Fock exchange interaction

We briefly present a heuristic argument how quark matter becomes ferromagnetic by the use of above definition [8], The Fock exchange interaction, /kc.qC between two quarks is defined by... 

In this talk we have discussed a magnetic aspect of quark matter based on QCD. First, we have introduced ferromagnetism (FM) in QCD, where the Fock exchange interaction plays an important role. Presence of the axial-vector mean-field (AV) after the Fierz transformation is essential to give rise to FM, in the context of self-consistent framework. As one of the features of the relativistic FM, we have seen that the Fermi sea is deformed in the presence of... 

Akama, T, Kobayashi, M., Nakai, H. Implementation of divide-and-conquer method including Hartree-Fock exchange interaction. J. Comput. Chem. 28, 2003-2012 (2007)... 

A I lai lcee-Fock calculation provides a set of orbital energies, e,. What is the significance oi these The energy of an electron in a spin orbital is calculated by adding the core inleraclion to the Coulomb and exchange interactions with the other electrons in the svstein ... 

The intraatomic d-d electron-electron interaction includes Coulomb and exchange interactions, and it is responsible for orbital and spin polarization. To account for orbital polarization, the idea of the LDA + U method was followed.70 A generalized Hartree-Fock approximation including all possible pairings was then used to write... 

About the same time, Douglas Hartree, along with other members of the informal club for theoretical physics at Cambridge University called the Del-Squared Club, began studying approximate methods to describe many-electron atoms. Hartree developed the method of the self-consistent field, which was improved by Vladimir Fock and Slater in early 1930, so as to incorporate the Pauli principle ab initio.37 Dirac, another Del-Squared member, published a paper in 1929 which focused on the exchange interaction of identical particles. This work became part of what soon became called the Heisenberg-Dirac approach.38... 

The Xa multiple scattering method generates approximate singledeterminant wavefunctions, in which the non-local exchange interaction of the Hartree-Fock method has been replaced by a local term, as in the Thomas-Fermi-Dirac model. The orbitals are solutions of the one-electron differential equation (in atomic units)... 

The sum over coulomb and exchange interactions in the Fock operator runs only over those spin-orbitals that are occupied in the trial VF. Because a unitary transformation among the orbitals that appear in F leaves the determinant unchanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to choose such a unitary transformation to make the Ey matrix diagonal. Upon so doing, one is left with the so-called canonical Hartree-Fock equations ... 

Calculations based on the continuum dielectric model have been performed by the hydrated electron in the limit of zero cavity size (19). The general treatment is based on a variational calculation using hydrogenic type wave functions for the ground and the first excited states. This treatment is based on a Hartree Fock scheme, where the Coulomb and exchange interaction of the excess electron with the medium are replaced by the polarization energy of a continuous dielectric. The results obtained are summarized in Table V. The fair agreement obtained with... 

The physical origin of correlation energy is in the nature of the Hartree-Fock equations. The inter-electronic interaction is represented by coulombic and exchange terms each electron has a direct interaction with the average charge of all the others obtained by squaring the one-electron functions (the molecular orbitals), but an exchange interaction only with elections of the same... 

Current hybrid functionals do not improve this situation. Their non-local component (Hartree-Fock exchange) cannot give rise to any attraction. To describe quantitatively the long-range interactions, either a non-local approximation to Exc[p] must be applied within the Kohn-Sham framework or methods using other-than-Kohn-Sham formalism should be used. Some of such approaches will be discussed in the last section of this review. Here, we mention an especially promising combination of symmetry adapted perturbation theory with of the Kohn-Sham orbitals.125... 

Figure 5.4 Diagrammatic representation of the first-order Hartree-Fock exchange diagram (a) no interaction with the leads is included, (b) the interaction with the leads is considered in the second order with respect to V2 for each G (c) the interaction with the lead electrons is considered in all orders to V2, and (d) the diagram (c) is presented according to Dyson equation (62) where the zeroth Green s functions, G , have been substituted by the renormalized zeroth Green s functions, G ...

The spin contribution to the magnetic susceptibility comes from the exchange term of the Coulomb interaction. If we take into account the exchange interaction in the Hartree-Fock approximation, the ground state of an electron gas is tht ferromagnetic state, in which all spins are parallel. 

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