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This paper investigates the augmented plane wave methods which are widely used in full-potential electronic structure calculations. These methods introduce basis functions that describe different regions using different discretization schemes. We construct a nonconforming method based on this idea and present an a priori error analysis for both linear Schrodinger type equations and nonlinear Kohn−Sham equations. Some numerical experiments are presented to support our theory.

The Coupled Cluster (CC) method is a widely used and highly successful high precision method for the solution of the stationary electronic Schrodinger equation, with its practical convergence properties being similar to that of a corresponding Galerkin (CI) scheme. This behaviour has for the discrete CC method been analyzed with respect to the discrete Galerkin solution (the "full-CI-limit") in (Schneider, 2009). Recently, we globalized the CC formulation to the full continuous space, giving a root equation for an infinite dimensional, nonlinear Coupled Cluster operator that is equivalent the full electronic Schrodinger equation (Rohwedder, 2011). In this paper, we combine both approaches to prove existence and uniqueness results, quasi-optimality estimates and energy estimates for the CC method with respect to the solution of the full, original Schrodinger equation. The main property used is a local strong monotonicity result for the Coupled Cluster function, and we give two characterizations for situations in which this property holds.

The Hartree-Fock equation is widely accepted as the basic model of electronic structure calculation which serves as a canonical starting point for more sophisticated many-particle models. We have studied the s ∗ -compressibility for Galerkin discretizations of the Hartree-Fock equation in wavelet bases. Our focus is on the compression of Galerkin matrices from nuclear Coulomb potentials and nonlinear terms in the Fock operator which hitherto has not been discussed in the literature. It can be shown that the s ∗ -compressibility is in accordance with convergence rates obtained from best N -term approximation for solutions of the Hartree-Fock equation. This is a necessary requirement in order to achieve numerical solutions for these equations with optimal complexity using the recently developed adaptive wavelet algorithms of Cohen, Dahmen and DeVore.

We present a novel application of best N -term approximation theory in the framework of electronic structure calculations. The paper focusses on the description of electron correlations within a Jastrow-type ansatz for the wavefunction. As a starting point we discuss certain natural assumptions on the asymptotic behaviour of two-particle correlation functions near electron-electron and electron-nuclear cusps. Based on Nitsche's characterization of best N -term approximation spaces , we prove that for q>1 and with respect to a certain class of anisotropic wavelet tensor product bases. Computational arguments are given in favour of this specific class compared to other possible tensor product bases. Finally, we compare the approximation properties of wavelet bases with standard Gaussian-type basis sets frequently used in quantum chemistry.

We discuss best N -term approximation spaces for one-electron wavefunctions and reduced density matrices ρ emerging from Hartree-Fock and density functional theory. The approximation spaces for anisotropic wavelet tensor product bases have been recently characterized by Nitsche in terms of tensor product Besov spaces. We have used the norm equivalence of these spaces to weighted spaces of wavelet coefficients to proof that both and ρ are in for all with . Our proof is based on the assumption that the possess an asymptotic smoothness property at the electron-nuclear cusps.