“Know how to solve every problem that has been solved.” “What I cannot create, I do not understand.” — Richard Feynman

Geometry optimization in PySCF, by the diff

Quantum Chemistry

A single-point energy tells you what one fixed shape costs. Geometry optimization answers the question a chemist actually cares about: what shape does the molecule adopt? The route there is short — compute the forces on the nuclei, then walk downhill until they vanish — and in PySCF it's a handful of one-line additions on top of the basics. As before, every program runs and every number below is what PySCF actually printed.

The chain: a single point at a deliberately-bad guess → the gradient (forces) → optimize() → confirm the minimum → the effect of the level of theory. Step with the arrows; hover a marked line for its rationale.

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Single point — one fixed shape

starting point

 from pyscf import gto, scf
  
 mol = gto.M(
     atom="O 0 0 0; H 0 -0.90 0.70; H 0 0.90 0.70",
     basis="sto-3g",
 )
  
 mf = scf.RHF(mol)
 e = mf.kernel()
 print(f"E = {e:.6f} Ha")

the starting point

One energy at one fixed geometry — a guess, with the O–H bonds stretched to 1.14 Å (real water is about 0.96). This is the same single-point Hartree-Fock from the basics tutorial. The energy is just a number for this one shape. Geometry optimization answers the next question: which shape does the molecule actually want?

hover a marked line (left bar) for its rationale

what it prints

E = -74.925328 Ha

Everything follows from the gradient

Three things to take from the diffs, all visible in the output panels:

A geometry is optimized exactly when the gradient is zero (steps 2 vs 4). Same calculation, before and after: max|grad| falls from 0.122 to 0.000014 Ha/bohr. The gradient is the force; no force means no incentive to move; that's the minimum. The optimizer just chases the gradient to zero.
Optimization returns a new molecule, the energy drops (step 3). −74.925 → −74.966 Ha as the stretched 1.14 Å guess relaxes to 0.989 Å. You always re-attach a method to the optimized Mole to read properties at the new geometry.
The minimum lives on the surface you chose (step 5). RHF/sto-3g bottoms out at 0.989 Å / 100.0°; B3LYP/cc-pVDZ at 0.969 Å / 102.7°, against experiment's 0.957 Å / 104.5°. Optimization is faithful to the surface; whether the surface is faithful to reality is a separate choice.

As a citeable, falsifiable claim: at a converged geometry optimum the maximum nuclear gradient is zero (to the convergence threshold), because the optimum is where the forces vanish. The experiment is steps 2 and 4; it would be falsified by a structure the optimizer reports as converged while a fresh gradient comes back large — which would mean a broken convergence test, not a real minimum.

Geometry optimization in PySCF, by the diff

Everything follows from the gradient

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