---
title: "Map Electronic-Structure Errors Along a Bond Interactive Lab | Sandbox Physics"
description: "Does a small energy error also guarantee a good bond force? A force depends on how quickly the energy changes with separation."
canonical_url: "https://sandboxphysics.com/labs/electronic-method-error-map/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/electronic-method-error-map/"
---

FR13-C · A method needs more than one test

# Map Electronic-Structure Errors Along a Bond

**Does a small energy error also guarantee a good bond force?** A force depends on how quickly the energy changes with separation. A small energy offset can have a steep slope, and an accurate-looking energy at one geometry can hide a poor dissociation curve. Compare energies and their derivatives across several geometries.

Compare several electronic methods against same-basis FCI along a fixed hydrogen stretching grid. Inspect energy errors, force errors and the occupations that signal a changing electronic state.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

The map compares deterministic electronic-structure calculations for fixed hydrogen geometries in STO-3G. Its energy and force data come from saved PySCF calculations for RHF, UHF, MP2, FCI and restricted PBE, with software versions recorded. Force differences use finite differences. The scene freshly calculates FCI reference density for every method. DFT energies are not variational upper bounds to FCI.

Saved PySCF calculations · STO-3G

## Where each method begins to struggle

RHF UHF FCI MP2 PBE · offline

A force tests the slope

Energy and force have different units. Read each comparison against its own axis.

Units, density and calculation details

$\operatorname{Tr}(PS)=2$

All total energies include nuclear repulsion. FCI solves the same finite Hamiltonian as HF. The saved calculation includes its geometry, method, density matrices and numerical diagnostics.

[Learn how the calculation works, from orbitals to a broken bond](https://sandboxphysics.com/stories/molecular-electronic/)

Physics tutorial

## Map Electronic-Structure Errors Along a Bond

**Background** An electronic method predicts a whole potential-energy curve. Its slope sets the force on the nuclei, so one accurate energy cannot certify the curve.

**Why it matters** Comparing the same geometries and basis isolates differences between electronic methods.

Start with the essentials

- **Focus question**: Does a small energy error also guarantee a good bond force?
- **One-sentence intuition**: Force error is the negative slope of energy error along the same geometric coordinate.

### Core mathematical model

#### Compare the slope as well as the height

$$
\Delta E_m(R)=E_m(R)-E_{\rm FCI}(R),\qquad \Delta F_m(R)=-\frac{d\,\Delta E_m}{dR}
$$

All methods use the same nuclear coordinate and include the same nuclear repulsion. Energy and force errors therefore answer different questions.

#### A controlled numerical derivative

$$
F(R)\approx-\frac{E(R+h)-E(R-h)}{2h}
$$

The reference record fixes the derivative step and stores the method and software version. Decreasing the step checks numerical sensitivity.

### Common difficulties

#### One accurate point

**Typical misconception** A small energy error at equilibrium guarantees an accurate dissociation curve.

**Better mental model** A whole curve can bend incorrectly despite passing through one accurate point. Forces and occupations reveal changes that a single energy misses.

### Run the experiment

1. 01

   #### Predict before stretching

   Choose Near the energy minimum, then Correlation grows. Say what you expect the density and the energy difference to do before changing the preset.

   **What to observe:** The molecular density separates into two lobes, and the restricted description loses accuracy as the bond stretches.

2. 02

   #### Follow the second diagnostic

   Select each method in turn. Compare the energy-error map with the force-error graph at the same geometry.

   **What to observe:** A method can have a modest energy error and a noticeable force error. The latter follows the slope of the former.

3. 03

   #### Save a reproducible comparison

   Use Save calculation at two separations. Compare the saved basis, geometry, density matrices and total energies.

   **What to observe:** Every record identifies the finite electronic model. Comparing it with a larger basis tests a separate source of error.

Physical conclusion

Judge a molecular method across geometries, energies, forces and occupations.

## Related experiments and articles

Molecular electrons and breaking bonds · Reading path

### Orbitals, self-consistency, correlation and dissociation

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **When a small correlation correction becomes too large**](https://sandboxphysics.com/stories/molecular-electronic/perturbation-correlation/)
2. [Article **What density functional theory changes**](https://sandboxphysics.com/stories/molecular-electronic/density-functional-theory/)
3. [Article **An accurate energy can still give the wrong bond force**](https://sandboxphysics.com/stories/molecular-electronic/energies-forces-and-bases/)
4. [Article **A fractional-spin test for a broken bond**](https://sandboxphysics.com/stories/molecular-electronic/fractional-spin-paper/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)