The Hohenberg-Kohn Theorems

The crucial fact which makes this possible is that knowledge of n0(r) implies implicit knowledge of much more than that of an arbitrary function /(r). The ground-state wave function T0 must not only reproduce the ground-state density, but also minimize the energy. For a given ground-state density n0(r), we can write this requirement as 

9 The use of functionals and their derivatives is not limited to density-functional theory, or even to quantum mechanics. In classical mechanics, e.g., one expresses the Lagrangian C in terms of of generalized coordinates q(x,t) and their temporal derivatives q(x,t), and obtains the equations of motion from extremizing the action functional 4[g] = J C q, q t)dt. The resulting equations of motion are the well-known Euler-Lagrange equations 0 = = fy — which are a special case of Eq. (14). 

Equations (15) to (17) constitute the constrained-search proof of the Hohenberg-Kohn theorem, given independently by M. Levy [22] and E. Lieb 

Since 1964, the HK theorem has been thoroughly scrutinized, and several alternative proofs have been found. One of these is the so-called strong form of the Hohenberg-Kohn theorem , based on the inequality [26, 27, 28] 

10Note that this is exactly the opposite of the conventional prescription to specify the Hamiltonian via v(r), and obtain I d from solving Schrodinger s equation, without having to specify n(r) explicitly. 

In the proof essential use is made of the fact that the density and the potential are conjugate variables. For the same reason we can, for instance, prove that the two-particle interaction is a unique functional of the diagonal two-particle density matrix. The general mapping between /-particle density matrices and /V-body potentials is discussed by De Dominicis and Martin [6]. 

If Vj — v2 is not constant in some region then XP must vanish in this region for the above equation to be true. However, if v, v2 L + L3/2 then I XP) cannot vanish on an open set (a set with nonzero measure) by the unique continuation theorem [1]. So we obtain a contradiction, and hence we must have made a wrong assumption. Therefore, I %) l1 ) and we obtain the result that different potentials (differing more than a constant) give different wavefunctions. Consequently, we find that the map C is invertible. 

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 

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

A theorem due to Hohenberg and Kohn points to the central role of the electron density in representing the properties of a system. In 1964, Hohenberg and Kohn (1964) proved that the properties of a system with a nondegenerate ground state are unique functionals of the electron density. 

The proof of the Hohenberg-Kohn theorem is quite straightforward. Excluding nucleus-nucleus interactions and the nuclear kinetic energy, the Hamiltonian may be written as 

Since this is a contradiction, it follows that p(r) p (r). Thus, there is a one-to-one correspondence between the local potential F(r) and the electron density p(r). This implies that V, T, and are uniquely determined by the electron density, and therefore are functionals of the electron density. If is a unique functional of the electron density p, the kinetic and exchange-correlation energies, T and Exc must be functionals of p also. 

The Hohenberg-Kohn theorem does not go beyond this point it offers no guidance on the nature of the functionals that it shows must exist. 

In their article from 1964, Hohenberg and Kohn proved that the density uniquely determines the potential up to a constant, which does not matter since the potential is always determined up to a constant in any way. This means that the density can be used as the basic variable of the problem, since the potential determines all ground state properties of the system, as can be seen from the Schrodinger equation. In the article, the proof is very simple, and done by contradiction  

Suppose that there are two different potentials, v(r) and v (r) with ground states T(r) and IGr) respectively, which yield the same density. Then, unless f (r) — w(r) = const., (r) is different from T(r) since they solve different Schrodinger equations. So, if the Hamiltonians and energies associated with T fr) 

Changing from primed quantities to unprimed quantities gives us  

Another thing can also be noticed If we write the energy functional of n(r) as  

the exact ground state density can be found by minimizing the functional EVo [n] over all n. But we can write EVo [n as  

At the heart of DFT is the Hohenberg-Kohn theorem, which states that for ground states (8) can be inverted given a ground-state density noir) it is possible, in principle, to calculate the corresponding 

If is a density different from the ground-state density no in potential v(r), then the E that produce this n are different from the ground-state wave function Eo, and according to the variational principle the minimum obtained from [ ] is higher than (or equal to) the ground-state energy Ey = Ey[no]- Thus, the functional Ey[n] is minimized by the ground-state density no, and its value at the minimum is Eyfi. 

For future reference we now provide a commented summary of the content of the Hohenberg-Kohn theorem. This summary consists of four statements  

The nondegenerate ground-state wave function is a unique functional of the ground-state density  

Slater s Xa method is now regarded as so much history, but it gave an important stepping stone towards modem density functional theory. In Chapter 12, I discussed the free-electron model of the conduction electrons in a solid. The electrons were assumed to occupy a volume of space that we identified with the dimensions of the metal under smdy, and the electrons were taken to be non-interacting. 

1 explained how the model could be extended to allow for the fermion nature of electrons (that is, the Pauli principle). 

Electrons do of course interact with each other through their mutual Coulomb electrostatic potential, so an alternative step to greater sophistication might be to allow electron repulsion into the free-electron model. We therefore start again from the free-electron model but allow for the Coulomb repulsion between the electrons. We don t worry about the fermion nature of electrons at this point. 

