First-order function for semirelativistic atomic equation

The first-order correction in the electron-electron interaction to the solation of the two-electron semirelativistic atomic equation is represented in, the form
$$ \Psi^{(1)}(r_1,r_2)=\dfrac 12 S_k[\psi_k(r_1)\Phi_k(r_2)-\psi_k(r_2)\Phi_k(r_1)], $$
where $\psi_k(r)$ are solutions of the one-electron Dirac equation. Analytic expressions are obtained for $\Phi_k(r)$ that are convenient for calculations of matrix elements.