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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.[1]
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. 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:
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 terms are duplicated and the exchange terms are null between different spins.
For the restricted case which assumes closed-shell orbitals and single-
determinantal wavefunctions, the Fock operator for the i-th electron is given by:[2]
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 number of electrons and is the number of occupied orbitals in the closed-shell system,
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 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.
^Callaway, J. (1974). Quantum Theory of the Solid State. New York: Academic Press. ISBN 9780121552039.
In the Hartree–Fock method of quantum mechanics, the Fockmatrix is a matrix approximating the single-electron energy operator of a given quantum system...
Vladimir Aleksandrovich Fock (or Fok; Russian: Влади́мир Алекса́ндрович Фок) (December 22, 1898 – December 27, 1974) was a Soviet physicist, who did foundational...
the Fockmatrix (which depends on the coefficients C due to electron-electron interactions), C is a matrix of coefficients, S is the overlap matrix of...
Vladimir Fock: FockmatrixFock operator Fock model Fock representation Fock space Bargmann–Fock space Fock state Fock symmetry Fock–Lorentz symmetry Fock–Schwinger...
be applied, where the states summed over to make the density matrix are drawn from a Fock space.: 174 Quantum decoherence theory typically involves non-isolated...
physics, within the Hartree–Fock theory, the atomic and molecular orbitals can be defined by the eigenvectors of the Fock operator. The corresponding...
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"Linear scaling computation of the Fockmatrix. II. Rigorous bounds on exchange integrals and incremental Fock build". The Journal of Chemical Physics...
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually...
correlation effects on the fragment can be accurately modeled by a post-Hartree–Fock solver. This method has shown high-quality results in 1D- and 2D- Hubbard...
notably, by Pascual Jordan and Vladimir Fock. In this approach, the quantum many-body states are represented in the Fock state basis, which are constructed...
Moreover, the matrix is always positive definite; that is to say, the eigenvalues are all strictly positive. Roothaan equations Hartree–Fock method Pi bond...
y}\,} The R matrix is the probability per unit time for the particle to make a transition from x to y. The condition that the K matrix elements add up...
energy bound state coherent state squeezed coherent state density state Fock state, Fock space vacuum state quasinormal mode no-cloning theorem quantum entanglement...
simpler electron functions with the dominant function being the Hartree-Fock function. Each of these simple functions are then approximated using only...
{E} (u\mathbf {r} ,t)du.} The gauge condition of the Fock–Schwinger gauge (named after Vladimir Fock and Julian Schwinger; sometimes also called the relativistic...
other mode". Finally, there also exist twin Fock states for bosonic modes, which can be created by feeding a Fock state into two arms leading to a beam splitter...
canonical quantization of a Yang–Mills theory and its correct application to the Fock space of instantaneous field configurations were elucidated by Taichiro Kugo...
the electric fields are operators as explained by second quantization and Fock states. Each electrical field operator can further be expressed in terms...
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![{\displaystyle {\hat {F}}(i)={\hat {h}}(i)+\sum _{j=1}^{N}[{\hat {J}}_{j}(i)-{\hat {K}}_{j}(i)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55aa35162bbddbdedba26a16c33d1ddb05841919)
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