Recursive Linear Tensor Expansion with Natural Occupation Analysis.

We introduce an innovative recursive tensor decomposition method that draws inspiration from quantum chemical theories. This approach integrates ideas such as natural occupation numbers and natural basis, much like natural orbitals, and employs truncations that parallel the excitation-level truncations in the linear expansions of configuration interaction theory. The framework features recursive algorithms that combine linear expansion with natural basis transformations at each step, ensuring convergence to the original tensor. Consequently, a numerical technique is developed that reconstructs the initial tensor with precision within a predetermined tolerance, using only subtensors of limited dimension and a series of matrix transformations. An initial Python implementation has been created for the 3D tensor scenario where 3D tensors are decomposed to be represented using vectors and matrices alone. We illustrate the behavior of the final Recursive Linear Tensor Expansion in Natural basis algorithm in processing random data sets, experimental data sets from diverse sources with both real and complex tensors, and data sets representing both time-independent and time-dependent anharmonic vibrational wave functions of water. Finally, the systematic accuracy control is illustrated for density fitting two-electron repulsion integrals and tested for the second-order correlation energy of molecular nitrogen and benzene.