X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=posic%2Fthesis%2Fbasics.tex;h=f78de50d1f0a2b0a9f88d3991f402dd3cf64eb6f;hp=fcf920c3938a01027f2aafea98b6c2b9ef704894;hb=f70b30c1ee10f4b89912c3a0e7760f5d5d184cbc;hpb=ecfd4c7626a37a0607f9e1239d5e98f322776a49

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index fcf920c..f78de50 100644
--- a/posic/thesis/basics.tex
+++ b/posic/thesis/basics.tex
@@ -50,15 +50,17 @@ 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{ .}
+\label{eq:basics:newton}
 \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{.}
+\label{eq:basics:force}
 \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 +184,7 @@ 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}
@@ -358,7 +360,7 @@ The respective Kohn-Sham equations for the effective single-particle wave functi
 \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{eff}}(\vec{r})=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}
@@ -369,7 +371,7 @@ n(\vec{r})=\sum_{i=1}^N |\Phi_i(\vec{r})|^2
 \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.
+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 $V_{\text{H}}(\vec{r})$ 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})}
@@ -417,7 +419,7 @@ 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}
@@ -438,13 +440,13 @@ Local basis set functions usually are atomic orbitals, i.e. mathematical functio
 Molecular orbitals can be represented by linear combinations of atomic orbitals (LCAO).
 By construction, only a small number of basis functions is required to represent all of the electrons of each atom within reasonable accuracy.
 Thus, local basis sets enable the implementation of methods that scale linearly with the number of atoms.
-However, these methods rely on ...
+However, these methods rely on the fact that the wave functions are localized and exhibit an exponential decay resulting in a sparse Hamiltonian.
 
 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 basis set.
+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 plane-wave 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}|<G_{\text{cut}}}
@@ -455,34 +457,134 @@ Thus, the one-electron wave function $\Phi_i(\vec{r})$ associated with the wave
 \end{equation}
 The basis set, which in principle should be infinite, can be truncated to include only plane waves that have kinetic energies $\hbar^2|\vec{k}+\vec{G}|^2/2m$ less than a particular cut-off energy $E_{\text{cut}}$.
 Although coefficients $c_{i,\vec{k}+\vec{G}}$ corresponding to small kinetic energies are typically more important, convergence with respect to the cut-off energy is crucial for the accuracy of the calculations.
-Convergence, however, is easily achieved by increasing $E_{\text{cut}}$ until the differences in total energy approximate zero.
-
-There are several advantages of plane waves.
-
-
-Disadvantage ... periodic system required, but escapable by respective choice of the supercell.
-
-
-very popular and most natural choice ...
-plane wave, natural ... choice in periodic systems
-can be thought of a fourier series ...
-constructed this way ...
-by definition orthonormal ...
-indeed it has been shown that accuracy ...
-
-
+Convergence with respect to the basis set, however, is easily achieved by increasing $E_{\text{cut}}$ until the respective differences in total energy approximate zero.
+
+Next to their simplicity, plane waves have several advantages.
+The basis set is orthonormal by construction and, as mentioned above, it is simple to check for convergence.
+The biggest advantage, however, is the ability to perform exact calculations by a discrete sum over a numerical grid.
+This is due to the related construction of the grid and the PW basis.
+Ofcourse, exactness is restricted by the fact that the PW basis set is finite.
+The simple form of the PW representation of the KS equations
+\begin{equation}
+\sum_{\vec{G}'} \left[
+ \frac{\hbar^2}{2m}|\vec{k}+\vec{G}|^2 \delta_{\vec{GG}'}
+ + \tilde{V}(\vec{G}-\vec{G}')
+ + \tilde{V}_{\text{H}}(\vec{G}-\vec{G}')
+ + \tilde{V}_{\text{xc}}(\vec{G}-\vec{G}')
+\right] c_{i,\vec{k}+\vec{G}} = \epsilon_i c_{i,\vec{k}+\vec{G}}
+\label{eq:basics:pwks}
+\end{equation}
+reveals further advantages.
+The various potentials are described in terms of their Fourier transforms.
+Equation \eqref{eq:basics:pwks} is solved by diagonalization of the Hamiltonian matrix $H_{\vec{k}+\vec{G},\vec{k}+\vec{G}'}$ given by the terms in the brackets.
+The gradient operator is diagonal in reciprocal space whereas the exchange-correlation potential has a diagonal representation in real space.
+This suggests to carry out different operations in real and reciprocal space, which requires frequent Fourier transformations.
+These, however, can be efficiently achieved by the fast Fourier transformation (FFT) algorithm.
+
+There are likewise disadvantages associated with the PW representation.
+By construction, PW calculations require a periodic system.
+This does not pose a severe problem since non-periodic systems can still be described by a suitable choice of the simulation cell.
+Describing a defect, for instance, requires the inclusion of enough bulk material in the simulation to prevent or reduce the interaction with its periodic, artificial images.
+As a consequence the number of atoms involved in the calculations are increased.
+To describe surfaces, sufficiently thick vacuum layers need to be included to avoid interaction of adjacent crystal slabs.
+Clearly, to appropriately approximate the wave functions and the respective charge density of a system composed of vacuum in addition to the solid in a PW basis, an increase of the cut-off energy is required.
+According to equation \eqref{eq:basics:pwks} the size of the Hamiltonian depends on the cut-off energy and, therefore, the computational effort is likewise increased.
+For the same reason, the description of tightly bound core electrons and the respective, highly localized charge density is hindered.
+However, a much more profound problem exists whenever wave functions for the core as well as the valence electrons need to be calculated within a PW basis set.
+Wave functions of the valence electrons exhibit rapid oscillations in the region occupied by the core electrons near the nuclei.
+The oscillations maintain the orthogonality between the wave functions of the core and valence electrons, which is compulsory due to the exclusion principle.
+Accurately approximating these oscillations demands for an extremely large PW basis set, which is too large for practical use.
+Fortunately, the impossibility to model the core in addition to the valence electrons is eliminated in the pseudopotential approach discussed in the next section.
 
