lattice, reciprocal lattice, OUTCAR

Lattice

1. some definitions

real space lattice, \(\overrightarrow{a_1}\), \(\overrightarrow{a_2}\), \(\overrightarrow{a_3}\); reciprocal space lattice, reclat, \(\overrightarrow{rec_1}\), \(\overrightarrow{rec_2}\), \(\overrightarrow{rec_3}\)
\(\overrightarrow{rec_1}=2\pi\frac{\overrightarrow{a_2}\times\overrightarrow{a_3}}{V}, \overrightarrow{rec_2}=2\pi\frac{\overrightarrow{a_3}\times\overrightarrow{a_1}}{V}, \overrightarrow{rec_2}=2\pi\frac{\overrightarrow{a_3}\times\overrightarrow{a_1}}{V}\)
However, in this article, we use same definition as reciprocal lattice in OUTCAR.
\(\overrightarrow{b_1}=\frac{\overrightarrow{a_2}\times\overrightarrow{a_3}}{V}, \overrightarrow{b_2}=\frac{\overrightarrow{a_3}\times\overrightarrow{a_1}}{V}, \overrightarrow{b_2}=\frac{\overrightarrow{a_3}\times\overrightarrow{a_1}}{V}\\lat=\left[\begin{matrix}\overrightarrow{a_1}\\\overrightarrow{a_2}\\\overrightarrow{a_3}\\\end{matrix}\right],reclat=\left[\begin{matrix}\overrightarrow{b_1}\\\overrightarrow{b_2}\\\overrightarrow{b_3}\\\end{matrix}\right],\)
considering \([\overrightarrow{a_1}][\overrightarrow{b_1}]^T=\overrightarrow{a_1}\cdot\overrightarrow{b_1}=a_{11}b_{11}+a_{12}b_{12}+a_{13}b_{13}\)
Here, we can easily get,
\(lat\times reclat^T==\left[\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}\right]\)

2. OUTCAR

grep -A3 "reciprocal lattice vectors" OUTCAR | sed -n 2,4p
3.945159882 0.000000000 0.000000000 0.253475152 0.000000000 0.000000000
0.000000000 3.921179652 0.000000000 0.000000000 0.255025296 0.000000000
0.000000000 0.000000000 23.273546976 0.000000000 0.000000000 0.042967237

the OUTCAR give us
\(lat=\left[\begin{matrix}3.945&0&0\\0&3.921&0\\0&0&23.274\end{matrix}\right]\)
\(reclat=\left[\begin{matrix}0.253&0&0\\0&0.255&0\\0&0&0.0430\end{matrix}\right]\)

3. Coordinate transformation

\(\left[\begin{matrix}kc_1&kc_2&kc_3\end{matrix}\right]=\left[\begin{matrix}kf_1&kf_2&kf_3\end{matrix}\right]\left[\begin{matrix}\overrightarrow{b_1}\\\overrightarrow{b_2}\\\overrightarrow{b_3}\\\end{matrix}\right]\)

\(\left[\begin{matrix}xc_1&xc_2&xc_3\end{matrix}\right]=\left[\begin{matrix}xf_1&xf_2&xf_3\end{matrix}\right]\left[\begin{matrix}\overrightarrow{a_1}\\\overrightarrow{a_2}\\\overrightarrow{a_3}\\\end{matrix}\right]\)