Development and Application of Scaling Correction Methods in Density Functional Theory

Density functional theory (DFT) has become the main working horse for performing electronic structure calculations for chemical and physical systems nowadays. The theory is exact in principle, however, a density functional approximation (DFA) to the unknown exchange-correction energy $E_{\rm{xc}}$ in DFT has to be used in practice. Conventional DFAs have gained much success, while they usually possess intrinsic error and fail to describe some critical physical properties. In this dissertation, we focus on the delocalization error, a key concept to understand the systematic error existing in conventional DFAs, and we present the scaling correction methods which are designed to systematically and effectively reduce the delocalization error. The applications and improvements of two recently developed scaling correction methods, namely the the global scaling correction (GSC) method and the localized orbital scaling correction (LOSC) method, are mainly discussed in this dissertation. First, we demonstrate that the scaling corrections is capable of accurately predicting the quasiparticle energies and photoemission spectra from the orbital energies of the (generalized) Kohn-Sham DFT calculations. Second, we present a new method called QE-DFT, which is developed based on the connection between the orbital energies and the quasiparticle energies, to describe the difficult excited-state problems, including the low-lying, Rydberg and charge transfer excitations and conical intersections. We further derive the analytical gradients of the QE-DFT method, and demonstrate the application of QE-DFT for describing the potential energy surface and geometry optimization of excited states. Third, we show the application of LOSC to describe the polymer polarizability, which is a challenging problem for conventional DFAs. Fourth, we present the recent development for both GSC and LOSC methods. Specifically, we developed the analytic and exact second-order correction under the framework of GSC and achieved much improved accuracy compared to the original work of GSC. We also developed a new and robust self-consistent approach for LOSC method to avoid the convergence difficulties in the original LOSC work, which comes from using the approximate LOSC effective Hamiltonian. Finally, we developed the implementation of the scaling correction methods as a library with the supports to multiple programming languages. In summary, we demonstrated with extensive results that the GSC and LOSC are powerful and effective scaling correction methods to conventional DFAs to largely reduce the delocalization error. With the further developments to GSC and LOSC, they should be of great potential for broad application to describing challenging electronic structure problems of complex systems with high accuracy in the future.