Most derivations I have seen of the Born-Oppenheimer approximation are made using wave-functions. To understand it better, I was trying to write a derivation using Dirac notation, but I am stuck. I am going to post what I have done so far so you guys can help me out.
The hamiltonian of the molecule can be written as the sum of two parts, $H_\text{mol} = H_\text{el} + H_\text{nuc}$, where $H_\text{nuc}$ is the hamiltonian of the nuclei by themselves, and $H_\text{el}$ is the hamiltonian of the electrons interacting with the nuclei:
$$H_\text{el} = T_\text{el} + V_\text{el-el} + V_\text{el-nuc}$$ $$H_\text{nuc} = T_\text{nuc} + V_\text{nuc-nuc}$$
We want to find the energy levels of the molecule. That is, we want to solve $H_{\text{mol}} |\mathcal{E}\rangle =\mathcal{E} |\mathcal{E}\rangle$, where $\mathcal{E}$ and $|\mathcal{E}\rangle$ are the eigenenergy and corresponding eigenket of the molecule.
The state space of the molecule can be separated into electronic and nuclear parts: $\mathcal{S}_\text{mol} = \mathcal{S}_\text{el} \otimes \mathcal{S}_\text{nuc}$. Let $|R\rangle \in \mathcal{S}_\text{nuc}$ be the position basis of the nuclei, where $R$ denotes the coordinates of all the nuclei.
Define $H_{\text{el}}(R) = \langle R| H_{\text{el}} |R\rangle$, which is an operator in $\mathcal{S}_{\text{el}}$. Let $E_a(R)$ and $|E_a(R)\rangle$ be the eigenvalues and corresponding eigenkets of $H_{\text{el}}(R)$ in $\mathcal{S}_{\text{el}}$, so that $H_{\text{el}}(R) |E_a(R)\rangle = E_a(R) |E_a(R)\rangle$. For each $R$, the kets $|E_a(R)\rangle$ make a basis for $\mathcal{S}_{\text{el}}$.
The set of kets $|E_a(R)\rangle |R\rangle \in \mathcal{S}_{\text{mol}}$ for all $R$ and $a$ then make a basis for $\mathcal{S}_{\text{mol}}$. Using this basis, the state of the molecule can be written: $$|\psi \rangle =\sum _a \int \chi_a(R) |E_a(R)\rangle |R\rangle dR$$
where $\chi_a(R)$ is an amplitude function.
Note that:
$$H_\text{el} |E_a(R)\rangle |R\rangle = [ H_\text{el}(R) |E_a(R)\rangle ] |R\rangle = E_a(R) |E_a(R)\rangle |R\rangle $$
Therefore, the molecular eigenproblem $H_{\text{mol}} |\psi \rangle =\mathcal{E} |\psi \rangle$ can be written:
$$\sum_a \int (E_a + T_\text{nuc} + V_\text{nuc-nuc} - \mathcal{E}) \chi_a(R) |E_a(R)\rangle |R\rangle dR = 0$$
Multiplying by $\langle E_a(R)|$ on the left:
$$\int (E_a + T_\text{nuc} + V_\text{nuc-nuc} - \mathcal{E}) \chi_a(R) |R\rangle dR = 0$$
At last, we define a ket $|\chi_a\rangle \in \mathcal{S}_\text{nuc}$ such that $\chi_a(R) = \langle R | \chi_a \rangle$:
$$|\chi_a\rangle := \int \chi_a(R) |R\rangle dR$$
Then we can write:
$$(T_\text{nuc} + V_\text{nuc-nuc} + E_a - \mathcal{E}) | \chi_a \rangle = 0$$
HERE ENDS MY DERIVATION SO FAR.
I must have done something wrong somewhere, because the final equation that I obtain is, as far as I can tell, the Born-Oppenheimer approximation, but I am obtaining it here as an exact equation. What did I do wrong?
Also, if anyone knows of some reference, textbook or paper, that deals with the Born-Oppenheimer approximation in Dirac notation, please post it.