Centrifugal repulsion

As the kinetic energy operator has no off-diagonal elements in the space-fixed axis representation we may more easily correct for the fact that the analysis line is not tmly in the asymptotic region as far as the centrifugal representation is concerned. We do this first by subtracting the residual centrifugal repulsion, I I + l)/(2p / ), from the radial kinetic energy at the analysis line in the product channel. Thus we use the expression... 

Fig. 4.6 Effective potential given by eqn 421. (a) orbital with /= 0 (b) /> 0 showing effect of centrifugal repulsion term.

Figure 4.7 shows the forms of other radial wavefunctions for orbitals with n up to three. It can be seen that functions for r orbitals are non-zero at the nucleus, whereas the other orbitals are zero there because of the centrifugal repulsion term. The other obvious feature is that the number of nodes in these functions increases with n. In fact, the number of radial nodes is equal to n - / — 1. If we recall that the angular functions can also have nodes, the number being /, then it is apparent that the total number of nodes in any atomic orbital is equal to n -1 this increases with the energy, as we have found with the wavefunctions for other systems. 

The physical origin of the double well is also readily understood. In H (as opposed to many-electron atoms), if the centrifugal repulsive term is included within an effective potential then, since it grows as ( + l)h2/2mr2, there is a repulsive potential at small radius (r < ro, say) can only expel wavefunctions of high angular momentum into the outer Coulombic reaches. Note that, at all radii, the effective potential is the net... 

The requisite value of Zoo is readily obtained by inverting Eq.(14) e.., for helium we find 2.343. The increase above the actual charge, AZ = Zoo — offsets the enhanced centrifugal repulsion of the D oo limit. 

The concept of an extended 4f state is better understood by invoking a quasi-atomic model based on the principles described in sect. 2 of this chapter. The general idea is that a barrier is formed, owing to the combined effects of the Coulomb interaction in a many-electron atom and the centrifugal repulsion for a 4f electron. This gives rise to a double-well potential in which the outer well is shallow and extended. We discuss this idea in more detail below, in connection with the XAS of higher oxides of Ce, Pr and Tb. 

The role of the effective attraction increases with M/m. As a result, the decrease of the relaxation rate with increasing a becomes weaker. The exponent s in Equation 10.33 continuously decreases with increasing M/m and becomes zero for M/m = 12.33 (see Figure 10.9). In the Bom-Oppenheimer picture this means that at this point one has a balance between the mediated attraction and the centrifugal repulsion. A further increase in M/m makes s negative and it reaches the value = — 1 for the critical mass ratio M/m = 13.6. Thus, in the range 12.33 < M/m < 13.6 the relaxation rate increases with a. 

This so-called orbiYing happens when there can be more flian one solution for the turning point of the trajectory and is taken up in Chapter 4. This is of importance at low collision energies, typically lower than the well depth. In classical mechanics the signature of orbiting is the deflection due to the top of the barrier in the effective potential, the point where the centrifugal repulsion is just balanced by the attraction due to the potential V R). 

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