The radial Kohn-Sham (KS) equation for the H-like atom has a form [1, p.9, formula (8)]:
- $$\left( { - {1 \over 2}{{{d^2}} \over {d{r^2}}} + {{l\left( {l + 1} \right)} \over {2{r^2}}} + {V^{KS}}\left( r \right)} \right){P_{nl}}\left( r \right) = {\varepsilon _{nl}}{P_{nl}}\left( r \right)$$
where ${P_{nl}}\left( r \right) = r{\Psi _{nl}}\left( r \right)$ is the radial wave function,
whereas ${V^{KS}}\left( r \right) = {V^{NUC}}\left( r \right) + {V^{ee}}\left( r \right) + {V^{XC}}\left( r \right)$ is the KS potential,
${V^{XC}}\left( r \right)$ is the exchange-correlation potential,
the Coulomb potential from the nucleus
- $${V^{NUC}}\left( r \right) = - {Z \over r}$$
the electron-electron repulsion potential (the Hartree potential)
- $${V^{ee}}\left( r \right) = 4\pi \int {{{\rho \left( {r'} \right)} \over {\left| {r - r'} \right|}}} {r'^2}dr'$$
here $\rho \left( r \right) = {\left( {{{{P_{nl}}\left( r \right)} \over r}} \right)^2}$ is the electron density.
For $1s$ state, $n=1$, $l=0$, the KS equation has a form:
- $$\left( { - {1 \over 2}{{{d^2}} \over {d{r^2}}} + {V^{KS}}\left( r \right)} \right){P_{10}}\left( r \right) = {\varepsilon _{10}}{P_{10}}\left( r \right)$$
References:
[1] Hartree, D.R. The calculation of atomic structures. Chapman & Hall, Ltd., London, 1957
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