Algebraic variational approach

D. W. Schwenke, M. Mladenovic, M. Zhao, D. G. Truhlar, Y. Sun, and D. J. Kouri, Computational strategies and improvements in the linear algebraic variational approach to rearrangement scattering, Supercomputer Algorithms for Reactivity Dynamics and Kinetics of Small Molecules (A. Lagana, ed.), Kluwer, Dordrecht, 1989, p. 131. 

COMPUTATIONAL STRATEGIES AND IMPROVEMENTS IN THE LINEAR ALGEBRAIC VARIATIONAL APPROACH TO REARRANGEMENT SCATTERING... 

Summary. The main approaches to time-independent quantum reactive scattering, linear algebraic variational approaches using Jacobi coordinates and propagation methods using hyperspherical coordinates, are discussed in detail. Recent developments in computer implementations and applications are briefly outlined. 

The present second part of these notes is based mostly on the work of Jaquet et al. [1], Launay [2], Kouri and Hoffmann [3], and Kuppermann [4]. We will discuss in some detail the two main approaches, (a) linear algebraic variational approaches using Jacobi coordinates (Section 2) and (b) propagation methods using hyperspherical coordinates (Section 3). Recent developments in computer implementations and a few applications will be briefly mentioned in the flnal sections. 

Linear algebraic variational approach with Jacobi coordinates versus propagation methods with hyperspherical coordinates... 

Variational methods - theoretically the variational approach offers the most powerful procedure for the generation of a computational grid subject to a multiplicity of constraints such as smoothness, uniformity, adaptivity, etc. which cannot be achieved using the simpler algebraic or differential techniques. However, the development of practical variational mesh generation techniques is complicated and a universally applicable procedure is not yet available. 

This variation (electrical anharmonicity) can be taken into account, within the algebraic approach, by expanding the operator as... 

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... 

For simple geometries-, the variation of b from, element to element can be approximated by a simple mathematical expression and the total intensity found by integration. Otherwisie the easiest approach is to measure the actual thickness of shield. material traversed along a straight line from each source element and add algebraically the separate intensity contributions. The total intensity should be multiplied by a fi factor to allow for tbe forward scattering in the shield. This is usually about a factor of 10 for thick shields. 

The extension of the approach to constrained multibody systems and differential-algebraic equations affects the formulation of the multiple shooting method and the computation of sensitivity matrices. The former requires a more sophisticated treatment because variations of initial values and parameters may no longer be consistent with the algebraic equations. The latter can be done efficiently by exploiting the fact that the number of degrees of freedom of the system is reduced due to the presence of constraints. 

In the Real-Time (time-domain dynamics) approach, the same set of algebraic and differential equations are encountered as in the frequency domain. However, the major advantage of solving these equations in real time is the ability to observe the interactions of the process, control scheme and load variables much as the operator of a plant observes the behaviour of an actual plant. Dynamic simulation allows for the comparison of several candidate control strategies and assesses the propagation of variation through a process/plant. In other words, dynamic simulation allows for the evaluation of plant-wide versus single-loop control schemes. 

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