The Hohenberg-Kohn Theorems

The Hohenberg-Kohn theorems relate to any system consisting of electrons moving under the influence of an external potential $v_{\hbox{{ext}}}({\bf r})$. Stated simply they are as follows:

Theorem 1.
The external potential $v_{\hbox{{ext}}}({\bf r})$, and hence the total energy, is a unique functional of the electron density $n({\bf r})$.

The energy functional $E[n({\bf r})]$ alluded to in the first Hohenberg-Kohn theorem can be written in terms of the external potential $v_{\hbox{{ext}}}({\bf r})$ in the following way,

$$E[n({\bf r})] = \int n({\bf r}) \, v_{\hbox{{ext}}}({\bf r}) \, d{\bf r} \, + \, F[n({\bf r})] \, ,$$ (1.28)

where $F[n({\bf r})]$ is an unknown, but otherwise universal functional of the electron density $n({\bf r})$ only. Correspondingly, a Hamiltonian for the system can be written such that the electron wavefunction $\Psi$ that minimises the expectation value gives the groundstate energy (1.30) (assuming a non-degenerate groundstate),

$$E[n({\bf r})] = \langle \Psi \vert \hat{H} \vert \Psi \rangle \, .$$ (1.29)

The Hamiltonian can be written as,

$$\hat{H} = \hat{F} + \hat{V}_{\hbox{{ext}}} \, ,$$ (1.30)

where $\hat{F}$ is the electronic Hamiltonian consisting of a kinetic energy operator $\hat{T}$ and an interaction operator $\hat{V}_{\hbox{{ee}}}$,

$$\hat{F} = \hat{T} + \hat{V}_{\hbox{{ee}}} \, .$$ (1.31)

The electron operator $\hat{F}$ is the same for all $N$-electron systems, so $\hat{H}$ is completely defined by the number of electrons $N$, and the external potential $v_{\hbox{{ext}}}({\bf r})$.

The proof of the first theorem is remarkably simple and proceeds by reductio ad absurdum. Let there be two different external potentials, $v_{\hbox{{ext}},1}({\bf r})$ and $v_{\hbox{{ext}},2}({\bf r})$, that give rise to the same density $n_0({\bf r})$. The associated Hamiltonians, $\hat{H}_1$ and $\hat{H}_2$, will therefore have different groundstate wavefunctions, $\Psi_1$ and $\Psi_2$, that each yield $n_0({\bf r})$. Using the variational principle [19], together with (1.31) yields,

$$E_1^0 < \langle \Psi_2\vert \hat{H}_1 \vert \Psi_2 \rangle = \langle \Psi_2\vert \hat{H}_2 \vert \Psi_2 \rangle + \langle \Psi_2\vert \hat{H}_1 - \hat{H}_2 \vert \Psi_2 \rangle$$ (1.32)

$$= E_2^0 + \int n_0({\bf r}) [v_{\hbox{{ext}},1}({\bf r}) - v_{\hbox{{ext}},2}({\bf r})] \, d{\bf r}$$ (1.33)

where $E_1^0$ and $E_2^0$ are the groundstate energies of $\hat{H}_1$ and $\hat{H}_2$ respectively. It is at this point that the Hohenberg-Kohn theorems, and therefore DFT, apply rigorously to the groundstate only. An equivalent expression for (1.34) holds when the subscripts are interchanged. Therefore adding the interchanged inequality to (1.35) leads to the result:

$$E_1^0 + E_2^0 < E_2^0 + E_1^0$$ (1.34)

which is a contradiction, and as a result the groundstate density uniquely determines the external potential $v_{\hbox{{ext}}}({\bf r})$, to within an additive constant. Stated simply, the electrons determine the positions of the nuclei in a system, and also all groundstate electronic properties, because as mentioned earlier, $v_{\hbox{{ext}}}({\bf r})$ and $N$ completely define $\hat{H}$.

Theorem 2.
The groundstate energy can be obtained variationally: the density that minimises the total energy is the exact groundstate density.

The proof of the second theorem is also straightforward: as just shown, $n({\bf r})$ determines $v_{\hbox{{ext}}}({\bf r})$, $N$ and $v_{\hbox{{ext}}}({\bf r})$ determine $\hat{H}$ and therefore $\Psi$. This ultimately means $\Psi$ is a functional of $n({\bf r})$, and so the expectation value of $\hat{F}$ is also a functional of $n({\bf r})$, i.e.

$$F[n({\bf r})] = \langle \psi \vert \hat{F} \vert \psi \rangle \, .$$ (1.35)

A density that is the ground-state of some external potential is known as $v$-representable. Following from this, a $v$-representable energy functional $E_v[n({\bf r})]$ can be defined in which the external potential $v({\bf r})$ is unrelated to another density $n'({\bf r})$,

$$E_v[n({\bf r})] = \int n'({\bf r}) \, v_{\hbox{{ext}}}({\bf r}) \, d{\bf r} + F[n'({\bf r})] \, ,$$ (1.36)

and the variational principle asserts,

$$\langle \psi' \vert \hat{F} \vert \psi' \rangle + \langle \psi' \... ...ert \psi \rangle + \langle \psi \vert \hat{V}_{\hbox{{ext}}} \vert \psi \rangle$$ (1.37)

where $\psi$ is the wavefunction associated with the correct groundstate $n({\bf r})$. This leads to,

$$\int n'({\bf r}) \, v_{\hbox{{ext}}}({\bf r}) \, d{\bf r} + F[n'(... ...; \int n({\bf r}) \, v_{\hbox{{ext}}}({\bf r}) \, d{\bf r} + F[n({\bf r})] \, ,$$ (1.38)

and so the variational principle of the second Hohenberg-Kohn theorem is obtained,

$$E_v[n'({\bf r})] \, > \, E_v[n({\bf r})] \, .$$ (1.39)

Although the Hohenberg-Kohn theorems are extremely powerful, they do not offer a way of computing the ground-state density of a system in practice. About one year after the seminal DFT paper by Hohenberg and Kohn, Kohn and Sham [9] devised a simple method for carrying-out DFT calculations, that retains the exact nature of DFT. This method is described next.