Every Density Functional Theory calculation ever run rests on two theorems published by Pierre Hohenberg and Walter Kohn in 1964 (Phys. Rev. 136, B864). They answer a startling question: can the humble electron density — a function of just three coordinates — really contain all the information locked inside the fantastically complex many-body wavefunction? This post explains both theorems, their proofs in plain language, and why they matter for practical calculations.
The Motivation: Escaping the Many-Body Wavefunction
For a system of \(N\) interacting electrons, the ground state is described by a wavefunction that depends on the coordinates of every electron:
$$ \Psi = \Psi(\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N) $$This object lives in a \(3N\)-dimensional space and is impossible to store or manipulate for more than a handful of electrons. The electron density, by contrast, is defined by integrating the wavefunction over all electrons but one:
$$ n(\mathbf{r}) = N \int \lvert\Psi(\mathbf{r}, \mathbf{r}_2, \ldots, \mathbf{r}_N)\rvert^2\, d\mathbf{r}_2 \cdots d\mathbf{r}_N $$It is always a function of only three spatial coordinates, regardless of \(N\). The physical intuition that the density might be enough came from the Thomas-Fermi model of the 1920s, but that was a plausible approximation, not a proof. Hohenberg and Kohn supplied the rigorous justification.
What Defines an Electronic System
To appreciate the theorems, note what actually specifies a many-electron problem. Fix the number of electrons \(N\) and the external potential \(v_{\mathrm{ext}}(\mathbf{r})\) — for a molecule or solid, this is simply the electrostatic potential of the nuclei. Together these determine the Hamiltonian, hence the wavefunction, hence every ground-state property.
The conventional logic flows in one direction:
flowchart LR A["v_ext r"] --> B["H"] B --> C["Psi"] C --> D["n r and all properties"]
The external potential is the input; the density is one of many outputs. Hohenberg and Kohn’s astonishing claim is that this arrow can be reversed.
The First Hohenberg-Kohn Theorem: Density Determines Everything
The first theorem states:
The external potential \(v_{\mathrm{ext}}(\mathbf{r})\) is determined, up to an additive constant, uniquely by the ground-state electron density \(n(\mathbf{r})\).
In other words, the logical arrow runs both ways:
flowchart LR A["n r"] <--> B["v_ext r"] B --> C["H"] C --> D["Psi"] D --> E["everything"]
Since \(n(\mathbf{r})\) fixes \(v_{\mathrm{ext}}(\mathbf{r})\), and \(v_{\mathrm{ext}}(\mathbf{r})\) (with \(N\), which is just the integral of the density) fixes the Hamiltonian, the ground-state density determines the full Hamiltonian and therefore every ground-state property of the system — total energy, forces, band structure, magnetic moments, all of it.
The Proof by Contradiction
The original proof is elegant and short. Assume the theorem is false: suppose two different external potentials, \(v_1\) and \(v_2\) (differing by more than a constant), both give the same ground-state density \(n(\mathbf{r})\). They correspond to different Hamiltonians \(H_1\) and \(H_2\) with different ground states \(\Psi_1\) and \(\Psi_2\) and energies \(E_1\) and \(E_2\).
Using the variational principle, \(\Psi_2\) is not the ground state of \(H_1\), so:
$$ E_1 < \langle\Psi_2\lvert H_1\rvert\Psi_2\rangle = E_2 + \int (v_1 - v_2)\, n(\mathbf{r})\, d\mathbf{r} $$By symmetry, swapping the labels gives:
$$ E_2 < E_1 + \int (v_2 - v_1)\, n(\mathbf{r})\, d\mathbf{r} $$Adding these two strict inequalities produces the contradiction:
$$ E_1 + E_2 < E_1 + E_2 $$which is impossible. Therefore no two distinct external potentials can share the same ground-state density. The mapping from density to potential is unique.
The Universal Functional
Because \(n(\mathbf{r})\) determines the Hamiltonian, it also determines the kinetic energy and the electron-electron interaction energy — and crucially, those two pieces do not depend on the specific external potential. Hohenberg and Kohn collected them into a single object:
$$ F[n] = T[n] + V_{\mathrm{ee}}[n] $$This \(F[n]\) is the universal functional. The word “universal” is the key: it is the same functional for every electronic system in the universe — a hydrogen atom, a protein, a slab of copper. The only thing that changes from system to system is the external potential. The total energy is then written:
$$ E[n] = F[n] + \int v_{\mathrm{ext}}(\mathbf{r})\, n(\mathbf{r})\, d\mathbf{r} $$If we knew \(F[n]\) exactly, we could compute the ground state of any material without ever touching a wavefunction. The catch, of course, is that its exact form is unknown — the entire enterprise of building exchange-correlation functionals is the ongoing effort to approximate the difficult parts of \(F[n]\).
