The electronic structure calculation of atoms is one of most fundamental bases in not only
understanding electronic structures of molecules and solids, but also developing efficient and
accurate electronic structure methods. For the latter case, a highly accurate method
should be used to distinguish the intrinsic error produced by the theoretical framework
itself from that caused by the other numerical problems such as incompleteness of basis set
and inaccurate numerical integration. As a such method, we developed an accurate finite element
method for atomic calculations based on density functional theory (DFT) within local density
approximation (LDA) and Hartree-Fock (HF) method.
The radial wave functions are expanded by cubic Hermite spline functions on a uniform mesh for
x=√ r
,
and all the associated integrals are analytically evaluated in conjunction with fitting procedures of
the hartree and the exchange-correlation potentials to the same cubic Hermite spline functions
using a set of recurrence formulas. The total energy of atoms systematically converges from above,
and the error algebraically decays as the mesh spacing decreases.
When the mesh spacing is taken to be
0.025/√ Z
bohr, the total energy for an atom of atomic number Z can be calculated within error of
10^{-7}
hartree for both the LDA and HF methods.
The equal applicability of the method to DFT and the HF method with a similar degree of high accuracy
enables the method to be a reliable platform for development of new functionals in DFT such as
hybrid functionals.

- "Accurate finite element method for atomic calculations based on density functional theory and Hartree-Fock method", T. Ozaki and M. Toyoda, Comp. Phys. Comm. 182, 1245 (2011).
- "Exchange functional by a range-separated exchange hole", M. Toyoda and T. Ozaki, Phys. Rev. A 83, 032515 (2011).