Total Energy

The total HF energy is:

$E = \frac{1}{2} Tr [P (H + F)] + E_{nuc}$

where:

$P$ is the density matrix
$H$ is the core Hamiltonian matrix
$F$ is the Fock matrix
$E_{nuc} = \sum_{A < B} \frac{Z _{A} Z _{B}}{R _{A B}}$ is the nuclear repulsion energy

Derivation

Starting from the expectation value of the Hamiltonian for a single Slater determinant:

$E = i \sum ⟨ i ∣ \hat{h} ∣ i ⟩ + \frac{1}{2} i, j \sum [⟨ ij ∣ ij ⟩ - ⟨ ij ∣ ji ⟩]$

Substituting the LCAO expansion and density matrix definition yields the trace formula above.

The total energy is partitioned into:

Component	Formula	Description
Electronic	$\frac{1}{2} Tr [P (H + F)]$	Sum of one- and two-electron contributions
Nuclear	$E_{nuc}$	Classical repulsion between nuclei

The nuclear repulsion energy is a simple pairwise sum:

$E_{nuc} = A < B \sum \frac{Z _{A} Z _{B}}{R _{A B}}$

where $Z_{A}$ are atomic numbers and $R_{A B}$ are inter-nuclear distances.

The SCF procedure monitors:

Criterion	Default Threshold	Description
$Δ E$	10⁻⁸ Hartree	Energy change between iterations
$∥Δ P ∥_{RMS}$	10⁻⁸	Density matrix RMS change
$∥ g ∥_{max}$	10⁻⁸	Gradient (commutator) max norm

These can be tightened with the Conver keyword in the SCF section (Conver 10 for 10⁻¹⁰, etc.).