Density Functional Theory

Date:

28 July 2019

Dependency

What is the problem ?

Calculate the approximate wavefunction and energy for atoms

Derivation

Step 1 Hohenberg Kohn theorem

\(n(r)\): groundstate electron density

\(v(r)\): external potential

Then Functional \(v=v[n]\), meaning if we observe some \(n(r)\), then we know exactly what potential \(v(r)\) generates such density distribution

Step 2 Proof of Step 1

TO BE ADDED

Step 3 The energy

\[\begin{split}\begin{align*} E[n] & = T+U+V = T[n]+U[n]+\int n(r)v(r) \; d^3r\\ & where\\ & T = \text{Kinetic energy of electrons}\\ & U = \text{Electric potential}\\ & V = \text{Total external potential} \end{align*}\end{split}\]

Step 4 Kohn-Sham approach to evaluate The energy

Introduce single particle orbitals \(\chi_i\)

\[\begin{split}\begin{align*} & T=T_{\text{single particle}} + T_{\text{correlation}} = T_s+T_c \\ & T_s = T_s\{\chi_i[n]\}\\ & \text{Introduce:}\\ & E_{xc} = (T-T_s)+(U-U_H)\\ & \Rightarrow\\ E[n] & = T_s\{\chi_i[n]\}+U_H+E_{xc}+V \end{align*}\end{split}\]

The idea above is to leave all the non-trivial parts to \(E_{xc}=E_{x}+E_{c}\)

  • \(x\) stands for exchange energy due to the Pauli Principle

  • \(c\) stands for correlation energy, \(T_c\) is part of \(E_c\)

We have

\[\begin{split}\begin{align*} & T_s = ...\text{ trivial expression we know} \\ & U_H = ...\text{ trivial expression we know} \\ & E_x = ???\text{ can only be calculated after approximation(step 5)} \\ & E_c = ???\text{ can only be calculated after approximation(step 5)} \end{align*}\end{split}\]

Step 5 Approximations to find \(E_{xc}\)

TO BE ADDED

Step 6 Get the groundstate energy by minimizing \(E[n]\) wrt \(n\)

Kohn-Sham Equations

TO BE CONTINUED