Exchange term

In the case of adjacent atoms, a bond exists characterized by a bond energy analogous to the p of Huckel theory but modified by an election exchange term... 

Fig. 2. Fluid temperature profiles in (a) a parallel flow heat exchanger and (b) a counterflow heat exchanger. Terms are defined in text.

In order to describe the incoherent exchange, in the pulse collision model [Lynden-Bell 1964 Johnson 1967 Stunzes and Benderskii 1971] an effective exchange term, whose real part is... 

Hartree-Fock theory also includes an exchange term as part of its formulation. Recently, Becke has formulated functionals which include a mixture of Hartree-Fodc and DFT exchange along with DFT correlation, conceptually defining... 

The Kohn-Sham equations look like standard HF equations, except that the exchange term is replaced with an exchange-correlation potential whose form is unknown. 

It has one adjustable parameter which was chosen so that the sum of the LDA and Becke exchange terms accurately reproduce the exchange energies of six noble gas atoms. 

Implementation of the Kohn-Sham-LCAO procedure is quite simple we replace the standard exchange term in the HF-LCAO expression by an appropriate Vxc that will depend on the local electron density and perhaps also its gradient. The new integrals involved contain fractional powers of the electron density and cannot be evaluated analytically. There are various ways forward, all of which... 

Although the exchange term in principle is short-ranged, and thus should benel significantly from integral screening, this is normally not observed in practic ... 

For closed-shell systems LSDA is equal to LDA, and since this is the most common case, LDA is often used interchangeably with LSDA, although this is not true in the general case (eqs. (6.16) and (6.17)). The method proposed by Slater in 1951 can be considered as an LDA mediod where die correlation energy is neglected and the exchange term is given as... 

The electron-electron exchange term, Hex In equation (16) it is necessary to consider only He . As has been discussed, the energy difference between T and S states is equal to Je . With a minimal overlap integral due to a relatively large inter-radical separation. Hex can be given by the Dirac exchange operator [equation (18)],... 

Within the hydration process in Eqn. (8.8), a spherical ion becomes a (hydrated) octahedral ion, [M(H20)6]. Part of the Coulomb energy of the free ion concerns repulsion and exchange terms within the d" configuration. This is replaced by equivalent repulsion and exchange terms within the configuration. Let us estimate the trends in these quantities separately. 

A heat exchange term is added to the energy balance. Equation (5.30), to give... 

Table 1.2 Key data for production of polymers in the compact reactor/mixer/heat exchanger, termed micro reactor, by cationic polymerization yielding propylene, piperylene, butylenes, etc. [50].

Nuclear attraction, electron-electron repulsion, and exchange terms. 

Nuclear attraction, electron-electron repulsion and exchange terms. Using the Eqs. 6 and 7, these contributions are respectively written in terms of the quantity Wij(q) previously defined in Eq. 11 ... 

The first term is the familiar one-electron operator, the second term represents the Coulomb potential, and the third term is called exchange-correlation potential. HF and DFT differ only in this last term. In HF theory there is only a nonlocal exchange term, while in DFT the term is local and supposed to cover both exchange and correlation. It arises as a functional derivative with respect to the density ... 

There are several things known about the exact behavior of Vxc(r) and it should be noted that the presently used functionals violate many, if not most, of these conditions. Two of the most dramatic failures are (a) in HF theory, the exchange terms exactly cancel the self-interaction of electrons contained in the Coulomb term. In exact DFT, this must also be so, but in approximate DFT, there is a sizeable self-repulsion error (b) the correct KS potential must decay as 1/r for long distances but in approximate DFT it does not, and it decays much too quickly. As a consequence, weak interactions are not well described by DFT and orbital energies are much too high (5-6 eV) compared to the exact values. 

What does this mean We have replaced the non-local and therefore fairly complicated exchange term of Hartree-Fock theory as given in equation (3-3) by a simple approximate expression which depends only on the local values of the electron density. Thus, this expression represents a density functional for the exchange energy. As noted above, this formula was originally explicitly derived as an approximation to the HF scheme, without any reference to density functional theory. To improve the quality of this approximation an adjustable, semiempirical parameter a was introduced into the pre-factor Cx which leads to the Xa or Hartree-Fock-Slater (HFS) method which enjoyed a significant amount of popularity among physicists, but never had much impact in chemistry,... 

One obvious drawback of the LDA-based band theory is that the self-interaction term in the Coulomb interaction is not completely canceled out by the approximate self-exchange term, particularly in the case of a tightly bound electron system. Next, the discrepancy is believed to be due to the DFT which is a ground-state theory, because we have to treat quasi-particle states in the calculation of CPs. To correct these drawbacks the so-called self-interaction correction (SIC) [6] and GW-approximation (GWA) [7] are introduced in the calculations of CPs and the full-potential linearized APW (FLAPW) method [8] is employed to find out the effects. No established formula is known to take into account the SIC. 

Qi and Qj are the net charges of atoms i and j Nvai(i) and Nvai(j) their number of valence electrons. Cexch and pexCh are empirical parameters. Some additional refinements exist within SIBFA as explicit addition of lone pairs for the exchange term [50],... 

The first term is a Coulomb term and the second is an exchange term. The exchange term, as we will see in the following section on exchange transfer, is a short-range interaction. 

---