamical analogue of the form defined by equations (5.4) and (5.7). Two additional rules, 51 and 204, both of class-2, yield pii= 0 and 1 , respectively, so that their q-behavior remains essentially classical, (pn is the a= - a = transition probability which defines the quantum operator 4).

Table II must be used with care in anomalous cases in which the transition probability for ionization of the molecule is very low in some energy ranges (e.g., acetylene, benzene, methylamine). In such cases higher RE s, not included in the table and normally of small importance, may be responsible for the charge exchange processes although with small cross-sections (cf. 9, 13).

Table 2 shows transition moments calculated by the different EOM-CCSD models. As has been discussed above, the right-hand transition moment 9 is size intensive but the left-hand transition moment 9 in model I and model II is not size intensive. Model II is much improved as far as size intensivity is concerned because of the elimination of the apparent unlinked terms. The apparent unlinked terms are a product of the size-intensive quantity ro and size-extensive quantities and therefore are size extensive. The difference between the values of model I and model II, as summarized in the fifth column, reveals strict size extensivity. Complete elimination of unlinked diagrams by using A amplitudes brings strict size intensivity for the transition moment and therefore the transition probabilities calculated by model III are strictly size intensive. 

Table 1. Rigid-rotator transition probabilities for four...

Under some circumstances the rotationally anisotropy may be even further simplified for T-R energy transfer of polar molecules like HF (41). To explore this quantitatively we performed additional rigid-rotator calculations in which we retained only the spherically symmetric and dipole-dipole terms of the AD potential, which yields M = 3 (see Figures 1, 3, and 4). These calculations converge more rapidly with increasing N and usually yield even less rotationally inelastic scattering. For example Table 2 compares the converged inelastic transition probabilities... 

Table 2. Inelastic T R transition probabilities for full (M = 9) and truncated (M = 3) AD potentials as functions of relative translational energy...

Comparing the vibrational A lifetimes issued from both decay mechanisms (Tables 7 - 8), it is readily seen that the eleetric dipolar transition decay is always slightly favoured. A similar eonclusion holds for the A n state but, as expected, the vibrational transition probabilities are much larger for the dipolar decay which lead to mueh smaller vibrational lifetimes with respect to those via the cascade mode of decay, the differences amounting to five to six powers of ten (Table 7 - 8). 

From (4.56) and Table 4.3, we derive the relative intensity ratios 3 2 1 1 2 3 for the hyperfine components of a Zeeman pattern of a powder sample. The transition probability for the case of the polar angle 6 = Oq can readly be calculated by integrating (4.56) only over the azimuthal angle (j). One obtains a factor (1 + cos 0o)/2 and sin 0o for m = 1 and m = 0, respectively, which are multiplied by the square of the Clebsch-Gordan coefficients. As a consequence of the angular correlation of the transition probabilities the second and fifth hyperfine components (Fig. 4.17) disappear if the direction k of the y-rays and the magnetic field H are parallel (0q = 0). 

A different situation is found for a dipolar hfs tensor. The relative nuclear transition probabilities for B0 parallel to Ap or Aj are given in Table 2.2 and 2.3. As in the case of an isotropic hfs tensor, again Beff is oriented parallel or antiparallel to B0. Since the enhancement factor E is isotropic for B0 parallel to Ay, the net circularly polarized field rotates in the same sense as the applied field. In the case of B0 parallel to Ai however, the enhancement factor E is anisotropic. This anisotropy of E is responsible for the generation of counter rotating fields which induce residual lines. 

Table 2. Relative transition probabilities for circularly polarized fields. S = 1/2,1 = 1/2 1) isotropic hyperfine tensor ai 0 > 0 2) dipolar hyperfine tensor B0 A 3) dipolar hyperfine tensor B0 Ax...

The hfs tensors of ligand nuclei in the first coordination sphere of a metal complex are usually dominated by the isotropic interaction, i.e. the transition probabilities may be approximated by the formulae given in Table 2.1 for aiso > 2v . 

Table 2 shows the evolution of the ionic random walk with unequal transition probabilities, movement to active centre si being twice more probable than to active centre s5. 

Table 15 illustrates the development of the probability distribution of a solid particle being in one of the three characteristic zones in a cell, assuming that the particle residence has an equiprobable initial state, and that the transitional probabilities are as follows. Zone 1 —>2 0.4 Zone 2 —>1 0.4 Zone 1—>3 0.1 Zone 3->l 0.2 Zone 2 3 0.3 Zone 3->2 0.2. ... 

Probability Distribution in Table 16 Modified by Additional Transitional Probability for Zone 2—> 1 0.1... 

In Tables -A, we report oscillator strengths for some fine structure transitions in neutral fluorine, chlorine, bromine and iodine, respectively. Two sets of RQDO/-values are shown, those computed with the standard dipole length operator g(r) = r, and those where core-valence correlation has been explicitly introduced, Eq. (10). As comparative data, we have included in the tables /-values taken from critical compilations [15,18], results of length and velocity /-values by Ojha and Hibbert [17], who used large configuration expansions in the atomic structure code CIVS, and absolute transition probabilities measured through a gas-driven shock tube by Bengtson et al. converted... 

Table 1 Transition probabilities for a configuration change among the six Slater groups, induced by a visit of an unpaired Takagi group Tl or T3...

Table I Comparison of quantum and semiclassical transition probabilities for two surface model problem.

In order to test the small x assumptions in our calculations of condensed phase vibrational transition probabilities and rates, we have performed model calculations, - for a colinear system with one molecule moving between two solvent particles. The positions ofthe solvent particles are held fixed. The center of mass position of the solute molecule is the only slow variable coordinate in the system. This allows for the comparison of surface hopping calculations based on small X approximations with calculations without these approximations. In the model calculations discussed here, and in the calculations from many particle simulations reported in Table II, the approximations made for each trajectory are that the nonadiabatic coupling is constant that the slopes of the initial and final... 

Arnold et al.24 have calculated radiative lifetimes for the various collision complexes of singlet molecular oxygen on the basis of a collision time of 10"13 sec. The data for wavelengths and transition probabilities are presented in Table III. A recent paper25 describes the theory of double electronic transitions, and gives calculated oscillator strengths for the oxygen systems. 

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