In the Hartree-Fock method of 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 *i*-th electron is given by:

where:

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

- is the one-electron hamiltonian for the
*i*-th electron,

- is the total number of occupied orbitals in the system (equal to, where is the number of electrons),

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

- is the exchange operator, defining the quantistic effect produced by exchanging two electrons.

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

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“As all historians know, the past is a great darkness, and filled with echoes. Voices may reach us from it; but what they say to us is imbued with the obscurity of the *matrix* out of which they come; and try as we may, we cannot always decipher them precisely in the clearer light of our day.”

—Margaret Atwood (b. 1939)