Hohenberg-Kohn (HK) Theorems

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, i.e. \(\rho(r)\), 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:

\[ \begin{align} \hat{H} =& \hat{T}+\hat{V}_{ee}+\hat{V}_{ext}\\ \hat{H}\Psi =& E\Psi \end{align} \]

where

\(\hat{T}\) is the kinetic energy operator 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 \(\rho(\mathbf{r})\) is the number density of electrons at position \(\mathbf{r}\) — not the probability density \(|\Psi|^2\) itself, but the marginal single-particle density obtained by integrating out the coordinates of all other electrons and multiplying by \(N\) to account for the fact that any of the \(N\) indistinguishable electrons can be found at \(\mathbf{r}\):

\begin{equation} \rho(\mathbf{r}) = N\int |\Psi(\mathbf{r},\mathbf{r}_2,\ldots,\mathbf{r}_N)|^2\,d\mathbf{r}_2\,d\mathbf{r}_3\cdots d\mathbf{r}_N. \label{eq:charge-wf} \end{equation}

By construction \(\int \rho(\mathbf{r}),d\mathbf{r} = N\): the density integrates to the total number of electrons, not unity. \(\rho(\mathbf{r})\) is the central variable of DFT.

The First Hohenberg–Kohn 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:

\[ \begin{aligned} \hat{H} & = \hat{T} + \hat{V}_{\rm ee} + \hat{V}_{\rm ext}, & \hat{H}|\Psi\rangle = E_0|\Psi\rangle, \\ \hat{H}’ & = \hat{T} + \hat{V}_{\rm ee} + \hat{V}‘_{\rm ext}, & \hat{H}’|\Psi’\rangle = E’_0|\Psi’\rangle. \end{aligned} \]

We apply the Rayleigh–Ritz variational principle (Chapter 1.4), 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 \leq& \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 \leq \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}:

\[ \begin{align} E_0 + E’_0 & \leq E’_0 + E_0 + \int \underbrace{\left[ V_{\rm ext} - V’_{\rm ext} + V’_{\rm ext} - V_{\rm ext} \right]}_{=,0} \rho(\mathbf{r}) , d\mathbf{r}, \\ E_0 + E’_0 & \leq E_0 + E’_0. \end{align} \]

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

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 of the 2nd HK Theorem

By the First HK Theorem, every v-representable density \(\tilde{\rho}\) uniquely determines an external potential \(\tilde{V}_{\rm ext}\) (up to a constant), and hence a Hamiltonian \(\tilde{H}\) and a ground-state wavefunction \(\tilde{\Psi}\). The ground-state energy functional is therefore well-defined:

\[ E[\tilde{\rho}] = \langle \tilde{\Psi}[\tilde{\rho}] | \hat{T} + \hat{V}_{ee} + \hat{V}_{\rm ext} | \tilde{\Psi}[\tilde{\rho}] \rangle = F[\tilde{\rho}] + \int V_{\rm ext}(\mathbf{r})\tilde{\rho}(\mathbf{r}) d\mathbf{r} \]

Now consider the true ground state \(|\Psi_0\rangle\) with density \(\rho_0\) and energy \(E_0\). For any trial density \(\tilde{\rho} \neq \rho_0\), the associated wavefunction \(\tilde{\Psi}\) is different from \(\Psi_0\) (again by the First HK Theorem). Since \(\Psi_0\) is the true ground state of \(\hat{H}\), the Rayleigh–Ritz variational principle (Chapter 1.4) gives:

\[ E_0 = \langle \Psi_0 | \hat{H} | \Psi_0 \rangle \leq \langle \tilde{\Psi} | \hat{H} | \tilde{\Psi} \rangle. \]

The right-hand side is precisely \(E[\tilde{\rho}]\) as defined above (using \(\hat{V}_{\rm ext}\) of the original system). Therefore:

\[ E_0 \leq E[\tilde{\rho}], \quad \forall, \tilde{\rho} \neq \rho_0, \]

with equality if and only if \(\tilde{\Psi} = \Psi_0\), i.e. \(\tilde{\rho} = \rho_0\). \(\blacksquare\)

The key logical chain is:

The First HK Theorem guarantees the map \(\tilde{\rho} \mapsto \tilde{\Psi}\) is one-to-one, so \(\tilde{\rho} \neq \rho_0 \Rightarrow \tilde{\Psi} \neq \Psi_0\).
The Rayleigh–Ritz principle then directly delivers the inequality \(E[\tilde{\rho}] \geq E_0\).

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 \(\rho\).

The Levy constrained-search formulation. The original HK derivation defines \(F[\rho]\) only for densities that are ground-state densities of some external potential — the \(v\)-representable densities. Not every smooth non-negative \(\rho\) with \(\int\rho,d\mathbf{r} = N\) satisfies this; characterising the set of \(v\)-representable densities is in fact an open problem. Levy (1979) and Lieb (1983) eliminated this concern by recasting the universal functional as a constrained search over wavefunctions:

\begin{equation} F[\rho] = \min_{\Psi \to \rho}\,\langle\Psi|\hat{T} + \hat{V}_{ee}|\Psi\rangle, \label{eq:Levy} \end{equation}

where the minimisation runs over all antisymmetric \(N\)-electron wavefunctions \(\Psi\) that yield the given density \(\rho(\mathbf{r})\) via equation \eqref{eq:charge-wf}. Two features make this definition powerful:

It is well-defined for any non-negative \(\rho\) integrating to \(N\) — the larger class of \(N\)-representable densities — sidestepping the \(v\)-representability question entirely.
The minimisation principle of the second HK theorem follows directly: for any trial density \(\tilde\rho\), \(E[\tilde\rho] = F[\tilde\rho] + \int V_{\rm ext}\tilde\rho,d\mathbf{r} \geq E_0\), because the inner search over \(\Psi\) is itself a Rayleigh–Ritz minimisation.

In the Kohn–Sham scheme (Chapter 3), an analogous non-interacting constrained search defines \(T_s[\rho] = \min_{\Psi \to \rho}\langle\Psi|\hat{T}|\Psi\rangle\) over Slater determinants yielding \(\rho\). The existence of a Slater determinant for any reasonable \(\rho\) is called non-interacting \(v\)-representability and is believed to hold for all physically relevant ground-state densities, though a general proof remains an open problem.

Practical takeaway. The universal functional \(F[\rho]\) is well-defined and unique, but its explicit form is unknown. Computing it 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.

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