Born-Oppenheimer approximation

Question

In B-O approximation, one of the basic assumptions is that the total many-body wavefunction can be expanded as the following: $$\Psi(\bf{r},\bf{R})=\sum_n\phi_n(\bf{R})\psi_n(\bf{r;\bf{R}})$$ where $\phi_n(\bf{R})$ is the wavefunction for the ions alone and the index $n$ can be thought of as the index for energy levels for the ions. $\psi(\bf{r};\bf{R})$ is the electron wavefunction for fixed ion position $\bf{R}$. The electron wavefunction $\psi$ depends parametrically on $\bf{R}$ and we can assume that electrons adjust themselves very quickly to the ion movement due to the huge difference in their velocities.

Now my question is why we can assign the same index $n$ to the electron part of the wavefunction? My reasoning is the following: For each fixed $\bf{R}$, we have a fixed background potential for the electrons, then we can in principle solve for electronic energy eigenstates, indexed by $m$. This $m$, under the aforementioned conditions, should be independent from the level index $n$ of the ions. This seems to invalidate the above expansion.

Answer 1

This is the result of Schmidt decompositions:

$|A_n\rangle $ is the complete basis of Hilbert space $H_A$

$|B_n\rangle $ is the complete basis of Hilbert space $H_B$

Then, for any pure state $|\Psi\rangle=\sum_{nm} c_{nm}|A_n\rangle |B_m\rangle$ in the total Hilbert space $H_A \otimes H_B$ , we can always find a new basis in each of the sub-Hilbert space, that such

$|\Psi\rangle = \sum_n c_n |\phi_n\rangle |\psi_n\rangle$

the new basis is some unitary rotation of old basis.

$|\phi_n\rangle =\sum_m U_{nm}|A_m\rangle $

$|\psi_n\rangle =\sum_m V_{nm}|B_m\rangle $

---

Going back to the spacial representation.

$$\langle R,r|\Psi\rangle = \sum_n c_n \langle R |\phi_n\rangle \langle r |\psi_n\rangle$$

$$ \Psi(R,r)=\sum_n c_n \phi_n(R) \psi_n(r)$$

However, the final expression looks different from $\psi_n(r;R)$

Since the ion is heavier:

(1) the wavefunction of $\phi_n(R)$ are more localized than $\psi_n(r)$

(2) those $\phi_n(R)$ are similar in shape for different $n$

---

Then we can perform a coarse grain of those similar $\phi_n(R)$

$$ \sum_n = \sum_N \sum_{n \in [N]} $$

$[N]$ is a set, in which the shape of $\phi_n(R)$ are similar to each other.

$$ \sum_n c_n \phi_n(R) \psi_n(r) = \sum_{N} \tilde{\phi}_N(R) \frac{ \sum_{n \in [N]} c_n \phi_n(R) \psi_n(r)}{\tilde{\phi}_N(R)} = \sum_{N} \tilde{\phi}_N(R) \tilde{\psi}_N(r;R)$$

where the newly defined electron function is $$ \tilde{\psi}_N(r;R) := \frac{ \sum_{n \in [N]} c_n \phi_n(R) \psi_n(r)}{\tilde{\phi}_N(R)}$$

Answer 2

First of all, it is not an assumption that you can decompose the wave function in this basis, you can decompose any wave function in $L^2(\mathbb{R}^{dN} \times \mathbb{R}^{dK}) \cong L^2(\mathbb{R}^{dN}) \otimes L^2(\mathbb{R}^{dK})$ as a linear combination of product states.

In case of Born-Oppenheimer systems, this basis is useful, because to leading order the time evolution maps product states of the above form onto product states if you are away from band crossings. And when you get close to band crossings, only the bands that cross will be involved. Note that once you include higher-order corrections to the Born-Oppenheimer approximation, it gets more complicated, but also here it is true that as long as you stay away from band crossings, the electronic state remains in the $n$th band.

An intuitive explanation, which can be made rigorous is as follows: because of the difference in mass between electron and nuclei, the electrons move much more quickly than the nuclei. So on the time scale of the nuclei, the electrons adjust instantaneously to the motion of the nuclei. As long as you are away from band crossings, band transitions are exponentially suppressed and therefore nothing to worry about. The first- and higher-order corrections then include the backreaction from the nucleonic dynamics onto the electrons.

Justifying the Born-Oppenheimer approximation and deriving higher-order corrections has a rich literature; notable contributions were made by Littlejohn and Hagedorn in the 1980s, although there are more modern approaches nowadays. Notable rigorous works here are e. g. by Hagedorn and Joye as well as Panati, Spohn and Teufel.