Building the Fock matrix, term by term

In the basis, the Fock matrix is the core Hamiltonian plus a two-electron part G built by contracting the ERI tensor with the density matrix P:

The first term in the bracket is Coulomb: the electron in μν feels the repulsion of the whole charge density. The second is exchange, and its ½ is bookkeeping with physical content — exchange only operates between same-spin electrons, and in a closed-shell density exactly half the electrons share any given spin.

Stare at the formula and the circularity of Hartree-Fock stares back: F is built from P, but P comes from the orbitals you get by diagonalizing F. Hold that thought — it becomes the SCF loop two lessons from now.

One trap worth springing on yourself deliberately: the orbital energies ε that come out of F each contain that orbital's repulsion against everyone else. Sum the occupied ε's and every pair interaction gets counted twice.