started basis set

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index 2aa0e3e..3bc9527 100644 (file)
--- 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}
 \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})}
 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.
 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.
 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}
 \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}
 \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.
 
 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}
 \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}
 \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 $|\nabla n|$.
 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}
 \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.
 \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}
 
 
 \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}
 \subsection{Pseudopotentials}
 
 \subsection{Brillouin zone sampling}