An energy-minimizing mesh for the Schrödinger equation
A variational formulation is used to derive the mesh density that is optimal in the sense of minimizing the error in energy for a given number of degrees of freedom in a finite difference or finite element approximation. In d dimensions it is given by \[ \rho_{opt}(r)\sim (\sum_{| \beta | =N+1}| \frac{\partial^{| \beta |}\psi_ 0(r)}{\partial^{\beta_ 1}...\partial^{\beta_ d}}|^ 2)^{\frac{1}{2N/d+1}},\quad where\quad | \beta | \equiv \sum^{d}_{i=1}\beta_ i \] and \(\psi_ 0(r)\) is the exact normalized wavefunction being sought. N is the order of polynomial completeness to which the approximate wavefunction is assumed to have an expression in terms of a finite element basis, and \(\psi_ 0(r)\) is assumed to be differentiable through at least order N except at isolated points (i.e. Coulomb singularities). In 1 dimension \[ \rho_{opt}(x)\sim | \frac{\partial^{N+1}\psi_ 0(x)}{\partial x^{N+1}}|^{\frac{1}{N+1/2}}. \] This result compares well with the empirically optimized mesh used for the ground state of the hydrogen radial equation by \textit{F. Hermann} and \textit{S. Skillman}, Atomic structure calculations. Prentice-Hall, Englewood Cliffs, NJ, (1963). Finite element computations for a \(\delta\)-function potential show sharp minima in excess energy at the predicted optimal mesh spacings, and experiments with a square-well potential also support the theory. The positioning of mesh points in relation to potential singularities and dependence on the number of degrees of freedom of the approximating basis are studied numerically and agree with the theory. The singularity at the nucleus poses a serious problem in 3 dimensions because it cannot easily be approximated by polynomial functions of cartesian coordinates, and a uniform mesh spacing seriously reduces the convergence rate. The appropriate mesh density for the hydrogen ground state is \[ \rho_{opt}(r)\sim (e^{-2r}\sum^{2N}_{k=0}c_ kr^{- k})^{-\frac{3}{2N+3}}, \] where values for \(c_ k\) are tabulated. The singularity at \(r=0\) is integrable.
scientific article; zbMATH DE number 4131602
scientific article