Repulsion potential, variation with

Centrifugal barrier effects have their origin in the balance between the repulsive term in the radial Schrodinger equation, which varies as 1/r2, and the attractive electrostatic potential experienced by an electron in a many-electron atom, whose variation with radius differs from atom to atom because of screening effects. In order to understand them properly, it is necessary to appreciate the different properties of short and of long range potential wells in quantum mechanics. 

One need only consider the energy dependence of the differential Franck-Condon factor. For small He, this factorization may be justified in the same way as in the case of second-order energy shifts of bound states. The variation of T and 5E with v and J provides information about the initially unknown shape of the repulsive potential curve. 

For a crossing on the left side (repulsive branch) of the bound potential, an inner crossing or one of Mulliken s a-, c, or b predissociation cases, a single oscillation of r will extend over many vibrational levels so that, in practice, only a small variation with v (and J) can be sampled (see Fig. 7.23). The reason for this is that the slopes of the bound and repulsive curves at Rc are much larger and more nearly equal for an inner rather than an outer wall crossing. Consequently, as EVtj—Ec increases, the innermost maximum of Xv,j(R) moves to smaller R much more slowly than the outermost maximum sweeps to larger R. 

Fig. 3.14. Variation of the attractive and repulsive components and the total potential energy with the separation distance between two Pd nanocrystals of 4.5 nm diameter coated with (a) octanethiol and (b) dodecanethiol (reproduced with permission from [585])...

As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. 

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