The Second Hohenberg-Kohn Theorem: The Variational Principle
The first theorem establishes existence but gives no way to find the right density. The second theorem supplies the search principle:
For any trial density \(n'(\mathbf{r})\) that is non-negative and integrates to \(N\), the energy functional satisfies \(E[n'] \geq E_0\), with equality if and only if \(n'(\mathbf{r})\) is the true ground-state density.
In symbols:
$$ E[n'] \geq E_0 \quad \text{for all valid } n'(\mathbf{r}) $$$$ E[n_0] = E_0 \quad (\text{the true ground-state density}) $$This is a variational principle in the density. It means we can find the ground state by minimizing the energy functional over densities, rather than over the vastly larger space of wavefunctions:
$$ E_0 = \min_{n(\mathbf{r})} E[n] $$The proof follows directly from the ordinary Rayleigh-Ritz variational principle for wavefunctions: each valid density corresponds to a wavefunction, and minimizing over densities is a constrained minimization over those wavefunctions, which can never dip below the true ground-state energy.
Comparing the Two Theorems
| Feature | First theorem | Second theorem |
|---|---|---|
| What it establishes | Density determines the potential (uniqueness) | Energy is variational in the density |
| Type of statement | Existence / one-to-one mapping | Optimization principle |
| Practical role | Justifies using \(n(\mathbf{r})\) as the variable | Tells us how to find the ground state |
| Proof technique | Reductio ad absurdum | Rayleigh-Ritz variational principle |
Why These Theorems Justify Using n(r)
Before Hohenberg-Kohn, using the density as the fundamental variable was a hopeful approximation. After 1964 it was rigorous. The theorems collapse a \(3N\)-dimensional problem to a 3-dimensional one in principle exactly, not approximately. This is the philosophical bedrock of the entire field.
Two important caveats keep the theory honest:
- v-representability. Not every mathematically valid density corresponds to some external potential’s ground state. The Levy constrained-search formulation (1979) later generalized the theorems to sidestep this, defining \(F[n]\) as a minimization over all wavefunctions yielding a given density.
- Ground state only. The theorems apply strictly to the non-degenerate ground state. Excited states require extensions such as time-dependent DFT.
From Theorems to Practical Calculations
The Hohenberg-Kohn theorems are pure existence results — they prove \(F[n]\) exists but not what it is, and the direct minimization of \(E[n]\) fails because the kinetic-energy functional is unknown. The bridge to real computation is the Kohn-Sham construction of 1965 (Phys. Rev. 140, A1133), which reintroduces orbitals to capture the kinetic energy accurately. We cover that machinery in our companion post on the Kohn-Sham equations, and the choice of engine and functional in our computational engines overview.
In a modern plane-wave code such as Quantum ESPRESSO, the abstract variational principle becomes a concrete self-consistent minimization of the total energy, launched with a command as simple as:
pw.x -in scf.in > scf.out
Every such run is, at bottom, an exercise in the two theorems described here: search over densities (second theorem) for the one that determines the true ground state (first theorem).
The Levy Constrained-Search Formulation
The original Hohenberg-Kohn proof has a subtle gap: it assumes every density we minimize over is the ground-state density of some external potential — the v-representability problem. Mel Levy (1979) and Elliott Lieb (1983) removed this restriction with a more general definition of the universal functional:
$$ F[n] = \min_{\Psi \to n} \langle\Psi\lvert T + V_{\mathrm{ee}}\rvert\Psi\rangle $$In words: search over all antisymmetric wavefunctions that yield the given density \(n(\mathbf{r})\), and take the one that minimizes the kinetic plus interaction energy. This constrained search only requires the density to be N-representable (obtainable from some antisymmetric wavefunction), a much weaker and easily satisfied condition. The Levy formulation put DFT on fully rigorous footing and clarified exactly what the universal functional means.
Spin-DFT and Practical Extensions
Real calculations almost always use spin-density functional theory, which promotes the theorems to work with two densities — spin-up \(n_\uparrow(\mathbf{r})\) and spin-down \(n_\downarrow(\mathbf{r})\) — or equivalently the total density and the magnetization density. This is essential for magnetic materials, radicals, and any open-shell system. Analogous existence and variational theorems hold. Further generalizations extend the Hohenberg-Kohn logic to finite temperature (Mermin), to time-dependent potentials (Runge-Gross, the basis of TDDFT for excited states), and to current-carrying systems (current-DFT), showing how far the original insight reaches.
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