Editing Fock matrix

{{Refimprove|date=February 2013}}
In the [[Hartree–Fock method]] of [[quantum mechanics]], the '''Fock matrix''' is a [[matrix (mathematics)|matrix]] approximating the single-electron [[energy operator]] of a given [[Quantum mechanics|quantum]] system in a given set of [[basis set (chemistry)|basis]] vectors.<ref>{{Cite book|first=J.|last= Callaway|title=Quantum Theory of the Solid State|publisher= Academic Press|location= New York|year=1974|url=https://books.google.com/books?id=HKTvAAAAMAAJ|isbn=9780121552039}}</ref>
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 (quantum theory)|Hamiltonian]] [[Operator (mathematics)|operator]] of the quantum system. It includes the effects of [[electron|electron-electron]] [[Coulomb force|repulsion]] only in an average way. 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.  In its general form the Fock operator writes:
:<math>\hat F(i) = \hat h(i)+\sum_{ j=1 }^{N} [\hat J_j(i)-\hat K_j(i)]</math>
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 <math>\hat J</math> terms are duplicated and the exchange terms are null between different spins.
For the restricted case which assumes [[closed-shell]] [[atomic orbital|orbitals]] and single-
determinantal wavefunctions, the Fock operator for the ''i''-th electron is given by:<ref>Levine, I.N. (1991) ''Quantum Chemistry'' (4th ed., Prentice-Hall), p.403</ref>

:<math>\hat F(i) = \hat h(i)+\sum_{ j=1 }^{n/2}[2 \hat J_j(i)-\hat 
K_j(i)]</math>

where:

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

:<math>{\hat h}(i)</math> is the one-electron [[Hamiltonian (quantum mechanics)|Hamiltonian]] for the ''i''-th electron,

:<math>n</math> is the number of electrons and <math> \frac{n}{2} </math> is the number of occupied orbitals in the closed-shell system,

:<math>\hat J_j(i)</math> is the [[Coulomb operator]], defining the repulsive force between the ''j''-th and ''i''-th electrons in the system,

:<math>\hat K_j(i)</math> 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.

==See also==
*[[Hartree–Fock method]]
*[[Unrestricted Hartree–Fock]]
*[[Restricted open-shell Hartree–Fock]]

==References==
{{Reflist}}

{{Matrix classes}}

{{DEFAULTSORT:Fock Matrix}}
[[Category:Atomic, molecular, and optical physics]]
[[Category:Quantum chemistry]]
[[Category:Matrices (mathematics)]]


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