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.

Selected energy error $0.183$

Selected force error $-0.0788$

Second natural occupation $0.644$

Separation in bohr $4$

Hydrogen · STO-3G

Compare the whole curve

Nuclear separation

R/a_0

4.00

Method to inspect

Drag a nucleus to select a saved geometry. The scene always shows the FCI reference density.

What the cloud means

$n(\mathbf r)=\sum_{\mu\nu}P_{\mu\nu}\chi_\mu(\mathbf r)\chi_\nu(\mathbf r)$

The cloud uses the same density matrix as the calculation. Its translucent surfaces join points of equal density. Gold spheres mark enlarged nuclei. Cyan marks total density; in UHF, coral marks alpha-spin excess and cyan marks beta-spin excess.

Total-density surfaces use the two levels below. In UHF, the inner surfaces show the signed spin-density difference. The section view uses continuously shaded density.

$n=0.017,\ 0.075\;a_0^{-3}$

$n_\alpha-n_\beta=\pm0.020\;a_0^{-3}$

$a_0=0.529177210903\,\mathrm{\mathring A}$ $E_{\rm h}=27.211386246\,\mathrm{eV}$

PBE is the saved restricted Kohn–Sham reference. Select its fixed geometries to compare energy and force errors.

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

RHFUHFFCIMP2PBE · 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

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 misconceptionA small energy error at equilibrium guarantees an accurate dissociation curve.

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

Run the experiment

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.
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.
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.

Map Electronic-Structure Errors Along a Bond

Where each method begins to struggle