---
title: "How electrons create potential-energy surfaces"
description: "Use a vibrating bond to understand energy surfaces, nuclear wave packets and reaction exits before adding a second electronic state."
canonical_url: "https://sandboxphysics.com/stories/nonadiabatic-dynamics/potential-energy-surfaces/"
language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/nonadiabatic-dynamics/potential-energy-surfaces/"
last_modified: "2026-10-28"
---

[Nonadiabatic molecular dynamics](https://sandboxphysics.com/stories/nonadiabatic-dynamics/)

L1 · Chapter 1

# How electrons create potential-energy surfaces

Sandbox Physics · 2026-10-28

Electronic energy changes when the nuclei move, and its slope supplies a force. Near a stable bond the energy looks like a bowl. Exciting the electrons can change that landscape and give the same nuclear preparation a different future.

## Before you begin

You need kinetic and potential energy and a graph of a function. A derivative is a local slope. A wavefunction’s squared magnitude is a probability density. We begin with one bond coordinate before adding dimensions and electronic states.

- Distinguish nuclear configuration, electronic state and potential surface.
- Estimate vibration from the curvature of an energy curve.
- Separate probability beyond a finite distance from a final reaction yield.

## 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.

$$
\hat H_e(R)\phi_n(r;R)=E_n(R)\phi_n(r;R)
$$

Nuclear configuration is a parameter in this electronic eigenvalue problem.

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.

$$
E(x)\simeq E(0)+\frac12kx^2,\qquad \mu\ddot x=-kx,\qquad \omega=\sqrt{k/\mu}
$$

The quadratic expansion assumes the wave packet remains close enough to the minimum.

## 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.

$$
P_{R>R_c}(t)=\int_{R_c}^{\infty}|\chi(R,t)|^2\,dR
$$

The threshold and observation time belong to the definition of this finite-time probability.

## 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.

## Try the prediction

### [Compare a returning packet with an escaping component](https://sandboxphysics.com/labs/morse-molecular-bond-breaking/)

1. Predict which part of the packet will leave the well after a stronger momentum kick.
2. Change only the kick and compare the Morse potential with the ideal spring.
3. Pause at the same time and distinguish stretched, returning and outward-moving probability.

## Exercises and explanations

1. Both nuclear masses double without changing the energy curvature. What happens to the small-vibration period?

   Read the explanation

   The reduced mass doubles. Frequency falls by the square root of two and period grows by the square root of two.

2. Does a higher point on the plotted surface mean an atom is higher above the floor?

   Read the explanation

   The vertical coordinate denotes electronic energy. A physical molecular geometry must be supplied separately from this configuration-space graph.

[Next: What changes when the electronic basis moves](https://sandboxphysics.com/stories/nonadiabatic-dynamics/moving-electronic-basis/)

## Sources and calculation scope

1. [John C. Tully, Molecular dynamics with electronic transitions, J. Chem. Phys. 93, 1061–1071 (1990)](https://doi.org/10.1063/1.459170)

   Published primary method paper. Sections II–III define fewest switches and its one-dimensional test models. Our three-method comparison uses the simple avoided-crossing Hamiltonian; the nuclear packet, finite windows and numerical settings are stated with each saved run.

2. [Curchod, Agostini & Gross, An Exact Factorization Perspective on Quantum Interferences in Nonadiabatic Dynamics, arXiv:1604.02351v2](https://arxiv.org/html/1604.02351v2)

   Fixed 30 June 2016 author version; published in J. Chem. Phys. 145, 034103 (2016), DOI 10.1063/1.4958637. Reading scope: equations (5)–(6), (14)–(18), and Figures 3–7. Its Shin–Metiu electron–ion calculation differs from the Tully and conical teaching models.