Hohenberg and Kohn s 1964 paper was widely regarded by physicists, but its tme importance in chemistry has only become apparent during the last decade 

The paper deals with a gas of interacting metallic electrons, and examines the ground state of such a gas in the presence of an external potential U(r). 

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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.. 

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. 

According to the Hohenberg-Kohn theorem of the density functional theory, the ground-state electron density determines all molecular properties. E. Bright Wilson [46] noticed that Kato s theorem [47,48] provides an explicit procedure for constructing the Hamiltonian of a Coulomb system from the electron density ... 

Secondly, information is obtained on the nature of the nuclei in the molecule from the cusp condition [11]. Thirdly, the Hohenberg-Kohn theorem points out that, besides determining the number of electrons, the density also determines the external potential that is present in the molecular Hamiltonian [15]. Once the number of electrons is known from Equation 16.1 and the external potential is determined by the electron density, the Hamiltonian is completely determined. Once the electronic Hamiltonian is determined, one can solve Schrodinger s equation for the wave function, subsequently determining all observable properties of the system. In fact, one can replace the whole set of molecular descriptors by the electron density, because, according to quantum mechanics, all information offered by these descriptors is also available from the electron density. 

This chapter has dealt with introducing the main concepts within a theory called MQS. It has discussed the different steps to be taken to evaluate and quantify a degree of similarity between molecules in some molecular set but also fragments in molecules. QSM provides a scheme that relieves the arbitrariness of molecular similarity by using the electron density function as the sole descriptor, in agreement with the Hohenberg-Kohn theorems. It also addressed the different pitfalls that are present, for example the dependence on proper molecular alignment. 

According to the Hohenberg-Kohn theorem [48], the properties of a system of electrons and nuclei in its ground state are determined entirely by p(r). Thus the total energy, for example, is a functional of p(r), E = 9[p(r)]. 

The preceding theorem falls well short of the Hohenberg-Kohn theorem because it is restricted to Coulombic external potentials. The theorem is not true for all external potentials. In fact, for any Coulombic system, there always exists a one-electron system, with external potential,... 

The basic theorem in DFT is the Hohenberg-Kohn theorem, where the ground-state energy is defined as a functional of electron density and is given by [10]... 

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)]. 

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

Within the Hohenberg-Kohn approach [17, 18], the possibility of transforming density functional theory into a theory fully equivalent to the Schrodinger equation hinges on whether the elusive universal energy functional can ever be found. Unfortunately, the Hohenberg-Kohn theorem, being just an existence theorem, does not provide any indication of how one should proceed in order to find this functional. Moreover, the contention that such a functional should exist - and that it should be the same for systems that have neither the same number of particles nor the same symmetries (for an atom, for example, those symmetries are defined by U, L, S, and the parity operator ft) -certainly opens the door to dubious speculation. 

P. W. Ayers, S. Golden, and M. Levy, Generalizations of the Hohenberg—Kohn theorem I. Legendre transform constructions of variational principles for density matrices and electron distribution functions. J. Chem. Phys. 124, 054101 (2006). 

Density functional theory is originally based on the Hohenberg-Kohn theorem [105, 106]. In the case of a many-electron system, the Hohenberg-Kohn theorem establishes that the ground-state electronic density p(r), instead of the potential v(r), can be used as the fundamental variable to describe the physical properties of the system. In the case of a Hamiltonian given by... 

The Hohenberg-Kohn theorem can be used to redefine entanglement measures in terms of new physical quantities expectation values of observables, ai, instead of external control parameters, li. Consider an arbitrary entanglement measure M for the ground state of Hamiltonian (85). For a bipartite entanglement, one can prove a central lemma, which very generally connects M and energy derivatives. 

A useful way to write down the functional described by the Hohenberg-Kohn theorem is in terms of the single-electron wave functions, vji (r). Remember from Eq. (1.2) that these functions collectively define the electron density, (r). The energy functional can be written as... 

In fact, the true form of the exchange-correlation functional whose existence is guaranteed by the Hohenberg-Kohn theorem is simply not known. Fortunately, there is one case where this functional can be derived exactly the uniform electron gas. In this situation, the electron density is constant at all points in space that is, n(r) = constant. This situation may appear to be of limited value in any real material since it is variations in electron density that define chemical bonds and generally make materials interesting. But the uniform electron gas provides a practical way to actually use the Kohn-Sham equations. To do this, we set the exchange-correlation potential at each position to be the known exchange-correlation potential from the uniform electron gas at the electron density observed at that position ... 

It should not surprise you that the LDA is not the only functional that has been tried within DFT calculations. The development of functionals that more faithfully represent nature remains one of the most important areas of active research in the quantum chemistry community. We promised at the beginning of the chapter to pose a problem that could win you the Nobel prize. Here it is Develop a functional that accurately represents nature s exact functional and implement it in a mathematical form that can be efficiently solved for large numbers of atoms. (This advice is a little like the Hohenberg-Kohn theorem—it tells you that something exists without providing any clues how to find it.)... 

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