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Atomic Reference Data
for Electronic Structure Calculations

Radial grids

Suitable choice of a radial grid is key to obtaining accurate numerical solutions of the integro-differential equations of density-functional theory. The codes in our test suite make different choices for the radial grid. Two codes make perhaps the simplest choice, an exponentially increasing grid

eq22 (Eq. 22)

with three parameters: the minimum radius, rmin, the maximum radius, rmax, and the number of intervals, N. The application of the exponential grid to the atomic Schrödinger equation has been discussed by Desclaux [8]. For one code we used N = 15788, rmin = 1/(160 Z), and rmax = 50. (All distances are in atomic units.) Another code used N less than equal 8000, rmin = 10-6/Z, and rmax = 800 Z-1/2; in this case, the energies were extrapolated to n right arrow infinity using an N-2 or N-4 dependence, depending on the quantity in question.

Another code chooses a grid which is nearly linear near the origin, and exponentially increasing at large r,

eq23 (Eq. 23)

which again is determined by three parameters, a, b, and N. This grid includes the origin explicitly as r0. In this case, we took a = 4.34 10-6/Z, b = 0.002 304, and rmax = 50, leading to N = 7058 for H, increasing to N = 9021 for U, and to r1 = 10-7 for H, decreasing to 1.1 10-9 for U.

A fourth code uses a change of variable technique:

$$\rho = \ln r $$. (Eq. 24)

A uniform grid is taken in the transformed variable from rho(rmin) to rho(rmax) where the parameters are taken to be rmin = 0.01 e-4/Z, for atomic number Z, and rmax = 50. The number of points increased from N = 2113 for H to N = 2837 for U. The density of points chosen in the latter two codes, linear near the origin and exponentially increasing at large r, is similar to that suggested from theoretical considerations [9].

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