Log-derivative propagation

The standard log-derivative propagator now corrects for the difference between U and Uprising a Simpson-nile integration. The specific fomuilas are... 

Johnson, B. R. (1973) The Multichannel Log-Derivative Method for Scattering Calculations, J. Comp. Phys. 13, 445-9 Manolopoulos, D. E. (1986) An improved log derivative propagator for inelastic scattering, J. Chem. Phys. 85. 6425-9. 

Here we derive boundary conditions for the HBU model using the log-derivative propagation mcthod[29] and hypcrsphcrical coordinates. For the log-dcrh tivc propagation wc have h = Y(p,)h, where Y(pi) is the log-derh tive matrix, h is the derivative matrix of h with respect to p and h has elements defined... 

There are many different propagators that have been developed to take account of the special properties of Equation 1.42, and a detailed discussion is beyond the scope of this chapter. Among those commonly used in cold molecule collisions are the renormalized Numerov approach [16,17] and several different log-derivative propagators [15,18,19]. The latter actually propagate the log-derivative matrix, defined as... 

Propagating wavefunctions explicitly in the presence of deeply closed channels is notoriously numerically unstable. It is much more satisfactory to use a propagator that is stable in the presence of closed channels, such as one of the log-derivative propagators described above. The multichannel matching condition can be expressed very simply in terms of the log-derivative matrix F(r) of Equation 1.52. If E is an eigenvalue of the coupled equations, there must exist a wavefunction vector tn( mid) = (fmid) = t IT ( mid) for which... 

One way to avoid the stabilization problem just mentioned is to propagate the log derivative matrix Y(R) [43]. This is defined by... 

Close-coupling equations were solved using the hybrid modified log-derivative / Airy propagator of Alexander and Manolopulos [67]. The agreement... 

The propagation of the solutions to the CC equations are carried out entirely in the fully-uncoupled, body-frame basis of eq. 2. At the end of the propagation, but before extraction of the S matrix, we transform the log-derivative matrix into the partially coupled basis discussed above. Here, the atomic states in the reactant arrangement are labeled by the total electronic angular momentum of the atom, ja, and... 

As discussed above, for each value of the hyperradius r, we expand the total scattering wavefunction in a set of previously determined orthogonal surface functions. A log-derivative method [47] is used to propagate the solution numerically, from small r to large r. The parameters which control the accuracy of the integration are (a) the number of sectors, (b) the number of vibration-rotation states included for each electronic state in each arrangement, and (c) the maximum value of the total projection quantum number K. These are increased until the desired quantities (integral and/or... 

It should also be clear that a similar approach could be employed, if desired, to propagate overlaps of bound state wavefunctions with the gradient of the scattering wavefunction. Note from Eqs. (93)-(94) that the propagation of overlaps requires at each step an inversion of the R matrix if one propagates the log-derivative matrix instead of the R matrix, then the propagation of overlaps by this scheme requires only a few additional matrix multiplications at each step. 

One apparent limitation of using the R-matrix propagation is that only (inverse) log derivative information is carried along—the wave-function is only determined asymptotically with the imposition of b.c. s. This makes the accumulation of overlap integrals for photodissociation, for example, somewhat more complicated. At this Workshop, however,... 

R. B. Walker [24] showed that the log derivative matrix can be propagated using the information evaluated in the R-matrix propagation, with a very similar algorithm. This permits the efficient propagation of overlap information in a relatively simple fashion analogous to that used by Kulander, et al. for collinear reactive photodissociation [25] and by Heather, et al. for 3-D triatomic photodissociation [26]. 

Much attention has been diverted to the numerical solution of the coupled-channel nations (3). Two of the most widely used and stable methods are the log-derivative approach introduced originally by Johnson[8] and the R-matrix propagator method developed by Light and Walker[9]. The log-derivative approach propagates the ratio between the derivative of the wavefunction (with respect to R) and the wavefonction, while the R matrix propagates the inverse of this quantity. Thus the log derivative matrix is defined by... 

Thus we have obtained a set of coupled second-order differential equations in the diffraction channels. The equations (5.5) may be solved efficiently by a number of methods, such as the log-derivative method, in which 4>c = (d In c/dz) = (d G/dz) G is propagated instead of g [126]. Thus substitution transforms the second-order differential equation to a first-order nonlinear Ricatti equation. 

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