User:Brainandforce/sandbox/Hohenberg-Kohn theorems

The Hohenberg-Kohn theorems are two theorems which form the foundation of density functional theory (DFT). The first theorem proves the existence of a functional which relates the electron density to the total energy of an electronic system, known as the density functional. The second theorem asserts that there is a ground state electron density which minimizes the total energy of the system. These theorems were proven in 1964 by Pierre Hohenberg and Walter Kohn.

The density functional takes the general form $$E\left[\rho\right] = T\left[\rho\right] + U\left[\rho\right] + V\left[\rho\right]$$ where $$T\left[\rho\right]$$ is the kinetic energy functional, $$U\left[\rho\right]$$ is the electron-electron interaction energy functional, and $$V\left[\rho\right]$$ is the nuclear interaction energy functional. Although $$V\left[\rho\right]$$ is dependent on the chemical system, the sum $$F\left[\rho\right] = T\left[\rho\right] + U\left[\rho\right]$$, known as the Hohenberg-Kohn functional, holds for any system containing electrons, and is thus considered a "universal" functional.

Although the Hohenberg-Kohn theorems prove that a density functional exists and that the associated ground state electron density associated with a potential may be found through a variational method, it does not assist with the construction of the density functional, and the exact form of the functional is unknown. Practical implementations of DFT use approximations, primarily of the electron-electron interaction term $$U\left[\rho\right]$$: most common is Kohn-Sham DFT, which treats the fermions comprising the density as non-interacting, and treating the electron-electron interactions with an exchange-correlation functional which may be approximated with a variety of methods.

First theorem
The statement of the first Hohenberg-Kohn theorem is given below:

If two systems of electrons, one trapped in a potential $$v_1\left(\vec{r}\right)$$, and the other in a potential $$v_2\left(\vec{r}\right)$$, admit identical ground state electron densities $$\rho\left(\vec{r}\right)$$, then the difference between the potentials, $$v_2\left(\vec{r}\right) - v_1\left(\vec{r}\right)$$, is a constant function of $$\vec{r}$$.

Proof
We start by working with the time-independent Schrödinger equation for a chemical system, whose Hamiltonian operator $$\hat{H}$$ is the sum of a kinetic energy operator $$\hat{T}$$, a nuclear potential opeator $$\hat{V}$$, and an electron-electron interaction operator $$\hat{U}$$, which acts upon a wavefunction $$\psi$$:

$$\hat{H}\psi = \left[\hat{T} + \hat{V} + \hat{U}\right] \psi$$

Corollaries
The first Hohenberg-Kohn theorem has multiple important implications. Critically, the form of the Hamiltonian operator $$\hat{H}$$ can be derived from the ground state electron density, and this means that the entire spectrum of energy eigenvalues associated with the operator may be derived from the density alone.

Second theorem
The statement of the second Hohenberg-Kohn theorem is given below:

If and only if the ground-state electron density of a system, $$\rho_{0}$$, is known, the total energy functional of the system evaluated at the density, $$E\left[\rho_{0}\right]$$, returns the lowest possible value of the total energy.

Extensions of the theorems
The Hohenberg-Kohn theorems have been extended so that they apply to chemical systems beyond the ground state.