Local Potentials Reconstructed within Linearly Independent Product Basis Sets of Increasing Size

Given a matrix representation of a local potential v(r) within a one-electron basis set of functions that form linearly independent products (LIP), it is possible to construct a well-defined local potential $$ that is equivalent to v(r) within that basis set and has the form of an expansion in basis function products. Recently, we showed that for exchange-correlation potentials v_XC(r) defined on the infinite-dimensional Hilbert space, the potentials $$ reconstructed from matrices of v_XC(r) within minimal LIP basis sets of occupied Kohn-Sham orbitals bear only qualitative resemblance to the originals. Here, we show that if the LIP basis set is enlarged by including low-lying virtual Kohn-Sham orbitals, the agreement between $$ and v_XC(r) improves to the extent that the basis function products are appropriate as a basis for v_XC(r). These findings validate the LIP technology as a rigorous potential reconstruction method.