Hohenberg-Kohn theorems exchange correlation functional energy

Dalton s atomic theory, overview, 1 De Broglie equation, 23 Delocalization energy, definition, 174 Density functional theory chemical potential, 192 computational chemistry, 189-192 density function determination, 189 exchange-correlation potential and energy relationship, 191-192 Hohenberg-Kohn theorem, 189-190 Kohn-Sham equations, 191 Weizsacker correction, 191 Determinism, concept, 4 DFT, see Density functional theory Dipole moment, molecular symmetry, 212-213... 

The Hohenberg-Kohn theorem says nothing specific about the form of Thk[w], and therefore the utility of the density functional theory depends on the choice of sufficiently accurate approximations for it. In order to do this, the unknown functional, E[n], is rewritten as a sum of the Hartree total energy and another but presumably smaller unknown functional called the exchange-correlation functional, Exc[n]. 

The resulting single-particle eigenvalue equations are the Kohn-Sham equations. The Hohenberg-Kohn theorems ensure that the exchange-correlation energy in Eq. (9) is a functional of the electron density. 

As in the Hartree-Fock method, there is no problem with ncoui but a serious difficulty arises with the exchange-correlation operator v c, or (equivalent) with the energy E c- The second Hohenberg-Kohn theorem says that the functional E [p] exists, but it does not guarantee that it is simple. For now, we will worry about this potential, but we will go ahead anyway. 

From what has been said already with respect to the variational collapse and the minimax principle, it is clear from the beginning that the standard derivation of the Hohenberg-Kohn theorems [386], which are the fundamental theorems of nonrelativistic DFT and establish a variational principle, must be modified compared to nonrelativistic theory [383-385]. Also, we already know that the electron density is only the zeroth component of the 4-current, and we anticipate that the relativistic, i.e., the fundamental, version of DFT should rest on the 4-current and that different variants may be derived afterwards. The main issue of nonrelativistic DFT for practical applications is the choice of the exchange-correlation energy functional [387], an issue of equal importance in relativistic DFT [388,389] but beyond the scope of this book. 

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