Kohn Sham Equations

Date:

13 Sep 2019

Derivation

Step 1 Hatree Effective Potential for Single Particle

\[\begin{align*} & v_{\text{HAeffectiveSP}} = \underbrace{-{Z\over r}}_{\text{Nucleus}} + \underbrace{ {1\over 2}\int {n(r') \over |r-r'|} \; dr' }_{\text{Other }e^-} \end{align*}\]

Step 2 Kohn Sham Effective Potential for Single Particle

\[\begin{split}\begin{align*} & \text{Rewrite } F[n]\equiv \min\limits_{i} \langle \psi_n^{(i)}|T+U_{\text{Interaction}}|\psi_n^{(i)} \rangle \\ & F[n] \equiv T[n]+{1\over 2}\int {n(r)n(r') \over |r-r'|} \; dr dr' +\underbrace{E_{XC}[n]}_{\text{eXchange-Correlation}}\\ & \text{Now } E[n]\equiv T[n]+{1\over 2}\int {n(r)n(r') \over |r-r'|} \; dr dr' + E_{XC}[n] +\int v_{\text{External}}(r) n(r)dr \\ & \text{To find } E_{GS}, \text{ Use Euler-Lagrange Equation:}\\ & \delta E[n] = \int \delta n \left\{ v_{\text{KSeffectiveSP}}(r)+ {\delta \over \delta n}T[n]-\epsilon \right\} dr\overset{!}{=}0\\ & where \\ & \epsilon \text{ is Lagrange Multiplier to assure Particle Conservation (Foot Note 1)}\\ & v_{\text{KSeffectiveSP}}(r) \equiv v_{\text{External}}(r) + {1\over 2}\int {n(r') \over |r-r'|} \; dr' +\underbrace{ {\delta \over \delta n}E_{XC}[n] }_{\equiv v_{XC}} \end{align*}\end{split}\]

• Foot Note 1: see DFT In Momentum Space for details

Step 3 Self-Consistent Loop

\[\begin{split}\begin{align*} & \text{Step 1: Initialize }n(r) \text{ with Educated Guess} \\ & \text{Step 2: Use }n(r) \text{ to calculate } v_{\text{__effectiveSP}}\\ & \text{Step 3: Use Single Particle Schrodinger Equation:} \\ & \;\;\;\; (-{1\over 2}\nabla^2 + v_{\text{__effectiveSP}})\psi_i=\psi_i \text{ to calculate } \psi_i \\ & \text{Step 4: Use } \psi_i \text{ to calculate new } n(r)=\sum\limits_i |\psi_i|^2 \\ & \text{Step 5: Go to Step 2 unless }n(r) \text{ converges} \end{align*}\end{split}\]