Log-derivative methods

Manolopoulos D E 1986 An improved log derivative method for inelastic scattering J. Chem. Phys. 85 6425-9... 

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. 

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... 

Johnson, B. R. (1973) The Multichannel Log-Derivative Method for Scattering Calculations,... 

When the basis size is small enough to store the Hamiltonian matrix in the computer core memory, two things can be said with confidence. First, the method presented in Sec. II based on Eq. (1) and Eqs. (2) and (9) (or better to avoid anomalies, (1) and (21)) are very easy to comprehend and implement. This is especially true when the diagonalization of the full Hamiltonian is the key computational step. Second, there are many other approaches, such as the Kohn variational principle (21), the / -matrix theory (35), and the closely related, log-derivative methods (22, 23), that are easy to implement and anomaly free. The methods which use absorbing potentials clearly have a disadvantage relative to the above methods in the sense that they require larger than minimal basis sets and involve non-Hermitian matrices. 

R. Johnson, The multichannel log-derivative method for scattering calculations, J. Comput. Phys. 73 445 (1973). 

The additional problem present in the bound state case, at energies below the dissociation energy of the complex, is that of locating energies which are eigenvalues of the coupled equations, where a solution may be found that satisfies bound state boundary conditions. There are several procedures available for doing this, "" but the stablest are the log-derivative methods.In the many-channel case, the log-derivative matrix Y(R) is defined by ... 

One apparent disadvantage of the log-derivative methods is that they do not directly give explicit wavefunctions, which are needed to calculate molecular properties (via expectation values) and spectroscopic intensities (via off-diagonal matrix elements). However, the restriction is not as serious as it might appear a finite-difference approach for extracting expectation values from coupled channel calculations has been described by Hutson, and is available as an option in the BOUND program. 

Table 1. Photodissociation probabilities[20] for C I- CFa+I with laser frequency of co=40323cm . For more details see Refs 18 and 14. RM=R matrix, LD=Log-derivative methods.

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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