Mathematical and Numerical Methods for Complex Quantum Systems

Fractional Perspectives of DFT and Exchange-Correlation Energies/Electronic Excitation Energies from Pairing Matrix Fluctuations

Weitao Yang

Duke University

Density functional theory of electronic structure is widely and successfully applied in simulations throughout engineering and sciences. However, there are major failures for many predicted properties. These errors can be characterized and understood through the perspective of fractional charges and fractional spins. The fractional perspectives offer a possible pathway forward. Recently, we developed an adiabatic connection to formulate the ground-state exchange-correlation energy in terms of pairing matrix linear fluctuations. This formulation of the exchange-correlation energy opens new a channel for density functional approximations based on the many-body perturbation theory. We illustrate the potential of such approaches with an approximation based on the particle-particle Random Phase Approximation (pp-RPA). This resulting method has many highly desirable properties. It has minimal delocalization error with a nearly linear energy behavior for systems with fractional charges, describes van der Waals interactions similarly and thermodynamic properties significantly better than the conventional RPA, and eliminates static correlation error for single bond systems. Most significantly, it is the first known functional with an explicit and closed-form dependence on the occupied and unoccupied orbitals, which captures the energy derivative discontinuity in strongly correlated systems. We also adopted pp-RPA and the particle-particle Tamm-Dancoff approximation (pp-TDA) to approximate the pairing matrix fluctuation and then determine excitation energies by the differences of two-electron addition/removal energies. This approach captures all types of interesting excitations: single and double excitations are described accurately, Rydberg excitations are in good agreement with experimental data and CT excitations display correct 1/R dependence. Furthermore, the pp-RPA and the pp-TDA have a computational cost similar to TDDFT and consequently are promising for practical calculations.