mirror of https://github.com/abinit/abinit.git
180 lines
5.4 KiB
Markdown
180 lines
5.4 KiB
Markdown
---
|
|
authors: M. Giantomassi, X. Gonze, Y. Suzukawa, M. Mikami
|
|
---
|
|
|
|
# Geometric considerations
|
|
|
|
## Real space
|
|
|
|
The three primitive translation vectors are $\RR_1$, $\RR_2$, $\RR_3$.
|
|
Their representation in Cartesian coordinates (atomic units) is:
|
|
|
|
$$ \RR_1 \rightarrow {rprimd(:, 1)} \,, $$
|
|
|
|
$$ \RR_2 \rightarrow {rprimd(:, 2)} \,, $$
|
|
|
|
$$ \RR_3 \rightarrow {rprimd(:, 3)} \,. $$
|
|
|
|
Related input variables: [[acell]], [[rprim]], [[angdeg]].
|
|
|
|
The atomic positions are specified by the coordinates ${\bf x}_{\tau}$
|
|
for $\tau=1 \dots N_{atom}$ where $N_{atom}$ is the number of atoms [[natom]].
|
|
|
|
Representation in reduced coordinates:
|
|
|
|
\begin{eqnarray*}
|
|
{\bf x}_{\tau} &=& x^{red}_{1\tau} \cdot {\bf R}_{1}
|
|
+ x^{red}_{2\tau} \cdot {\bf R}_{2}
|
|
+ x^{red}_{3\tau} \cdot {\bf R}_{3} \,, \\
|
|
\text{where} \,
|
|
\tau &\rightarrow& {iatom} \,, \\
|
|
N_{atom} &\rightarrow& {natom} \,, \\
|
|
x^{red}_{1\tau} &\rightarrow& {xred(1,iatom)} \,, \\
|
|
x^{red}_{2\tau} &\rightarrow& {xred(2,iatom)} \,, \\
|
|
x^{red}_{3\tau} &\rightarrow& {xred(3,iatom)} \,.
|
|
\end{eqnarray*}
|
|
|
|
Related input variables: [[xcart]], [[xred]].
|
|
|
|
The volume of the primitive unit cell (called *ucvol* in the code) is
|
|
|
|
\begin{eqnarray*}
|
|
\Omega &=& {\bf R}_1 \cdot ({\bf R}_2 \times {\bf R}_3) \,.
|
|
\end{eqnarray*}
|
|
|
|
The scalar products in the reduced representation are valuated thanks to
|
|
|
|
$$
|
|
{\bf r} \cdot {\bf r'} =\left(
|
|
\begin{array}{ccc}
|
|
r^{red}_{1} & r^{red}_{2} & r^{red}_{1}
|
|
\end{array}
|
|
\right)
|
|
\left(
|
|
\begin{array}{ccc}
|
|
{\bf R}_{1} \cdot {\bf R}_{1} & {\bf R}_{1} \cdot {\bf R}_{2} &
|
|
{\bf R}_{1} \cdot {\bf R}_{3} \\
|
|
{\bf R}_{2} \cdot {\bf R}_{1} & {\bf R}_{2} \cdot {\bf R}_{2} &
|
|
{\bf R}_{2} \cdot {\bf R}_{3} \\
|
|
{\bf R}_{3} \cdot {\bf R}_{1} & {\bf R}_{3} \cdot {\bf R}_{2} &
|
|
{\bf R}_{3} \cdot {\bf R}_{3}
|
|
\end{array}
|
|
\right)
|
|
\left(
|
|
\begin{array}{c}
|
|
r^{red \prime}_{1} \\
|
|
r^{red \prime}_{2} \\
|
|
r^{red \prime}_{3}
|
|
\end{array}
|
|
\right) \,,
|
|
$$
|
|
|
|
that is
|
|
|
|
$$ {\bf r} \cdot {\bf r'} = \sum_{ij} r^{red}_{i} {\bf R}^{met}_{ij} r^{red \prime}_{j} \,, $$
|
|
|
|
where ${\bf R}^{met}_{ij}$ is the metric tensor in real space stored in `rmet` array:
|
|
|
|
$$ {\bf R}^{met}_{ij} \rightarrow {rmet(i,j)} \,. $$
|
|
|
|
## Reciprocal space
|
|
|
|
The three primitive translation vectors in reciprocal space are
|
|
$\GG_1$, $\GG_2$,$\GG_3$
|
|
|
|
\begin{eqnarray*}
|
|
{\bf G}_{1}&=&\frac{2\pi}{\Omega}({\bf R}_{2}\times{\bf R}_{3}) \rightarrow {2\pi\, gprimd(:,1)} \,, \\
|
|
{\bf G}_{2}&=&\frac{2\pi}{\Omega}({\bf R}_{3}\times{\bf R}_{1}) \rightarrow {2\pi\, gprimd(:,2)} \,, \\
|
|
{\bf G}_{3}&=&\frac{2\pi}{\Omega}({\bf R}_{1}\times{\bf R}_{2}) \rightarrow {2\pi\, gprimd(:,3)} \,.
