$U_2$ is a two body pair potential which only depends on the distance $r_{ij}$ between the two atoms $i$ and $j$.
If not only pair potentials are considered, three body potentials $U_3$ or many body potentials $U_n$ can be included.
Usually these higher order terms are avoided since they are not easy to model and it is rather time consuming to evaluate potentials and forces originating from these many body terms.
-Ordinary pair potentials have a close-packed structure like face-centered cubic (FCC) or hexagonal close-packed (HCP) as a ground state.
-A pair potential is, thus, unable to describe properly elements with other structures than FCC or HCP.
+Ordinary pair potentials have a close-packed structure like face-centered cubic (fcc) or hexagonal close-packed (hcp) as a ground state.
+A pair potential is, thus, unable to describe properly elements with other structures than fcc or hcp.
Silicon and carbon for instance, have a diamond and zincblende structure with four covalently bonded neighbors, which is far from a close-packed structure.
A three body potential has to be included for these types of elements.
Tersoff applied the potential to silicon \cite{tersoff_si1,tersoff_si2,tersoff_si3}, carbon \cite{tersoff_c} and also to multicomponent systems like silicon carbide \cite{tersoff_m}.
The basic idea is that, in real systems, the bond order, i.e. the strength of the bond, depends upon the local environment \cite{abell85}.
Atoms with many neighbors form weaker bonds than atoms with only a few neighbors.
-Although the bond strength intricately depends on geometry the focus on coordination, i.e. the number of neighbors forming bonds, is well motivated qualitatively from basic chemistry since for every additional formed bond the amount of electron pairs per bond and, thus, the strength of the bonds is decreased.
+Although the bond strength intricately depends on geometry, the focus on coordination, i.e. the number of neighbors forming bonds, is well motivated qualitatively from basic chemistry since for every additional formed bond the amount of electron pairs per bond and, thus, the strength of the bonds is decreased.
If the energy per bond decreases rapidly enough with increasing coordination the most stable structure will be the dimer.
In the other extreme, if the dependence is weak, the material system will end up in a close-packed structure in order to maximize the number of bonds and likewise minimize the cohesive energy.
This suggests the bond order to be a monotonously decreasing function with respect to coordination and the equilibrium coordination being determined by the balance of bond strength and number of bonds.
\subsubsection{Improved analytical bond order potential}
-Although the Tersoff potential is one of the most widely used potentials there are some shortcomings.
+Although the Tersoff potential is one of the most widely used potentials, there are some shortcomings.
Describing the Si-Si interaction Tersoff was unable to find a single parameter set to describe well both, bulk and surface properties.
Due to this and since the first approach labeled T1 \cite{tersoff_si1} turned out to be unstable \cite{dodson87}, two further parametrizations exist, T2 \cite{tersoff_si2} and T3 \cite{tersoff_si3}.
While T2 describes well surface properties, T3 yields improved elastic constants and should be used for describing bulk properties.
\label{section:dft}
Dirac declared that chemistry has come to an end, its content being entirely contained in the powerful equation published by Schr\"odinger in 1926 \cite{schroedinger26} marking the beginning of wave mechanics.
-Following the path of Schr\"odinger the problem in quantum-mechanical modeling of describing the many-body problem, i.e. a system of a large amount of interacting particles, is manifested in the high-dimensional Schr\"odinger equation for the wave function $\Psi({\vec{R}},{\vec{r}})$ that depends on the coordinates of all nuclei and electrons.
+Following the path of Schr\"odinger, the problem in quantum-mechanical modeling of describing the many-body problem, i.e. a system of a large amount of interacting particles, is manifested in the high-dimensional Schr\"odinger equation for the wave function $\Psi({\vec{R}},{\vec{r}})$ that depends on the coordinates of all nuclei and electrons.
The Schr\"odinger equation contains the kinetic energy of the ions and electrons as well as the electron-ion, ion-ion and electron-electron interaction.
This cannot be solved exactly and finding approximate solutions requires several layers of simplification in order to reduce the number of free parameters.
Approximations that consider a truncated Hilbert space of single-particle orbitals yield promising results, however, with increasing complexity and demand for high accuracy the amount of Slater determinants to be evaluated massively increases.
\subsection{Hohenberg-Kohn theorem and variational principle}
Investigating the energetics of Cu$_x$Zn$_{1-x}$ alloys, which for different compositions exhibit different transfers of charge between the Cu and Zn unit cells due to their chemical difference and, thus, varying electrostatic interactions contributing to the total energy, the work of Hohenberg and Kohn had a natural focus on the distribution of charge.
-Although it was clear that the Thomas Fermi (TF) theory only provides a rough approximation to the exact solution of the many-electron Schr\"odinger equation the theory was of high interest since it provides an implicit relation of the potential and the electron density distribution.
+Although it was clear that the Thomas Fermi (TF) theory only provides a rough approximation to the exact solution of the many-electron Schr\"odinger equation, the theory was of high interest since it provides an implicit relation of the potential and the electron density distribution.
This raised the question how to establish a connection between TF expressed in terms of $n(\vec{r})$ and the exact Schr\"odinger equation expressed in terms of the many-electron wave function $\Psi({\vec{r}})$ and whether a complete description in terms of the charge density is possible in principle.
The answer to this question, whether the charge density completely characterizes a system, became the starting point of modern DFT.
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