Block tensor decomposition: A dual-grid scheme with a formal <i>O</i> ( <i>N</i> 3) scale for THC decomposition of molecular systems

The accurate and efficient treatment of electron-electron interactions remains a central challenge in electronic structure theory. Post-Hartree-Fock (HF) methods are often hindered by high computational costs, primarily due to the need to compute four-index electron repulsion integrals. To address this issue, low-rank approaches, such as tensor hyper-contraction (THC) and interpolative separable density fitting, have been developed to accelerate the computation of HF exchange and dynamic correlation energies in post-HF frameworks. Nevertheless, these methods remain inefficient for molecular systems, mainly because of the quartic-scaling computational cost associated with constructing the THC kernel with respect to the number of basis functions. In this work, we present a new algorithm, named block tensor decomposition (BTD), based on a dual-grid scheme. By integrating Hilbert sorting with pivoted Cholesky decomposition, BTD generates compact and non-redundant interpolative grids, achieving formal O(N3) scaling for kernel building. Key parameters of the method are optimized via differential evolution, ensuring an effective balance between computational efficiency and accuracy. We further demonstrate the application of BTD in scaled opposite-spin second-order Møller-Plesset perturbation theory, where sparse mapping in real space enables O(N2) scaling for both electron correlation and exchange evaluations. Test examples show that BTD is a robust, low-scaling framework for accurate electronic structure calculations in molecular systems.