Solution of KS equation for H-like atom using Green’s function

KS equation for H-like atom for $1s$ state

  1. $$\left( { - {1 \over 2}{{{d^2}} \over {d{r^2}}} + {V^{KS}}\left( r \right)} \right){P_{10}}\left( r \right) = {\varepsilon _{10}}{P_{10}}\left( r \right)$$

where ${V^{KS}}\left( r \right) = {V^{NUC}}\left( r \right) + {V^{ee}}\left( r \right) + {V_{XC}}\left( r \right)$ is the KS potential.

We move ${V^{ee}}\left( r \right)$ and ${V_{XC}}\left( r \right)$ to the right

  1. $$\left( { - {1 \over 2}{{{d^2}} \over {d{r^2}}} - {Z \over r} - {\varepsilon _{10}}} \right){P_{10}}\left( r \right) =  - \left( {{V^{ee}}\left( r \right) + {V_{XC}}\left( r \right)} \right){P_{10}}\left( r \right)$$

and write ${P_{10}}\left( r \right)$ as

  1. $${P_{10}}\left( r \right) = P_{10}^0\left( r \right) + g\left( r \right)$$

where $P_{10}^0\left( r \right)$ is the solution of Schrödinger equation for H-like atom.

For function $g\left( r \right)$ we can write

  1. $$\left( { - {1 \over 2}{{{d^2}} \over {d{r^2}}} - {Z \over r} - {\varepsilon _{10}}} \right)g\left( r \right) =  - \left( {{V^{ee}}\left( r \right) + {V_{XC}}\left( r \right)} \right)\left( {P_{10}^0\left( r \right) + g\left( r \right)} \right)$$

because

  1. $$\left( { - {1 \over 2}{{{d^2}} \over {d{r^2}}} - {Z \over r} - {\varepsilon _{10}}} \right)P_{10}^0\left( r \right) = 0$$

Eq. (4) can be considered as equation

  1. $${\bf{H}}g\left( {\bf{r}} \right) = f\left( {\bf{r}} \right)$$

The solution for Eq. (6) can be written in the form

  1. $$g\left( {\bf{r}} \right) = A{g^0}\left( {\bf{r}} \right) + \int {G\left( {{\bf{r}},{\bf{r'}}} \right)} f\left( {{\bf{r'}}} \right)d{\bf{r'}}$$

where ${g^0}\left( {\bf{r}} \right)$ is the solution of the uniform equation

  1. $${\bf{H}}{g^0}\left( {\bf{r}} \right) = 0$$

and ${G\left( {{\bf{r}},{\bf{r'}}} \right)}$ is the Green’s function satisfying the equation

  1. $${\bf{H}}G\left( {{\bf{r}},{\bf{r'}}} \right) =  - \delta \left( {{\bf{r}} - {\bf{r'}}} \right)$$

Thus the solution of Eq. (4) can be written as

  1. $$g\left( r \right) =  - \int {G_0^{\rm{H}}\left( {r,r'} \right)\left( {{V^{ee}}\left( {r'} \right) + {V_{XC}}\left( {r'} \right)} \right)\left( {P_{10}^0\left( {r'} \right) + g\left( {r'} \right)} \right)dr'} $$

The Green’s function $G_0^{\rm{H}}\left( {r,r'} \right)$ of Eq. (5) is known, see Task 1.

The Eq. (10) can be solved self-consistently.