# Born-Oppenheimer approximation confusion

• VortexLattice
In summary, the Born-Oppenheimer approximation for a solid states solves for the potential energy between the ions and electrons by considering them as stationary with respect to each other. Core electrons are treated separately from valence electrons, and the Hamiltonian is written as a sum of four terms. The first term is the potential energy between the ions and the core electrons, the second term is the potential energy between the core electrons and the nuclei, the third term is the potential energy between the nuclei and the electrons, and the fourth term is the potential energy between the electrons and everything else.
VortexLattice
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!

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.

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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.

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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.

## 1. What is the Born-Oppenheimer approximation?

The Born-Oppenheimer approximation is a theoretical concept in quantum mechanics that assumes that the motion of atomic nuclei and electrons in a molecule can be separated. This allows for easier calculations of the electronic structure of a molecule by treating the nuclei as fixed points and solving for the electronic wave function.

## 2. Why is the Born-Oppenheimer approximation important?

The Born-Oppenheimer approximation is important because it simplifies the calculations involved in determining the electronic structure of a molecule. By separating the motion of the nuclei and electrons, it allows for more accurate and efficient predictions of molecular properties, such as bond lengths and energies.

## 3. How does the Born-Oppenheimer approximation work?

The Born-Oppenheimer approximation works by assuming that the nuclei in a molecule move much slower than the electrons. This allows for the nuclei to be treated as stationary and their effect on the electronic wave function to be calculated separately. The electronic wave function is then used to determine the overall properties of the molecule.

## 4. What are the limitations of the Born-Oppenheimer approximation?

The Born-Oppenheimer approximation is a simplification of the complex interactions between electrons and nuclei in a molecule. It assumes that the nuclei are fixed points and does not take into account any nuclear motion. This can lead to errors in calculations for highly reactive molecules or those with large nuclear motion.

## 5. How is the Born-Oppenheimer approximation related to molecular spectroscopy?

The Born-Oppenheimer approximation is essential in understanding and interpreting molecular spectroscopy. It allows for the prediction of molecular properties, such as bond energies and vibrational frequencies, which are crucial in interpreting spectroscopic data. Without this approximation, it would be much more difficult to analyze and understand the spectra of molecules.

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