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BornOppenheimer approximation confusion 
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#1
Jan2113, 12:35 AM

P: 146

Hi, I'm reading about the BornOppenheimer 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.
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! 


#2
Jan2113, 01:30 AM

Sci Advisor
P: 1,658

If you want a more intuitive explanation, what you do is not solving the coupled system of ions and electrons, but getting a solution for the electron system for a fixed set of ion positions. 


#3
Jan2113, 01:42 AM

Sci Advisor
P: 3,596

No, up to what VortexLattice has described, the whole wavefunction is still completely general and the BO approximation has not been invoked, yet.
V_ei clearly acts also on the nuclear wavefunction, but as it is a multiplicative operator, it does not matter whether it appears in front or after ##\Phi_n##. 


#4
Jan2113, 01:52 AM

Sci Advisor
P: 1,658

BornOppenheimer approximation confusion
Oh sorry. Maybe I misread and should stop posting after midnight.



#5
Jan2113, 12:27 PM

P: 146

So why isn't that in the second term as well? Thank you! 


#6
Jan2213, 01:33 AM

Sci Advisor
P: 3,596

But the ordering chosen is not a matter of mathematical necessity but depends on the approximations which will probably introduced in the continuation of the argument. 


#7
Jan2213, 09:42 AM

P: 146

##(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## and after doing many manipulations, get: ##(T_i + V_{ii} + E_{core} + E_{e,n})\Phi_n = E_n\Phi_n## and ##(T_e + V_{ee} + V_{ei})\Psi_{e,n}(r,R) = E_{e,n}\Psi_{e,n}(r,R)## So, the placement of ##V_{ei}## makes it end up in either the nuclear or electronic equation. 


#8
Jan2213, 10:27 AM

Sci Advisor
P: 3,596

V_ei depends on both nuclear and electronic coordinates. You want an equation for ##\Phi_n## which does not contain electronic coordinates. Likewise you want an equation for ##\Psi_{e,n}## which does not contain derivatives with respect to R.



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