Hohenberg and Kohn theorems

The field of rigorous density functional theory was born in 1964 with the publication of the Hohenberg and Kohn paper (1964). They proved the following:

I.

Every observable of a stationary quantum mechanical system (including energy), can be calculated, in principle exactly, from the ground-state density alone, i.e., every observable can be written as a functional of the ground-state density.

II.

The ground state density can be calculated, in principle exactly, using the variational method involving only density,

How these theorems were derived? By quite an original logic. Within a Born-Oppenheimer approximation, the ground state of the system of electrons is a result of positions of nuclei. This is obvious, if we look at the hamiltonian in equation (20). In this hamiltonian, the kinetic energy of electrons ( ) and the electron-electron interaction ( ) ``adjust'' themselves to the external (i.e., coming from nuclei) potential . Once the is in place, everything else is, including electron density, which simply adjusts itself to give the lowest possible total energy of the system. The external potential is the only variable term in this equation, and everything else depends indirectly on it.

Hohenberg and Kohn posed a more interesting question, which is quite opposite to the traditional logic: Is uniquely determined from the knowledge of electron density ? Can we find out (in principle, though it need not be easy) where and what the nuclei are, if we know the density in the ground state? Is there a precise mapping from to ? The answer to this question is: Yes!. Actually the mapping is accurate within a constant, which would not change anything, since Schrödinger equations with and yields exactly the same eigenfunctions, i.e., states (it is easy to prove based on the linear property of the hamiltonian), and the energies will be simply elevated by the value of this const. Note that all energies are known only within some constant, which establishes the frame of reference (e.g., we do not include electron-Mars gravitational attraction within most calculations).

Why was this question so important? Because, if this is true, the knowledge of density provides total information about the system, and formally if we know the density, we know everything there is to known. Since determines number of electrons, N:

and determines the , the knowledge of total density is as good, as knowledge of , i.e., the wave function describing the state of the system. They proved it through a contradiction:

1.

Assume that we have an exact ground state density ,

2.

Assume that the ground state is nondegenerate (i.e., there is only one wave function for this ground state (though HK theorems can be easily extended for degenerate ground states, see, e.g., Dreizler & Gross, 1990; Kohn, 1985),

3.

Assume that for the density there are two possible external potentials: and , which obviously produce two different hamiltonians: and , respectively. They obviously produce two different wave functions for the ground state: and , respectively. They correspond to energies: and , respectively.

4.

Now, let us calculate the expectation value of energy for the with the hamiltonian and use variational theorem:

 

5.

Now let us calculate the expectation value of energy for the with the hamiltonian and use varational theorem:

 

6.

By adding equations (61) and (62) by sides we obtain:

and it leads to a contradiction.

Additionally, HK grouped together all functionals which are secondary (i.e., which are responses) to the :

The functional operates only on density and is universal, i.e., its form does not depend on the particular system under consideration (note that N-representable densities integrate to N, and the information about the number of electrons can be easily obtained from the density itself).

The second HK theorem provides variational principle in electron density representation . For a trial density such that and for which ,

where is the energy functional. In other words, if some density represents the correct number of electrons N, the total energy calculated from this density cannot be lower than the true energy of the ground state.

As to the necessary conditions for this theorem, there is still some controversy concerning the, so called, representability of density. The N-representability, i.e., the fact that the trial has to sum up to N electrons is easy to achieve by simple rescaling. It is automatically insured if can be mapped to some wave function (for further discussion see: Parr & Yang, 1989; Gilbert, 1975; Lieb, 1982; and Harriman, 1980). Assuring that the trial density has also -representability (usually denoted in the literature as v-representability) is not that easy. Levy (1982) and Lieb (1983) have shown, that there are some ``reasonable'' trial densities, which are not the ground state densities for any possible potential, i.e., they do not map to any external potential. Such densities do not correspond therefore to any ground state, and their optimization will not lead to a ground state. Moreover, during energy minimization, we may take a wrong turn, and get stuck into some non v-representable density and never be able to converged to a physically relevant ground state density. For an interesting discussion, see Hohenberg et al. (1990). Assuming that we restrict ourselved only to trial densities which are both N and v representable, the variational principle for density is easly proven, since each trial density defines a hamiltonian . From the hamiltonian we can derive the corresponding wave function for the ground state represented by this hamiltonian. And according to the traditional variational principle, this wave function will not be a ground state for the hamiltonian of the real system :

where is the true ground state density of the real system.

The condition of minimum for the energy functional: needs to be constrained by the N-representability of density which is optimized. The Lagrange's method of undetermined multipliers is a very convenient approach for the constrained minimization problems. In this method we represent constraints in such a way that their value is exactly zero when they are satisfied. In our case, the N representability constraint can be represented as:

These constraints are then multiplied by an undetermined constants and added to a minimized function or functional.

where is yet undetermined Lagrange multiplier. Now, we look for the minimum of this expression by requiring that its differential is equal to zero (a necessary condition of minimum).

Solving this differential equation will provide us with a prescription of finding a minimum which satisfies the constraint. In our case it leads to:

since and N are constants. Using the definition of the differential of the functional (see, e.g., Parr & Yang, 1989):

and the fact that differential and integral signs may be interchanged, we obtain

Since integration runs over the same variable and has the same limits, we can write both expressions under the same integral:

which provides the condition for constrained minimisation and defines the value of the Lagrange multiplier at minium. It is also expressed here via external potential from equation (65):

Density functional theory gives a firm definition of the chemical potential , and leads to several important general conclusions. For review, please refer to Parr & Yang (1989), chapters 4 and 5.

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