Mathematical statement of the Born-Oppenheimer approximation

Question

I have been looking up a formal mathematical definition of the Born-Oppenheimer approximation. I have thus far come across two (my wording):

Definition 1

The Born-Oppenheimer approximation is given by: $$\nabla^2\psi_e(\vec r, R)\chi(\vec R) \approx \psi_e(\vec r, \vec R) \nabla^2\chi(\vec R)$$ (from: Linne, M. A. (2002). Spectroscopic measurement: an introduction to the fundamentals. Academic Press. (p224))

Definition 2

The Born-Oppenheimer approximation is given by: $$\psi=\psi_e \chi$$ (from: Das, I. et al. (2005) An Introduction to Physical Chemistry. New Age International.(p105))

In both cases $\psi_e$ is the electron wave function and $\chi$ is the nuclear wave function. My question is are these definitions equivalent. If so how can it be proved and if not which is taken to be the standard definition. (A source describing both would also be great).

Answer 1

The two expressions are clearly not equivalent: the first one assumes the second one, and extends it. Note that writing $f(x,y)=g(x)h(x,y)$ is not an approximation at all, so the second expression is meaningless on its own. The true approximation is $h'\ll g'$, so that you can write $f'\approx g' h$. Therefore, the second expression is just notation; it is the first expression what really introduces an approximation.

Answer 2

$\textbf{General Question}:$ We may write the nonrelativistic Hamiltonian for a molecule as a sum of five terms:

$$H = T_N(\textbf{R}) + T_e(\textbf{r}) + V_{eN}(\textbf{r},\textbf{R}) + V_{NN}(\textbf{R})+V_{ee}(\textbf{r}),$$

where $\textbf{R}$ is the set of nuclear coordinates and $\textbf{r}$ is the set of electronic coordinates.(Here we have ignored the spin-orbit effects.)

$\textbf{Problem}:$ Unfortunately, the $V_{eN} (\textbf{r};\textbf{R})$ term prevents us from separating H into nuclear and electronic parts, which would allow us to write the molecular wavefunction as a product of nuclear and electronic terms, $\Psi(\textbf{r};\textbf{R} ) = \Psi(\textbf{r})\Psi(\textbf{R})$.

$\textbf{Solution}:$ We thus introduce the Born-Oppenheimer approximation, by which we conclude that this nuclear and electronic separation is approximately correct.The term $V_{eN}(\textbf{r};\textbf{R})$ is large and cannot be neglected; however, we can make the $\textbf{R}$ dependence parametric, so that the total wavefunction is given as $\Psi(\textbf{r};\textbf{R})\Psi(\textbf{R})$. The Born-Oppenheimer approximation rests on the fact that the nuclei are much more massive than the electrons, which allows us to say that the nuclei are nearly fixed with respect to electron motion. We can fix $\textbf{R}$, the nuclear configuration, at some value $\textbf{R}_a$, and solve for the electronic wavefunction $\Psi(\textbf{r};\textbf{R}_a)$, which depends only parametrically on $\textbf{R}$. If we do this for a range of R, we obtain the potential energy curve along which the nuclei move.

$\textbf{Mathematical Details}:$ You could refer this link,which will support a nice derivation for the mathematical statement of this approximation.

Hope it helps.