Hohenberg theorems

In some ways, DFT may be more easily understood. According to the Kohn-Hohenberg theorem,the energy is minimized when the calculated and true electron densities are equal. One important consequence of DFT calculations is that their accuracy corresponds to standard post Hartree-Fock methods. Another consequence of DFT calculations of more practical importance is the reduction in the computation time required to complete a calculation. The time required to complete Hartree-Fock calculations is a function of the number of electrons in the system being examined, and it is proportional to n , where n represents the... 

The main diffieulty with DFTs is that the Hohenberg-Kolm theorem shows that tire ground-state values of T, Fgg, V, ete are all unique fiinetionals of the ground-state p (i.e. that they ean, in prineiple, be detemiined onee p is given), but it does not tell us what these fimetioiial relations are. 

The first Hohenberg-Kohn theorem states that, for a nondegenerate ground state, there is a one-to-one mapping among p. V. and iq... 

The so-ealled Hohenberg-Kohn theorem states that the ground-state eleetron density p(r) deseribing an N-eleetron system uniquely determines the potential V(r) in the Hamiltonian... 

The main problem relating to practical applications of the Hohenberg and Kohn theorems is obvious the theorems are existence theorems and do not give us any clues as to the calculation of the quantities involved. 

The fact that an exact density functional exists is known from a theorem proved by Hohenberg and Sham and Kohn. However, this is a non-constructive proof since it does not actually give the form of the exact functional. DFT theorists must try to approximate this functional as well as they can. 

The First Hohenberg-Kohn Theorem Proof of Existence... 

The Second Hohenberg-Kohn Theorem Variational Principle... 

Let us recall that the Hohenberg-Kohn theorems allow us to construct a rigorous many-body theory using the electron density as the fundamental quantity. We showed in the previous chapter that in this framework the ground state energy of an atomic or molecular system can be written as... 

Gorling, A., 1999, Density-Functional Theory Beyond the Hohenberg-Kohn Theorem , Phys. Rev. A, 59, 3359. 

The electron density of a non-degenerate ground state system determines essentially all physical properties of the system. This statement of the Hohenberg-Kohn theorem of Density Functional Theory plays an exceptionally important role among all the fundamental relations of Molecular Physics. 

As a consequence of the Hohenberg-Kohn theorem [14], a non-degenerate ground state electron density p(r) determines the Hamiltonian H of the system within an additive constant, implying that the electron density p(r) also determines all ground state and all excited state properties of the system. 

The original Hohenberg-Kohn theorem was directly applicable to complete systems [14], The first adaptation of the Hohenberg-Kohn theorem to a part of a system involved special conditions the subsystem considered was a part of a finite and bounded entity regarded as a hypothetical system [21], The boundedness condition, in fact, the presence of a boundary beyond which the hypothetical system did not extend, was a feature not fully compatible with quantum mechanics, where no such boundaries can exist for any system of electron density, such as a molecular electron density. As a consequence of the Heisenberg uncertainty relation, molecular electron densities cannot have boundaries, and in a rigorous sense, no finite volume, however large, can contain a complete molecule. 

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