The mixed regularity of electronic wave functions in fractional order and weighted Sobolev spaces

Abstract

We continue the study of the regularity of electronic wave functions in Hilbert spaces of mixed derivatives. It is shown that the eigenfunctions of electronic Schrödinger operators and their exponentially weighted counterparts possess, roughly speaking, square integrable mixed weak derivatives of fractional order ϑ for ϑ < 3/4. The bound 3/4 is best possible and can neither be reached nor surpassed. Such results are important for the study of sparse grid-like expansions of the wave functions and show that their asymptotic convergence rate measured in terms of the number of ansatz functions involved does not deteriorate with the number of electrons. Mathematics Subject Classification (2000)35J10 · 35B65 · 41A25 · 41A63

Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation

“…Remark 3.4. As to be expected in view of Proposition 2.3, the proof of Theorem 3.1 does not carry over to the adjoint problem (2.5), but following the same reasoning as in [37], one obtains that the solution of the adjoint problem is in H s,1 (R 3 ; 2) for s < 3 4 , i.e., has the same regularity as the eigenfunctions of the unmodified bilinear form a.…”

“…For u, it is shown in [48] by similar arguments as adapted here that u ∈ H 1/2,1 (R 3 ; 2); in [37], this is improved to the sharp result u ∈ H s,1 (R 3 ; 2) for s < 3 4 . On the spaces H 1,k (R 3 ; 2), k = 0, 1, which can be identified with closures of D(R 6 ) under the norms defined in (3.1), we make use of the equivalent norms and seminorms…”

“…For comparison, we can apply Theorem 4.1 to a direct hyperbolic wavelet discretization of the standard formulation (3.2): it has been shown in [48] that the corresponding eigenfunctions are in H 1/2,1 (R 3 ; 2), and one obtains a convergence rate of almost (#Λ) −1/6 . This regularity result has recently been sharpened to H s,1 (R 3 ; 2) for s < 3 4 in [37], corresponding to almost (#Λ) −1/4 . The regularity result for the explicitly correlated formulation cannot be expected to be sharp.…”

“…In the regularity results for n-electron eigenfunctions of (1.1) collected in [49], and the recent improvement in [37], the obstacle for higher regularity -and thus for a further reduction of approximation complexity -lies in the electron-electron cusp. In the present work, we use an explicitly correlated ansatz to arrive at a modified problem where the electron-electron singularities are eliminated.…”

On the approximation of electronic wavefunctions by anisotropic Gauss and Gauss–Hermite functions

“…Basically we show that electronic wavefunctions can be approximated with arbitrary order in the number of the involved terms by linear combinations of such Gauss functions. This behavior is in contrast to approximation results that are based on the mixed regularity of the wavefunctions; see [19,20,21] and [13]. These regularity properties together with the antisymmetry of the wavefunctions enforced by the Pauli principle suffice to construct approximations by linear combinations of Slater determinants composed of a fixed set of basis functions that converge with an order in the number of the involved terms that does not deteriorate with the number of electrons.…”