QPMS
/
qpms
1
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

Rewrite Ewald intro. 

Former-commit-id: df9c666b9cb34b45fab00155b0ee2a89f0c7c1e8
This commit is contained in:
Marek Nečada 2020-06-22 16:29:11 +03:00
parent f756592bc5
commit a5cf8505f7
2 changed files with 34 additions and 7 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

32
lepaper/infinite.lyx
View File

@ -953,8 +953,8 @@ literal "false"
basic idea can be used as well, resulting in exponentially convergent summation basic idea can be used as well, resulting in exponentially convergent summation
formulae, but the technical details are considerably more complicated than formulae, but the technical details are considerably more complicated than
in electrostatics. in electrostatics.
For the scalar Helmholtz equation in three dimensions, the formulae were For the scalar Helmholtz equation in three dimensions, the formulae for
developed by Ham & Segall lattice Green's functions were developed by Ham & Segall
\begin_inset CommandInset citation \begin_inset CommandInset citation
LatexCommand cite LatexCommand cite
key "ham_energy_1961" key "ham_energy_1961"
@ -988,10 +988,28 @@ literal "false"
\end_inset \end_inset
. .
We will not rederive the formulae here, but for reference, we restate the
results in a form independent upon the normalisation and phase conventions \end_layout
for spherical harmonic bases (pointing out some errors in the aforementioned
literature) and discuss some practical aspects of the numerical evaluation. \begin_layout Standard
For our purposes we do not need directly the lattice Green's functions but
rather the related lattice sums of spherical wavefunctions defined in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:sigma lattice sums"
plural "false"
caps "false"
noprefix "false"
\end_inset
, which can be derived by an analogous procedure.
Below, we state the results in a form independent upon the normalisation
and phase conventions for spherical harmonic bases (pointing out some errors
in the aforementioned literature) and discuss some practical aspects of
the numerical evaluation.
The derivation of the somewhat more complicated 1D and 2D periodicities
is provided in the Supplementary Material.
\begin_inset Note Note \begin_inset Note Note
status open status open
@ -1001,7 +1019,7 @@ Tady ještě upřesnit, co vlastně dělám.
\end_inset \end_inset
\end_layout \end_layout
\begin_layout Standard \begin_layout Standard

9
lepaper/symmetries.lyx
View File

@ -596,6 +596,10 @@ noprefix "false"
\end_inset \end_inset
, we have , we have
\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset Marginal \begin_inset Marginal
status open status open
@ -607,6 +611,11 @@ Check this carefully.
\end_inset \end_inset
\end_layout
\end_inset
\begin_inset Formula \begin_inset Formula
\begin{multline} \begin{multline}
\left(\groupop g\vect E\right)\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(\kappa\left(\vect r-R_{g}\vect r_{p}\right)\right)\right.+\\ \left(\groupop g\vect E\right)\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(\kappa\left(\vect r-R_{g}\vect r_{p}\right)\right)\right.+\\