Hohenberg-Kohn theorem

The Hohenberg-Kohn theorem (of Walter Kohn and Pierre Hohenberg ) is in the Quantum Chemistry a proof that important characteristics such as the energy of a system of electrons (e.g., as a single molecule or a solid ) directly from the electron density of the System. The theorem is thus the basis for modern density functional theory (DFT).

The theorem consists of two parts (sometimes referred to as Hohenberg-Kohn theorems , HK1 and HK2 ). The first part consists of the proof that the underlying electrical potential can be clearly deduced from a certain electron density (except for an additive constant ) , whereby the Hamilton operator in the Schrödinger equation and thus all other properties of the system are determined. The second part is a proof that, for a given system, the electron density of the ground state leads to the lowest value for the energy of the ground state, whereby it is possible to approximate it.

In this simple formulation the theorems only apply if the basic state of the system is not degenerate - in this case, however, they can be extended. The case of electron densities that can be changed over time is covered by the Runge-Gross theorem .

proof

The following proof is valid for non-degenerate ground states and proves part 1 of the theorem about a reductio ad absurdum . The first assumption: the ground state is given by the wave function . Furthermore, the system has the potential and thus the Hamilton operator . ${\displaystyle \Psi _{1}}$ ${\displaystyle V_{1}({\vec {r}})}$ ${\displaystyle {\hat {H}}_{1}={\hat {V}}_{1}+{\hat {U}}+{\hat {T}}}$

Then applies in Dirac notation

${\displaystyle E_{1}=\langle \Psi _{1}|{\hat {H}}_{1}|\Psi _{1}\rangle =\int V_{1}({\vec {r}})\rho ({\vec {r}})\,\mathrm {d^{3}} r+\langle \Psi _{1}|({\hat {T}}+{\hat {U}})|\Psi _{1}\rangle }$

With

• ${\displaystyle E}$: the total electronic energy of the system,
• ${\displaystyle \rho ({\vec {r}})}$: the electron density at the location , ${\displaystyle {\vec {r}}}$
• ${\displaystyle {\hat {T}}}$: kinetic energy of electrons and
• ${\displaystyle {\hat {U}}}$: electrostatic repulsion of electrons among each other.

Now it is assumed that there is a second potential , with the Hamilton operator and the associated ground-state wave function , which leads to the same density . ${\displaystyle V_{2}({\vec {r}})\neq V_{1}({\vec {r}})}$ ${\displaystyle {\hat {H}}_{2}={\hat {V}}_{2}+{\hat {U}}+{\hat {T}}}$ ${\displaystyle \Psi _{2}}$ ${\displaystyle \rho ({\vec {r}})}$

Then it follows with the Rayleigh-Ritz principle , i.e. that the wave function of the ground state always leads to the lowest energy, for ${\displaystyle E_{1}}$

${\displaystyle E_{1}<\langle \Psi _{2}|{\hat {H}}_{1}|\Psi _{2}\rangle =\langle \Psi _{2}|{\hat {H}}_{2}|\Psi _{2}\rangle +\langle \Psi _{2}|{\hat {H}}_{1}-{\hat {H}}_{2}|\Psi _{2}\rangle =E_{2}+\int (V_{1}({\vec {r}})-V_{2}({\vec {r}}))\rho ({\vec {r}})\,\mathrm {d^{3}} r}$

In reverse composition, it follows for : ${\displaystyle E_{2}}$

${\displaystyle E_{2}<\langle \Psi _{1}|{\hat {H}}_{2}|\Psi _{1}\rangle =E_{1}+\int (V_{2}({\vec {r}})-V_{1}({\vec {r}}))\rho ({\vec {r}})\,\mathrm {d^{3}} r}$

By adding the two inequalities it follows:

${\displaystyle E_{1}+E_{2}<E_{1}+E_{2}}$

So the assumption was wrong and the theorem is thus proven.

literature

• P. Hohenberg and W. Kohn: Inhomogeneous Electron Gas . Phys. Rev. 136 (1964) B864-B871, doi : 10.1103 / PhysRev.136.B864

Individual evidence

1. ^ Wolfram Koch, Max C. Holthausen: Density Functional Theory for Chemists; Wiley-VCH, 2001