Continuous approximation for linear combination coefficients: Exploring a neglected concept

An alternative approach is introduced for accurately describe atomic orbitals using an integral transform of Gaussian-type basis functions (GTO) combined with continuous and analytical weight functions. The functional form of these weight functions was optimized using the variational criterion, achieving μHartree-level precision in energy calculations for atoms, monovalent ions up to the third period of the periodic table, and the H₂ molecule. However, errors increased for heavier atoms, such as sulfur and chlorine, indicating the need for further refinement of the weight functions. This approach enhances the accuracy of electronic structure calculations, offering a rigorous alternative for deriving one-electron wave functions from discrete basis sets and vice-versa for atoms and molecules.