From 602832efd59018f1de63a29e6f9cdddaf5afd5da Mon Sep 17 00:00:00 2001 From: hackbard Date: Mon, 16 May 2011 19:57:48 +0200 Subject: [PATCH] started lda --- bibdb/bibdb.bib | 14 +++++++++ posic/thesis/basics.tex | 64 +++++++++++++++++++++++++++++++++++++++-- 2 files changed, 75 insertions(+), 3 deletions(-) diff --git a/bibdb/bibdb.bib b/bibdb/bibdb.bib index 012acb0..81c6b2f 100644 --- a/bibdb/bibdb.bib +++ b/bibdb/bibdb.bib @@ -2919,6 +2919,20 @@ doi = "10.1017/S0305004100011919", } +@Article{slater29, + title = {The Theory of Complex Spectra}, + author = {Slater, J. C.}, + journal = {Phys. Rev.}, + volume = {34}, + number = {10}, + pages = {1293--1322}, + numpages = {29}, + year = {1929}, + month = {Nov}, + doi = {10.1103/PhysRev.34.1293}, + publisher = {American Physical Society} +} + @Article{kohn65, title = "Self-Consistent Equations Including Exchange and Correlation Effects", diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex index 6c03878..b9ea381 100644 --- a/posic/thesis/basics.tex +++ b/posic/thesis/basics.tex @@ -329,6 +329,7 @@ E_0=\min_{n(\vec{r})} F[n(\vec{r})] + \int n(\vec{r}) V(\vec{r}) d\vec{r} \right\} \text{ ,} +\label{eq:basics:hkm} \end{equation} where $F[n(\vec{r})]$ is a universal functional of the charge density $n(\vec{r})$, which is composed of the kinetic energy functional $T[n(\vec{r})]$ and the interaction energy functional $U[n(\vec{r})]$. The challenging problem of determining the exact ground-state is now formally reduced to the determination of the $3$-dimensional function $n(\vec{r})$, which minimizes the energy functional. @@ -346,14 +347,71 @@ U=\frac{1}{2}\int\frac{n(\vec{r})n(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}d\vec{r \subsection{Kohn-Sham system} -Now find $F[n]$ ... - -As in the last section, the complex many-electron effects are relocated, this time into the exchange-correlation functional. +Inspired by the Hartree equations, i.e. a set of self-consistent single-particle equations for the approximate solution of the many-electron problem \cite{hartree28}, which describe atomic ground states much better than the TF theory, Kohn and Sham presented a Hartree-like formulation of the Hohenberg and Kohn minimal principle \eqref{eq:basics:hkm} \cite{kohn65}. +However, due to a more general approach, the new formulation is formally exact by introducing the energy functional $E_{\text{xc}}[n(vec{r})]$, which accounts for the exchange and correlation energy of the electron interaction $U$ and possible corrections due to electron interaction to the kinetic energy $T$. +The respective Kohn-Sham equations for the effective single-particle wave functions $\Phi_i(\vec{r})$ take the form +\begin{equation} +\left[ + -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(\vec{r}) +\right] \Phi_i(\vec{r})=\epsilon_i\Phi_i(\vec{r}) +\label{eq:basics:kse1} +\text{ ,} +\end{equation} +\begin{equation} +V_{\text{eff}}=V(\vec{r})+\int\frac{e^2n(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}' + + V_{\text{xc}(\vec{r})} +\text{ ,} +\label{eq:basics:kse2} +\end{equation} +\begin{equation} +n(\vec{r})=\sum_{i=1}^N |\Phi_i(\vec{r})|^2 +\text{ ,} +\label{eq:basics:kse3} +\end{equation} +where the local exchange-correlation potential $V_{\text{xc}}(\vec{r})$ is the partial derivative of the exchange-correlation functional $E_{\text{xc}}[n(vec{r})]$ with respect to the charge density $n(\vec{r})$ for the ground-state $n_0(\vec{r})$. +The first term in equation \eqref{eq:basics:kse1} corresponds to the kinetic energy of non-interacting electrons and the second term of equation \eqref{eq:basics:kse2} is just the Hartree contribution to the interaction energy. +%\begin{equation} +%V_{\text{xc}}(\vec{r})=\frac{\partial}{\partial n(\vec{r})} +% E_{\text{xc}}[n(\vec{r})] |_{n(\vec{r})=n_0(\vec{r})} +%\end{equation} + +The system of interacting electrons is mapped to an auxiliary system, the Kohn-Sham (KS) system, of non-interacting electrons in an effective potential. +The exact effective potential $V_{\text{eff}}(\vec{r})$ may be regarded as a fictious external potential yielding a gound-state density for non-interacting electrons, which is equal to that for interacting electrons in the external potential $V(\vec{r})$. +The one-electron KS orbitals $\Phi_i(\vec{r})$ as well as the respective KS energies $\epsilon_i$ are not directly attached to any physical observable except for the ground-state density, which is determined by equation \eqref{eq:basics:kse3} and the ionization energy, which is equal to the highest occupied relative to the vacuum level. +The KS equations may be considered the formal exactification of the Hartree theory, which it is reduced to if the exchange-correlation potential and functional are neglected. +In addition to the Hartree-Fock (HF) method, KS theory includes the difference of the kinetic energy of interacting and non-interacting electrons as well as the remaining contributions to the correlation energy that is not part of the HF correlation. + +The self-consistent KS equations \eqref{eq:basics:kse1,eq:basics:kse2,eq:basics:kse3} may be solved numerically by an iterative process. +Starting from a first approximation for $n(\vec{r})$ the effective potential $V_{\text{eff}}(\vec{r})$ can be constructed followed by determining the one-electron orbitals $\Phi_i(\vec{r})$, which solve the single-particle Schr\"odinger equation for the respective potential. +The $\Phi_i(\vec{r})$ are used to obtain a new expression for $n(\vec{r})$. +These steps are repeated until the initial and new density are equal or vary only slightly. + +Again, it is worth to note that the KS equations are formally exact. +Assuming exact functionals $E_{\text{xc}}[n(vec{r})]$ and potentials $V_{\text{xc}}(\vec{r})$ all many-body effects are included. +Clearly, this directs attention to the functional, which now contains the costs involved with the many-electron problem. \subsection{Approximations for exchange and correlation} +As discussed above, the HK and KS formulations are exact and so far no approximations except the adiabatic approximation entered the theory. +However, to make concrete use of the theory, effective approximations for the exchange and correlation energy functional $E_{\text{xc}}[n(vec{r})]$ are required. + +Most simple and at the same time remarkably useful is the approximation of $E_{\text{xc}}[n(vec{r})]$ by a function of the local density +\begin{equation} +E_{\text{xc}}[n(vec{r})]=\int\epsilon_{\text{xc}}(n(\vec{r}))n(\vec{r}) d\vec{r} +\text{ .} +\label{eq:basics:xca} +\end{equation} +Here, the exchange-correlation energy per particle of the uniform electron gas of constant density $n$ is used for $\epsilon_{\text{xc}}(n(\vec{r}))$. +This is called the local density approximation (LDA). + +\subsection{Plane-wave basis set} + \subsection{Pseudopotentials} +\subsection{Brillouin zone sampling} + +\subsection{Hellmann-Feynman forces} + \section{Modeling of defects} \label{section:basics:defects} -- 2.20.1