Ground-state electron density 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)]. 

We will now derive a condition for an electronic density no (r) to minimize E [n] (37), that is to be the ground state electronic density of the system in the external potential vq. The variational principle states that if we assume a small variation of n, Sn(r), which conserves the number of electrons (f d rSn(r) = 0),... 

The Variational Principle for the Ground-State Electron Density... 

The second H-K theorem defines the energy functional for the system and proves that the correct ground-state electron density minimizes this energy functional. This is important since it allows using the variational principle as is done for the HF method. If p(r) is the exact density, the energy functional E[/ (r)] is minimum, and we search for p by minimizing E[p(r)] with J p(r)dr = (A is the total number of electrons). p(r) is a priori unknown. 

The basic lemma of Hohenberg and Kohn [4] states that the ground state electron density of a system of interacting electrons in an arbitrary external potential determines this potential uniquely. The proof is given by the variational principle. If we consider a Hamiltonian Hi of an external potential Vi as... 

There exists a variational principle in terms of the electron density which determines the ground state energy and electron density. Further, the ground state electron density determines the external potential, within an additive constant. 

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... 

The last term in Eq. (2) represents the main problem of the DFT. In DFT methodologies the optimal electron density (p ) is computed following the variational principle. Kohn and Sham proposed that a real electron density can be represented by a fictitious noninteracting reference system. Electrons in the latter system do not interact, but its ground-state electron density distribution is exactly the same as corresponding to the real system under consideration. The deviation in the behavior of noninteracting electrons from that of the real ones is then taken into account by the unknown XC functional that is included in DFT methods in an approximate form. The development of such approximate functional is a very active research topic in theoretical chemistry. 

More precisely the second theorem of Hohenberg and Kohn tells us that the ground state electron density can be calculated, in principle exactly, using the variational method involving only the electron density. 

Since the exact ground-state electronic wave function and density can only be approximated for most A-electron systems, a variational theory is needed for the practical case exemplified by an orbital functional theory. As shown in Section 5.1, any rule T 4> defines an orbital functional theory that in principle is exact for ground states. The reference state for any A-electron wave function T determines an orbital energy functional E = Eq + Ec,in which E0 = T + Eh + Ex + V is a sum of explicit orbital functionals, and If is aresidual correlation energy functional. In practice, the combination of exchange and correlation energy is approximated by an orbital functional Exc. 

All of this development certainly has problems associated with the N-and u-representability of the electron density and they are the subject of the chapter by Ludeiia et al in this book. Here it will be enough to state that the variational principle in eq. (23) requires that the trial p be such that there exists a local external potential v> which will yield a Hamiltonian whose ground state density is p. ... 

Part I of the paper develops an exact variational principle for the ground-state energy, in which the density (r) is the variable function (i.e. the one allowed to vary). The authors introduce a universal functional F[n(r)] which applies to all electronic systems in their ground states no matter what the external potential is. This functional is used to develop a variational principle. 

Among various theories of electronic structure, density functional theory (DFT) [1,2] has been the most successful one. This is because of its richness of concepts and at the same time simplicity of its implementation. The new concept that the theory introduces is that the ground-state density of an electronic system contains all the information about the Hamiltonian and therefore all the properties of the system. Further, the theory introduces a variational principle in terms of the ground-state density that leads to an equation to determine this density. Consider the expectation value (H) of the Hamiltonian (atomic units are used)... 

Equation 24.17 shows that the energy gained by the system when a field E is applied is a function of the electronic density represented by p. According to the variational principle of DFT, the energy in the ground state (in the absence of a field) is minimum [1,2]. 

Another model that describes the electronic structure of a system is provided by density functional theory (DFT). In DFT the electron density p of the system in the ground state plays the role of the many-electron wavefunction T in the wavefunction model because it uniquely defines all ground state properties of a system.An advantage of DFT is that T, which is a function of both spatial and spin coordinates of all electrons in the system, is replaced by a function that depends only on a position in Cartesian space p = p(r). The electron density can be obtained by using the variational principle... 

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