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 $\Psi$.

This is a remarkable result: it implies that a function of three spatial variables $\rho(r)$ — rather than a wavefunction in $3N$ dimensions — contains all the information needed to determine the ground state of an $N$-electron system. The HK theorems provide the rigorous justification for replacing the many-body Schrödinger equation with a variational problem over electron densities, forming the conceptual basis for all practical implementations of DFT.

Assumptions

We consider a system of $N$ interacting electrons subject to an external scalar potential $V_{\rm ext}(r)$, at zero temperature, in its ground state, with no time-dependent fields. The many-body Hamiltonian is:

where

• $\hat{T}$ is the kinetic energy operator of the electrons,
• $\hat{V}_{ee}$ is the Coulomb electron–electron repulsion operator,
• $\hat{V}{ext} = \sum{i=1}^N V_{\rm ext}(r_i)$ is the external potential due to the nuclei or any applied field — it is the same for all systems considered in the proofs below.

The electron density $\rho(r)$ is defined as the probability of finding any one electron at position $r$, integrated over all other electron coordinates:

Note that $\int \rho(r),dr = N$.

The First Hohenberg–Kohn Theorem

Theorem: For any system of interacting particles in an external potential $V_{\rm ext}(r)$, the external potential is uniquely determined — up to a constant — by the ground-state electron density $\rho_0(r)$.

Because $V_{\rm ext}(r)$ uniquely determines $\hat{H}$ (and hence all eigenstates and eigenvalues), the ground-state density $\rho_0(r)$ alone determines everything about the system. The wavefunction $\Psi_0$ is therefore a functional of $\rho_0$, written $\Psi_0 = \Psi[\rho_0]$.

Proof (by contradiction)

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

\begin{equation} V_{\rm ext}(\mathbf{r}) \neq V’_{\rm ext}(\mathbf{r}) + \text{const.}, \end{equation}

that nevertheless yield the same ground-state density: $\rho(\mathbf{r})=\rho’(\mathbf{r}) \equiv \rho(\mathbf{r})$.

Let the two Hamiltonians and their ground states be: $$ \begin{align} \hat{H} & = \hat{T} + \hat{V}_{\rm ee} + \hat{V}_{\rm ext}, & \hat{H}|\Psi\rangle = E_0|\Psi\rangle, \\ \hat{H}' & = \hat{T} + \hat{V}_{\rm ee} + \hat{V}'_{\rm ext}, & \hat{H}'|\Psi'\rangle = E'_0|\Psi'\rangle. \end{align} $$

We apply the Rayleigh–Ritz variational principle, which states that the true ground-state energy minimises the energy expectation value: any trial state $|\tilde{\Psi}\rangle \neq |\Psi_0\rangle$ satisfies $\langle\tilde{\Psi}|\hat{H}|\tilde{\Psi}\rangle > E_0$.

Using $\Psi’$ as a trial state for $\hat{H}$: $$ \begin{align} E_0 <& \langle \Psi' | \hat{H} | \Psi' \rangle = \langle \Psi' | \hat{H}' | \Psi' \rangle + \langle \Psi' | \hat{H} - \hat{H}' | \Psi' \rangle \notag\\ =& \; E'_0 + \int \left[ V_{\rm ext}(\mathbf{r}) - V'_{\rm ext}(\mathbf{r}) \right] \rho(\mathbf{r}) \, d\mathbf{r}. \label{eq:ineq1} \end{align} $$

Symmetrically, using $\Psi$ as a trial state for $\hat{H}’$: $$ \begin{equation} E'_0 < \langle \Psi | \hat{H}' | \Psi \rangle = E_0 + \int \left[ V'_{\rm ext}(\mathbf{r}) - V_{\rm ext}(\mathbf{r}) \right] \rho(\mathbf{r}) \, d\mathbf{r}. \label{eq:ineq2} \end{equation} $$

Adding the two inequalities \eqref{eq:ineq1} and \eqref{eq:ineq2}: $$ \begin{align} E_0 + E'_0 & < E'_0 + E_0 + \int \underbrace{\left[ V_{\rm ext} - V'_{\rm ext} + V'_{\rm ext} - V_{\rm ext} \right]}_{=\,0} \rho(\mathbf{r}) \, d\mathbf{r}, \\ E_0 + E'_0 & < E_0 + E'_0. \end{align} $$

This is a contradiction. Our assumption must therefore be false: no two different external potentials (differing by more than a constant) can share the same ground-state density. The map $\rho_0(r) \mapsto V_{\rm ext}(r)$ is therefore one-to-one (up to a constant). $\blacksquare$

Consequence: the universal energy functional

Since $\rho_0 \to V_{\rm ext} \to \hat{H} \to \Psi_0$, the ground-state energy can be written as a functional of the density: $$ \begin{align*} E[\rho] =& F[\rho]+\int V_{\rm ext}(r)\rho(r)\,dr\\ F[\rho] =& \langle\Psi[\rho]|\, \hat{T}+\hat{V}_{ee}\,| \Psi[\rho]\rangle \end{align*} $$

The quantity $F[\rho]$ is called the universal functional of the density. It is “universal” because it depends only on the electron density — not on the specific external potential — making it, in principle, applicable to any system of interacting electrons.

The Second Hohenberg–Kohn Theorem

Theorem: The true ground-state density $\rho_0(\mathbf{r})$ minimises the total energy functional $E[\rho]$. That is, for any trial density $\tilde{\rho}(\mathbf{r})$ satisfying \begin{equation} \tilde{\rho}(\mathbf{r}) \geq 0, \quad \int \tilde{\rho}(\mathbf{r}) , d\mathbf{r} = N, \end{equation} we have \begin{equation} E_0 \leq E[\tilde{\rho}], \end{equation} with equality if and only if $\tilde{\rho}(\mathbf{r}) = \rho_0(\mathbf{r})$.

The functional is: \begin{equation} E[\rho] = F[\rho] + \int V_{\rm ext}(\mathbf{r}) \rho(\mathbf{r}) , d\mathbf{r}, \end{equation} where $F[\rho]$ is the universal functional defined above.

Proof sketch: By the first HK theorem, any trial density $\tilde{\rho} \neq \rho_0$ corresponds to some other external potential and thus to some other wavefunction $\tilde{\Psi} \neq \Psi_0$. By the Rayleigh–Ritz principle applied to $\hat{H}$, $\langle\tilde{\Psi}|\hat{H}|\tilde{\Psi}\rangle \geq E_0$, which translates directly to $E[\tilde{\rho}] \geq E_0$. $\blacksquare$

The second theorem transforms the problem of solving the many-body Schrödinger equation into a variational problem over electron densities — a function in three dimensions rather than $3N$ dimensions. This is the promise of DFT.

Outlook

While the HK theorems establish that the ground-state density contains all physical information and that a universal energy functional $E[\rho]$ exists, they do not provide an explicit form for the universal functional $F[\rho]$. In particular, the kinetic energy $T[\rho]$ and the electron–electron interaction $V_{ee}[\rho]$ are unknown functionals of $\rho$.

Computing $F[\rho]$ accurately is the central challenge of DFT in practice. The Kohn–Sham equations, developed in the next chapter, provide a practical path forward by introducing a fictitious system of non-interacting electrons designed to reproduce the exact ground-state density — sidestepping the need to evaluate $F[\rho]$ directly.