Atomic exchange energies

The particular functional form of Equation 10 is certainly not the only conceivable dimensionally-consistent possibility, but was chosen on the basis of algebraic simplicity (12). Viewed in a semi-empirical spirit, it is found that Equation To, with suitable values of the constants b and c, fits exact atomic exchange energies throughout the periodic table with extremely high accuracy (i.e. roughly 0.2% error compared with 5-10% error for the exchange-only LDA). It should be noted in this connection that constants b and c are "universal" constants, and not adjustable from atom to atom as is the a parameter in so-called "muffin-tin" Xa schemes. 

Furthermore, the rather subtle cancellation observed for the exact AE and j is not found in the RLDA. The RLDA thus completely misrepresents the total relativistic correction to atomic exchange energies [36] (see Table 5.5). 

It is not obvious what the optimal form for the E functional is. Usually, it is broken into two parts, the exchange functional and the correlation functional (Eq. 14.55). A number of different forms have been suggested for each. A currently popular form for the exchange functional is one developed by Becke in 1988. While the exact form of this functional is not given here, it is important to realize that it contains a parameter (y) that is chosen to fit experimental data related to atomic exchange energies. As such, DFT methods have a semi-empirical flavor, in that there is a fit to experimental data (see Section 14.2.3). DFT as typically implemented is therefore not an ab initio method. 

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. 

With a = 2/3 this is identical to the Dirac expression. The original method used a = 1, but a value of 3/4 has been shown to give better agreement for atomic and molecular systems. The name Slater is often used as a synonym for the L(S)DA exchange energy involving die electron density raised to the 4/3 power (1/3 power for the energy density). 

Generally, in a system that is energetically and materially isolated from the environment without a change in volume (a closed system), the entropy of the system tends to take on a maximum value, so that any macroscopic structures, except for the arrangement of atoms, cannot survive. On the other hand, in a system exchanging energy and mass with the environment (an open system), it is possible to decrease the entropy more than in a closed system. That is, a macroscopic structure can be maintained. Usually such a system is far from thermodynamic equilibrium, so that it also has nonlinearity. 

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