Scattering theory matrix variational method

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 ... 

The great populariQr of variational methods in scattering theory has always been attributed to the fact that they achieve a higher level of accuracy for a given basis size N. Their superiority over nonvariational methods have been, as al dy mention explained by the fact that e variational functionals for quantities such as K, S or T matrix elements are... 

The three variational principles in common use in scattering theory are due to Kohn [9], Schwinger [11] and Newton [12]. Two of these variational principles, those due to Kohn and Newton, have been successfully developed and applied to reactive scattering problems in recent years there is the S-matrix Kohn method of Zhang, Chu, and Miller, the related log derivative Kohn method of Manolopoulos, D Mello, and Wyatt and the L - Amplitude Density Generalized Newton Variational Principle (L -AD GNVP) method of Schwenke, Kouri, and Truhlar. 

Another approach for developing approximations to CC and CS reactive scattering calculations is to use distorted wave theory. In this approach, one considers that reaction is only a small perturbation on the nonreactive collision dynamics. As a result, the reactive scattering matrix can be approximated by the matrix element of a perturbative Hamiltonian operator using reagent and product nonreactive wavefunctions. Variations on this idea can be developed by using different approximations to the nonreactive wavefunctions. At the top of the hierachy of these methods is the coupled channel distorted wave (CCDW) method, followed by coupled states distorted wave (CSDW). 

Theory. Usually we do not solve the fundamental equations directly. We use a theory, for example, Har-tree-Fock theory [3], Moller-Plesset perturbation theory [4], coupled-cluster theory [5], Kohn s [6, 7], Newton s [8], or Schlessinger s [9] variational principle for scattering amplitudes, the quasiclassical trajectory method [10], the trajectory surface hopping method [11], classical S-matrix theory [12], the close-coupling approximation... 

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