|
|
\end{eqnarray*}
|
|
|
|
This definition is such that $\GG_i \cdot \RR_j = 2\pi\delta_{ij}$ .
|
|
|
|
!!! warning
|
|
For historical reasons, the internal implementation uses the convention
|
|
$\GG_i \cdot \RR_j = \delta_{ij}$. This means that a factor $2\pi$ must be taken into account
|
|
in the Fortran code.
|
|
We don't use this convention in the theory notes to keep the equations as simple as possible.
|
|
|
|
Reduced representation of vectors (K) in reciprocal space
|
|
|
|
$$
|
|
{\bf K}=K^{red}_{1}{\bf G}_{1}+K^{red}_{2}{\bf G}_{2}
|
|
+K^{red}_{3}{\bf G}^{red}_{3} \rightarrow
|
|
(K^{red}_{1},K^{red}_{2},K^{red}_{3})
|
|
$$
|
|
|
|
e.g. the reduced representation of ${\bf G}_{1}$ is (1, 0, 0).
|
|
|
|
!!! important
|
|
|
|
The reduced representation of the vectors of the reciprocal space
|
|
lattice is made of triplets of integers.
|
|
|
|
The scalar products in the reduced representation are evaluated thanks to
|
|
|
|
$$
|
|
{\bf K} \cdot {\bf K'}=\left(
|
|
\begin{array}{ccc}
|
|
K^{red}_{1} & K^{red}_{2} & K^{red}_{1}
|
|
\end{array}
|
|
\right)
|
|
\left(
|
|
\begin{array}{ccc}
|
|
{\bf G}_{1} \cdot {\bf G}_{1} & {\bf G}_{1} \cdot {\bf G}_{2}
|
|
& {\bf G}_{1} \cdot {\bf G}_{3} \\
|
|
{\bf G}_{2} \cdot {\bf G}_{1} & {\bf G}_{2} \cdot {\bf G}_{2}
|
|
& {\bf G}_{2} \cdot {\bf G}_{3} \\
|
|
{\bf G}_{3} \cdot {\bf G}_{1} & {\bf G}_{3} \cdot {\bf G}_{2}
|
|
& {\bf G}_{3} \cdot {\bf G}_{3}
|
|
\end{array}
|
|
\right)
|
|
\left(
|
|
\begin{array}{c}
|
|
K^{red \prime}_{1} \\
|
|
K^{red \prime}_{2} \\
|
|
K^{red \prime}_{3}
|
|
\end{array}
|
|
\right) \,,
|
|
$$
|
|
|
|
that is
|
|
|
|
$$ {\bf K} \cdot {\bf K'} = \sum_{ij} K^{red}_{i}{\bf G}^{met}_{ij}K^{red \prime}_{j} \,, $$
|
|
|
|
where ${\bf G}^{met}_{ij}$ is the metric tensor in reciprocal space called `gmet` inside the code.
|
|
Taking into account the internal conventions used by the code, we have the correspondence:
|
|
|
|
$$ {\bf G}^{met}_{ij} \rightarrow {2\pi\,gmet(i,j)} \,. $$
|
|
|
|
## Fourier series for periodic lattice quantities
|
|
|
|
Any function with the periodicity of the lattice i.e. any function fullfilling the property
|
|
|
|
$$ u(\rr + \RR) = u(\rr) $$
|
|
|
|
can be represented with the discrete Fourier series:
|
|
|
|
$$ u(\rr)= \sum_\GG u(\GG)e^{i\GG\cdot\rr} \,, $$
|
|
|
|
where the Fourier coefficient, $u(\GG)$, is given by:
|
|
|
|
$$ u(\GG) = \frac{1}{\Omega} \int_\Omega u(\rr)e^{-i\GG\cdot\rr}\dd\rr \,. $$
|
|
|
|
<!--
|
|
This appendix reports the conventions used in this work for the Fourier
|
|
representation in frequency- and momentum-space.
|
|
The volume of the unit cell is denoted with $\Omega$, while $V$ is the
|
|
total volume of the crystal simulated employing Born-von K\'arman periodic boundary condition~\cite{Ashcroft1976}.
|
|
|
|
\begin{equation}\label{eq:IFT_2points_convention}
|
|
f(\rr_1,\rr_2)= \frac{1}{V} \sum_{\substack{\qq \\ \GG_1 \GG_2}}
|
|
e^{i (\qq +\GG_1) \cdot \rr_1}\,f_{\GG_1 \GG_2}(\qq)\,e^{-i (\qq+\GG_2) \cdot \rr_2}
|
|
\end{equation}
|
|
|
|
\begin{equation}\label{eq:FT_2points_convention}
|
|
f_{\GG_1\GG_2}(\qq) = \frac{1}{V} \iint_V
|
|
e^{-i(\qq+\GG_1) \cdot \rr_1}\,f(\rr_1, \rr_2)\,e^{i (\qq+\GG_2) \cdot \rr_2}\dd\rr_1\dd\rr_2
|
|
\end{equation}
|
|
-->
|