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The Fock matrix is defined by the Fock operator. In its general form the Fock operator writes:

${\displaystyle \hat{F}(i)=\hat{h}(i)+\sum _{j=1}^N[{\hat{J}}_j(i)-{\hat{K}}_j(i)]}$

Wherei runs over the totalN spin orbitals. In the closed-shell case, it can be simplified by considering only the spatial orbitals. Noting that the${\displaystyle \hat{J}}$ terms are duplicated and the exchange terms are null between different spins.For the restricted case which assumesclosed-shellorbitals and single-determinantal wavefunctions, the Fock operator for thei-th electron is given by:^[1]

${\displaystyle \hat{F}(i)=\hat{h}(i)+\sum _{j=1}^{n/2}[2{\hat{J}}_j(i)-{\hat{K}}_j(i)]}$

where:

${\displaystyle \hat{F}(i)}$ is the Fock operator for thei-th electron in the system,

${\displaystyle \hat{h}(i)}$ is the one-electronHamiltonian for thei-th electron,

${\displaystyle n}$ is the number of electrons and${\displaystyle \frac{n}{2}}$ is the number of occupied orbitals in the closed-shell system,

${\displaystyle {\hat{J}}_j(i)}$ is theCoulomb operator, defining the repulsive force between thej-th andi-th electrons in the system,

${\displaystyle {\hat{K}}_j(i)}$ is theexchange operator, defining the quantum effect produced by exchanging two electrons.

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 thei-th electron.

For systems with unpaired electrons there are many choices of Fock matrices.

• Hartree–Fock method
• Unrestricted Hartree–Fock
• Restricted open-shell Hartree–Fock

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