In the

Hartree–Fock method In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state. The Hartree–Fock method often ...

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...

, the Fock matrix is a matrix approximating the single-electron energy operator of a given quantum system in a given set of

basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...

vectors. It is most often formed in computational chemistry when attempting to solve the

Roothaan equations The Roothaan equations are a representation of the Hartree–Fock equation in a non orthonormal basis set which can be of Gaussian-type or Slater-type. It applies to closed-shell molecules or atoms where all molecular orbitals or atomic orbitals ...

for an atomic or molecular system. The Fock matrix is actually an approximation to the true

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

operator of the quantum system. It includes the effects of electron-electron repulsion only in an average way. Because the Fock operator is a one-electron operator, it does not include the

electron correlation Electronic correlation is the interaction between electrons in the electronic structure of a quantum system. The correlation energy is a measure of how much the movement of one electron is influenced by the presence of all other electrons. Ato ...

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:Levine, I.N. (1991) ''Quantum Chemistry'' (4th ed., Prentice-Hall), p.403 :

\hat F(i) = \hat h(i)+\sum_^\hat J_j(i)-\hat 
K_j(i) /math>

where:

: \hat F(i) is the Fock operator for the ''i''-th electron in the system,

: (i) is the one-electron

for the ''i''-th electron, :

n

is the number of electrons and

\frac

is the number of occupied orbitals in the closed-shell system, :

\hat J_j(i)

is the

Coulomb operator The Coulomb operator, named after Charles-Augustin de Coulomb, is a quantum mechanics, quantum mechanical operator (mathematics), operator used in the field of quantum chemistry. Specifically, it is a term found in the Hartree–Fock, Fock operato ...

, 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 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 the ''i''-th electron. For systems with unpaired electrons there are many choices of Fock matrices.

References

Atomic, molecular, and optical physics Quantum chemistry Matrices {{quantum-chemistry-stub

See also

References