Integration of products of Gaussians and wavelets with applications to electronic structure calculations (Q2870614)

The author considers the numerical evaluation of integrals of the form: NEWLINE\[NEWLINE m_{\alpha,\nu\mu}^{(2)} := \int_{\mathbb{R}^2} e^{-\alpha(x_1-x_2)^2}(\psi_{\nu_1}\otimes\psi_{\nu_2})(\psi_{\mu_1}\otimes\psi_{\mu_2})dx NEWLINE\]NEWLINE for Daubechies wavelets \(\psi_\nu,\nu\in\nabla\), which by the Plancherel theorem is equivalent to NEWLINE\[NEWLINE m_{\alpha,\nu\mu}^{(2)} =\sqrt{\frac{\pi}{\alpha}} \int_{\mathbb{R}} e^{-\xi^2/(4\alpha)}\overline{\mathcal{F}(\psi_{\nu_1}\psi_{\mu_1})(\xi)}\mathcal{F}(\psi_{\nu_2}\psi_{\mu_2})(\xi)d\xi=:\int_{\mathbb{R}} u(\xi) d\xi. NEWLINE\]NEWLINE For a given step size \(h\), the author first analyzes the approximation error of \(m_{\alpha,\nu\mu}^{(2)}\) by discretizing the integral of the form \(\int u(\xi)d\xi \) to \(h\sum_{k\in\mathbb{Z}} u(kh)\). The author gives the dependence of \(h\) in terms of \(\alpha\) and the required error \(\delta\). Then, the approximation of \(h\sum_{k\in\mathbb{Z}} u(kh)\) by finitely many terms \(h\sum_{|k|<N_h} u(kh)\) is analyzed. With the error analysis, the author then proceeds to the estimation of \(u(\xi)\), which involves computation of the Fourier transform of wavelets. Using the properties of wavelets, such as vanishing moments, refinement equation, etc., the computation of \(u(\xi)\) can be boiled down to the computation only involving with scaling function \(\varphi\). Numerical experiments are given showing the advantage of the proposed scheme compared to a reference scheme by a triple product using biorthogonal wavelets. The numerical experiments show that the proposed scheme is efficient, accurate, and far less storage demanding.