From: hackbard Date: Tue, 17 May 2011 15:51:57 +0000 (+0200) Subject: started basis set X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=3ea37b08b6020dd776a7ec70acd4ef7e47c2d8f1;p=lectures%2Flatex.git started basis set --- diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex index 2aa0e3e..3bc9527 100644 --- a/posic/thesis/basics.tex +++ b/posic/thesis/basics.tex @@ -368,7 +368,7 @@ 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 $n_0(\vec{r})$. +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})} @@ -387,18 +387,18 @@ 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 reasonably converged. 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. +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} \label{subsection:ldagga} 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. +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 \cite{kohn65} +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 \cite{kohn65} \begin{equation} -E^{\text{LDA}}_{\text{xc}}[n(vec{r})]=\int\epsilon_{\text{xc}}(n(\vec{r}))n(\vec{r}) d\vec{r} +E^{\text{LDA}}_{\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} @@ -411,17 +411,17 @@ Although LDA is known to overestimate the cohesive energy in solids by \unit[10- More accurate approximations of the exchange-correlation functional are realized by the introduction of gradient corrections with respect to the density \cite{kohn65}. Respective considerations are based on the concept of an exchange-correlation hole density describing the depletion of the electron density in the vicinity of an electron. -The averaged hole density can be used to give a formally exact expression for $E_{\text{xc}}[n(vec{r})]$ and an equivalent expression \cite{kohn96,kohn98}, which makes use of the electron density distribution $n(~\vec{r})$ at positions $~\vec{r}$ near $\vec{r}$, yielding the form +The averaged hole density can be used to give a formally exact expression for $E_{\text{xc}}[n(\vec{r})]$ and an equivalent expression \cite{kohn96,kohn98}, which makes use of the electron density distribution $n(\tilde{\vec{r}})$ at positions $\tilde{\vec{r}}$ near $\vec{r}$, yielding the form \begin{equation} -E_{\text{xc}}[n(vec{r})]=\int\epsilon_{\text{xc}}(\vec{r};[n(~\vec{r})])n(\vec{r}) d\vec{r} +E_{\text{xc}}[n(\vec{r})]=\int\epsilon_{\text{xc}}(\vec{r};[n(\tilde{\vec{r}})])n(\vec{r}) d\vec{r} \end{equation} -for the exchange-correlation energy, where $\epsilon_{\text{xc}}(\vec{r};[n(~\vec{r})])$ becomes a nearsighted functional of $n(~\vec{r})$. -Expressing $n(~\vec{r})$ in a Taylor series, $\epsilon_{\text{xc}}$ can be thought of as a function of coefficients, which correspond to the respective terms of the expansion. +for the exchange-correlation energy, where $\epsilon_{\text{xc}}(\vec{r};[n(\tilde{\vec{r}})])$ becomes a nearsighted functional of $n(\tilde{\vec{r}})$. +Expressing $n(\tilde{\vec{r}})$ in a Taylor series, $\epsilon_{\text{xc}}$ can be thought of as a function of coefficients, which correspond to the respective terms of the expansion. Neglecting all terms of order $\mathcal{O}(\nabla n(\vec{r})$ results in the functional equal to LDA, which requires the function of variable $n$. Including the next element of the Taylor series introduces the gradient correction to the functional, which requires the function of variables $n$ and $|\nabla n|$. -This is called the generalized gradient approximation (GGA), which expresses the exchange-correlation energy density as a function of the local density and the local gradient of the density. +This is called the generalized gradient approximation (GGA), which expresses the exchange-correlation energy density as a function of the local density and the local gradient of the density \begin{equation} -E^{\text{GGA}}_{\text{xc}}[n(vec{r})]=\int\epsilon_{\text{xc}}(n(\vec{r}),|\nabla n(\vec{r})|)n(\vec{r}) d\vec{r} +E^{\text{GGA}}_{\text{xc}}[n(\vec{r})]=\int\epsilon_{\text{xc}}(n(\vec{r}),|\nabla n(\vec{r})|)n(\vec{r}) d\vec{r} \text{ .} \end{equation} These functionals constitute the simplest extensions of LDA for inhomogeneous systems. @@ -429,6 +429,10 @@ At modest computational costs gradient-corrected functionals very often yield mu \subsection{Plane-wave basis set} +Practically, the KS equations are non-linear partial differential equations that are iteratively solved. +The one-electron KS wave functions can be represented in different basis sets. + + \subsection{Pseudopotentials} \subsection{Brillouin zone sampling}