Born-Oppenheimer or adiabatic approximation
The Born-Oppenheimer or adiabatic approximation is based on the observation, that the electrons are much lighter particles than the nuclei. The electron-nucleus mass ration $M_I/m_\mathrm{e}$ amounts to 1836 for a proton and to order of $10^4$ for typical elements such as carbon or silicon. Due to their smaller mass electrons bound in the potential of the nuclei move more rapidly than the heavy nuclei themselves
and on the average follow their motion. This can be expressed by making an ansatz for the wave function $\Psi\left(\{\mathbf{r}_j\},\{\mathbf{R}_J\}\right)$ in terms of a wave function of the nuclei $\chi$ and of the electron $\Phi$ that adiabatically follow them
\begin{equation}
\Psi\left(\{\mathbf{r}_j\},\{\mathbf{R}_J\}\right)=
\chi\left(\{\mathbf{R}_J\}\right)\,\Phi\left(\{\mathbf{r}_j\};\{\mathbf{R}_J\}\right)
\end{equation}
For simplicity, the set of all electron coordinates $\{\mathbf{r}_{j}\}$ will be denoted $\mathbf{q}$ in the following and correspondingly the set of all nuclear coordinate $\{\mathbf{R}_{J}\}$ by $\mathbf{Q}$. insertion of this ansatz and neglecting the $\mathbf{Q}$-dependence of $\Phi(\mathbf{q},\mathbf{Q})$ with respect to the kinetic energy operator $\mathbf{Q}$ allows the separation into an entirely electronic problem for each momentary set of nuclear coordinates and a Schrödinger equation for the nuclei that move on an effective potential surface that accounts for their interaction with the electrons
\begin{align}
\left(\hat{T}_{\mathrm{e}}+\hat{V}_{\mathrm{ee}}+\hat{V}_{\mathrm{ek}}(\mathbf{Q})\right)
\,\Phi_n(\mathbf{q};\mathbf{Q})&=E_n(\mathbf{Q})\,\Phi_n(\mathbf{q};\mathbf{Q})\\
\left(\hat{T}_{\mathrm{k}}+\hat{V}_{\mathrm{kk}}+E_n(\mathbf{Q})\right)
\,\chi_{n \nu}(\mathbf{Q})&=E_{n \nu}\,\chi(\mathbf{Q})
\end{align}
\begin{align}
\hat{H}\,\chi(\mathbf{Q})\,\Phi(\mathbf{q},\mathbf{Q})
= &\Phi(\mathbf{q};\mathbf{Q})\left(\hat{T}_{\mathrm{k}}+\hat{V}_{\mathrm{kk}}\right)
\,\chi(\mathbf{Q})\\
&+\chi(\mathbf{Q})\left(\hat{T}_{\mathrm{e}}+\hat{V}_{\mathrm{ee}}+\hat{V}_{\mathrm{ek}}\right)
\,\Phi(\mathbf{q};\mathbf{Q})\\
&+\hat{H}_{\mathrm{ek}}\,\chi(\mathbf{Q})\,\Phi(\mathbf{q};\mathbf{Q})\quad.
\end{align}
Here the term with operator $\hat{H}_{\mathrm{ek}}$ was introduced to collect the terms of the non-adiabatic coupling of electronic and nuclear motion. It resurects the terms that are missing for ignoring the $\mathbf{Q}$-dependence of $\Phi(\mathbf{q};\mathbf{Q})$ in the first line of the above equation
\begin{align}
\hat{H}_{\mathrm{ek}}\,\chi(\mathbf{Q})\,\Phi(\mathbf{q},\mathbf{Q})&=
\hat{T}_{\mathrm{k}}\chi(\mathbf{Q})\,\Phi(\mathbf{q},\mathbf{Q})-
\Phi(\mathbf{q};\mathbf{Q})\hat{T}_{\mathrm{k}}
\,\chi(\mathbf{Q})\\
&=-\frac{1}{2}\sum_{J}\frac{1}{M_J}\,\frac{\partial}{\partial \mathbf{R}_J}\left[
\frac{\partial \chi(\mathbf{Q})}{\partial \mathbf{R}_J}\,\Phi(\mathbf{q},\mathbf{Q})
+\chi(\mathbf{Q})\frac{\partial \Phi(\mathbf{q},\mathbf{Q})}{\partial \mathbf{R}_J}
\right]\\
&\ -\Phi(\mathbf{q};\mathbf{Q})\hat{T}_{\mathrm{k}}
\,\chi(\mathbf{Q})\\
&=-\frac{1}{2}\sum_{J}\frac{1}{M_J}\,\left[
\frac{\partial^2 \chi(\mathbf{Q})}
{\partial \mathbf{R}_J^2}\,\Phi(\mathbf{q};\mathbf{Q})
+2\,\frac{\partial \chi(\mathbf{Q})}{\partial \mathbf{R}_J}
\,\frac{\partial \Phi(\mathbf{q};\mathbf{Q})}{\partial \mathbf{R}_J}
\right.\\
&\hphantom{=-\frac{1}{2}\sum_{J}\frac{1}{M_J}\ \ }\ \left. +\chi(\mathbf{Q})\,\frac{\partial^2 \Phi(\mathbf{q};\mathbf{Q})}
{\partial \mathbf{R}_J^2}\right]
-\Phi(\mathbf{q};\mathbf{Q})\hat{T}_{\text{k}}\,\chi(\mathbf{Q})\\
&=-\frac{1}{2}\sum_{J}\frac{1}{M_J}\,\left[
2\,\frac{\partial \chi(\mathbf{Q})}{\partial \mathbf{R}_J}
\,\frac{\partial \Phi(\mathbf{q};\mathbf{Q})}{\partial \mathbf{R}_J}
+\chi(\mathbf{Q})\,\frac{\partial^2 \Phi(\mathbf{q};\mathbf{Q})}
{\partial \mathbf{R}_J^2}\right]
\end{align}
The non-adiabatic interaction term contains terms proportinal to the gradient of the electonic wave function with respect to the nuclear coordinate and the kinetic energy induced by the nuclear motion.
The parametric dependence $\hat{V}_{\mathrm{ek}}$ on $\mathbf{Q}$ requires that the electronic eigenvalues are also functions of $\mathbf{Q}$. $n$ and $\nu$ are the quantum numbers of electron and nuclei.
The $E_n(\mathbf{Q})$ as function of the coordinates act as an effective potential of the adiabatic electron-nuclei interaction, which depends on the quantum state the electronic subsystem occupies. Together with the repulsive Coulomb interaction between the nuclei it provides an effective potential in which the electrons move
\begin{equation}
V_n(\mathbf{Q})=V_\mathrm{kk}(\mathbf{Q})+E_n(\mathbf{Q})
\end{equation}
The effective potential $V_0(\mathbf{Q})$ associated with the electronic ground state is called the Born-Oppenheimer surface. As indicated in Fig. 1.1 the electronic terms are often well separated near the equilibrium positions and the vibrational motion in solids or the vibration and rotational motion in molecules leads to a finely spaced energy spectrum associated with each electronic quantum states.
Fig. 1.1: Illustration of the Born-Oppenheimer surface $V_{0}(\mathbf{Q})$ and the potential surface of the first excited state $V_{1}(\mathbf{Q})$ for one effective coordinate $Q$. The quantization of the motion of the nuclei on the two potential surfaces is indicated by the first four levels. Note that the separation between the ground states $E_{00}$ and $E_{01}$ is much larger than that between the other excited states $E_{0\nu}$.