Born-Oppenheimer approximation I'm studying about formulation of Born-Oppenheimer approximation from Atkins, Molecular quantum mechanics, 5th edition. In chapter 8.1 I found the following: 

I keep wondering where did this W come from in equation 8.4.? It says later in text that it is responsible for non-adiabatic effects, but I don't see how it just appeared here. If someone is willing to explain, I'd be greatful.

Also, if anyone has tips on where to find more nice and clean text about more general formulation of Born-Oppenheimer approximation, since this one is confined to particular system. 

Thanks!
Atkins did a good job on explaining what parametric dependence of the electronic wave function (and electronic energy) on nuclear coordinates is, but failed to mention that in the derivation of the Born-Oppenheimer approximation this dependence is assumed to be continuous and differentiable and that both first and second derivatives of this quantities with respect to nuclear coordinates are in general non-zero.
In particular, for the system described in the text, we have
$$
  \frac{\partial \psi}{\partial Z_j} \neq 0 \, , \quad
  \frac{\partial^2 \psi}{\partial Z_j^2} \neq 0\, .
$$

Now, when the solution of the form (8.3) is substituted into the Schrödinger equation (8.2), the term involving $T_\mathrm{e}$ is trivial: since the nuclear wave function $\chi$ is not a function of electronic coordinates it is just a constant when differentiating with respect to them, so we get
$$
  T_\mathrm{e} (\psi \chi) = \chi T_\mathrm{e} \psi \, .
$$
But the term involving $T_\mathrm{N}$ does not trivially transform in a similar way,
$$
  T_\mathrm{N} (\psi \chi) \neq \psi T_\mathrm{N} \chi
$$
since both $\psi$ and $\chi$ depend on the nuclear coordinates. Rather, applying the product rule twice, we get
\begin{align}
  \frac{\partial^2}{\partial Z_j^2} (\psi \chi)
  &amp;=
  \frac{\partial}{\partial Z_j} \left( \frac{\partial}{\partial Z_j} (\psi \chi) \right) \\
  &amp;=
  \frac{\partial}{\partial Z_j} \left( \psi \frac{\partial \chi}{\partial Z_j} + \chi \frac{\partial \psi}{\partial Z_j} \right) \\
  &amp;=
  \psi \frac{\partial^2 \chi}{\partial Z_j^2}
  +
  2 \frac{\partial \psi}{\partial Z_j} \frac{\partial \chi}{\partial Z_j} 
  +
  \chi \frac{\partial^2 \psi}{\partial Z_j^2} \, ,
\end{align}
so that 
$$
  T_\mathrm{N} (\psi \chi)
  =
  - \psi \sum\limits_{j=1,2} \frac{\hbar^2}{2 m_j} \frac{\partial^2 \chi}{\partial Z_j^2}
  -
  \sum\limits_{j=1,2} \frac{\hbar^2}{2 m_j} \left(
  2 \frac{\partial \psi}{\partial Z_j} \frac{\partial \chi}{\partial Z_j} 
  +
  \chi \frac{\partial^2 \psi}{\partial Z_j^2} \right) \, ,
$$
where the first term is nothing but $\psi T_\mathrm{N} \chi$ and the remainings are designated as $W$.