Buttle correction

The first view has been addressed in two ways. Buttle, in 1967 [9], proposed that a complete basis could be used, albeit with an approximate Hamiltonian. The so-called Buttle correction does help a great deal... 

A Buttle correction (for lack of completeness) is determined by adding R° to R and subtracting the contribution of basis functions already used. 

As indicated in the last section, serious problems exist with the standard R-matrix with respect to slow convergence of the phase shifts (or S-matrix) as the number of translational basis functions is increased. This has been amply demonstrated in a number of papers (e.g.. Refs. 8-12). In all of these cases an orthogonal translational basis is used which satisfies fixed log derivative boundary conditions at the R-matrix boundary, R = A. As indicated above, the Buttle correction [9] can be added to the R-matrix to account, in an approximate fashion, for the members of the complete basis not included in the explicit R-matrix evaluation. An additional variational correction [10,11] was proposed to improve the results further. Although these procedures help a great deal, they are both expensive, at least in terms of programming effort for complex systems. 

It was, however, only shown in the very recent article of Bocchetta and Gerratt [16] that the Buttle correction itself was unnecessary if Mori s prescription is followed, i.e., if a non-orthogonal (on O R A) basis which does not satisfy the log derivative b.c. s at A is used. 

TABLE II. Phase Shifts for N=10. Columns A, distributed Gaussian basis B, non-orthogonal sines C, orthogonal sines D, orthogonal sines with Buttle correction E, Exact. 

Although the R-matrix method for scattering was proposed many years ago, its implementation has been quite restricted because of the slow convergence of the unadorned R-matrix to numerically adequate results. For inelastic scattering the use of a simple zero order Hamiltonian with an analytic solution to generate the Buttle correction makes it a viable option for some systems. In particular the increase in memory and speed of large computers make large basis calculations feasible. 

For reactive scattering, however, the Buttle correction cannot be applied easily since there exists no reasonable zero order Hamiltonian with two or more chemical channels for which analytic solutions are known. Although a Buttle type correction may be determined in a basis, it adds substantially to the computational effort required. 

The root of this problem appears to be that the true wavefunction matches the log derivative b.c. s of the chosen translational basis only at an isolated set of energies. At these energies (near the of the l2 expansion of the Greenes function for a basis satisfying the Bloch operator b.c. s, Lq0= 0) the R-matrix results are very accurate for a given basis size, even without the Buttle or variational correction. However, between these energies, the results are very poor. 

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