An application of double exponential formula to radial quadrature grid in density functional calculation

We report an application of the double exponential formula to the numerical integration of the radial electron distribution function for atomic and diatomic molecular systems with a quadrature grid. Three types of mapping transformation in the double exponential formula are introduced into the radial quadrature scheme to generate new radial grids. The double exponential grids are examined for the electron-counting integrals of He, Ne, Ar, and Kr atoms which include occupied orbitals up to the 4p shell. The performance of radial grid is compared for the double exponential formula and the formulas proposed in earlier studies. We mainly focus our attention on the behavior of accuracy by the quadrature estimation for each radial grid with varying the mapping parameter and the number of grid points. The convergence behavior of the radial grids with high accuracy for atomic system are also examined for the electron-counting integrals of LiH, NaH, KH, Li(2), Na(2), K(2), HF, HCl, HBr, F(2), Cl(2), Br(2), LiF, NaCl, KBr, [ScH](+), [MnH](+), and [CuH](+) molecules. The results reveal that fast convergence of the integrated values to the exact value is achieved by applying the double exponential formula. It is demonstrated that the double exponential grids show similar or higher accuracies than the other grids particularly for the Kr atom, Br(2) molecule, alkali metal hydrides, alkali metal halogenides, and transition metal hydride cations, suggesting that the double exponential transformations have potential ability to improve the reliability and efficiency of the numerical integration for energy functionals.