Fock matrix

In quantum mechanics, the Fock matrix is a matrix approximating the single-electron energy operator of a given quantum system in a given set of basis vectors.

It is most often formed in computational chemistry when attempting to solve the Roothaan equations for an atomic or molecular system. The Fock matrix is actually an approximation to the true Hamiltonian operator of the quantum system. It includes the effects of electron-electron repulsion only in an average way. Importantly, because the Fock operator is a one-electron operator, it does not include the electron correlation energy.

The Fock matrix is defined by the Fock operator. For the restricted case which assumes closed-shell orbitals and single-determinantal wavefunctions, the Fock operator for the first electron is given by:

${\hat {F}}(1)={\hat {H}}^{core}(1)+\sum _{j=1}^{n}[2{\hat {J}}_{j}(1)-{\hat {K}}_{j}(1)]$

where:

${\hat {F}}(i)$

is the Fock operator for the i-th electron in the system,

${\hat {H}}^{core}(i)$

is the core Hamiltonian for the i-th electron,

$n$

is the total number of orbitals in the system (equal to half the number of electrons),

${\hat {J}}_{j}(i)$

is the Coulomb operator, defining the repulsive force between the j-th and i-th electrons in the system,

${\hat {K}}_{j}(i)$

is the exchange operator, defining the effect of exchanging the two electrons.

See also

• Roothaan equations
• Hartree-Fock