Good afternoon,
I did one more very simple check.
1st run:
POSCAR
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test
1.0
4.0 0.0 0.0
0.0 4.0 0.0
0.0 0.0 4.0
Si
1
Direct
0.0 0.0 0.0
KPOINTS
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k-points along high symmetry lines
10
Line-mode
reciprocal
0.0 0.0 0.0 ! G
0.5 0.0 0.0 ! X
k-points found in OUTCAR
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Following cartesian coordinates:
Coordinates Weight
0.000000 0.000000 0.000000 0.100000
0.013889 0.000000 0.000000 0.100000
0.027778 0.000000 0.000000 0.100000
0.041667 0.000000 0.000000 0.100000
0.055556 0.000000 0.000000 0.100000
0.069444 0.000000 0.000000 0.100000
0.083333 0.000000 0.000000 0.100000
0.097222 0.000000 0.000000 0.100000
0.111111 0.000000 0.000000 0.100000
0.125000 0.000000 0.000000 0.100000
The last k-point should be \(\frac{2\pi}{4}\frac12\) but it is \(\frac18\), that is the previous one in units of \(2\pi\).
2d run:
POSCAR
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test
4.0
1.0 0.0 0.0
0.0 1.0 0.0
0.0 0.0 1.0
Si
1
Direct
0.0 0.0 0.0
KPOINTS: same as above
k-points found in OUTCAR
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Following cartesian coordinates:
Coordinates Weight
0.000000 0.000000 0.000000 0.100000
0.055556 0.000000 0.000000 0.100000
0.111111 0.000000 0.000000 0.100000
0.166667 0.000000 0.000000 0.100000
0.222222 0.000000 0.000000 0.100000
0.277778 0.000000 0.000000 0.100000
0.333333 0.000000 0.000000 0.100000
0.388889 0.000000 0.000000 0.100000
0.444444 0.000000 0.000000 0.100000
0.500000 0.000000 0.000000 0.100000
The last k-point should be \(\frac{2\pi}{4}\frac12\) but it is \(\frac12\), that is the previous one in units of \(\frac{2\pi}4\).
So, this confirms that k-points labelled as "cartesian coordinates" are in units of \(\frac{2\pi}s\), where \(s\) is the scaling factor of the second line of POSCAR. As such, should one need k-points in cartesian coordinates, if taken from OUTCAR, they should be multiplied by \(\frac{2\pi}s\).
Similarly, should one need G-vectors, as taked from this secion of OUTCAR
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direct lattice vectors reciprocal lattice vectors
4.000000000 0.000000000 0.000000000 0.250000000 0.000000000 0.000000000
0.000000000 4.000000000 0.000000000 0.000000000 0.250000000 0.000000000
0.000000000 0.000000000 4.000000000 0.000000000 0.000000000 0.250000000
they are always given in units of \(2\pi\).