Reactance matrix

The reactance matrix K is ai Q. Exact solutions require the matrix to be of rank n0, implying n linearly independent null-vectors as solutions of the homogeneous equations ma = 0. Because this algebraic condition is not satisfied in general by approximate wave functions, a variational method is needed in order to specify in some sense an optimal approximate solution matrix a. 

At pa the XJMlU (p, Hx) are matched to asymptotic atom-diatom wave functions expressed in the usual mass-scaled Jacobi coordinates Rk, yk, iK, R 2, 0, asymptotic analysis one obtains the reactance matrix R7 1 and from it the scattering matrix S7111. This is done for all T and both parities II = 0 and 1 and a sufficient number of partial waves (i.e., values of J) for the resulting cross sections of interest have converged to the desired degree of accuracy. 

These integrals all consist of known functions and can be performed by quadrature. It is also convenient in these formulas to now set p equal to p everywhere p occurs. Then, from Eq. (23) the reactance matrix is... 

The GNVP may be applied to calculate the reactance matrix K, the scattering matrix S, or the transition matrix, T, from any of which one may... 

Final steps. Once all of the integrals have been evaluated, we solve for the reactance matrix by means of Eqs. (3), (3a), and (7). Then the scattering matrix is determined from... 

The final use of symmetry concerns the solution of Eq. (7) for the correction to the reactance matrix. So fax in the discussion we have emphasized aspects of the calculation prior to this step, but when the basis set is large enough, most of... 

In order to ve the relations between the real and complex quantities, it is desirable to partition matrices into parts corresponding to open (i.e., energetically allowed) and closed (i.e., classically forbidden) channels. This is because of the different boimdary conditions for these cases. Thus distortion potential block has a reactance matrix which we write, similarly to Eq. (3), as... 

Multichannel variants of the phase equation can also be defined, and this is where supercomputers would become attractive for implementing the method. Calogero defines a first order differential equation for the reactance matrix, R, and the scattering matrix, S. It is more enlightening to consider the radial evolution of S. S is complex, but it is bounded and S(r) shows directly the evolution of the transition probabilities as r is increased. This can display the radial regions where coupling is strong. The differential equation for S(r) is Eqn (7) ... 

Scattering-based approaches, as, for instance, the reactance matrix or K matrix [5] projected on an I basis sets, are found more appropriate for computing such kinds of properties. 

The scattering and reactance matrices and boundary conditions. Before attempting to solve (5.1), subject to the boundary conditions of (5.5), it is convenient to define the scattering matrix S and the reactance matrix R. This permits us to decouple the problem of obtaining arbitrary solutions of the Schrodinger equation from the problem of imposing asymptotic conditions appropriate for collision processes on these solutions. 

The reactance matrix R (R-matrix, sometimes called the K-matrix) is defined by the relation ... 

A = A ) ones, it suffices to obtain the reactance matrix R. This in term can be determined by calculating a sufficiently large number of linearly independent solutions of the Schrbdinger equation, and putting the associated g matrix in the form of (5.17). From this, the square coefficient matrices C and D can be obtained, and R calculated from (5.20) as long as care has been taken to as certain that C is nonsingular. 

The BFH reactance matrix R has the following very important properties, analogous to those for the collinear case ... 

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