Separation of the Molecular Hamiltonian

The full molecular Hamiltonian is $\hat{H} = \hat{T}N + \hat{T}_e + \hat{V}{ee} + \hat{V}{eN} + \hat{V}{NN}$. The BO ansatz factorises the total wavefunction as

\[\Psi(\mathbf{r}, \mathbf{R}) = \psi_{el}(\mathbf{r};\mathbf{R})\,\chi_{nuc}(\mathbf{R})\]

where $\mathbf{r}$ denotes electronic and $\mathbf{R}$ nuclear coordinates. The electronic Schrödinger equation is solved for fixed nuclei:

\[\hat{H}_{el}\psi_{el}(\mathbf{r};\mathbf{R}) = E_{el}(\mathbf{R})\,\psi_{el}(\mathbf{r};\mathbf{R})\]

Potential Energy Surface

The electronic energy $E_{el}(\mathbf{R})$ plus nuclear repulsion $V_{NN}(\mathbf{R})$ defines the PES:

\[U(\mathbf{R}) = E_{el}(\mathbf{R}) + V_{NN}(\mathbf{R})\]

Nuclei then move on this surface according to

\[\left[\hat{T}_N + U(\mathbf{R})\right]\chi_{nuc}(\mathbf{R}) = E_{tot}\,\chi_{nuc}(\mathbf{R})\]

Critical points on the PES — minima (stable structures), saddle points (transition states), and conical intersections — determine reaction pathways and rates.

Adiabatic and Non-Adiabatic Effects

The BO approximation neglects the nuclear kinetic energy operator acting on $\psi_{el}$, specifically the non-adiabatic coupling terms

\[d_{IJ}(\mathbf{R}) = \langle\psi_I|\nabla_{\mathbf{R}}|\psi_J\rangle\]

These become large near conical intersections where two PES cross. At such points, the single-surface BO picture breaks down and non-adiabatic dynamics — such as photochemical processes and ultrafast spectroscopy — must treat multiple coupled surfaces simultaneously using methods like surface hopping or MCTDH.

Validity and Limitations

Regime	BO Valid?	Reason
Ground-state thermochemistry	Yes	Large energy gaps
Excited-state photochemistry	Often No	Near-degenerate surfaces
Proton transfer / tunnelling	Partial	Light nucleus mass
Heavy nuclei, low temperature	Yes	Classical nuclear motion