Fock matrix
The Fock matrix is defined by the Fock operator. In its general form the Fock operator writes:
Where i runs over the total N spin orbitals. In the closed-shell case, it can be simplified by considering only the spatial orbitals. Noting that the terms are duplicated and the exchange terms are null between different spins.
For the restricted case which assumes closed-shell orbitals and single-
determinantal wavefunctions, the Fock operator for the i-th electron is given by:
where:
The Coulomb operator is multiplied by two since there are two electrons in each occupied orbital. The exchange operator is not multiplied by two since it has a non-zero result only for electrons which have the same spin as the i-th electron.
For systems with unpaired electrons there are many choices of Fock matrices.