Hi, I'm reading about the Born-Oppenheimer approximation for a solid and they're doing the formalism of it. They say that we can basically consider the ions stationary with respect to the electrons because they move so little and so slowly in comparison to them.(adsbygoogle = window.adsbygoogle || []).push({});

They say that ##R_i## are the positions of the ions and ##r_j## are the positions of the electrons, ##P_i## are the momenta of the ions, ##p_j## are the momenta of the electrons (all vectors but I'm just writing them like this here). Then they say that we'll look at "core" electrons separately from "valence" electrons, because core ones just hang out by the nuclei while valence ones move around. Given all this, the hamiltonian is:

##H = \sum\limits_i \frac{P_i^2}{2M} + \sum\limits_{j = cond. elecs} \frac{p_j^2}{2m} + \sum\limits_{i,i'} V_{i,i'}(|R_i - R_{i'}|) + (e^2/2) \sum\limits_{j,j'=cond. elecs} \frac{1}{|r_j - r_{j'}|} + \sum\limits_{i,j} V_{ei}(|r_j - R_i|) + E_{core}##

(where ##E_{core}## is the energy of the "core" electrons that are "attached" to the nuclei.)

Then they rewrite this as:

##H = T_i + T_e + V_{ii} + V_{ee} + V_{ei} + E_{core}##

Then they say that we can write the full wavefunction as a combination of two functions (here, ##r## and ##R## are the sets of the positions of all the electrons/ions, not single ones):

##\Psi(r,R) = \sum\limits_n \Phi_n(R) \Psi_{e,n}(r,R)##

Then, they just do the eigenvalue equation, ##H\Psi = E\Psi##:

##(T_i + V_{ii} + E_{core})\Psi + \sum\limits_n \Phi_n (T_e + V_{ee} + V_{ei})\Psi_{e,n}(r,R) = E\Psi##

In the second term, the part with the explicit sum, they put ##\Phi_n## out in front because the operators directly following it "only operate on the electron part of the product wavefunction", according to my book. But here's my confusion: doesn't ##V_{ei}## act on the ion part of the wave function? It was defined as ##\sum\limits_{i,j} V_{ei}(|r_j - R_i|)##, which has that ##R_i## in it. What am I missing?

Thank you!

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# Born-Oppenheimer approximation confusion

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