The Hohenberg-Kohn Theorems

DFT was given a formal footing by the two theorems introduced by Hohenberg and Kohn in 1965. It is said that after the two theorems were introduced, the spectroscopist E. Bright Wilson [69] stood up and gave a much more conceptual overview of the theory.

If the exact electron density is known, then the cusps in $\rho({\bf r})$ will provide the positions of the nuclei. The slope of $\rho({\bf r})$ at the nucleus A must obey
$$\begin{align}\left. \frac{\partial}{\partial{\bf r}_{A}}\bar{\rho}({\bf r}_{A})\right\vert _{{\bf r}_{A}=0} = -2 Z_{A} \bar{\rho}(0) \end{align}$$
(where $\bar{\rho}$ denotes the spherical average of the density) giving the charge at the nucleus, Z_A. Thus the full Schrödinger Hamiltonian is known, as it is completely defined by the nuclear charges and position. Therefore the wavefunction and energy can be found, and hence, the system can be completely described by the electron density.

A Hamiltonian of the form
 $$\begin{align} \hat{H} = -\frac{1}{2} \sum_{i}^{N} \nabla_{i}^{2} + \sum_{i}^{N}v... ...um_{i}^{N}\sum_{j>i}^{N}\frac{1}{\vert{\bf r}_{i}-{\bf r}_{j}\vert} \end{align}$$
is completely determined by the external potential, $v({\bf r})$. The first Hohenberg and Kohn theorem [70] states that, for non-degenerate ground states, the external potential $v({\bf r})$ is determined, to within an additive constant, by the electron density, $\rho({\bf r})$. The theorem has since been extended to include degenerate ground states [71].

The proof is based on the minimum energy principle and begins by considering two external potentials, $v_{1}({\bf r})$ and $v_{2}({\bf r})$ arising from the same density. There will be two Hamiltonians, $\hat{H}_{1}$ and $\hat{H}_{2}$ with the same density, but different wavefunctions, $\Psi_{1}$ and $\Psi_{2}$. Now, using the variational principle,
$$\begin{align}E_{1}^{0} < \langle\Psi_{2}\vert\hat{H}_{1}\vert\Psi_{2}\rangle &= ... ...2}^{0} + \int\rho({\bf r})[v_{1}({\bf r})-v_{2}({\bf r})]\,d{\bf r}. \end{align}$$
Similarly
$$\begin{align}E_{2}^{0} < E_{1}^{0} - \int\rho({\bf r})[v_{1}({\bf r})-v_{2}({\bf r})]\,d{\bf r}, \end{align}$$
which leads to the contradiction
$$\begin{align}E_{1}^{0} + E_{2}^{0} < E_{2}^{0} + E_{1}^{0}. \end{align}$$
Hence, the external potential is determined by the density and we may thus represent the energy as a functional of the density
 $$\begin{align} E[\rho]=\int\rho({\bf r})v({\bf r})\,dr+T[\rho]+V_{ee}[\rho] \end{align}$$
where $T[\rho]$ is the kinetic energy and $V_{ee}[\rho]$ is the electron-electron repulsion, including the Coulombic interaction, $J[\rho]$:
$$\begin{align}J[\rho]=\frac{1}{2}\iint\frac{\rho({\bf r}_{1})\rho({\bf r}_{2})}{\vert{\bf r}_{1}-{\bf r}_{2}\vert}\,d{\bf r}_{1}d{\bf r}_{2}. \end{align}$$

The second Hohenberg-Kohn theorem [70] introduces the variational principle into DFT; for a trial density $\tilde{\rho}({\bf r})$, such that $\tilde{\rho}({\bf r})\ge0$ and $\int\tilde{\rho}({\bf r})\,d{\bf r}=N$,
 $$\begin{align} E_{0} \le E[\tilde{\rho}] \end{align}$$
where $E[\tilde{\rho}]$ is the energy functional from equation (2.7). The proof is as follows: the first Hohenberg-Kohn theorem allows $\tilde{\rho}$ to determine its own potential $\tilde{v}$, Hamiltonian $\hat{H}$ and wavefunction $\tilde{\Psi}$, which can be used as a trial wavefunction for the problem with external potential v. Therefore,
$$\begin{align}\langle\tilde{\Psi}\vert\hat{H}\vert\tilde{\Psi}\rangle = \int\tild... ...}+T[\tilde{\rho}]+V_{ee}[\tilde{\rho}] = E[\tilde{\rho}]\ge E[\rho]. \end{align}$$

Assuming that $E[\rho]$ is differentiable, equation (2.9) requires that the ground state density be stationary, subject to the constraint that the integral of the density gives the number of electrons,
$$\begin{align}\delta E[\rho]-\mu\delta\left[\int\rho({\bf r})\,d{\bf r}-N\right]=0 \end{align}$$
which leads to the Euler-Lagrange equation
 $$\begin{align} \mu = v({\bf r}) + \frac{\delta T[\rho]}{\delta \rho({\bf r})} + \frac{\delta V_{ee}[\rho]}{\delta \rho({\bf r})} \end{align}$$
where a familiar property to chemists, the chemical potential $\mu$, has been introduced.

Equation (2.12) would be an exact equation for $\rho({\bf r})$ if the exact form of $T[\rho]$ and $V_{ee}[\rho]$ were known. Unfortunately the Hohenberg-Kohn theorems do not provide this, only that they exist. Also $T[\rho]$ and $V_{ee}[\rho]$ are defined independently of $v({\bf r})$, so once we have a form for these functionals they can be applied to any system.