Hohenberg and Kohn theorem

The main problem relating to practical applications of the Hohenberg and Kohn theorems is obvious the theorems are existence theorems and do not give us any clues as to the calculation of the quantities involved. 

The Hohenberg and Kohn theorem applies to ground states at the absolute zero of temperature. Fortunately there is a finite-temperature version of DFT, first proved by Mermin. The equilibrium properties of a grand canonical ensemble are determined by the grand potential, Q, which is defined as follows ... 

The ultimate goal of computational chemistry is to understand chemical reactivity and to predict the outcome of molecular interactions. Although this is a complicated problem, some simple considerations rooted in the first principles of physics can go a long way. This does, at least, allow us to ask the right questions and to define the relevant properties. In this section it is demonstrated how the Hohenberg and Kohn theorems [7] can qualitatively elucidate the concept of chemical reactivity our definition of sensitivity analysis is introduced. 

Which fundamental properties X could we be interested in Realizing that the electron density distribution function contains all the information about the system in the ground state (Hohenberg and Kohn theorems), its response to several perturbations is certainly of fundamental importance. Other properties also provide valuable information, such as the energy and the electronic chemical potential of the system. We will consider all of these and try to find analytical expressions for their response to, or resistance against, changes in N or v(r). 

Due to the Hohenberg and Kohn theorems (1964), all properties of the ground state are functions only on iVand V(r). Therefore, much chemistry is comprised in above EN, as /r = /lc[N, E(r )] measures the escaping tendency of an electronic cloud from the equilibrium system. 

Many theorems concerning the ground state, such as the Hohenberg and Kohn theorem, are based on the minimum principle of the ground state. Most numerical solutions are also based on this principle. Fortunately, in an indirect way, all eigenstates of the Hamiltonian can be derived by a minimum principle. For this purpose, it is necessary to consider a subspace of the square integrable functions, that is, a subspace S of dimension M of the Hilbert space of square integrable functions and the subspace functional... 

Suppose now that the density of = lOJ is p(r). Then, according to the Hohenberg and Kohn theorem [1], there is an external potential V that reproduces the above density. 

By the above inequality derived initially in reference [3], it is easy to get the Hohenberg and Kohn theorem because, if it is supposed that the two densities are equal for all r, a contradiction is obtained. Thus,... 

By using subspace minimization of E(S), one can derive in a straightforward way the Hohenberg and Kohn theorem [1] for subspaces, that is, the one-to-one correspondence between subspace density and minimizing subspace of certain dimension M. The next step is to derive the equation for the minimizing subspace of the... 

According to the Hohenberg and Kohn theorem, the ground state energy of a system is expressed as a unique functional of electron density ... 

The advantage offered by DFT as compared to conventional, wave function, quantum chemistry, lies in the fundament role attributed to the electron density fimction p(r) of which the electronic energy of an atomic or molecular system is a fimctional as elegantly shown by the famous Hohenberg and Kohn theorems [2]... 

Consequently, from the density the Hamiltonian can be readily obtained, and then every property of the system can be determined by solving the Schrodinger equation to obtain the wave function. One has to emphasize, however, that this argument holds only for Coulomb systems. By contrast, the density functional theory formulated by Hohenberg and Kohn is valid for any external potential. Kato s theorem is valid not only for the ground state but also for the excited states. Consequently, if the density n, of the f-th excited state is known, the Hamiltonian H is also known in principle and its eigenvalue problem ... 

As will be developed in more detail below, the paper by Hohenberg and Kohn (1964) [7], which proved the existence theorem that the ground state energy is a functional of n(r), but now without the approximations (valid for large N) in the explicit energy functional (1), formally completed the TFD theory. The work of Kohn and Sham (1965) [8] similarly gave the formal completion of Slater s 1951 proposal. 

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. 

---