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 v~(r) 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 vXC(r) defined on the infinite-dimensional Hilbert space, the potentials v~XC(r) reconstructed from matrices of vXC(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 v~XC(r) and vXC(r) improves to the extent that the basis function products are appropriate as a basis for vXC(r). These findings validate the LIP technology as a rigorous potential reconstruction method.

Multiplicative potentials for kinetic energy and exact exchange.

Harriman showed that within finite basis sets of one-electron functions that form linearly independent products (LIP), differential and integral operators can be represented exactly and unambiguously by multiplicative (local) potentials. Although almost no standard basis sets of quantum chemistry form LIPs in a numerical sense, occupied self-consistent field (SCF) orbitals routinely do so. Using minimal LIP basis sets of occupied SCF orbitals, we construct multiplicative potentials for electronic kinetic energy and exact exchange that reproduce the Hartree-Fock and Kohn-Sham Hamiltonian matrices and electron densities for atoms and molecules. The results highlight fundamental differences between local and nonlocal operators and suggest a practical possibility of developing exact kinetic energy functionals within finite basis sets by using effective local potentials.

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.

QSARs for phenols and phenolates: oxidation potential as a predictor of reaction rate constants with photochemically produced oxidants.

Quantitative structure-activity relationships (QSARs) for prediction of the reaction rate constants of phenols and phenolates with three photochemically produced oxidants, singlet oxygen, carbonate radical, and triplet excited state sensitizers/organic matter, are developed. The predictive variable is the one-electron oxidation potential (E1), which is calculated for each species using density functional theory. The reaction rate constants are obtained from the literature, and for singlet oxygen, are augmented with new experimental data. Calculated E1 values have a mean unsigned error compared to literature values of 0.04-0.06 V. For singlet oxygen, a single linear QSAR that includes both phenols and phenolates is developed that predicts experimental rate constants, on average, to within a factor of three. Predictions for only 6 out of 87 compounds are off by more than a factor of 10. A more limited data set for carbonate radical reactions with phenols and phenolates also gives a single linear QSAR with prediction of rate constant being accurate to within a factor of three. The data for the reactions of phenols with triplet state sensitizers demonstrate that two sensitizers, 2-acetonaphthone and methylene blue, most closely predict the reactivity trend of triplet excited state organic matter with phenols. Using sensitizers with stronger reduction potentials could lead to overestimation of rate constants and thus underestimation of phenolic pollutant persistence.