The Tersoff potential explicitly incorporates the dependence of bond order on local environments, permitting an improved description of covalent materials.
Due to the covalent character Tersoff restricted the interaction to nearest neighbor atoms accompanied by an increase in computational efficiency for the evaluation of forces and energy based on the short-range potential.
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}.
+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.
\label{subsection:statistical_ensembles}
Using the above mentioned algorithms the most basic type of MD is realized by simply integrating the equations of motion of a fixed number of particles ($N$) in a closed volume $V$ realized by periodic boundary conditions (PBC).
-Providing a stable integration algorithm the total energy $E$, i.e. the kinetic and configurational energy of the particles, is conserved.
+Providing a stable integration algorithm the total energy $E$, i.e.\ the kinetic and configurational energy of the particles, is conserved.
This is known as the $NVE$, or microcanonical ensemble, describing an isolated system composed of microstates, among which the number of particles, volume and energy are held constant.
However, the successful formation of SiC dictates precise control of temperature by external heating.
Using this method the system does not behave like a true $NpT$ ensemble.
On average $T$ and $p$ correspond to the expected values.
-For large enough time constants, i.e. $\tau > 100 \delta t$, the method shows realistic fluctuations in $T$ and $p$.
+For large enough time constants, i.e.\ $\tau > 100 \delta t$, the method shows realistic fluctuations in $T$ and $p$.
The advantage of the approach is that the coupling can be decreased to minimize the disturbance of the system and likewise be adjusted to suit the needs of a given application.
It provides a stable algorithm that allows smooth changes of the system to new values of temperature or pressure, which is ideal for the investigated problem.
\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{Kohn-Sham system}
-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}.
+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}
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.
-Convergence 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 calculations.
+Convergence 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 calculations.
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.
+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.
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.
+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}
This way, the pseudo wave functions become smooth near the nuclei.
Most PPs satisfy 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.
+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.
V_{\text{nl}}(\vec{r}) = \sum_{lm} | lm \rangle V_l(\vec{r}) \langle lm |
\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 pseudopotential $V_l(\vec{r})$ for angular momentum $l$.
+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 pseudopotential $V_l(\vec{r})$ for angular momentum $l$.
The standard generation procedure of pseudopotentials proceeds by varying its parameters until the pseudo eigenvalues are equal to the all-electron valence eigenvalues and the pseudo wave functions match the all-electron valence wave functions beyond a certain cut-off radius determining 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}.
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 accurate results.
+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 accurate 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 fictitious anyway.
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.
+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.
+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}}=
If an additional atom is incorporated into the perfect crystal, this is called interstitial defect.
A substitutional defect exists, if an atom belonging to the perfect crystal is replaced with an atom of another species.
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 likewise determining its relative stability.
+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 likewise determining its relative stability.
The formation energy of a defect is defined by
\begin{equation}
\label{eq:basics:ef2}
\end{equation}
where $E$ is the total energy of the interstitial structure involving $N_i$ atoms of type $i$ with chemical potential $\mu_i$.
-Here, the chemical potentials are determined by the chemical potential of the respective equilibrium bulk structure, i.e. the cohesive energy per atom for the fully relaxed structure at zero temperature and pressure.
+Here, the chemical potentials are determined by the chemical potential of the respective equilibrium bulk structure, i.e.\ the cohesive energy per atom for the fully relaxed structure at zero temperature and pressure.
Considering C interstitial defects in Si, the chemical potential for C could also be determined by the cohesive energies of Si and SiC according to the relation $\mu_{\text{C}}=\mu_{\text{SiC}}-\mu_{\text{Si}}$ of the chemical potentials.
In this way, SiC is chosen as a reservoir for the C impurity.
For defect configurations consisting of a single atom species the formation energy reduces to
\end{equation}
where the formation energies $E_{\text{f}}^{\text{comb}}$, $E_{\text{f}}^{1^{\text{st}}}$ and $E_{\text{f}}^{2^{\text{nd}}}$ are determined as discussed above.
Accordingly, energetically favorable configurations result in binding energies below zero while unfavorable configurations show positive values for the binding energy.
-The interaction strength, i.e. the absolute value of the binding energy, approaches zero for increasingly non-interacting isolated defects.
+The interaction strength, i.e.\ the absolute value of the binding energy, approaches zero for increasingly non-interacting isolated defects.
Thus, $E_{\text{b}}$ indeed can be best thought of a binding energy, which is required to bring the defects to infinite separation.
The methods presented in the last two chapters can be used to investigate defect structures and energetics.
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