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Extend k-point symmetry backends to closed GENERAL sets (#5173)

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Co-authored-by: Thomas D. Kuehne <tkuehne@cp2k.org>
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# K-Points
Unfortunately no one has gotten around to writing this page yet :-(
Periodic Quickstep calculations can sample the Brillouin zone through the `&DFT%KPOINTS` section.
The most common setup is a Monkhorst-Pack mesh:
In the meantime, the following links might be helpful:
```text
&DFT
&KPOINTS
SCHEME MONKHORST-PACK 6 6 6
WAVEFUNCTIONS COMPLEX
&END KPOINTS
&END DFT
```
`SCHEME GAMMA` uses only the Gamma point. `SCHEME MONKHORST-PACK` and `SCHEME MACDONALD` generate
regular meshes. `SCHEME GENERAL` uses explicitly listed k-points:
```text
&KPOINTS
SCHEME GENERAL
KPOINT 0.0 0.0 0.0 1.0
KPOINT 0.5 0.0 0.0 1.0
&END KPOINTS
```
K-points are given in reciprocal lattice-vector coordinates by default (`UNITS B_VECTOR`). Cartesian
coordinates can be selected with `UNITS CART_BOHR` or `UNITS CART_ANGSTROM`.
## Symmetry Reduction
Atomic symmetry can reduce the number of k-points that have to be solved explicitly:
```text
&KPOINTS
SCHEME MONKHORST-PACK 8 8 8
SYMMETRY ON
SYMMETRY_BACKEND SPGLIB
WAVEFUNCTIONS COMPLEX
&END KPOINTS
```
`SYMMETRY_BACKEND` selects the backend that provides and applies atom and k-point symmetry
operations:
- `K290`: the established CP2K default.
- `SPGLIB`: use the space-group operations returned by SPGLIB, including fractional translations.
`SYMMETRY_REDUCTION_METHOD` selects the method used to build the irreducible k-point set. If
`SYMMETRY_BACKEND` is set explicitly and `SYMMETRY_REDUCTION_METHOD` is omitted, the reduction
method follows the backend. `SYMMETRY_REDUCTION_METHOD SPGLIB` with `SYMMETRY_BACKEND K290` can be
used as a comparison mode: SPGLIB proposes the k-point orbits, while only mappings represented by
the K290 backend are used for the actual transformations.
`INVERSION_SYMMETRY_ONLY ON` restricts the reduction to time-reversal/inversion symmetry.
`FULL_GRID ON` disables symmetry reduction while still generating the regular mesh.
For general atomic k-point symmetry, use complex wavefunctions. Real wavefunctions are only valid
for Gamma and special k-points with real Bloch phases.
## Explicit K-Point Sets
`SCHEME GENERAL` can be used with symmetry reduction when the explicit k-point set is equally
weighted and closed under the requested symmetry operations:
```text
&KPOINTS
SCHEME GENERAL
SYMMETRY ON
SYMMETRY_BACKEND SPGLIB
KPOINT -0.25 -0.25 -0.25 1.0
KPOINT -0.25 -0.25 0.25 1.0
KPOINT -0.25 0.25 -0.25 1.0
KPOINT -0.25 0.25 0.25 1.0
KPOINT 0.25 -0.25 -0.25 1.0
KPOINT 0.25 -0.25 0.25 1.0
KPOINT 0.25 0.25 -0.25 1.0
KPOINT 0.25 0.25 0.25 1.0
&END KPOINTS
```
The same closure requirement applies to K290, SPGLIB, and mixed SPGLIB-reduction/K290-backend
setups. For band paths or other intentionally nonuniform explicit lists, keep `SYMMETRY OFF`,
because the order and weights of the points are part of the requested property.
## Cell Convention
CP2K's k-point machinery assumes the standard CP2K cell convention: vector `A` lies along the
Cartesian X axis and vector `B` lies in the XY plane. Using `ABC` together with `ALPHA_BETA_GAMMA`,
or reading a CIF file, lets CP2K construct the cell in that convention from orientation-independent
lattice parameters.
## Moving Geometries
For moving geometries (`GEO_OPT`, `CELL_OPT`, `MD`, and related run types), atomic k-point symmetry
is rebuilt from the current cell and coordinates instead of reusing operations from the initial
geometry. This applies to regular Monkhorst-Pack/MacDonald meshes and to closed `GENERAL` k-point
sets. If the current geometry no longer supports a symmetry operation, the reduced set changes
accordingly or the explicit `GENERAL` set is rejected if it is no longer symmetry-closed.
`KEEP_SPACE_GROUP T` is the safest way to combine geometry optimization with full atomic k-point
symmetry. `KEEP_SYMMETRY T` constrains the cell metric, but does not by itself keep atoms on
space-group-related positions.
## Wannier90
The Wannier90 interface already has its own k-point path controlled by
`DFT%PRINT%WANNIER90%MP_GRID`. It builds and diagonalizes a full Monkhorst-Pack grid for the
Wannier90 export. Atomic k-point symmetry from the SCF setup is not reused, because Wannier90
expects the full mesh and its nearest-neighbour connectivity. Explicit `SCHEME GENERAL` lists and
symmetry-reduced SCF k-point sets are therefore not exported directly to Wannier90.
## Related Pages
- <https://www.cp2k.org/faq:kpoints>
- [](hartree-fock/ri_kpoints)