Problems with exchange-correlation

The DFT models can be tested when applied to exactly solvable problems with electronic correlation (like the harmonium, as discussed in Chapter 4). It turns out that despite the exchange and correlation DFT potentials deviating from the exact ones, the total energy is quite accurate. 

The situation is still more complicated for open-shell (radical) conjugated oligomers. In neutral polyene radicals, even the most precise and expensive ab initio correlated methods, such as CCSD(T), can give at best a qualitative prediction of the spin density distributions, and the behavior of common DFT schemes is also mediocre [42]. The problems with exchange and correlation are augmented in this case by the spin-restricted/umestricted ansatz dilemma discussed above. An efficient practical solution is, somewhat unexpectedly, provided by correlated semiempirical methods with a simple Hamiltonian, such as PPP [53]. We therfore also often use semiempirical methods in our studies of conjugated oligomer radical-ions. 

Vgiec and Vxc represent the electron-nuclei, electron-electron and exchange-correlation dionals, respectively. The delta function is zero unless G = G, in which case it has lue of 1. There are two potential problems with the practical use of this equation for a croscopic lattice. First, the summation over G (a Fourier series) is in theory over an rite number of reciprocal lattice vectors. In addition, for a macroscropic lattice there effectively an infinite number of k points within the first Brillouin zone. Fortunately, e are practical solutions to both of these problems. 

There is one more problem which is typical for approximate exchange-correlation functionals. Consider the simple case of a one electron system, such as the hydrogen atom. Clearly, the energy will only depend on the kinetic energy and the external potential due to the nucleus. With only one single electron there is absolutely no electron-electron interaction in such a system. This sounds so trivial that the reader might ask what the point is. But... 

A second major problem connected to the use of finite grids for the evaluation of the exchange-correlation energy is associated with the determination of derivatives of the energy, such as the gradients used in geometry optimizations. We use... 

If the capital cost of new heat transfer area is expressed in the form of Equation 18.6, then this will lead to poor retrofit projects. The problem with Equation 18.6 is that the optimization is likely to spread the new heat transfer area in the network in many locations, without incurring a cost penalty associated with the many modifications that would result. To ensure that new heat transfer area is not spread around throughout the existing heat exchanger network, a capital cost correlation should be used that is of the form ... 

Thus, we have come a long way from the exactly soluble problems of quantum mechanics, the free-electron gas and the hydrogen atom. The concept of the exchange-correlation hole linked with the LDA has allowed... 

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