Hohenberg Kohn Variational Principle

Date:

13 Sep 2019

Derivation

Step 1 Recall the textbook Variational Principle to find Ground State energy \(E\)

\[\begin{align*} & E_{GS} = \min\limits_{\psi} \langle \psi|H|\psi \rangle \end{align*}\]

Step 2 Hohenberg Kohn Variational Principle (Constrained Search Method by Levy and Lieb)

\[\begin{split}\begin{align*} & \underbrace{n(r)}_{\text{Electron Density}} = \underbrace{|\psi(r)|^2}_{\text{Probability to find it at } r} \\ & \text{EACH wavefunction }\psi(r) \text{ gives ONE density distribution }n(r)\\ & \text{But EACH }n(r) \text{ may be generated by MANY }\psi(r) \\ & \Rightarrow \psi_n^{(i)}(r) \text{ so that } n(r)= |\psi_n^{(1)}(r)|^2 = |\psi_n^{(2)}(r)|^2 = ...\\ & \text{Thus first fix }n, \text{ find } F[n]\equiv \min\limits_{i} \langle \psi_n^{(i)}|T+U_{\text{Interaction}}|\psi_n^{(i)} \rangle \\ & \text{Then find } E_{GS} = \min\limits_{n} F[n]+\int \underbrace{v_{\text{External}}(r)}_{\text{Energy per particle}} n(r)dr \end{align*}\end{split}\]