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Discrete variable representation (DVR) basis sets on the radial half-line, based on Bessel functions, are presented. These are Hankel transforms of the eigenfunctions of the particle in a spherical box in k space, but there is no box or bound on the radial variable r. The grid points extend to infinity on the r axis. The DVR functions are exactly orthonormal and exactly satisfy the interpolation properties usually associated with DVR functions. The exact matrix elements of the kinetic energy are computed, and the use of the Bessel DVR functions in radial eigenvalue problems is illustrated. The phase space or semiclassical interpretation of the Bessel DVR functions is presented, and variations on these functions, corresponding to alternative boundary conditions in k space, are discussed. An interesting feature of Bessel DVR functions is that they are based on a finite basis representation that is continuously infinite. [ABSTRACT FROM AUTHOR]

This paper explores the difficulties of constructing multidimensional discrete variable representation (DVR) basis sets and the strategies that can be used to overcome them. A parameter count shows that the conditions on a DVR basis set cannot be satisfied on most spaces of wave functions. One-dimensional, orthogonal polynomials are an exception, but the Y[sub lm]'s on the sphere only go 3/4 of the way, in a certain sense, toward supplying enough parameters to satisfy the DVR conditions. It is shown that DVR sets involving rotationally invariant subspaces of wave functions on the sphere (consisting of complete subshells only) exist only for small values of the angular momentum cutoff. However, an exploration of DVR sets invariant under subgroups of the full rotation group leads to the discovery of a 12-point DVR set that is invariant under the tetrahedral group, whose grid points are the vertices of an icosahedron. [ABSTRACT FROM AUTHOR]

A framework for discrete variable representation (DVR) basis sets is developed that is suitable for multidimensional generalizations. Those generalizations will be presented in future publications. The new axiomatization of the DVR construction places projection operators in a central role and integrates semiclassical and phase space concepts into the basic framework. Rates of convergence of basis set expansions are emphasized, and it is shown that the DVR method gives exponential convergence, assuming conditions of analyticity and boundary conditions are met. A discussion of nonorthogonal generalizations of DVR functions is presented, in which it is shown that projected δ-functions and interpolating functions form a biorthogonal basis. It is also shown that one of the generalized DVR proposals due to Szalay [J. Chem. Phys. 105, 6940 (1996)] gives exponential convergence.© 2002 American Institute of Physics. [ABSTRACT FROM AUTHOR]