Integral-transform trial functions: a numerical study of generalized scaling, iterated transforms, and the scale space integration

Two conceptually important and computationally useful aspects of the integral-transform (IT) approach have been tested on the He sequence. Both generalized scaling and the iteration approach enable one to optimize the shape function; a priori discretization of the IT trial functions by M-point integration rules is computationally expedient; coupled with extrapolation procedures it can alleviate the integration difficulties inherent in the IT approach. By optimizing a mapping (generalized scaling) the initial choice of a shape function becomes less crucial; for discretized IT functions it results in the reduction of the number of simultaneous parameter optimizations. The iteration approach transforms the nonlinear parameter optimization into a multistage process, with its attendant advantages. It also minimizes the importance of the initial choice of the cubature rule (discretization). A judicious blend of mapping and iteration, coupled with a priori discretization and subsequent extrapolation seems to be the way to combine the conceptual advantages of the IT approach with computational feasibility. The IT orbitals were of the form φ1(r)= ∫0∞ dt tνexp(−q t) exp(−tλr2), they were discretized by Gauss-Laguerre-type M-point integration rules, and extrapolation studies indicated that as M → ∞, λ(∞)≈−1.5 and E(∞)≈EH.F. for He.