X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=posic%2Fthesis%2Fbasics.tex;h=231345a8f6d7fe5d48e68ea91c1f8f4d51441650;hp=6c03878e0cd080000c83be9732848b0983616bf7;hb=fcc70f48c064efc50d86a560245aac02789dfe39;hpb=2cfbb5d8e375f3f740f35b6f02dc45a45a236a26 diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex index 6c03878..231345a 100644 --- a/posic/thesis/basics.tex +++ b/posic/thesis/basics.tex @@ -329,6 +329,7 @@ E_0=\min_{n(\vec{r})} 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. @@ -346,14 +347,179 @@ U=\frac{1}{2}\int\frac{n(\vec{r})n(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}d\vec{r \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}}(\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} +\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 $V_{\text{H}}(\vec{r})$ 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})} +%\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 (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 +\begin{equation} +\Phi_i(\vec{r})=\sum_{\vec{G} +%, |\vec{G}+\vec{k}|