Born–Oppenheimer separation and potential surfaces

Let r denote electronic coordinates and R nuclear coordinates after any needed overall-motion separation. Write the nonrelativistic molecular Hamiltonian as nuclear kinetic energy plus an electronic Hamiltonian plus nucleus-nucleus repulsion. Here $H_e$ includes electron kinetic energy, electron-nucleus attraction, and electron-electron repulsion, but excludes nucleus-nucleus repulsion. At clamped R, diagonalize this electronic operator. Add the repulsion once to form each nuclear potential surface. Some references include it inside the clamped Hamiltonian instead; both conventions work if they are not mixed.

A surface is a function of nuclear geometry, not a material membrane. A diatomic has a potential curve in its internuclear separation after translations and orientations are separated. A nonlinear molecule has many internal coordinates and can possess minima, saddle regions, and intersecting surfaces. An electronic surface calculated with an approximate basis or electronic method has its own numerical and model errors. Separating nuclear motion does not make an approximate electronic calculation exact, nor does an empirical fitted curve become an ab initio prediction merely because nuclei move on it.

Expand the total wavefunction as a sum of electronic functions multiplied by nuclear amplitudes. In a complete electronic basis this Born–Huang expansion is a representation rather than a single-surface approximation. Applying nuclear kinetic energy differentiates both factors. Besides the ordinary second derivative of each nuclear amplitude, this creates first and second derivatives of the electronic functions with respect to geometry. Projecting onto another electronic state produces couplings between nuclear channels.

The elementary Born–Oppenheimer approximation retains one surface and neglects those intersurface derivative terms; refined adiabatic treatments can retain a diagonal correction. The small electron-to-nuclear mass ratio motivates slow nuclear dynamics, but a small electronic energy denominator can offset that advantage. Electronic wavefunctions may rotate rapidly in state space near an avoided crossing or conical intersection. Thus masses, nuclear velocities, energy gaps, and derivative matrix elements must be considered together when estimating whether a surface can be followed independently.

For a diatomic radial vibration the relevant mass is the reduced mass $\mu=m_Am_B/(m_A+m_B)$, not either atomic mass alone. Near a surface minimum with curvature k, the harmonic angular frequency is $\sqrt{k/\mu}$. Isotope substitution therefore changes nuclear level spacings even when the leading clamped-electron surface stays the same. Small mass-dependent adiabatic and nonadiabatic corrections can still matter in precision spectroscopy, so the isotope-independent-surface statement is a leading approximation.

A complete interpretation should keep three questions separate: whether the electronic surface is accurate, whether nuclear dynamics can remain on one surface, and whether rotation and vibration can themselves be decoupled. They involve different approximations. A good single-surface result for low vibration need not survive dissociation or excitation into a crossing region. Conversely, a visible isotope shift is not automatically evidence of Born–Oppenheimer failure; most isotope shifts already follow from the reduced mass in an otherwise unchanged nuclear potential.