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.