Fisher Information-Based Optimization of Mapped Fourier Grid Methods

The mapped Fourier grid method (mapped-FGM) is a simple and efficient discrete variable representation (DVR) numerical technique for solving atomic radial Schr & ouml;dinger differential equations. It is set up on equidistant grid points, and the mapping, a suitable coordinate transformation to the radial variable, deals with the potential energy peculiarities that are incompatible with constant step grids. For a given constrained number of grid points, classical phase space and semiclassical arguments help in selecting the mapping function and the maximum radial extension, while the energy does not generally exhibit a variational extremization trend. In this work, optimal computational parameters and mapping quality are alternatively assessed using the extremization of (coordinate and momentum) Fisher information. A benchmark system (hydrogen atom) is employed, where energy eigenvalues and Fisher information are traced in a standard convergence procedure. High-precision energy eigenvalues exhibit a correlation with the extrema of Fisher information measures. Highly efficient mapping schemes (sometimes classically counterintuitive) also stand out with these measures. Same trends are evidenced in the solution of Dalgarno-Lewis equations, i.e., inhomogeneous counterparts of the radial Schr & ouml;dinger equation occurring in perturbation theory. A detailed analysis of the results, implications on more complex single valence electron Hamiltonians, and future extensions are also included.