# Born Oppenheimer approximation and screening

In the Born-Oppenheimer approximation we take the full Hamiltonian of a solid, given by $$H = T_e + T_{ion} + V_{ee} + V_{e-ion} + V_{ion-ion}$$ where the T are the kinetic energy of electrons and ions and the V are the interaction (Coulomb) potentials of electrons with other electrons, electrons with ions and ions with ions respectively. We then assume the eigenfunctions $$\xi(\vec{r_i},\vec{R_i})$$ of the electronic Hamiltonian ($$\vec{r_i}$$ just being a shorthand for all possible position vectors) $$H_e = T_e + V_{ee} + V_{e-ion} + V_{ion-ion}$$ are known, $$H_e \xi(\vec{r_i},\vec{R_i}) = E_e(\vec{R_i}) \xi(\vec{r_i},\vec{R_i})$$ make some approximations and find an effective Schrödinger equation for the wave function $$\phi(\vec{R})$$ of the ions, where the electron energie $$E_e(\vec{R_i})$$ acts as an effective potential: $$(T_{ion} + E_e(\vec{R_i})) \phi(\vec{R_i}) = E \phi(\vec{R_i}).$$ This effective potential contains the Coulomb interaction of the ions as an additive term, since it is just a constant in the electronic Schrödinger equation: $$E_e = V_{ion-ion} + ...$$ My question is how to take screening of the ion potentials by electrons into account. When reading Ashcroft & Mermin they seemed to indicate that one needs to replace the bare ion potential by the screened one. Here is for example problem 1 from Chapter 26 of Ashcroft & Mermin, where they treat phonons in a metal. The dynamical matrix mentioned comes from the harmonic approximation of $$E_e$$.

However I feel like the screening should somehow be taken into account automatically by the effective potential $$E_e$$, since this already includes the response of the electrons to the ions, which is exactly where the screening originates. So I don't understand why it needs to be added by hand.