Born-Oppenheimer approximation confusion

  • Context: Graduate 
  • Thread starter Thread starter VortexLattice
  • Start date Start date
  • Tags Tags
    Approximation Confusion
Click For Summary

Discussion Overview

The discussion revolves around the Born-Oppenheimer approximation in the context of quantum mechanics, specifically how it applies to the treatment of ions and electrons in a solid. Participants explore the formalism of the approximation, the role of various potential energy terms, and the implications for wavefunction separation.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant describes the Born-Oppenheimer approximation as treating ions as stationary relative to electrons due to their slower movement, leading to a separation of the wavefunction into ionic and electronic components.
  • Another participant argues that the potential energy term ##V_{ei}## acts on both the nuclear and electronic wavefunctions, suggesting that its placement in the equation does not affect the overall structure since it is a multiplicative operator.
  • A different participant questions the consistency of the treatment of the potential terms, specifically why ##V_{ii}## is not included in the same term as ##V_{ei}##, raising concerns about the implications for the final results derived from the equations.
  • One participant suggests that the ordering of terms is not a mathematical necessity but rather depends on the approximations made in the argument, indicating that different placements could lead to different outcomes.
  • Another participant emphasizes the need for equations that do not contain electronic coordinates when deriving the equation for ##\Phi_n##, indicating a desire for clarity in the separation of variables.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the Born-Oppenheimer approximation and the treatment of potential energy terms. There is no consensus on the correct interpretation of the placement of the terms in the equations or their impact on the final results.

Contextual Notes

Participants highlight the importance of assumptions in the Born-Oppenheimer approximation and the potential for different interpretations based on the ordering of terms in the equations. The discussion remains open regarding the implications of these choices.

VortexLattice
Messages
140
Reaction score
0
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.

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!
 
Physics news on Phys.org
VortexLattice said:
What am I missing?

You are more or less missing the Born-Oppenheimer approximation itself. Recall what you said in the beginning:

VortexLattice said:
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.

So your assumption is basically that your electron distribution does not drag your ions around. The distribution of the ions will stay as it is. Therefore you can treat the position of the ions as a parameter instead of a variable. So the potential will only depend on these positions, but not act on them.

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.
 
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##.
 
Oh sorry. Maybe I misread and should stop posting after midnight.
 
Last edited:
DrDu said:
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##.

Hmmm, this doesn't seem consistent though. If ##V_{ei} = \sum\limits_{i,j} V_{ei}(|r_j - R_i|)##, I see what you're saying about the order of the operator ##R_i## not mattering (that's what you're saying, right?), but then what about ##V_{ii}##? That also just has multiplicative factors of ##R##: ##V_{ii} = \sum\limits_{i,i'} V_{i,i'}(|R_i - R_{i'}|)##.

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

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

Feel free to put it there!
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.
 
Last edited:
DrDu said:
Feel free to put it there!
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.

Hmmm, but that changes the final result! The final result they reach for this is separating ##\Phi_n## and ##\Psi_{e,n}## by separating this equation:

##(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.
 
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.
 

Similar threads

Calculate reaction heat from quantum mechanics
  • · Replies 1 ·
Replies
1
Views
2K
Susceptibility of a simple metal (Problem 31.6 in Ashcroft's
  • · Replies 1 ·
Replies
1
Views
2K