Hohenberg-Kohn theorems uniqueness

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 difficulty with DFT is that the Hohenberg-Kohn theorem shows that the ground-state values of T,, V, etc. are all unique functionals of the ground-state p (i.e.. 

Density functional theory (DFT) uses the electron density p(r) as the basic source of information of an atomic or molecular system instead of the many-electron wave function T [1-7]. The theory is based on the Hohenberg-Kohn theorems, which establish the one-to-one correspondence between the ground state electron density of the system and the external potential v(r) (for an isolated system, this is the potential due to the nuclei) [6]. The electron density uniquely determines the number of electrons N of the system [6]. These theorems also provide a variational principle, stating that the exact ground state electron density minimizes the exact energy functional F[p(r)]. 

The DFT approach (for an excellent introduction, see Parr and Yang 1989) is different and somewhat simpler. The electron density p(r) has been recognized to be a feature that uniquely determines all properties of the electronic ground state (1st Hohenberg-Kohn theorem). Instead of minimizing E with respect to coefficients of the wave function as in HF, E is minimized with respect to the electron density p... 

So the highest occupied Kohn-Sham orbital has a fractional occupation number Hohenberg-Kohn theorem applied to the non-interacting system. The proof of... 

The spirit of the Hohenberg-Kohn theorem is that the inverse statement is also true The external potential v(r) is uniquely determined by the ground-state electron density distribution, n(r). In other words, for two different external potentials vi(r) and V2(r) (except a trivial overall constant), the electron density distributions ni(r) and 2(r) must not be equal. Consequently, all aspects of the electronic structure of the system are functionals of n(r), that is, completely determined by the function (r). 

The Hohenberg-Kohn theorems state that the electron density uniquely determines the external potential and the number of electrons of an atomic or molecular system. Since these determine in turn the Hamiltonian of the system, p(r) will ultimately detennine the energy of the system ... 

By addition these two inequalities yield E+Ex external potentials cannot generate the same electron density. But E is uniquely determined by the external potential and hence one deduces that the ground-state energy is a unique functional of the electron density p(r). This result, known as the Hohenberg-Kohn theorem, was assumed in all the early work on the density description (see refs. 4 and 16). Of course, the problem of finding the energy functional remains to date it is a matter of judicious approximation for the problem under consideration. 

By functional we understand a function which depends on the form of another function—loosely, a function of a function. In the present context, a functional can be considered a recipe for extracting a single number from a function. For example, the variational principle involves a functional of the wavefunction, E = E -. A more elegant formulation of the first Hohenberg-Kohn theorem is the statement the wavefunction is a unique functional of the density. 

For an external potential v r) there is a unique ground-state wavefunction and as a result it gives rise to a unique ground state density n°(r). The Hohenberg-Kohn theorem states that for electronic densities n(r) which are t -representable, that is which are the ground state of some external potential, this external potential which gives rise to them is unique. This can be expressed by the following map ... 

More generally, the Hohenberg-Kohn theorem of SDFT states that in the presence of a magnetic field B r) that couples only to the electron spin (via the familiar Zeeman term), the ground-state wave function and all ground-state observables are unique functionals of n and m or, equivalently, of n- and. In the particular field-free case, the SDFT HK theorem still holds and continues to be useful, e.g., for systems with spontaneous polarization. Almost the entire... 

If we assume that nx = n2 then we obtain the contradiction 0 < 0 and we conclude that different ground states must yield different densities. Therefore the map D is also invertible. Consequently, the map DC V — A is also invertible and the density therefore uniquely determines the external potential. This proves the Hohenberg-Kohn theorem. 

Let us now pick an arbitrary density out of the set A of densities of nondegenerate ground states. The Hohenberg-Kohn theorem then tells us that there is a unique external potential v (to within a constant) and a unique ground state wavefunction I W[ri]) (to within a phase factor) corresponding to this density. This also means that the ground state expectation value of any observable, represented by an operator O. can be regarded as a density functional... 

We are now ready to tackle the question of how to calculate functional derivatives with respect to the density. By the Hohenberg-Kohn theorem every density associated with a nondegenerate ground state uniquely determines that ground state and the external potential that produced it. Because the density n determines the external potential v and vice versa we can parametrize the ground state either by the density or the external potential. The same is, of course, true for the expectation value of any operator O that we calculate from the ground state. We will therefore write... 

We can ask ourselves the question whether a given first order variation 8m, uniquely determines the first order density change 8v. One can show from the Hohenberg-Kohn theorem for degenerate states that this is indeed the case. If we in equation (126) take v2 = v + e 8v(r) where 8v is not a constant function we obtain... 

This follows directly from the Hohenberg-Kohn theorem applied to a noninteracting system. The density n uniquely determines the Kohn-Sham potential vs (up to a constant) and therefore the also the orbitals (up to a phase factor) and eigenvalues (up to constant). The arbitrariness with respect to a constant shift and with respect to the phase factor cancels out in the energy expression and therefore the zth-order energy becomes a pure density functional. We therefore have the following series of... 

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