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}=2pifrac{overrightarrow{a_2} imesoverrightarrow{a_3}}{V}, overrightarrow{rec_2}=2pifrac{overrightarrow{a_3} imesoverrightarrow{a_1}}{V}, overrightarrow{rec_3}=2pifrac{overrightarrow{a_1} imesoverrightarrow{a_2}}{V})
However, in this article, we use same definition as reciprocal lattice in OUTCAR.
(overrightarrow{b_1}=frac{overrightarrow{a_2} imesoverrightarrow{a_3}}{V}, overrightarrow{b_2}=frac{overrightarrow{a_3} imesoverrightarrow{a_1}}{V}, overrightarrow{b_2}=frac{overrightarrow{a_3} imesoverrightarrow{a_1}}{V}\lat=left[egin{matrix}overrightarrow{a_1}\overrightarrow{a_2}\overrightarrow{a_3}\end{matrix}
ight],reclat=left[egin{matrix}overrightarrow{b_1}\overrightarrow{b_2}\overrightarrow{b_3}\end{matrix}
ight],)
considering ([overrightarrow{a_1}][overrightarrow{b_1}]^T=overrightarrow{a_1}cdotoverrightarrow{b_1}=a_{11}b_{11}+a_{12}b_{12}+a_{13}b_{13})
Here, we can easily get,
(lat imes reclat^T==left[egin{matrix}1&0&0\0&1&0\0&0&1end{matrix}
ight])
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[egin{matrix}3.945&0&0\0&3.921&0\0&0&23.274end{matrix}
ight])
(reclat=left[egin{matrix}0.253&0&0\0&0.255&0\0&0&0.0430end{matrix}
ight])
3. Coordinate transformation
(left[egin{matrix}kc_1&kc_2&kc_3end{matrix} ight]=left[egin{matrix}kf_1&kf_2&kf_3end{matrix} ight]left[egin{matrix}overrightarrow{b_1}\overrightarrow{b_2}\overrightarrow{b_3}\end{matrix} ight])
(left[egin{matrix}xc_1&xc_2&xc_3end{matrix} ight]=left[egin{matrix}xf_1&xf_2&xf_3end{matrix} ight]left[egin{matrix}overrightarrow{a_1}\overrightarrow{a_2}\overrightarrow{a_3}\end{matrix} ight])