Finite Element Approximations to the Discrete Spectrum of the Schrödinger Operator with the Coulomb Potential

Abstract

In the present paper, the authors consider the Schrödinger operator H with the Coulomb potential defined in R 3m , where m is a positive integer. Both bounded domain approximations to multielectron systems and finite element approximations to the helium system are analyzed. The spectrum of H becomes completely discrete when confined to bounded domains. The error estimate of the bounded domain approximation to the discrete spectrum of H is obtained. Since numerical solution is difficult for a higher-dimensional problem of dimension more than three, the finite element analyses in this paper are restricted to the S-state of the helium atom. The authors transform the six-dimensional Schrödinger equation of the helium S-state into a three-dimensional form. Optimal error estimates for the finite element approximation to the three-dimensional equation, for all eigenvalues and eigenfunctions of the three-dimensional equation, are obtained by means of local regularization. Numerical results are shown in the last section. 1. Introduction. The multielectron Coulomb problem in quantum mechanics cannot be solved in a finite form. Nevertheless it challenges and stimulates many mathematicians and physicists to devote themselves to developing efficient methods for solving the system. Several successful approximation techniques in quantum physics/chemistry have been developed for this problem. They include the Hartree-Fock method [15], the finite difference method [19], [35], the correlation-function hyperspherical-harmonic method [18], [24], and various variational approximations. For the Hartree-Fock method, every electron is considered independently to be in a central electric field formed by the nucleus and other electrons. The finite difference method needs a rectangular domain in R N and uniform grids. The double and triple basis set methods (which are variational methods indeed) are very powerful for the eigenvalue problem of the helium atom. Kono and Hattori (see [21], [22]) used two sets of basis functions r i 1 r j 2 r k 12 e −ξr1−ηr2 A ("ξ terms") and r i 1 r j 2 r k 12 e −ζ(r1+r2) A ("ζ terms") to calculate the energy levels for the S, P , and D states of the helium atom. (A is an appropriate angular factor.) The former set of functions is expected to describe the whole wave function roughly, while the latter is expected to describe the short-and middle-range correlation effects. Their calculations yield 9-10 significant digits for S states. Kleindienst, Lüchow, and Merckens [20] and Drake and Yan [12] applied the double basis set method to S-states of helium. Their basis functions are r i 1 r j 2 r k 12 e −ξ1r1−η1r2 and r i 1 r j 2 r k 12 e −ξ2r1−η2r2 . Drake and Yan employed truncations to ensure numerical stability and convergence. By complete optimization of the exponential scale factors ξ 1 ,

Pseudospectral calculation of the wave function of helium and the negative hydrogen ion

“…Finite difference methods (FDM) [45][46][47][48] and Finite element methods (FEM) [49][50][51][52][53][54] represent the solution and the differential equation on a discrete grid. The FDM grid is usually evenly spaced with derivatives calculated to some small (usually second) order.…”

“…FEM uses subdomains, concentrating grid points where more accuracy is needed. Recent work achieves as many as seven decimal places in the energy of the ground state but produces surprisingly nonsmooth wave functions [53,54]. The rate of convergence of these methods is limited by the order of the representation of derivatives and is always algebraic with some small index dependent on the order used for derivatives.…”

Application of the virial theorem for improving eigenvalue calculations of multiparticle systems

“…The abstract spectral approximation of variationally formulated eigenvalue problems can be found in [4,Section 8]. For the Schrödinger operator with the Coulomb potential, Zheng and Ying [31] proved a result similar to Theorem 5.1. Even though a general result like Theorem 5.1 is to be expected, but its proof does not seem to be available in the literature, to the best of our knowledge.…”

On Friedrichs–Poincaré-type inequalities

“…These inequalities ensure that the solution is in a more suitable space from a numerical viewpoint than the solution space itself. Most proofs of them in references are by reduction to absurdity [1,3,6]. The method of reduction to absurdity produces an controlling constant depending on the domain implicitly.…”