Eckart (see references in Notes and Bibliography at the end of this appendix) introduced a one-dimensional, analytically solvable, quantum mechanical model, which is applicable to the study of chemical reactionss. The potential is given by

where the parameter *κ* provides the shape of the transition from the initial situation with energy *U*(−∞) to the final one with energy *U*(∞) = 0. A *κ*-value on the interval [−1,1] gives a smooth step, while other values yield a barrier or a well in the transition region (see Figure F.1).

The Green’s function satisfies the equation ( = 1)

The introduction of dimensionless independent and dependent variables such that *x* = *bq*, *E* = *k*^{2}/2*Mb*^{2}, *G*(*x*, *x*′; *E*) = 2*Mbg*(*q*, *q*′; *k*^{2}), *U*(*x*) = *w*(*q*)/2*Mb*^{2}, *a* = 2*Mb*^{2}*U*(−∞), and *w*(*q*) = *a*[1 + *κ*/(1 + *e*^{−q})/(1 + *e*^{q}) results in the equation

A solution *u*(*z*) to the homogeneous equation corresponding to Eq. (F.3) can conveniently be expressed in terms of the Gauss hypergeometric series *F*(*z*) of the auxiliary variable *z* = 1/(1 + *e*^{−q}). Thus,

with

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