Propagators in Quantum Chemistry, 2nd Edition

Appendix F

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 + 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

Figure F.1:  Eckart potential for three ...