Density functional theory (DFT) is undoubtedly one of the most popular methods for electronic structure calculations. Experimentalists as well as theoreticians have been using DFT calculations to support their experimental results for various chemical applications such as elucidating chemical reactions, designing organic and inorganic materials, and identifying spectroscopic data. Theoretically, the exchange-correlation functional determines the accuracy of DFT. Since the exact exchange-correlation functional is unknown, approximate functionals have been used such as the local density approximation (LDA), the generalized gradient approximation (GGA), and the hybrid functionals. As a result, the same accuracy is not guaranteed by using the same functional to different systems, and the accuracy has to be tested depending on the properties concerned. In the field of physics mainly dealing with solids, the local functionals like LDA and GGA are quite enough to obtain accurate results. In contrast, these functionals are insufficient to chemists interested in molecular systems, since the variation of electron densities are more rapid than solids.
The emergence of hybrid functionals attracted great attention in chemistry. The hybrid functionals mix the Hartree-Fock (HF) exact exchange energy and the DFT exchange-correlation energy with a certain ratio, and depending on the way of mixing, they are further categorized into global-hybrid, double-hybrid, range-separated, and local-hybrid functionals. Compared to the local DFT functionals, the conventional hybrid functionals use the nonlocal HF exchange potential, so they get out of the standard Kohn-Sham (KS) framework, and are formally handled by the generalized KS scheme.
The local exact exchange potential with the optimized effective potential (OEP) method can be introduced to the hybrid functionals. In this way, the original KS formalism can be maintained. Moreover, it causes significant difference compared to the HF hybrid functionals--the meaning of unoccupied orbitals changes. The HF method yields unoccupied orbitals of (N+1)-electron systems, while unoccupied orbitals from the OEP exact exchange potential are N-electron orbitals. As a result, the unoccupied orbitals of the HF hybrid functionals are the mixing of N- and (N+1)-electron orbitals, whereas those of the OEP hybrid functionals are still N-electron ones.
Practically, basis functions are also important for accurate calculations. Historically, atom-centered basis sets and plane wave (PW) basis sets have been widely used in chemistry and physics, respectively. The atom-centered basis sets such as Gaussian- and Slater-type basis sets employ predefined parameters and systems can be depicted by small number of basis functions, but the proper choice of basis sets is necessary to guarantee the accuracy of calculations. The PW basis sets describe systems in the reciprocal space. They use larger number of basis functions compared to the atom-centered basis sets, but the accuracy can be easily controlled by a single parameter. Apart from the above basis sets, there are emerging basis sets in computational chemistry, called real-space grid-based basis sets, which can be further categorized into the finite difference method, the finite element method, and the localized basis representation method. The real-space grid-based basis sets are similar to the PW; they employ large number of basis functions and the accuracy can be easily handled. However, in contrast with the PW, the boundary condition can be changed and both periodic and isolated systems can be treated.
In this dissertation, the implementation of an electronic structure calculation program with Lagrange-sinc (LS) functions, which is one of the real-space grid-based methods, was examined, and the numerical accuracy of the LS functions was investigated. In addition, the difference between the local and the nonlocal exact exchange potentials was investigated using the LS basis set. The accuracy of the LS basis set was verified by comparing atomization energies, ionization energies, electron affinities, and static polarizabilities to Gaussian-type basis sets. The difference between the local and the nonlocal exact exchange potentials was investigated by comparing excitation energies and characteristics of unoccupied orbitals of molecules, and the local exact exchange potential was more effective than the HF one.