Hohenberg-Kohn Theorem

The Hohenberg–Kohn (HK) theorems, proposed in 1964 by Pierre Hohenberg and Walter Kohn, are the fundamental theoretical pillars of Density Functional Theory (DFT). They establish that the ground-state properties of a many-electron system are uniquely determined by its electron density $\rho(r)$, rather than by the complex many-body wavefunction $\psi$.

These theorems provide the justification for replacing the many-body Schrödinger equation with a variational problem over electron densities, forming the basis for all practical implementations of DFT.

Assumptions

For a system of $𝑁$ interacting electrons subject to an external scalar potential $𝑉_{ext}(r)$. The many-body Hamiltonian is given by:

$$ \begin{align*} \hat{H} =& \hat{T}+\hat{V}_{ee}+\hat{V}_{ext}\\ \hat{H}\psi =& E\psi \end{align*} $$ where

• $\hat{T}$ is the kinetic energy operator
• $\hat{V}_{ee}$ is the Coloumb interaction operator between two electrons
• $\hat{V}_{ext} = \sum_1^N V(r_i)$ is the external potential due to nuclei or applied fields.

The electron density $\rho(r)$ is given by $$ \rho(r)=N\int |\Psi(r_1,r_2\cdots r_N)|^2 dr_1dr_2\cdots dr_N $$

The First Hohenberg–Kohn Theorem

For any system of interacting particles in an external potential $V_{ext}(r)$, the potential is uniquely determined—up to a constant—by the ground-state electron density $\rho(r)$.

Proof (by contradiction)

Assume the contrary. Suppose there exist two different external potentials $V_{\text{ext}}(\mathbf{r})$ and $V’_{\text{ext}}(\mathbf{r})$, differing by more than a constant:

\begin{equation} V_{\text{ext}}(\mathbf{r}) \neq V’_{\text{ext}}(\mathbf{r}) + \text{const.} \end{equation} that yield the same ground-state density: $\rho(r)=\rho’(r)$

Let the Hamiltonians corresponding to these potentials be: $$ \begin{align} \hat{H} & = \hat{T} + \hat{V}_{\text{ee}} + \hat{V}_{\text{ext}}, \\ \hat{H}' & = \hat{T} + \hat{V}_{\text{ee}} + \hat{V}'_{\text{ext}}, \end{align} $$ and let the ground-state energies be ( E_0 ) and ( E’_0 ), respectively: $$ \begin{align} E_0 & = \langle \Psi | \hat{H} | \Psi \rangle, \\ E'_0 & = \langle \Psi' | \hat{H}' | \Psi' \rangle. \end{align} $$

Using the Rayleigh–Ritz variational principle, we evaluate the energy expectation of $\Psi’$ with respect to $\hat{H}$:

Similarly, evaluate $E’_0$ with respect to $\hat{H}’$ using $\Psi$: $$ \begin{equation} E'_0 < \langle \Psi | \hat{H}' | \Psi \rangle = E_0 + \int \left[ V'_{\text{ext}}(\mathbf{r}) - V_{\text{ext}}(\mathbf{r}) \right] \rho(\mathbf{r}) \, d\mathbf{r}. \end{equation} $$

Adding equations , we obtain: $$ \begin{align} E_0 + E'_0 & < E'_0 + E_0 + \int \left[ V_{\text{ext}}(\mathbf{r}) - V'_{\text{ext}}(\mathbf{r}) + V'_{\text{ext}}(\mathbf{r}) - V_{\text{ext}}(\mathbf{r}) \right] \rho(\mathbf{r}) \, d\mathbf{r}, \\ E_0 + E'_0 & < E_0 + E'_0. \end{align} $$

This is a contradiction. Hence, our initial assumption that two different external potentials can yield the same ground-state density must be false. Therefore, the external potential $V_{\text{ext}}(\mathbf{r})$ is uniquely determined (up to a constant) by the ground-state density $\rho_0(\mathbf{r})$.

This justifies the existence of a universal ground-state energy functional: $$ \begin{align*} E[\rho] =& F[\rho]+\int V_{ext}(r)\rho(r)dr\\ F[\rho] =& \langle\Psi[\rho]| T+V_{ee}| \Psi[\rho]\rangle \end{align*} $$ where $F[\rho]$ is called the universional functional of the density.

The Second Hohenberg–Kohn Theorem

Let $\tilde{\rho}(\mathbf{r})$ be any trial electron density that satisfies the following conditions: \begin{equation} \tilde{\rho}(\mathbf{r}) \geq 0, \quad \int \tilde{\rho}(\mathbf{r}) , d\mathbf{r} = N, \end{equation} where $N$ is the total number of electrons in the system.

Then, the total ground-state energy functional, \begin{equation} E[\rho] = F[\rho] + \int V_{\text{ext}}(\mathbf{r}) \rho(\mathbf{r}) , d\mathbf{r}, \end{equation} satisfies the variational principle: \begin{equation} E_0 \leq E[\tilde{\rho}], \end{equation} with equality if and only if $\tilde{\rho}(\mathbf{r}) = \rho_0(\mathbf{r})$, where $\rho_0(\mathbf{r})$ is the true ground-state density corresponding to the external potential $V_{\text{ext}}(\mathbf{r})$.

Here, $E_0$ is the exact ground-state energy of the system, and the universal functional $F[\rho]$ is defined as:

\begin{equation} F[\rho] = \langle \Psi[\rho] | \hat{T} + \hat{V}_{\text{ee}} | \Psi[\rho] \rangle, \end{equation}