Reconstruction of Exchange–Correlation Potentials from Their Matrix Representations

Within a basis set of one-electron functions that form linearly independent products (LIPs), it is always possible to construct a unique local (multiplicative) real-space potential that is precisely equivalent to an arbitrary given operator. Although standard basis sets of quantum chemistry rarely form LIPs in a numerical sense, occupied and low-lying virtual canonical Kohn–Sham orbitals often do so, at least for small atoms and molecules. Using these principles, we construct atomic and molecular exchange–correlation potentials from their matrix representations in LIP basis sets of occupied canonical Kohn–Sham orbitals. The reconstructions are found to imitate the original potentials in a consistent but exaggerated way. Since the original and reconstructed potentials produce the same ground-state electron density and energy within the associated LIP basis set, the procedure may be regarded as a rigorous solution to the Kohn–Sham inversion problem within the subspace spanned by the occupied Kohn–Sham orbitals.