Kohn-Sham equations exchange energy

In addition to the energy terms for the exchange-correlation contribution (which enables the total energy to be determined) it is necessary to have corresponding terms for the potential, Vxc[p(i )]/ which are used to solve the Kohn-Sham equations. These are obtained as the appropriate first derivatives using Equation (3.52). 

In this equation Exc is the exchange correlation functional [46], is the partial charge of an atom in the classical region, Z, is the nuclear charge of an atom in the quantum region, is the distance between an electron and quantum atom q, r, is the distance between an electron and a classical atom c is the distance between two quantum nuclei, and r is the coordinate of a second electron. Once the Kohn-Sham equations have been solved, the various energy terms of the DF-MM method are evaluated as... 

Given in Table 7.1 are the results [24] of the total energy of some atoms obtained by solving the Kohn-Sham equation self-consistently with the exchange potential Wx within the central field approximation. The energy is obtained from Equation 7.10... 

Most practical electronic structure calculations using density functional theory [1] involve solving the Kohn-Sham equations [2], The only unknown quantity in a Kohn-Sham spin-density functional calculation is the exchange-correlation energy (and its functional derivative) [2]... 

A critical feature of this quantity is that it is nonlocal, that is, a functional based on this quantity cannot be evaluated at one particular spatial location unless the electron density is known for all spatial locations. If you look back at the Kohn-Sham equations in Chapter 1, you can see that introducing this nonlocality into the exchange-correlation functional creates new numerical complications that are not present if a local functional is used. Functionals that include contributions from the exact exchange energy with a GGA functional are classified as hyper-GGAs. 

Kohn-Sham Equations. The set of equations obtained by applying the Local Density Approximation to a general multi-electron system. An Exchange/Correlation Functional which depends on the electron density has replaced the Exchange Energy expression used in the Hartree-Fock Equations. The Kohn-Sham equations become the Roothaan-Hall Equations if this functional is set equal to the Hartree-Fock Exchange Energy expression. 

Hohenberg-Kohn theorems, but use the Kohn-Sham construction and local approximations to such non-local potentials and often lump together the exchange and the correlation energies into an exchange-correlation energy Exc[n], This yields a local exchange-correlation potential vxc(t) in the Kohn-Sham equations that determine the Kohn-Sham spin orbitals j, i.e. 

In comparing Eq. (13) to the Kohn-Sham equations Eq. (3) one concludes that E(.r, x E), since it is derived from exact many-electron theory [22], is the exact Coulomb (direct) plus exchange-correlation potential. It is non-local and also energy-dependent. In view of this it is hard to see how the various forms of constructed local exchange correlation potentials that are in use today can ever capture the full details of the correlation problem. 

In most LDA studies reported in this article, the Ceperley-Alder exchange-correlation formula is used [10,11]. Also the norm-conserving pseudopotentials of Troullier and Martins are used [12]. Therefore, one only has to deal with the valence electrons in solving the self-consistent Kohn-Sham equations in the LDA. As for basis functions, plane waves with the cutoff energy of 50 Ryd are used. 

---