R-matrix propagation method

Kulander, K.C. and Light, J.C. (1980). Photodissociation of triatomic molecules Application of the R-matrix propagation methods to the calculation of bound-free Franck-Condon factors, J. Chem. Phys. 73, 4337-4346. 

The coupled-channel equations (Eq. (31), (32), or (33)), may be solved in a variety of ways, but we use the R matrix propagation method of Light and Walker.Wo will review this method briefly in this section, as applied to the coupled channel equations in Jacobi coor dinates. The approach is essentially the same in other coordinates. 

This gives a set of close-coupling equations coupled in the orbital quantum number Z and n. As before, we solve these equations using the R-matrix propagator method [22,26]. 

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... 

R-matrix propagation method is a very stable technique, even when, many closed channels are necessary for convergence of the S matrix. 

Green s function (on O R A), the connection between this "local" R-matrix and the S-matrix must be made. For A in the as)nTiptotic region, this relation is given in Eq, (II.11), but we want A to be as small as possible for efficiency. We therefore consider below two methods of connecting the R-matrix to the S-matrix using a small "R-matrix box," i.e., small value of A. The first is a simple WKB (adiabatic) connection, and the second the more accurate R-matrix propagation [20,21]. 

Using this approach, calibration can be performed knowing only the one component of interest in the system. Interfering compounds only have to be present, not quantified. They are implicitly modeled with this approach. The major implication of this technique is that application of the sensor array in remote environments is better facilitated. A restriction that this model imposes is that the number of sensors must be less than the number of calibration samples in order to perform the generalized inverse. This method usually has more error propagation due to the instability of the R matrix inversion. Collinearity plays an important role in this case. 

Using whatever propagation method, one has to evaluate the action of the Hamiltonian operator on the wavefunction P(r). This is normally carried out by expanding P(f) in a suitable basis set and then evaluates the operator action on basis functions. One can use the FFT (fast Fourier transform) techniques (7,14), discrete variable representation (DVR) (15,16) techniques, or simply calculate matrix elements of the operator in a given basis set. 

For the propagation of the multichannel wavefunction 4>(R), in real or complex-scaled coordinates, an efficient algorithm is furnished by fhe Fox-Goodwin-Numerov method [8, 44], which results from a discrefizafion of the differential operator appearing in Eqs. (39). Given adjacent points R — h, R, and R + h on the grid, we define an inward mafrix (labeled /) and an outward matrix (labeled o) as ... 

We may apply the split-operator method (Sec. 3.2.1) to the three matrices lijV) V/), and Vo for a short-time numerical propagation scheme. Ionization is now described by population of the neutral state Xn R,t) transferring to the ionized state partial-wave components Xc,kjix R,t) over time through the interaction represented by the matrix Vo(i ,f)-... 

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