Extracting eigenvalues for a certain high-symmetry path
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Hello all. I have a question about CRYSTAL23 and how it works when it comes to calculating eigenvalues for certain k-points. Let's look at the example of SiO2 cell with 12 ions:
*** K POINTS COORDINATES (OBLIQUE COORDINATES IN UNITS OF IS = 6)
1-R( 0 0 0) 2-C( 1 0 0) 3-C( 2 0 0) 4-R( 3 0 0)
5-C( 1 1 0) 6-C( 2 1 0) 7-C( 2 2 0) 8-C( 0 0 1)
9-C( 1 0 1) 10-C( 2 0 1) 11-C( 3 0 1) 12-C( 1 1 1)
13-C( 2 1 1) 14-C( 2 2 1) 15-C( 0 0 2) 16-C( 1 0 2)
17-C( 2 0 2) 18-C( 3 0 2) 19-C( 1 1 2) 20-C( 2 1 2)
21-C( 2 2 2) 22-R( 0 0 3) 23-C( 1 0 3) 24-C( 2 0 3)
25-R( 3 0 3) 26-C( 1 1 3) 27-C( 2 1 3) 28-C( 2 2 3)So here it shows 28 k-points for the 6x6x6 Monkhorst-Pack grid, and if I understand correctly, the reason why it's not showing all 216 points is because the program recognizes the crystal symmetry so it knows that the energy at (1 0 0) is the same as at (0 1 0), (0 0 1), (-1 0 0), etc?
If that is the case, then to build a high-symmetry path along Gamma-X-K-Gamma, for example... we could just take the points 4-R( 3 0 0) and 7-C( 2 2 0) and divide them by IS, which in this case is 6 to get the points (0.5, 0, 0) and (1/3, 1/3, 0). Would that be correct? Because that would make the extraction of eigenvalues quite simple for this path.
Thanks in advance,
Chris
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Hi Chris,
You’re right that the code uses the symmetry of the crystal to reduce the number of k-points it actually computes. That’s why you don’t see all 216 k-points from the 6×6×6 Monkhorst-Pack grid: only the irreducible k-points are listed, since the energies at the symmetry-equivalent points can be obtained by applying the crystal’s symmetry operations.
That said, one important distinction: the program does not literally “reuse” the eigenvalues from one representative k-point for its equivalents, it only knows that due to symmetry, those results will be the same, so it avoids recalculating them.
For band structure plots along a high-symmetry path (e.g. Gamma-X-K-Gamma), the reduced k-point mesh from the SCF run is not sufficient. Instead, you should perform a separate band structure calculation (BAND keyword in PROPERTIES, page 309 of the user manual). In this type of calculation, you explicitly define the path through reciprocal space, and PROPERTIES computes eigenvalues along that path directly, without relying on the Monkhorst-Pack grid.
Let me know if you have any truble running the properties calculation.
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Hi Giacomo,
That's a very useful answer, thank you!
Chris