X-Git-Url: https://hackdaworld.org/gitweb/?a=blobdiff_plain;f=posic%2Fthesis%2Fbasics.tex;h=249614ae173c0d53adf87671ee8925e332f4dec6;hb=0d3acfac2b23ed5b37c86a77118a7f467f48c0e8;hp=b095d71bec202dc3ed31b08d7da446ac179bcb19;hpb=ab861cbd757aa36afee0ecd21c1d88bc25f36f36;p=lectures%2Flatex.git

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index b095d71..249614a 100644
--- a/posic/thesis/basics.tex
+++ b/posic/thesis/basics.tex
@@ -526,7 +526,7 @@ Mathematically, a non-local PP, which depends on the angular momentum, has the f
 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 $| lm \rangle$, 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}.
 
@@ -558,15 +558,15 @@ However, moving an ion, i.e.\ altering its position, changes the wave functions
 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}}=
- \sum_j \Phi_j^* \frac{\partial H}{\partial \vec{R}_i} \Phi_j
-+\sum_j \frac{\partial \Phi_j^*}{\partial \vec{R}_i} H \Phi_j
-+\sum_j \Phi_j^* H \frac{\partial \Phi_j}{\partial \vec{R}_i}
+ \sum_j \langle \Phi_j | \frac{\partial H}{\partial\vec{R}_i} | \Phi_j \rangle
++\sum_j \langle \frac{\partial \Phi_j}{\partial\vec{R}_i} | H \Phi_j \rangle
++\sum_j \langle \Phi_j H | \frac{\partial \Phi_j}{\partial \vec{R}_i} \rangle
 \text{ ,}
 \end{equation}
 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 \Phi_j^*\Phi_j\frac{\partial V}{\partial \vec{R}_i}
+\vec{F}_i=-\sum_j \langle \Phi_j | \Phi_j\frac{\partial V}{\partial \vec{R}_i} \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.