 \subsection{Pseudopotentials}
 
-Since core electrons tend to be concentrated very close to the atomic nuclei, resulting in large wavefunction and density gradients near the nuclei which are not easily described by a plane-wave basis set unless a very high energy cutoff, and therefore small wavelength, is used.
+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 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 (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}
 
-\subsection{Hellmann-Feynman forces}
+Following Bloch's theorem only a finite number of electronic wave functions need to be calculated for a periodic system.
+However, to calculate quantities like the total energy or charge density, these have to be evaluated in a sum over an infinite number of $\vec{k}$ points.
+Since the values of the wave function within a small interval around $\vec{k}$ are almost identical, it is possible to approximate the infinite sum by a sum over an affordable number of $k$ points, each representing the respective region of the wave function in $\vec{k}$ space. 
+Methods have been derived for obtaining very accurate approximations by a summation over special sets of $\vec{k}$ points with distinct, associated weights \cite{baldereschi73,chadi73,monkhorst76}.
+If present, symmetries in reciprocal space may further reduce the number of calculations.
+For supercells, i.e. repeating unit cells that contain several primitive cells, restricting the sampling of the Brillouin zone (BZ) to the $\Gamma$ point can yield quite accurat results.
+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{Structural relaxation and Hellmann-Feynman theorem}
+
+Up to this point, the system is in the ground state with respect to the electronic subsystem, while the positions of the ions as well as size and shape of the supercell are fixed.
+To investigate equilibrium structures, however, the ionic subsystem must also be allowed to relax into a minimum energy configuration.
+Local minimum configurations can be easily obtained in a MD-like way by moving the nuclei over small distances along the directions of the forces, as discussed in the MD chapter above.
+Clearly, the conjugate gradient method constitutes a more sophisticated scheme, which will locate the equilibrium positions of the ions more rapidly.
+To find the global minimum, i.e. the absolute ground state, methods like simulated annealing or the Monte Carlo technique, which allow the system to escape local minima, have to be used for the search.
+
+The force on an ion is given by the negative derivative of the total energy with respect to the position of the ion.
+However, moving an ion, i.e. altering its position, changes the wave functions to the KS eigenstates corresponding to the new ionic configuration.
+Writing down the derivative of the total energy $E$ with respect to the position $\vec{R}_i$ of ion $i$
+\begin{equation}
+\frac{dE}{d\vec{R_i}}=
+ \sum_j \Phi_j^* \frac{\partial H}{\partial \vec{R}_i} \Phi_j
++\sum_j \frac{\partial \Phi_j^*}{\partial \vec{R}_i} H \Phi_j
++\sum_j \Phi_j^* H \frac{\partial \Phi_j}{\partial \vec{R}_i}
+\text{ ,}
+\end{equation}
+indeed reveals a contributon to the chnage in total energy due to the change of the wave functions $\Phi_j$.
+However, provided that the $\Phi_j$ are eigenstates of $H$, it is easy to show that the last two terms cancel each other and in the special case of $H=T+V$ the force is given by
+\begin{equation}
+\vec{F}_i=-\sum_j \Phi_j^*\Phi_j\frac{\partial V}{\partial \vec{R}_i}
+\text{ .}
+\end{equation}
+This is called the Hellmann-Feynman theorem \cite{feynman39}, which enables the calculation of forces, called the Hellmann-Feynman forces, acting on the nuclei for a given configuration, without the need for evaluating computationally costly energy maps.
 
-\section{Modeling of defects}
+\section{Modeling of point defects}
 \label{section:basics:defects}
 
+Point defects are defects that affect a single lattice site.
+At this site the crystalline periodicity is interrupted.
+An empty lattice site, which would be occupied in the perfect crystal structure, is called a vacancy defect.
+If an additional atom is incorporated into the perfect crystal, this is called interstitial defect.
+The disturbance caused by these defects may result in the distortion of the surrounding atomic structure and is accompanied by an increase in configurational energy.
+Thus, next to the structure of the defect, the energy needed to create such a defect, i.e. the defect formation energy, is an important value characterizing the defect and its probability of occurence.
+
+The methods presented in the last two chapters enable the investigation of defects.
+...
+constructing defect configuration intuitively followed by relaxation procedure
+
 \section{Migration paths and diffusion barriers}
 \label{section:basics:migration}
 
+% todo
+% advantages of pw basis with respect to hellmann feynman forces / pulay forces
+