silicon, si self interstitials, free energy",
}
+@Article{mattsson08,
+ title = "Electronic surface error in the Si interstitial
+ formation energy",
+ author = "Ann E. Mattsson and Ryan R. Wixom and Rickard
+ Armiento",
+ journal = "Phys. Rev. B",
+ volume = "77",
+ number = "15",
+ pages = "155211",
+ numpages = "7",
+ year = "2008",
+ month = apr,
+ doi = "10.1103/PhysRevB.77.155211",
+ publisher = "American Physical Society",
+ notes = "si self interstitial formation energies by dft",
+}
+
@Article{goedecker02,
title = "A Fourfold Coordinated Point Defect in Silicon",
author = "Stefan Goedecker and Thierry Deutsch and Luc Billard",
}
@Article{slater29,
- title = {The Theory of Complex Spectra},
- author = {Slater, J. C.},
- journal = {Phys. Rev.},
- volume = {34},
- number = {10},
- pages = {1293--1322},
- numpages = {29},
- year = {1929},
- month = {Nov},
- doi = {10.1103/PhysRev.34.1293},
- publisher = {American Physical Society}
+ title = "The Theory of Complex Spectra",
+ author = "J. C. Slater",
+ journal = "Phys. Rev.",
+ volume = "34",
+ number = "10",
+ pages = "1293--1322",
+ numpages = "29",
+ year = "1929",
+ month = nov,
+ doi = "10.1103/PhysRev.34.1293",
+ publisher = "American Physical Society",
}
@Article{kohn65,
notes = "dft, exchange and correlation",
}
+@Article{kohn96,
+ title = "Density Functional and Density Matrix Method Scaling
+ Linearly with the Number of Atoms",
+ author = "W. Kohn",
+ journal = "Phys. Rev. Lett.",
+ volume = "76",
+ number = "17",
+ pages = "3168--3171",
+ numpages = "3",
+ year = "1996",
+ month = apr,
+ doi = "10.1103/PhysRevLett.76.3168",
+ publisher = "American Physical Society",
+}
+
+@Article{kohn98,
+ title = "Edge Electron Gas",
+ author = "Walter Kohn and Ann E. Mattsson",
+ journal = "Phys. Rev. Lett.",
+ volume = "81",
+ number = "16",
+ pages = "3487--3490",
+ numpages = "3",
+ year = "1998",
+ month = oct,
+ doi = "10.1103/PhysRevLett.81.3487",
+ publisher = "American Physical Society",
+}
+
@Article{kohn99,
title = "Nobel Lecture: Electronic structure of matter---wave
functions and density functionals",
notes = "vasp pseudopotentials",
}
+@Article{ceperley80,
+ title = "Ground State of the Electron Gas by a Stochastic
+ Method",
+ author = "D. M. Ceperley and B. J. Alder",
+ journal = "Phys. Rev. Lett.",
+ volume = "45",
+ number = "7",
+ pages = "566--569",
+ numpages = "3",
+ year = "1980",
+ month = aug,
+ doi = "10.1103/PhysRevLett.45.566",
+ publisher = "American Physical Society",
+}
+
+@Article{perdew81,
+ title = "Self-interaction correction to density-functional
+ approximations for many-electron systems",
+ author = "J. P. Perdew and Alex Zunger",
+ journal = "Phys. Rev. B",
+ volume = "23",
+ number = "10",
+ pages = "5048--5079",
+ numpages = "31",
+ year = "1981",
+ month = may,
+ doi = "10.1103/PhysRevB.23.5048",
+ publisher = "American Physical Society",
+}
+
@Article{perdew86,
title = "Accurate and simple density functional for the
electronic exchange energy: Generalized gradient
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 is not part of the HF correlation.
+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,eq:basics:kse2,eq:basics:kse3} may be solved numerically by an iterative process.
+The self-consistent KS equations \eqref{eq:basics:kse1}, \eqref{eq:basics:kse2} and \eqref{eq:basics:kse3} 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 vary only slightly.
+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
+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{xc}}[n(vec{r})]=\int\epsilon_{\text{xc}}(n(\vec{r}))n(\vec{r}) d\vec{r}
-\text{ .}
+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 $\epsilon_{\text{xc}}(n(\vec{r}))$.
-This is called the local density approximation (LDA).
+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 $n(~\vec{r})$ at positions $~\vec{r}$ near $\vec{r}$, yielding the form
+\begin{equation}
+E_{\text{xc}}[n(vec{r})]=\int\epsilon_{\text{xc}}(\vec{r};[n(~\vec{r})])n(\vec{r}) d\vec{r}
+\end{equation}
+for the exchange-correlation energy, where $\epsilon_{\text{xc}}(\vec{r};[n(~\vec{r})])$ becomes a nearsighted functional of $n(~\vec{r})$.
+Expressing $n(~\vec{r})$ 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}
projects / lectures / latex.git / commitdiff