m p - ph ] 1 5 Ju n 20 05 1 An efficient numerical quadrature for the calculation of the potential energy of wavefunctions expressed in the Daubechies wavelet basis

An efficient numerical quadrature for the calculation of the potential energy of wavefunctions expressed in the Daubechies wavelet basis. Abstract. An efficient numerical quadrature is proposed for the approximate calculation of the potential energy in the context of pseudo potential electronic structure calculations with Daubechies wavelet and scaling function basis sets. Our quadrature is also applicable in the case of adaptive spatial resolution. Our theoretical error estimates are confirmed by numerical test calculations of the ground state energy and wave function of the harmonic oscillator in one dimension with and without adaptive resolution. As a byproduct we derive a filter, which, upon application on the scaling function coefficients of a smooth function, renders the approximate grid values of this function. This also allows for a fast calculation of the charge density from the wave function. Gaussians and plane waves are at present the most popular basis sets for density functional electronic structure calculations. Wavelets are a promising new basis set that combines most of the theoretical advantages of these two basis sets. In particular the Daubechies wavelets [1] have a larger number of interesting properties than any other basis set. They form a systematic orthogonal basis set that allows for adaptivity. The basis functions are localized both in real (compact support) and in Fourier space. The first attempts to use wavelets in the electronic structure calculations appeared more than 10 years ago. The first papers we are aware of used the Mexican hat wavelet [2], [3] and the Meyer wavelet [4]. However, these wavelet families were soon abandoned because they do not have compact support. Because of the above mentioned advantages Daubechies wavelets were then investigated in a series of publications [5]-[8]. To use them in the variational Galerkin method for the Schroedinger