F[n(\vec{r})] + \int n(\vec{r}) V(\vec{r}) d\vec{r}
\right\}
\text{ ,}
+\label{eq:basics:hkm}
\end{equation}
where $F[n(\vec{r})]$ is a universal functional of the charge density $n(\vec{r})$, which is composed of the kinetic energy functional $T[n(\vec{r})]$ and the interaction energy functional $U[n(\vec{r})]$.
The challenging problem of determining the exact ground-state is now formally reduced to the determination of the $3$-dimensional function $n(\vec{r})$, which minimizes the energy functional.
\subsection{Kohn-Sham system}
-Now find $F[n]$ ...
-
-As in the last section, the complex many-electron effects are relocated, this time into the exchange-correlation functional.
+Inspired by the Hartree equations, i.e. a set of self-consistent single-particle equations for the approximate solution of the many-electron problem \cite{hartree28}, which describe atomic ground states much better than the TF theory, Kohn and Sham presented a Hartree-like formulation of the Hohenberg and Kohn minimal principle \eqref{eq:basics:hkm} \cite{kohn65}.
+However, due to a more general approach, the new formulation is formally exact by introducing the energy functional $E_{\text{xc}}[n(vec{r})]$, which accounts for the exchange and correlation energy of the electron interaction $U$ and possible corrections due to electron interaction to the kinetic energy $T$.
+The respective Kohn-Sham equations for the effective single-particle wave functions $\Phi_i(\vec{r})$ take the form
+\begin{equation}
+\left[
+ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(\vec{r})
+\right] \Phi_i(\vec{r})=\epsilon_i\Phi_i(\vec{r})
+\label{eq:basics:kse1}
+\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{xc}(\vec{r})}
+\text{ ,}
+\label{eq:basics:kse2}
+\end{equation}
+\begin{equation}
+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 $n_0(\vec{r})$.
+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})}
+% E_{\text{xc}}[n(\vec{r})] |_{n(\vec{r})=n_0(\vec{r})}
+%\end{equation}
+
+The system of interacting electrons is mapped to an auxiliary system, the Kohn-Sham (KS) system, of non-interacting electrons in an effective potential.
+The exact effective potential $V_{\text{eff}}(\vec{r})$ may be regarded as a fictious external potential yielding a gound-state density for non-interacting electrons, which is equal to that for interacting electrons in the external potential $V(\vec{r})$.
+The one-electron KS orbitals $\Phi_i(\vec{r})$ as well as the respective KS energies $\epsilon_i$ are not directly attached to any physical observable except for the ground-state density, which is determined by equation \eqref{eq:basics:kse3} and the ionization energy, which is equal to the highest occupied relative to the vacuum level.
+The KS equations may be considered the formal exactification of the Hartree theory, which it is reduced to if the exchange-correlation potential and functional are neglected.
+In addition to the Hartree-Fock (HF) method, KS theory includes the difference of the kinetic energy of interacting and non-interacting electrons as well as the remaining contributions to the correlation energy that are not part of the HF correlation.
+
+The self-consistent KS equations \eqref{eq:basics:kse1}, \eqref{eq:basics:kse2} and \eqref{eq:basics:kse3} are non-linear partial differential equations, which may be solved numerically by an iterative process.
+Starting from a first approximation for $n(\vec{r})$ the effective potential $V_{\text{eff}}(\vec{r})$ can be constructed followed by determining the one-electron orbitals $\Phi_i(\vec{r})$, which solve the single-particle Schr\"odinger equation for the respective potential.
+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.
+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.
+
+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{ ,}
+\label{eq:basics:xca}
+\end{equation}
+which is, thus, called local density approximation (LDA).
+Here, the exchange-correlation energy per particle of the uniform electron gas of constant density $n$ is used for $\epsilon_{\text{xc}}(n(\vec{r}))$.
+Although, even in such a simple case, no exact form of the correlation part of $\epsilon_{\text{xc}}(n)$ is known, highly accurate numerical estimates using Monte Carlo methods \cite{ceperley80} and corresponding paramterizations exist \cite{perdew81}.
+Obviously exact for the homogeneous electron gas, the LDA was {\em a priori} expected to be useful only for densities varying slowly on scales of the local Fermi or TF wavelength.
+Nevertheless, LDA turned out to be extremely successful in describing some properties of highly inhomogeneous systems accurately within a few percent.
+Although LDA is known to overestimate the cohesive energy in solids by \unit[10-20]{\%}, the lattice parameters are typically determined with an astonishing accuracy of about \unit[1]{\%}.
+
+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 $n(\tilde{\vec{r}})$ 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(\tilde{\vec{r}})])n(\vec{r}) d\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 $n$.
+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}
+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.
+
+\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.
+Covergence 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 calulations.
+Two classes of basis sets, the plane-wave and local basis sets, exist.
+
+Local basis set functions usually are atomic orbitals, i.e. mathematical functions that describe the wave-like behavior of electrons, which are localized, i.e. centered on atoms or bonds.
+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 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.
+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
+\begin{equation}
+\Phi_i(\vec{r})=\sum_{\vec{G}
+%, |\vec{G}+\vec{k}|<G_{\text{cut}}}
+}c_{i,\vec{k}+\vec{G}} \exp\left(i(\vec{k}+\vec{G})\vec{r}\right)
+\text{ .}
+%E_{\text{cut}}=\frac{\hbar^2 G^2_{\text{cut}}}{2m}
+%\text{, }
+\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 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.
+matrix elements of the Hamiltonian have a simple form (pw rep of ks equations)
+As mentioned above ... simple to check for convergence.
+
+Disadvantage ... periodic system required, but escapable by respective choice of the supercell.
+size of matrix to diagonalize determined by cut-off energy, severe
\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.
+
+\subsection{Brillouin zone sampling}
+
+Due to the Bloch 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 an intergration over special sets of $\vec{k}$ points \cite{}.
+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, sampling of the Brillouin zone restricted to the $\Gamma$ point can be quite accurately used, which is equivalent to calculating a single primitive cell using multiple $\vec{k}$ points.
+
+\subsection{Hellmann-Feynman forces}
+
\section{Modeling of defects}
\label{section:basics:defects}
projects / lectures / latex.git / blobdiff