Hohenberg-Kohn relations

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. 

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. 

It is clear from Eqs. (18) and (19) that the number and momentum densities are not related by Fourier transformation. This is most readily understood for a one-electron system where the r-space density is just the squared magnitude of the orbital and the p-space density is the squared magnitude of the Fourier transform of the orbital. The densities are not Fourier transforms of one another because the operations of Fourier transformation and taking the absolute value squared do not commute. Moreover, there is no known direct and practical route from one density to the other even though the Hohenberg-Kohn theorem [32] guarantees that it must be possible to obtain the ground state n(p) from p(r). 

The main difficulty with DFTs is that the Hohenberg-Kohn theorem shows that the ground-state values of T, Fqq, y, etc are all unique functionals of the ground-state p (i.e. that they can, in principle, be determined onee p is given), but it does not tell us what these functional relations are. 

However, the absolute //, IP, and EA analytical density functionals based on (4.253)-(4.256) formulations, respectively, can be achieved within conceptual DFT chemistry, as will be further exposed. More, because of the Hohenberg-Kohn theorem prescription, the relations (4.253)-(4.256) open the possibility of the systematic treatment of the absolute j, //, IP, and EA indices when either or both functional dependences on N and V(r) of the E and are assumed such systematics will be next exposed. 

The first Hohenberg-Kohn (HKl) theorem gives space to the concept of electronic density of the system p r) in terms of the extensive relation... 

Once in game the external applied potential provides the second Hohenberg-Kohn (HK2) theorem, hr short, HK2 theorem says that the external applied potential is determirred up to an additive constant by the electronic density of the iV-electrorric system ground state . In mathematical terms, the theorem assures the validity of the variatiorral principle applied to the density functional (4.381) relation, i.e., (Emzerhof, 1994)... 

The electrostatic potential is related rigorously to the electronic density by Poisson s equation and is therefore, through the Hohenberg-Kohn theorem [40], a property of fundamental significance [41, 42]. For example, exact atomic and molecular energies can be formulated as functions of V(r) [43]. 

The univocity of the relation between the external potential applied to the electronic system and the electronic density is provided by the Hohenberg-Kohn Theorems, see the Volume I of this five-volume book dedicated to quantum nanochemistry (Putz, 2016a). Moreover, one of the theorems also states the inequality relation between a density functional energy of any electronic state, E[p and the true functional energy of the fundamental electronic state of the system, i [p, namely as... 

While considering the observability models in chemistry, the density functional theory has at its center the electronic density p, which helps in determining the atomic or molecular energy as a density functional as well as its properties. One of these properties is that for a state density coherently formulated in relation to an external V potential (Hohenberg-Kohn Theorem), the following succession of identities that define the chemical potential as the virtual flow of the driven electrons and therefore correlated to the electronegativity itself, % actually validate the reality of chemical reactivity... 

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