K-matrix element

Equation (B.IO) stands for the j,k) matrix element of the left-hand side of Eq. (B.7). Next, we consider the (j,k) element of the first term on the right-hand side of Eq. (B.7), namely. 

The K-matrix method is essentially a configuration interaction (Cl) performed at a fixed energy lying in the continuum upon a basis of "unperturbed funetions that (at the formal level) includes both diserete and eontinuous subsets. It turns the Schrodinger equation into a system of integral equations for the K-matrix elements, which is then transformed into a linear system by a quadrature upon afinite L basis set. 

It should be noted that the integral equations [2] determining the elements Kp B. aE derived as an energy-variational problem, correspond also to the stationary condition of the variational functional proposed by Newton (9). Thus the K-matrix elements obeying equation [2] guarantee a stationary value for the K-matrix on the energy shell. 

The differential cross section for positronium formation may be expressed in terms of the partial-wave K-matrix elements as... 

The diagonal (k = k) matrix elements xkKP Xk))(r) of the operator P vanish, because this operator is Hermitian and odd with respect to time reversal. The oflF-diagonal matrix elements satisfy... 

The Ku matrix element is called an exchange integral, and has no classical analogy. Note that the order of the MOs in the J and K matrix elements is according to the electron indices. The energy can thus be written as in eq. (3.31). 

Figure 1. Photodissofciation cross sections for the two lowest energy rotational predissociation processes in Ar-H2. The top two panels show the (slow) variation of the K matrix elements and of a quantity obtained...

The k k) matrix element of this transition operator is also a sum over the atoms in B, from which it follows that the collisional TCP is a sum over pairs b, b ) of atoms in B,... 

The k k) matrix elements of this two-atom operator can be readily calculated [30]. We change wavevector variables from k, kf,) for A and b to the total and relative-motion wavevectors given by... 

The Kato identity,[ll] which forms the basis for the three most commonly used variational methods in scattering theory, the KVP, the SVP and the NVP, is given as follows, for the K matrix element ... 

Thus we see that the error in the computed K matrix element is one order higher than the error in the corresponding wavefunction. From the Kato identity point of view, we see that the error term in the functional would be < Wl > for the KVP, for... 

From each of our methods, we extract four types of K matrix elements [Kmlf, which are ... 

The error in the wavefunctions is computed as Xir) =/ex(r)-/ >m(r). for different basis sizes N. We have found it most illumnating to present these as "error-scapes" or three-dimensional perspective plots in which m( ) is plotted as a function of the coordinate r, and the number of basis functions N. The error in the K matrix elements are presented as fractional errors, given by (Kex-[Km]f)/Kex. which are tabulated for each... 

The behavior of the error for the different methods as revealed in Figs. 1-3 for potential A, appears to be typical of the methods, at least for potential scattering, and similar patterns are observed for potentials B and C also. So, the information present and discussed above can be reliably extended to these cases dso. Instead of presenting the error-scapes for these cases, we now turn attention to a matter that, in the final count, is more important than the convergence of the wavefunctions themselves, viz., the convergence of the K matrix elements. Here we present the results for all three potentials, and discuss the three cases. 

This leads to another question why are the K matrix elements [KmmadId... 

KmmadIsvP inaccurate of the different computed K matrix elements, reach fairly... 

The weighting factors in each case play decisive roles in determining which regions of the wavefimction converge first, and to what extent. They also determine how sensitive the computed K matrix element would be, to large emn S in the unconverged regions of the wavefunction. 

---