Uniform electron gas limit of an exact expression for the Kohn–Sham exchange-correlation potential

Previously, we derived an exact formula for the Kohn–Sham exchange-correlation potential corresponding, in the basis-set limit, to the Hartree–Fock electron density of a given system. This formula expresses the potential in terms of the occupied Hartree–Fock and Kohn–Sham orbitals and orbital energies. Here, we show that, when applied to the Hartree–Fock description of a uniform electron gas, the formula correctly reduces to the exchange-only local density approximation

Noninteracting v-Representable Subspaces of Orbitals in the Kohn–Sham Method

The notion of noninteracting v-representability is extended from electron densities to finite-dimensional linear subspaces of orbitals. Unlike electron densities, orbital subspaces can be tested for noninteracting v-representability using a transparent necessary condition: the subspace must be invariant under the action of some one-electron Kohn–Sham Hamiltonian. This condition allows one to determine in principle, and sometimes in practice, whether a given one-electron basis set can represent an N-electron density within the Kohn–Sham method and to find the corresponding Kohn–Sham effective potential v if it exists. If the occupied Kohn–Sham orbitals form linearly independent products, then their subspace is determined by the corresponding ground-state electron density. This means that the Kohn–Sham effective potential corresponding to certain finite-basis-set electron densities can be deduced from the basis set itself

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

Do fractionally incremented nuclear charges improve time-dependent density functional theory excitation energies as reliably as fractional orbital populations?

Gaiduk et al. (Phys Rev Lett 108:253005, 2012) showed that one can improve local, semilocal, and hybrid approximations to the Kohn–Sham effective potentials of atoms and molecules by removing a system-independent fraction of electron charge from the highest occupied molecular orbital (HOMO); if the corrected Kohn–Sham potential is used for adiabatic linear-response time-dependent density functional theory (TDDFT) calculations, accurate Rydberg excitation energies are obtained. One may ask whether the same effect could also be achieved by fractionally increasing the positive charges of the nuclei. We investigate this question and find that a small increase in nuclear charges can indeed substantially reduce errors in TDDFT Rydberg excitation energies. However, the optimal magnitude of the charge increase is system-dependent. In addition, the procedure is ambiguous for molecules, where one has to decide how to distribute the additional charge among individual nuclei. These two drawbacks of the fractional nuclear charge method make it disadvantageous compared to the HOMO depopulation technique

Exact expressions for the Kohn–Sham exchange-correlation potential in terms of wave-function-based quantities

Several workers have deduced various exact expressions for the Kohn–Sham exchange-correlation potential in terms of quantities computable from the interacting and noninteracting wave functions of the system. We show that all these expressions can be obtained by one general method in which the interacting N-electron wave function is expanded in products of one- and (N − 1)-electron functions. Different expressions correspond to different choices of the latter functions. Our analysis unifies and clarifies the previously proposed exact treatments of the exchange-correlation potential, and suggests new ways of expressing this quantity

Asymptotic behavior of the average local ionization energy in finite basis sets

The average local ionization energy (ALIE) has important applications in several areas of electronic structure theory. Theoretically, the ALIE should asymptotically approach the first vertical ionization energy (IE) of the system, as implied by the rate of exponential decay of the electron density; for one-determinantal wavefunctions, this IE is the negative of the highest-occupied orbital energy. In practice, finite-basis-set representations of the ALIE exhibit seemingly irregular and sometimes dramatic deviations from the expected asymptotic behavior. We analyze the long-range behavior of the ALIE in finite basis sets and explain the puzzling observations. The findings have implications for practical calculations of the ALIE, the construction of Kohn–Sham potentials from wavefunctions and electron densities, and basis-set development

What Is the Accuracy Limit of Adiabatic Linear-Response TDDFT Using Exact Exchange–Correlation Potentials and Approximate Kernels?

Calculation of vertical excitation energies by the adiabatic linear-response time-dependent density-functional theory (TDDFT) requires static Kohn–Sham potentials and exchange–correlation kernels. When these quantities are derived from standard density-functional approximations (DFA), mean absolute errors (MAE) of the method are known to range from 0.2 eV to over 1 eV, depending on the functional and type of excitation. We investigate how the performance of TDDFT varies when increasingly accurate exchange–correlation potentials derived from Hartree–Fock (HF) and post-HF wavefunctions are combined with different approximate kernels. The lowest MAEs obtained in this manner for valence excitations are about 0.15–0.2 eV, which appears to be the practical limit of the accuracy of TDDFT that can be achieved by improving the Kohn–Sham potentials alone. These findings are consistent with previous reports on the benefits of accurate exchange–correlation potentials in TDDFT, but provide new insights and afford more definitive conclusions

Structurally Diverse Boron-Nitrogen Heterocycles from an N2O23− Formazanate Ligand

Five new compounds comprised of unprecedented boron-nitrogen heterocycles have been isolated from a single reaction of a potentially tetradentate N2O23− formazanate ligand with BF3­•OEt2 and NEt3. Optimized yields for each product were obtained through variation of experimental conditions and rationalized in terms of relative Gibbs free energies of the products as determined by electronic structure calculations. Chemical reduction of two of these compounds resulted in the formation of a stable anion, radical anion, and diradical dianion. Structural and electronic properties of this new family of redox-active heterocycles were characterized using UV-vis absorption spectroscopy, cyclic voltammetry and X-ray crystallography