Tensor numerical methods in quantum chemistry

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Conventional numerical methods for multidimensional problems face the "curse of dimensionality," which high-performance computing cannot fully mitigate. Novel tensor numerical methods utilize a "smart" rank-structured tensor representation of multivariate functions and operators on Cartesian grids, transforming the solution of multidimensional integral-differential equations into 1D calculations. This work explains basic tensor formats and algorithms, highlighting the revolutionary impact of orthogonal Tucker tensor decomposition from chemometrics on numerical analysis, supported by rigorous approximation theory. The advantages of the tensor approach are illustrated through ab-initio electronic structure calculations. For instance, computing 3D convolution integrals for functions with multiple singularities is simplified to a series of 1D operations, allowing accurate MATLAB calculations on laptops with 3D uniform tensor grids up to size 10^15. Additionally, a fast tensor-based Hartree-Fock solver, which incorporates low-rank factorization of two-electron integrals, lays the groundwork for efficient excitation energy calculations of molecules. The tensor approach also facilitates effective grid-based numerical treatment of long-range electrostatic potentials on large 3D finite lattices with defects. The innovative range-separated tensor format applies to interaction potentials in multi-particle systems, paving the way for new