Radial coupling matrix elements

The evaluation of the radial coupling matrix elements between molecular states of the same symmetry 3... 

The main features of the radial coupling matrix elements are presented in Fig. 2. In correspondence with the avoided crossings between the potential energy curves of singleelectron capture, sharp peaked functions appear at respectively 6.35, 7.50 and 8.30 a.u.. They are approximately 1.23, 2.53 and 12.21 a.u. high and respectively 0.75, 0.50 and less than 0.10 a.u. wide at half height. 

Fig. 2. a, b, c. Non-adiabatic radial coupling matrix elements for the states of single-electron capture. 

In relation with these avoided crossings, the radial coupling matrix elements present sharp peaks at respectively 5.4, 6.6, 7.55 and 9.5 a.u. (Fig. 5). We may notice that these radial couplings are almost insensitive to the choice of the origin of electronic coordinates. The most sensitive one is the g23 function at short internuclear distance range, but we may expect weak translational effects for such electron capture processes dominated by collisions at large distance of closest approach. 

Fig. 5.3, b. Non adiabatic radial coupling matrix elements between states, a) Origin N. b) Origin He. 

The radial coupling matrix elements between all pairs of states of same symmetry have been calculated by means of the finite difference technique ... 

Figure 2. (a) Radial coupling matrix elements between 3 + and 3A states of the S3+(3s23p)+H collision system (see labels in Figure la), (b) Radial coupling between 3n states of the S3+(3s23p) + H collision system (see labels in Figure lb). 

The potential energy curves and radial coupling matrix elements have been calculated for a distance Cq+-04 varying from 0.5 to 5 A. The charge transfer levels are presented in Figure 4 for the series q = (2-5). For the charges q = (2-4), the... 

Fig. 16. Calculated rotational and radial coupling matrix elements for various states in MgNa as a function of intemudear separation.

Figure 7. Radial coupling matrix elements derived from Eq. (26) using model matrix elements for the system B + C. The results labeled SHM, Demkov, and Nikitin refer to the matrix elements in Figures 5a, 5b, and 5c, respectively.

It should be added that the coupling radius Rc is about equal to the full width at half-maximum A/ (FWHM) of the radial coupling matrix element (Figure 7). For Demkov s model it follows that A/ (FWHM) = a In (2 + -v/S) 3a72, where a is replaced by ca (Table 1). Hence, the... 

To deduce radial coupling matrix elements, Eq. (14) is applied. It is expected that KL-vacancy sharing is primarily determined by the 3o--4model results compare well with the data from the more elaborate three-state calcula-... 

Figure 23. Radial coupling matrix element for the 3o- and 4a MO s in the system Ne + Kr. The curve labeled SHM refers to calculations using the model matrix elements (18) and the circles labeled VSM are calculated by Fritsch and Wille on the basis of the variable screening model/ ...

It has been pointed out by Fritsch and Wille that the functional form of from a three-state analysis differs considerably from that of the two-state model, which does not account for the 2s state. The shape of M30.40. from the three-state analysis may be understood from the contribution of the different atomic orbitals involved. At intemuclear distances R > 0.3 a.u. the interaction between the Is and 2p orbitals dominates, whereas at R < 0.3 a.u. the ls-2s interaction gains importance. Since the quantities W, 2, and which account for the atomic interactions have different signs, the radial coupling matrix element passes through zero near 0.3 a.u. (Figure 23). Thus, the 2s orbital has primarily the effect of reducing the 3o--4o- radial coupling. 

Fig. 20.5 (a) Variation of the radial coupling matrix element gyg for different geometries of the OH radical around the equilibrium distance roH = 1.834967 a.u. in the linear geometry, (b) Corresponding charge transfer cross sections for the C + + OH system... 

Fig. 20.6 (a) Radial coupling matrix elements between Z states of the +HF system in the... 

Similar correlations between non-adiabatic effect and charge transfer cross sections may be established for the collision of ions on HF. However, in that case, the analysis is more complex as several non-adiabatic interactions may interfere and the concomitant evolution of several radial coupling matrix elements has to be taken into account. 

The radial coupling matrix elements and charge transfer cross sections for different rnp values are presented respectively in Fig. 20.6 a, b. As in the previous C + + OH collision system, the cross sections show a regular increase when the vibration coordinate rnp is reduced from 2.0 to 1.5 a.u. around the equilibrium distance, in agreement with the increase of the radial coupling matrix element rad23 between the entry channel and the 2 X+ level. On the other hand, the radial coupling rad 12 between and 2 X+ exit channels decreases with... 

Fig. 20.9 (a) Radial coupling matrix elements between Z states of the +HF system at eqmlibiium, rnp = 1.738368 tcu., for different orientations. Same labels as in Kg. 20.2a. Upper curves, rad23 lower curves, tadl2. Dotted line, 9 = 90° dashed line, 9 = 160P thin full line, 9 = 0° full line, 6 = 180°. (b) Cotresponchng charge transfer cross-sections for the C + + HF system at equilibrium, for different orientations 9 from 0° to 180°... 

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