Kohn variational method scattering

Although the Kohn variational method is not a bounded method (except at zero energy when, subject to certain conditions, it can yield an upper bound on the scattering length), it is found in practice that the phase shift usually becomes more positive as the flexibility of the trial function is enhanced by increasing w it converges towards what is assumed to be the exact value according to a pattern which is quite accurately represented by... 

The scattering length can be calculated using the Kohn variational method in a similar manner to that employed for the phase shift, but the Kohn functional then becomes... 

Theoretical aspects of annihilation and scattering in gases 333 using the Kohn variational method with trial functions of the form... 

Armour, E.A.G. (1984). Application of a generalisation of the Kohn variational method to the calculation of cross sections for low-energy positron-hydrogen molecule scattering. J. Phys. B At. Mol. Opt. Phys. 17 L375-L382. 

J. Chang, N. J. Brown, M. D Mello, R. E. Wyatt, and H. Rabitz, Quantum functional sensitivity analysis within the log-derivative Kohn variational method for reactive scattering, J. Chem. Phys. 97 6240 (1992). 

Kohn variational method that has been described in detail in previous publications [5, 25, 26]. We briefly review this approach for quantum reactive scattering below [27]. In addition, we demonstrate how to apply functional sensitivity analysis within the 5—matrix Kohn framework. We report the cross section calculations resulting from the two PESs for total angular momenta J = 0 and 10. We find that the theoretical cross sections do not change significantly when the LSTH PES is replaced by the DMBE [28]. This suggests that the theoretical prediction — that there is no sharp resonance in the H+H2 integraJ cross section — may be correct. 

The Kohn variational method described above for potential scattering extends in a straightforward way to the collinear reactive scattering problem described in Section 2 without the need to introduce any special coordinates (i.e., one can continue to work with the optimum mass-scaled Jacobi coordinates of the reactant and product arrangements). Moreover, the extensions that are required are comparatively minor, and the overall structure of the method remains the same. 

Continuum wavefunctions can also be generated by solving the partial differential equation (3.3) directly without first transforming it into a set of ordinary differential equations. One possible scheme is the finite elements method (Askar and Rabitz 1984 Jaquet 1987). Another method, which has been applied for the calculation of multi-dimensional scattering wavefunctions, is the 5-matrix version of the Kohn variational principle (Zhang and Miller 1990). 

Kohn-variational (12), Schwinger-variational, (13) R-Matrix (14), and linear algebraic techniques (15,16) have been quite successful in calculating collisional and phH oTo nization cross sections in both resonant and nonresonant processes. These approaches have the advantage of generality at the cost of an explicit treatment of the continuous spectrum of the Hamiltonian and the requisite boundary conditions. In the early molecular applications of these scattering methods, a rather direct approach based on the atomic collision problem was utilized which lacked in efficiency. However in recent years important conceptual and numerical advances in the solution of the molecular continuum equations have been discovered which have made these approaches far more powerful than those of a decade ago... 

Variational methods are at present used extensively in the study of inelastic and reactive scattering involving atoms and diatomic molecules[l-5]. Three of the most commonly used variational methods are due to Kohn (the KVP)[6], Schwinger (the SVP)[7] and Newton (the NVP)[8]. In the applications of these methods, the wavefunction is typically expanded in a set of basis functions, parametrized by the expansion coefficients. These linear variational parameters are then determined so as to render the variational functional stationary. Unlike the variational methods in bound state calculations, the variational principles of scattering theory do not provide an upper or lower bound to the quantity of interest, except in certain special cases.[9] Neverthless, variational methods are useful because, the minimum basis size with which an acceptable level of accuracy can be achieved using a variational method is often much smaller than those required if nonvariational methods are used. The reason for this is generally explained by showing (as... 

The variational approach received a major boost also when it was realised [79] that the simplest variational method - the Kohn variational principle, which is essentially the Rayleigh-Ritz variational principle for eigenvalues modified to incorporate scattering boundary conditions - is free of anomalous (i.e., spurious, unphysical) singularities if it is formulated with S-matrix type boundary conditions rather than standing wave boundary conditions as had been typically used previously. It is useful first to state the Kohn variational approach for the general inelastic scattering. Thus the variational expression for the S-matrix is... 

Variational principles play an important role in the solution of the Schrbdinger equation, e.g. the Ritz-method [5] for the solution of partial differential equations and the Rayleigh-Ritz method [6] for the calculation of bound states of atoms and molecules. The oldest variational method for scattering problems was introduced 1944 by Hulthen [7]. Four years later Hulthen and Kohn [8, 9, 10] independently developed what is now known as the Hulthen-Kohn... 

In this article we focus on one of these approaches, the complex Kohn (CK) method, both because it is a singularly successful example of the variational methods and because it provides a particularly clear view of the central problem of electron scattering, namely the consistent treatment of electronic correlation in the target molecule and correlation involving the scattered electrons. The CK method has its origins in Kohs s 1948 paper on variational principles, and some connections between this method and various other approaches have been discussed by McCurdy, Rescigno, and Schneider. ... 

Mito, Y. and Kamimura, M. (1976). The generator coordinate method for composite-particle scattering based on the Kohn-Hulth6n variational principle,... 

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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