From: hackbard Date: Fri, 20 May 2011 17:51:30 +0000 (+0200) Subject: finsihed pp, starting hf theorem X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=commitdiff_plain;h=97fd88ef629e90e71b9de659a1824bbd85e41077 finsihed pp, starting hf theorem --- diff --git a/bibdb/bibdb.bib b/bibdb/bibdb.bib index 5c53f66..f2f91b3 100644 --- a/bibdb/bibdb.bib +++ b/bibdb/bibdb.bib @@ -3890,6 +3890,22 @@ notes = "paw method", } +@InCollection{cohen70, + title = "The Fitting of Pseudopotentials to Experimental Data + and Their Subsequent Application", + editor = "Frederick Seitz Henry Ehrenreich and David Turnbull", + booktitle = "", + publisher = "Academic Press", + year = "1970", + volume = "24", + pages = "37--248", + series = "Solid State Physics", + ISSN = "0081-1947", + doi = "DOI: 10.1016/S0081-1947(08)60070-3", + URL = "http://www.sciencedirect.com/science/article/B8GXT-4S9NXKG-9/2/ba01db77c8b19c2bab01506d4ea571b3", + author = "Marvin L. Cohen and Volker Heine", +} + @Article{hamann79, title = "Norm-Conserving Pseudopotentials", author = "D. R. Hamann and M. Schl{\"u}ter and C. Chiang", @@ -3905,6 +3921,21 @@ notes = "norm-conserving pseudopotentials", } +@Article{troullier91, + title = "Efficient pseudopotentials for plane-wave + calculations", + author = "N. Troullier and Jos\'e Luriaas Martins", + journal = "Phys. Rev. B", + volume = "43", + number = "3", + pages = "1993--2006", + numpages = "13", + year = "1991", + month = jan, + doi = "10.1103/PhysRevB.43.1993", + publisher = "American Physical Society", +} + @Article{vanderbilt90, title = "Soft self-consistent pseudopotentials in a generalized eigenvalue formalism", diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex index 231345a..f469237 100644 --- a/posic/thesis/basics.tex +++ b/posic/thesis/basics.tex @@ -50,15 +50,15 @@ A system of $N$ particles of masses $m_i$ ($i=1,\ldots,N$) at positions ${\bf r} %m_i \frac{d^2}{dt^2} {\bf r}_i = {\bf F}_i \Leftrightarrow %m_i \frac{d}{dt} {\bf r}_i = {\bf p}_i\textrm{ , } \quad %\frac{d}{dt} {\bf p}_i = {\bf F}_i\textrm{ .} -m_i \ddot{{\bf r}_i} = {\bf F}_i \Leftrightarrow -m_i \dot{{\bf r}_i} = {\bf p}_i\textrm{, } -\dot{{\bf p}_i} = {\bf F}_i\textrm{ .} +m_i \ddot{\bf r}_i = {\bf F}_i \Leftrightarrow +m_i \dot{\bf r}_i = {\bf p}_i\textrm{, } +\dot{\bf p}_i = {\bf F}_i\textrm{ .} \end{equation} The forces ${\bf F}_i$ are obtained from the potential energy $U(\{{\bf r}\})$: \begin{equation} {\bf F}_i = - \nabla_{{\bf r}_i} U({\{\bf r}\}) \, \textrm{.} \end{equation} -Given the initial conditions ${\bf r}_i(t_0)$ and $\dot{{\bf r}}_i(t_0)$ the equations can be integrated by a certain integration algorithm. +Given the initial conditions ${\bf r}_i(t_0)$ and $\dot{\bf r}_i(t_0)$ the equations can be integrated by a certain integration algorithm. The solution of these equations provides the complete information of a system evolving in time. The following sections cover the tools of the trade necessary for the MD simulation technique. Three ingredients are required for a MD simulation: @@ -182,7 +182,7 @@ The potential succeeds in the description of the low as well as high coordinated The description of elastic properties of SiC is improved with respect to the potentials available in literature. Defect properties are only fairly reproduced but the description is comparable to previously published potentials. It is claimed that the potential enables modeling of widely different configurations and transitions among these and has recently been used to simulate the inert gas condensation of Si-C nanoparticles \cite{erhart04}. -Therefore the Erhart/Albe (EA) potential is considered the superior analytical bond order potential to study the SiC precipitation and associated processes in Si. +Therefore, the Erhart/Albe (EA) potential is considered the superior analytical bond order potential to study the SiC precipitation and associated processes in Si. \subsection{Verlet integration} \label{subsection:integrate_algo} @@ -417,7 +417,7 @@ E_{\text{xc}}[n(\vec{r})]=\int\epsilon_{\text{xc}}(\vec{r};[n(\tilde{\vec{r}})]) \end{equation} 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 $n$. +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 \begin{equation} @@ -497,15 +497,37 @@ Fortunately, the impossibility to model the core in addition to the valence elec \subsection{Pseudopotentials} As discussed in the last part of the previous section, an extremely large basis set of plane waves would be required to perform an all-electron calculation and a vast amount of computational time would be required to calculate the electronic wave functions. -It is worth to stress out one more time, that this is due to the orthogonalization wiggles of the wave functions of valence electrons near the nuclei. +It is worth to stress out one more time, that this is mainly due to the orthogonalization wiggles of the wave functions of valence electrons near the nuclei. Thus, existing core states practically prevent the use of a PW basis set. However, the core electrons, which are tightly bound to the nuclei, do not contribute significantly to chemical bonding or other physical properties