Problems with exchange-correlation energy

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... 

The two delta terms which have been placed side by side encapsulate the main problem with DFT the sum of the kinetic energy deviation from the reference system and the electron-electron repulsion energy deviation from the classical system, called the exchange-correlation energy. In each term an unknown functional transforms electron density into an energy, kinetic and potential respectively. This exchange-correlation energy is a functional of the electron density function ... 

We have to consider the calculation of the fourth term, the problem term, in the KS operator of Eq. 7.23, the exchange-correlation potential vXc(r). This is defined as the functional derivative [36, 37] of the exchange-correlation energy functional, fsxc[p(r)], with respect to the electron density functional (Eq. 7.23). The exchange-correlation energy UX( lp(r)], a functional of the electron density function p(r), is a quantity which depends on the function p(r ) and on just what mathematical form the... 

In the earliest implementation applied to molecular problems, K. Johnson [39] used scattered-plane waves as a basis and the exchange-correlation energy was represented by (13). This SW-Xa method employed in addition an (muffin-tin) approximation to the Coulomb potential of (17) in which Vc is replaced by a sum of spherical potentials around each atom. This approximation is well suited for solids for which the SW-Xa method originally was developed [40]. However, it is less appropriate in molecules where the potential around each atom might be far from spherical. The SW-Xa method is computationally expedient compared to standard ab initio techniques and has been used with considerable success [41] to elucidate the electronic structure in complexes and clusters of transition metals. However, the use of the muffin-tin approximation precludes accurate calculations of total energies. The method has for this reason not been successful in studies involving molecular structures and bond energies [42]. 

This leaves us with the exchange-correlation energy functional, ExciPo) (Eq. (7.15)) as the only term for which some method of calculation must be devised. Devising accurate exchange-correlation functionals for calculating this energy term from the electron density function is the main problem in DFT research. This is discussed in section 7.2.3.4. 

The approach we have discussed here addresses both problems with comparable emphasis. The density functional formalism, with the LSD approximation for the exchange-correlation energy, provides us with an approximate method of calculating energy surfaces, and the results have predictive value in many contexts. DF can also be carried out with comparable ease for all elements. When coupled with MD at elevated temperatures (simulated annealing), it is possible to study cases where the most stable isomers are unknown, or where the energy surfat have many local minima. 

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. 

We are immediately confronted with the problem of how to find the unknown exchange-correlation energy Exc, which is replaced also by an unknown exchange-correlation potential in the form of a functional derivative Vxc = We obtain the Kohn-Sham equation (resembling the Fock equation) -jA -F ng (/>i = eifi, where wonder-potential t)g = t) -F Vcoyi -F i xc, fcoul stands for the sum of the usual Coulombic operators (as in the Hartree-Fock method) (built from the Kohn-Sham spinorbitals) and vxc is the potential to be found. 

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