The Hohenberg-Kohn Variational Theorem

The first theorem of Hohenberg and Kohn is an existence theorem. As such, it is provocative with potential, but altogether unhelpful in providing any indication of how to predict the density of a system. Just as with MO theory, we need a means to optimize our fundamental quantity. Hohenberg and Kohn showed in a second theorem that, also just as with MO theory, the density obeys a variational principle. 

To proceed, first, assume we have some well-behaved candidate density that integrates to the proper number of electrons, N. In that case, the first theorem indicates that this density determines a candidate wave function and Hamiltonian. That being the case, we can evaluate the energy expectation value 

The difficulty lies in the nature of the functional itself. Up to this point, we have indicated that there are mappings from the density onto the Hamiltonian and the wave function, and hence the energy, but we have not suggested any mechanical means by which the density can be used as an argument in some general, characteristic variational equation, e.g., with terms along the lines of Eqs. (8.5) and (8.7), to determine the energy directly without recourse to the wave function. Such an approach first appeared in 1965. 

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The Kohn-Sham orbitals are found as follows. The Hohenberg-Kohn variational theorem tells us that we can find the ground-state energy by varying p (subject to the constraint f p dr = n) so as to minimize the functional E [p]. Equivalently, instead of varying p, we can vary the KS orbitals dP, which determine p by (16.45). (In doing so, we must constrain the dP s to be orthonormal, since orthonormality was assumed when we evaluated T. )... 

This theorem means that the ground state electron density, as obtained from the Hohenberg-Kohn variational principle, uniquely determines the ground state properties of the system of interest. The electron density is obtained from the variational principle... 

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)]. 

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). 

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 ... 

For the GS, the two Hohenberg-Kohn (HK) theorems legitimize the density p(r) (a function of only 3 coordinates) as the basie variational variable henee, all terms in the GS eleetronie energy of a quantum system are frmetionals of the density ... 

As already stated in the preceding section, the PB equation neglects ion size effects and interparticle correlations. One route to improve the theory can be done on a density functional level. The PB equation can be derived via a variational principle out of a local density functional [25, 31]. This is also a convenient formulation to overcome its major deficiencies, namely the neglect of ion size effects and interparticle correlations. The Hohenberg-Kohn theorem gives an existence proof of a density functional that will produce the correct density profile upon variation. However, it does not specify its... 

In this work, the electronic kinetic energy is expressed in terms of the potential energy and derivatives of the potential energy with respect to nuclear coordinates, by use of the virial theorem (5-5). Thus, the results are valid for all bound electronic states. However, the functional derived for E does not obey a variational principle with respect to (pg(r)), even though x( ,r co)is in principle a functional of (p (r)), as implied by the Hohenberg-Kohn theorem 9-121... 

A totally different point of view is proposed by Time-Dependent Density Functional Theory [211-215] (TD-DFT). This important extension of DFT is based on the Runge-Gross theorem [216]. It extends the Hohenberg-Kohn theorem to time-dependent situations and states that there is a one to one map between the time-dependent external potential t>ea t(r, t) and the time-dependent charge density n(r, t) (provided we know the system wavefunction at t = —oo). Although it is linked to a stationary principle for the system action, its demonstration does not rely on any variational principle but on a step by step construction of the charge current. 

When we proved the Hohenberg-Kohn theorem above, we made the assumption that the density is v-representable. By this is meant that the density is a density associated with the anti-symmetric ground state wave function and some potential v(r). Why is this important The reason is that we want to use the variational character of the energy functional ... 

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... 

Undoubtedly, the Hohenberg-Kohn theorem has spurred much activity in density functional theory. In fact, most of the developments in this field are based on its tenets. Nevertheless, the approximate nature of all such developments, renders them functionally" non-jV-representable. This simple means that all approximate methods based on the Hohenberg-Kohn theorem are not in a one to one correspondence with either the Schrodinger equation or with the variational principle from which this equation ensues [21, 22], Thus, the specter of the 2-matrix N-representability problem creeps back in density functional theory. Unfortunately, the immanence of such a problem has not been adequately appreciated. It has been mistakenly assumed that this 2-matrix /V-representability condition in density matrix theory may be translocated into /V-representability conditions on the one-particle density [22], As the latter problem is trivially solved [23, 24], it has been concluded that /V-representability is of no account in the Hohenberg-Kohn-based versions of density functional theory. As discused in detail elsewhere [22], this is far from being the case. Hence, the lack of functional. /V-representability occurring in all these approximate versions, introduces a very serious defect and leads to erroneous results. 

According to the actual formulation of the Hohenberg-Kohn theorem 44], there exists a universal variational functional F(fi) of trial electron densities p, such that if o is the ground-state energy and Po(r) is the ground-state electron density that belongs to the given external potential V(r) specified in Eq. 15, then. 

The relativistic correction for the kinetic energy in the Dirac equation is naturally applicable to the Kohn-Sham equation. This relativistic Kohn-Sham equation is called the Dirac-KohnSham equation (Rajagopal 1978 MacDonald and Vosko 1979). The Dirac-Kohn-Sham equation is founded on the Rajagopal-Callaway theorem, which is the relativistic expansion of the Hohenberg-Kohn theorem on the basis of QED (Rajagopal and Callaway 1973). In this theorem, two theorems are contained The first theorem proves that the four-component external potential, which is the vector-potential-extended external potential, is determined by the four-component current density, which is the current-density-extended electron density. On the other hand, the second theorem establishes the variational principle for every four-component current density. See Sect. 6.5 for vector potential and current density. Consequently, the solution of the Dirac-Kohn-Sham equation is represented by the four-component orbital. This four-component orbital is often called a molecular spinor. However, this name includes no indication of orbital, which is the solution of one-electron SCF equations moreover, the targets of the calculations are not restricted to molecules. Therefore, in this book, this four-component orbital is called an orbital spinor. The Dirac-Kohn-Sham wavefunction is represented by the Slater determinant of orbital spinors (see Sect. 2.3). Following the Roothaan method (see Sect. 2.5), orbital spinors are represented by a linear combination of the four-component basis spinor functions, Xp, ... 

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