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:
\[ u(\GG) = \frac{1}{\Omega} \int_\Omega u(\rr)e^{-i\GG\cdot\rr}\dd\rr \,. \]