Functional, Hohenberg-Kohn

Note in particular that the exchange-correlation functional that emCTges here does not involve the kinetic energy. From the perspective of the DFT literature, (3.16) is a formulation of the Hohenberg-Kohn functional that is constructed to ensure that the functional derivatives required for variational minimization actually exist. We return to these issues in Sect. 3.3. Also note that in the time-dependent case the external potential V(r, )is often considered to be explicitly... 

Collecting the system independent parts into a new quantity, the Hohenberg-Kohn functional FHK[p0], we arrive at... 

The Levy construction [222] can be used to prove Hohenberg-Kohn theorems for the ground state of any such theory. It should be noted that any explicit model of the Hohenberg-Kohn functional F[p] implies a corresponding orbital functional theory. The relevant density function p(r) is that constructed from an OFT ground state. This has the orbital decomposition , as postulated by Kohn and Sham [205]. Unlike the density p,, for an exact A-electron wave function T, which cannot be determined for most systems of interest, the OFT ground-state density function is constructed from explicit solutions of the orbital Euler-Lagrange equations, and the theory is self-contained. 

Here tfo is the ground state wave function and Tno is any other wave function yielding the same density. We recognize this as the Hohenberg-Kohn functional, Fhk- It turns out that the ground state wave function of density n(r) can be defined as the wave function which yields n(r) and minimizes the Hohenberg-Kohn functional. 

In particular we can thus define the Hohenberg-Kohn functional HK on the set A as... 

Let us discuss some properties of LL- The functional Lll is an extension of the Hohenberg-Kohn functional /-nK. which is defined on A, to the larger set S, i.e.,... 

The core problem is finding adequate approximations for the Hohenberg-Kohn functional [cf. Eq. (21)]... 

The Hohenberg-Kohn functional E ipJ attains minimum Eq = F [poJ for the ideal density distribution. Now our job will be to find out what mathematical form the functional could have. And here we meet the basic problem of the DFT method nobody has so far been able to give such a formula. The best that has been achieved to date are some approximations. These approximations, however, are so good that they begin to supply results that satisfy chemists. 

The Hohenberg-Kohn theorem" states that for the ground-state, the first term of Eq. (5) is an universal functional, the Hohenberg-Kohn functional, of the electron density... 

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