Rydberg Transition Frequencies from the Local Density Approximation

A method is given that extracts accurate Rydberg excitations from density functional calculations in the local density approximation, despite the short-ranged potential. For the case of He and Ne, the asymptotic quantum defects predicted by the calculations are in less than 5% error, yielding transition frequency errors of less than 0.1 eV.

Figure 1

$s$-quantum defect as a function of $n$ for the potential shown in the inset. The quantum defect is a smooth function of $n$ and converges rapidly to its asymptotic value, ${\mu }_{\infty }=-0.441$.

Figure 2

Solution of Eq. (3) for $\mu$ as a function of $r$, for the potential of Fig. 1; the logarithmic derivative of the $n=20$ orbital was found numerically as a function of $r$, and at each location $r$, it was inserted into Eq. (3) to get $\mu (r)$.

Figure 3

$s$-quantum defect as a function of $n$ for the exact KS potential [17] of the He atom. The quantum defect converges rapidly to its asymptotic value, ${\mu }_{\infty }=0.213$.

Figure 4

He atom: solution of Eq. (3) for $\mu$ as a function of $r$; The $n=20$ orbital was used for the exact case, and the scattering orbital of energy $E=I+{ϵ}_{1s}^{\mathrm{LDA}}$ was used for the LDA.

Figure 5

Comparison of the exact KS potential of the He atom [17] (solid line), and the LDA potential (dashed line). The Coulomb potential is also shown (dotted line). At $r\sim 1$ the exact potential is almost Coulombic. The inset shows the potentials themselves.

Figure 6

Radial orbitals of He: LDA orbital of energy $E=0.333$ and exact 20 s orbital; the HOMO is also shown. The LDA scattering orbital and the exact KS Rydberg orbital are very close in the region $1<r<2$, where the quantum defect can be extracted (see text, and Fig. 4). The fact that the LDA orbital has an incorrect asymptotic behavior at large $r$ is irrelevant for the value of $\mu$, as well as for optical absorption [11].