Hohenberg-Kohn theorem, wave function

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. 

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... 

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... 

The density functional theory of Hohenberg, Kohn and Sham [173,205] has become the standard formalism for first-principles calculations of the electronic structure of extended systems. Kohn and Sham postulate a model state described by a singledeterminant wave function whose electronic density function is identical to the ground-state density of an interacting /V-clcctron system. DFT theory is based on Hohenberg-Kohn theorems, which show that the external potential function v(r) of an //-electron system is determined by its ground-state electron density. The theory can be extended to nonzero temperatures by considering a statistical electron density defined by Fermi-Dirac occupation numbers [241], The theory is also easily extended to the spin-indexed density characteristic of UHF theory and of the two-fluid model of spin-polarized metals [414],... 

The Levy construction [222] can be used to prove Hohenberg-Kohn theorems for the ground state of any such theory. It should be noted that any explicit model of the Hohenberg-Kohn functional F[p] implies a corresponding orbital functional theory. The relevant density function p(r) is that constructed from an OFT ground state. This has the orbital decomposition , as postulated by Kohn and Sham [205]. Unlike the density p,, for an exact A-electron wave function T, which cannot be determined for most systems of interest, the OFT ground-state density function is constructed from explicit solutions of the orbital Euler-Lagrange equations, and the theory is self-contained. 

The main difference between HF and DFT is that in the HF approach the objective is the determination of the wave function of the system while in DFT the quantity of interest is the charge density p in fact, the Hohenberg-Kohn theorem [65] ensures that the energy E of a system is a functional of p which takes its minimum value Eo for the ground state density po. Despite this profound conceptual difference, the two methods, HF-ncorrelation and DFT, provide in ultimate analysis very similar results and the choice of one approach or the other is largely matter of convenience. 

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... 

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 ... 

The Levy constrained search formulation of the Hohenberg-Kohn theorem [18, 20, 21] now states that we can divide our search for the ground state energy into two steps We minimize EVo [n] first over all wave functions giving a certain density, and then over all densities ... 

It is instructive to compare the Hohenberg-Kohn theorems to their counterparts in conventional wave function-based quantum theory. The wave function provides a... 

The Kohn-Sham wave function, KS, is not expected to be a good approximation to the exact wave function indeed, it is a worse approximation to the exact wave function than the Hartree-Fock wave function. Flowever, unlike the electron density obtained from the Flartree-Fock equations, the Kohn-Sham method yields, in principle, the exact electron density. Thus we do not need to use the Kohn-Sham wave function to compute the properties of chemical systems. Rather, motivated by the first Hohenberg-Kohn theorem, we compute properties directly from the Kohn Sham electron density. How one does this, for any given system and for any property of interest, is an active topic of research. 

The great surprise of density-functional theory is that in fact no information has been lost at all, at least as long as one considers the system only in its ground state according to the Hohenberg-Kohn theorem the ground-state density n0 ( / ) completely determines the ground-state wave function T0(xi,x2 -,xN).25... 

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