of the solid. -This fact is exploited in the pseudopotential approach \cite{} by removing the core electrons and replacing the atom and the associated strong ionic potential by a pseudoatom and a weaker pseudopotential that acts on a set of pseudo wave functions rather than the true valance wave functions. -Certain conditions need to be fulfilled by the constructed pseudopotentials and the resulting pseudo wave functions. -Outside the core region, the pseudo and real wafe functions as well as the generated charge densities need to be identical. -... -A pseudopotential is called norm-conserving if the pseudo and real charge contained within the core region match. -... +This fact is exploited in the pseudopotential (PP) approach \cite{cohen70} by removing the core electrons and replacing the atom and the associated strong ionic potential by a pseudoatom and a weaker PP that acts on a set of pseudo wave functions rather than the true valance wave functions. +This way, the pseudo wave functions become smooth near the nuclei. + +Most PPs statisfy four general conditions. +The pseudo wave functions generated by the PP should not contain nodes, i.e. the pseudo wave functions should be smooth and free of wiggles in the core region. +Outside the core region, the pseudo and real valence wave functions as well as the generated charge densities need to be identical. +The charge enclosed within the core region must be equal for both wave functions. +Last, almost redundantly, the valence all-electron and pseudopotential eigenvalues must be equal. +Pseudopotentials that meet the conditions outlined above are referred to as norm-conserving pseudopotentials \cite{hamann79}. + +%Certain properties need to be fulfilled by PPs and the resulting pseudo wave functions. +%The pseudo wave functions should yield the same energy eigenvalues than the true valence wave functions. +%The PP is called norm-conserving if the pseudo and real charge contained within the core region matches. +%To guarantee transferability of the PP the logarithmic derivatives of the real and pseudo wave functions and their first energy derivatives need to agree outside of the core region. +%A simple procedure was proposed to extract norm-conserving PPs obyeing the above-mentioned conditions from {\em ab initio} atomic calculations \cite{hamann79}. + +In order to achieve these properties different PPs are required for the different shapes of the orbitals, which are determined by their angular momentum. +Mathematically, a non-local PP, which depends on the angular momentum, has the form +\begin{equation} +V_{\text{nl}}(\vec{r}) = \sum_{lm} \mid lm \rangle V_l(\vec{r}) \langle lm \mid +\text{ .} +\end{equation} +Applying of the operator $V_{\text{nl}}(\vec{r})$ decomposes the electronic wave functions into spherical harmonics $\mid lm \rangle$, i.e. the orbitals with azimuthal angular momentum $l$ and magnetic number $m$, which are then multiplied by the respective psuedopotential $V_l(\vec{r})$ for angular momentum $l$. +The standard generation procedure of pseudopotentials proceeds by varying its parameters until the pseudo eigenvalues are eual to the all-electron valence eigenvalues and the pseudo wave functions match the all-electron valence wave functions beyond a certain cut-off radius detrmining the core region. +Modified methods to generate ultra-soft pseudopotentials were proposed, which address the rapid convergence with respect to the size of the plane wave basis set \cite{vanderbilt90,troullier91}. + +Using PPs the rapid oscillations of the wave functions near the core of the atoms are removed considerably reducing the number of plane waves necessary to appropriately expand the wave functions. +More importantly, less accuracy is required compared to all-electron calculations to determine energy differences among ionic configurations, which almost totally appear in the energy of the valence electrons that are typically a factor $10^3$ smaller than the energy of the core electrons. \subsection{Brillouin zone sampling} @@ -518,11 +540,20 @@ For supercells, i.e. repeating unit cells that contain several primitive cells, In fact, with respect to BZ sampling, calculating wave functions of a supercell containing $n$ primitive cells for only one $\vec{k}$ point is equivalent to the scenario of a single primitive cell and the summation over $n$ points in $\vec{k}$ space. In general, finer $\vec{k}$ point meshes better account for the periodicity of a system, which in some cases, however, might be fictious anyway. -\subsection{Hellmann-Feynman forces} +\subsection{Structural relaxation and the Hellmann-Feynman theorem} + +what is relaxed, ions! +if md based relaxaion followed the respective equations of motion are valid ... +of course better cg than simple moveing ions according to force ... +force as a derivative of total energy with respect to positions of the ions ... + \section{Modeling of defects} \label{section:basics:defects} +... +constructing defect configuration intuitively followed by relaxation procedure + \section{Migration paths and diffusion barriers} \label{section:basics:migration}