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

Watching it converge

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Lesson 17 of 24 standard ~5 min

Here is the entire convergence history of the H₂ calculation, verbatim:

iter 0    E_total = -1.79131    Δ = 0.603
iter 1    E_total = -1.11671    Δ = 4.4e-16    ← converged

Δ is the largest change in any density-matrix element between iterations.

Two things deserve a double-take. First, the speed: two basis functions give a 2×2 problem, and the fixed point is essentially reached in one rebuild. (Helium needs 7 iterations to settle at −2.8162; real molecules typically take 10–30, and codes accelerate the loop with DIIS.)

Second — and this one catches people — iteration 0's energy is below the converged answer, even below the exact energy. No alarm bells needed: with the core-Hamiltonian guess, P = 0, so the first pass has F = H_core and the energy contains no electron-electron repulsion at all. Electrons that don't repel bind absurdly tightly. The variational bound applies to the converged Hartree-Fock energy, which is an expectation value of the true Hamiltonian — not to half-built intermediates along the way.

standardMultiple choice

A common SCF starting guess is the core Hamiltonian guess. What does it ignore in the first step?

Go deeper ↓The SCF iteration

The energy at each step is assembled with the half-trace formula, and its ½ is the same double-counting correction you met in the Fock lesson:

E_{total} = \frac{1}{2} tr [P (H^{core} + F)] + E_{nuc}

hardMultiple choice

The converged SCF electronic energy is computed as E = ½ tr[P(h + F)] rather than tr[P h]. Why the factor of ½ and the (h + F)?

Go deeper ↓The SCF iteration