How electrons create potential-energy surfaces

Hold the nuclei still to calculate electronic energy

Let $r$ denote electronic coordinates and $R$ the nuclear configuration. For a fixed configuration, the electronic Hamiltonian $\hat H_e(R)$ determines allowed states $\phi_n(r;R)$ and energies $E_n(R)$. Include nuclear repulsion in the energy surface.

Repeat the electronic calculation at neighboring nuclear configurations. Joining the energies of a specified state gives a potential-energy surface. With one coordinate it is a curve. Its vertical axis measures energy. Drawing actual atomic positions requires a separate map from nuclear coordinates to molecular geometry.

Reading: Curchod · John C. Tully

Curvature sets the small-vibration frequency

For a bond, define the extension $x=R-R_e$, positive toward longer length. At equilibrium the first derivative of energy vanishes. Nearby, the leading change is quadratic, with curvature $k$. The relative motion of two nuclei uses the reduced mass $\mu=M_1M_2/(M_1+M_2)$.

Force is the negative energy slope. Substitution gives the harmonic-oscillator equation and its angular frequency. In SI units, curvature is measured in newtons per meter and angular frequency in inverse seconds. Large stretches require the full potential because a real bond can dissociate.

Worked example with a heavier isotope

Suppose isotopic substitution doubles the reduced mass and leaves the Born–Oppenheimer energy curve unchanged. The frequency becomes $1/\sqrt2$ times its original value and the period grows by $\sqrt2$. The change acts through nuclear inertia in this approximation.

The quantum ground state also becomes narrower. Its position standard deviation is $\sigma_x=\sqrt{\hbar/(2\mu\omega)}$. After substituting the frequency, this width scales as mass to the negative quarter power. A simulation should therefore show a slower vibration and a narrower initial distribution; they test different aspects of the model.

A nuclear packet can spread toward several outcomes

A nuclear wavefunction $\chi(R,t)$ is a complex amplitude. Its density $|\chi(R,t)|^2$ becomes a probability only after integration over a coordinate interval. The complex phase matters when two amplitudes overlap.

A momentum kick can send some amplitude outward and leave some bound. Probability beyond a chosen distance measures the distribution at that time. Calling it a final dissociation yield requires separating outgoing and returning components and checking stability when the final time and exit window move.

When a second surface matters

The Born–Oppenheimer approximation lets the electrons adjust to moving nuclei and follows nuclear motion primarily on one electronic state. Near a small energy gap, a rapidly changing electronic basis can transfer amplitude between states. Several nuclear components must then be evolved together.

Mass alone cannot decide whether this happens. We also need the gap, nuclear speed and the spatial rate at which the electronic states change. The next chapter puts those ingredients in the same equation.