1 \part{Theory of the solid state}
3 \chapter{Atomic structure}
5 \chapter{Electronic structure}
7 \section{Noninteracting electrons}
9 \subsection{Bloch's theorem}
11 \section{Nearly free and tightly bound electrons}
13 \subsection{Tight binding model}
15 \section{Interacting electrons}
17 \subsection{Density functional theory}
19 \subsubsection{Hohenberg-Kohn theorem}
21 The Hamiltonian of a many-electron problem has the form
27 T & = & \langle\Psi|\sum_{i=1}^N\frac{-\hbar^2}{2m}\nabla_i^2|\Psi\rangle\\
28 & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \,
29 \langle \Psi | \vec{r} \rangle \langle \vec{r} |
31 | \vec{r}' \rangle \langle \vec{r}' | \Psi \rangle\\
32 & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \,
33 \langle \Psi | \vec{r} \rangle \nabla_{\vec{r}_i}
34 \langle \vec{r} | \vec{r}' \rangle
35 \nabla_{\vec{r}'_i} \langle \vec{r}' | \Psi \rangle\\
36 & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \,
37 \nabla_{\vec{r}_i} \langle \Psi | \vec{r} \rangle
38 \delta_{\vec{r}\vec{r}'}
39 \nabla_{\vec{r}'_i} \langle \vec{r}' | \Psi \rangle\\
40 & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} \,
41 \nabla_{\vec{r}_i} \Psi^*(\vec{r}) \nabla_{\vec{r}_i} \Psi(\vec{r})
43 V & = & \int V(\vec{r})\Psi^*(\vec{r})\Psi(\vec{r})d\vec{r} \text{ ,} \\
44 U & = & \frac{1}{2}\int\frac{1}{\left|\vec{r}-\vec{r}'\right|}
45 \Psi^*(\vec{r})\Psi^*(\vec{r}')\Psi(\vec{r}')\Psi(\vec{r})
48 represent the kinetic energy, the energy due to the external potential and the energy due to the mutual Coulomb repulsion.
51 As can be seen from the above, two many-electron systems can only differ in the external potential and the number of electrons.
52 The number of electrons is determined by the electron density.
54 N=\int n(\vec{r})d\vec{r}
56 Now, if the external potential is additionally determined by the electron density, the density completely determines the many-body problem.
59 Considering a system with a nondegenerate ground state, there is obviously only one ground-state charge density $n_0(\vec{r})$ that corresponds to a given potential $V(\vec{r})$.
61 n_0(\vec{r})=\int \Psi_0^*(\vec{r},\vec{r}_2,\vec{r}_3,\ldots,\vec{r}_N)
62 \Psi_0(\vec{r},\vec{r}_2,\vec{r}_3,\ldots,\vec{r}_N)
63 d\vec{r}_2d\vec{r}_3\ldots d\vec{r}_N
65 In 1964, Hohenberg and Kohn showed the opposite and far less obvious result \cite{hohenberg64}.
67 \begin{theorem}[Hohenberg / Kohn]
68 For a nondegenerate ground state, the ground-state charge density uniquely determines the external potential in which the electrons reside.
72 The proof presented by Hohenberg and Kohn proceeds by {\em reductio ad absurdum}.
73 Suppose two potentials $V_1$ and $V_2$ exist, which yield the same electron density $n(\vec{r})$.
74 The corresponding Hamiltonians are denoted $H_1$ and $H_2$ with the respective ground-state wavefunctions $\Psi_1$ and $\Psi_2$ and eigenvalues $E_1$ and $E_2$.
75 Then, due to the variational principle (see \ref{sec:var_meth}), one can write
77 E_1=\langle \Psi_1 | H_1 | \Psi_1 \rangle <
78 \langle \Psi_2 | H_1 | \Psi_2 \rangle \text{ .}
81 Expressing $H_1$ by $H_2+H_1-H_2$, the last part of \eqref{subsub:hk01} can be rewritten:
83 \langle \Psi_2 | H_1 | \Psi_2 \rangle =
84 \langle \Psi_2 | H_2 | \Psi_2 \rangle +
85 \langle \Psi_2 | H_1 -H_2 | \Psi_2 \rangle
87 The two Hamiltonians, which describe the same number of electrons, differ only in the potential
89 H_1-H_2=V_1(\vec{r})-V_2(\vec{r})
93 E_1<E2+\int n(\vec{r}) \left( V_1(\vec{r})-V_2(\vec{r}) \right) d\vec{r}
97 By switching the indices of \eqref{subsub:hk02} and adding the resulting equation to \eqref{subsub:hk02}, the contradiction
99 E_1 + E_2 < E_2 + E_1 +
101 \int n(\vec{r}) \left( V_1(\vec{r})-V_2(\vec{r}) \right) d\vec{r} +
102 \int n(\vec{r}) \left( V_2(\vec{r})-V_1(\vec{r}) \right) d\vec{r}
105 is revealed, which proofs the Hohenberg Kohn theorem.% \qed