Exchange interaction approximation

These two expressions differ in a very important way the sum of occupied orbital energies double counts the Coulomb minus exchange interaction energies. Thus, within the FIE approximation, the sum of the occupied orbital energies is not equal to the total energy. 

The Lines approximation is expected to be quite accurate for the description of the exchange interaction between a strongly axial doublet and an arbitrary isotropic spin. For all other cases, the Lines model [84] is a reasonable approximation. Efficient implementation of the Lines model was done in the program POLY ANISO. 

In other words, in this approximation for electronic states given by Eqs. (3.87) and (3.88) the triplet-triplet transition is due to the exchange interaction. 

Ferromagnetic ordering of two-dimensional systems with dipole-dipole and exchange interactions in the approximation of a spherical model was examined in Ref. 73. The main simplifying assumption of the spherical model is the replacement of the condition er = 1 on the orientation vectors with the weaker condition... 

Note that the long-range classical electrostatic limit furnishes an excellent approximation for the electrical attraction of the end groups, since these are separated far outside the range of significant exchange interactions. 

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 exchange term represents the electrostatic interaction between the charge clouds. The transfer in fact occurs via overlap of the electron clouds and requires physical contact between D and A. The interaction is short range because the electron density falls off approximately exponentially outside the boundaries of the molecules. For two electrons separated by a distance in the pair D-A, the space part of the exchange interaction can be written as... 

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 decay time of the Cr " band of approximately 150 ns is very short for such emission. Radiative energy transfer may not explain it because in such a case the decay curves of each of the ions are independent of the presence of the other. Thus non-radiative energy transfer may also take part, probably via multipolar or exchange interactions. In such cases the process of luminescence is of an additive nature and the lifetime of the sensitizer from which the energy is transferred is determined, apart from the probability of emission and radiationless transitions, by the probability of the energy transfer to the ion activator. 

In a magnetic compound we have to consider exchange interactions in the molecular field approximation, the Curie law takes the Curie Weiss form... 

Low-dimensional Solids. - 2.10.1 Introduction. The magnetic properties of one and two-dimensional arrays of localized spins coupled by Heisenberg exchange interactions have been studied as a rather specialized branch of theoretical physics since the earliest days of quantum mechanics. However, recent advances in theory, and the preparation of real materials that are a good approximation to the theoretical models, have made low-dimensional systems much more central to condensed-matter science. There is enormous scope for synthetic chemistry in this area and, as will be seen later in this section, many new materials have been discovered recently. 

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