The Eckart Potential and its Propagator
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 = k2/2Mb2, G(x, x′; E) = 2Mbg(q, q′; k2), U(x) = w(q)/2Mb2, a = 2Mb2U(−∞), and w(q) = a[1 + κ/(1 + e−q)/(1 + eq) 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,