it's _a_ fit

[lectures/latex.git] / posic / thesis / basics.tex

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index 391af40..9a6eb02 100644 (file)
--- a/posic/thesis/basics.tex
+++ b/posic/thesis/basics.tex
@@ -151,7 +151,7 @@ This is illustrated in Fig.~\ref{img:tersoff_angle}.
 The angular dependence does not give a fixed minimum angle between bonds since the expression is embedded inside the bond order term.
 The relation to the above discussed bond order potential becomes obvious if $\chi=1, \beta=1, n=1, \omega=1$ and $c=0$.
 Parameters with a single subscript correspond to the parameters of the elemental system~\cite{tersoff_si3,tersoff_c} while the mixed parameters are obtained by interpolation from the elemental parameters by the arithmetic or geometric mean.
-The elemental parameters were obtained by fit with respect to the cohesive energies of real and hypothetical bulk structures and the bulk modulus and bond length of the diamond structure.
+The elemental parameters were obtained by a fit with respect to the cohesive energies of real and hypothetical bulk structures and the bulk modulus and bond length of the diamond structure.
 New parameters for the mixed system are $\chi$, which is used to finetune the strength of heteropolar bonds, and $\omega$, which is set to one for the C-Si interaction but is available as a feature to permit the application of the potential to more drastically different types of atoms in the future.
 
 The force acting on atom $i$ is given by the derivative of the potential energy.
@@ -566,7 +566,7 @@ Writing down the derivative of the total energy $E$ with respect to the position
 indeed reveals a contribution to the change in total energy due to the change of the wave functions $\Phi_j$.
 However, provided that the $\Phi_j$ are eigenstates of $H$, it is easy to show that the last two terms cancel each other and in the special case of $H=T+V$ the force is given by
 \begin{equation}
-\vec{F}_i=-\sum_j \langle \Phi_j | \Phi_j\frac{\partial V}{\partial \vec{R}_i} \rangle
+\vec{F}_i=-\sum_j \langle \Phi_j | \frac{\partial V}{\partial \vec{R}_i} \Phi_j \rangle
 \text{ .}
 \end{equation}
 This is called the Hellmann-Feynman theorem~\cite{feynman39}, which enables the calculation of forces, called the Hellmann-Feynman forces, acting on the nuclei for a given configuration, without the need for evaluating computationally costly energy maps.