X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=posic%2Fthesis%2Fbasics.tex;h=04542e23084efa8ec559331461b167ba92d4fc2c;hp=8d57d43d4619ffcedd0c31b48c68f510a8042cc5;hb=1de6d05da5be8de10b91e221d1de6580742e93f9;hpb=d02fc9175bf382d3fede6800f667cda3de7528b3 diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex index 8d57d43..04542e2 100644 --- a/posic/thesis/basics.tex +++ b/posic/thesis/basics.tex @@ -400,7 +400,7 @@ Clearly, this directs attention to the functional, which now contains the costs 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~\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} \text{ ,} @@ -428,13 +428,13 @@ This is called the generalized-gradient approximation (GGA), which expresses the 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. -At modest computational costs gradient-corrected functionals very often yield much better results than the LDA with respect to cohesive and binding energies. +This functional constitutes the simplest extension of LDA for inhomogeneous systems. +At modest computational costs, gradient-corrected functionals very often yield much better results than the LDA with respect to cohesive and binding energies. \subsection{Plane-wave basis set} Finally, a set of basis functions is required to represent the one-electron KS wave functions. -With respect to the numerical treatment it is favorable to approximate the wave functions by linear combinations of a finite number of such basis functions. +With respect to the numerical treatment, it is favorable to approximate the wave functions by linear combinations of a finite number of such basis functions. Convergence of the basis set, i.e.\ convergence of the wave functions with respect to the amount of basis functions, is most crucial for the accuracy of the numerical calculations. Two classes of basis sets, the plane-wave and local basis sets, exist. @@ -448,7 +448,7 @@ Another approach is to represent the KS wave functions by plane waves. In fact, the employed \textsc{vasp} software is solving the KS equations within a plane-wave (PW) basis set. The idea is based on the Bloch theorem~\cite{bloch29}, which states that in a periodic crystal each electronic wave function $\Phi_i(\vec{r})$ can be written as the product of a wave-like envelope function $\exp(i\vec{kr})$ and a function that has the same periodicity as the lattice. The latter one can be expressed by a Fourier series, i.e.\ a discrete set of plane waves whose wave vectors just correspond to reciprocal lattice vectors $\vec{G}$ of the crystal. -Thus, the one-electron wave function $\Phi_i(\vec{r})$ associated with the wave vector $\vec{k}$ can be expanded in terms of a discrete PW basis set +Thus, the one-electron wave function $\Phi_i(\vec{r})$ associated with the wave vector $\vec{k}$ can be expanded in terms of a discrete PW basis set: \begin{equation} \Phi_i(\vec{r})=\sum_{\vec{G} %, |\vec{G}+\vec{k}|