Crystal symmetry in ONETEP
- Author:
Chengcheng Xiao
- Date:
July 2025
Symmetry operations in ONETEP
Crystal symmetry operations can be defined using rotation and translation pairs. In ONETEP, symmetry operations (in fractional coordinates) are obtained by passing the crystal information (lattice, ionic positions, and types) to spglib <http://github.com/spglib/spglib>.
These operations can then be used to find symmetry-related k-points:
where \(\mathbf{R}\) is the rotation operator in reciprocal space, which is the inverse of the rotation operator in the real space.
Symmetry-related k-points have the same energy; hence, we can use only one k-point for each k-point group that is symmetry-related to achieve more efficient calculations. Note that the k-point weights are also adjusted based on the number of symmetry-related k-points in the group.
In addition to the crystal symmetry, ONETEP also supports time-reversal symmetry (TRS) to further reduce the number of k-points. In practice, this means that one can eliminate half of the k-points in the first Brillouin zone because TRS dictates that k-points (k) and their time-reversed counterparts (-k) have the same energy.
However, while the energy calculated by symmetry-related k-points is the same, the wavefunctions are not. Luckily, the wavefunctions of symmetry-related k-points are also symmetry-related, and we symmetrise the charge density (for each spin channel if spin-polarised) so that the charge density and energy is correct.
The symmetrisation of the charge density is done in reciprocal space. First, the charge density in the simulation cell is Fourier transformed to reciprocal space. Then, we apply the rotation operations in fractional coordinates to the indices of the G-vectors to find the indices of their symmetry-related counterparts,
where \(\mathbf{G}\) is the G-vector in reciprocal space. Finally, we extract the corresponding value of the charge density at the symmetry-related point in reciprocal space and perform a phase-weighted average. The phase is caused by the non-primitive cell translation part of the symmetry operation. And the whole process is repeated for each symmetry operation. Mathematically this can be expressed as:
We also leverage the fact that if the cell we are calculating is not primitive, we’ll have pure non-primitive translation operations, and if a G-vector is not commensurate with the total number of non-primitive translations, it will have zero amplitude so we can avoid applying symmetry operations to these G-vectors altogether.
Magnetic symmetry operations
Magnetic moments can affect the symmetry operations in a crystal. Specifically, the presence of certain magnetic moments can lead to the existence of additional symmetry operations - Time-reversal operations that swap the spin components of the wavefunctions.
In ONETEP, instead of using full time-reversal operations to symmetrise the spin polarised charge density, the entire crystal symmetry is lowered by treating the atoms that belong to the same species but have different initial magnetic moments as if they belong to different species.
Doing this allows the possibility for the lower symmetry magnetic structure (such as antiferromagnetic) to relax into a higher symmetry structure (such as ferrimagnetic).
This is done automatically by ONETEP when the use_symmetry keyword is set to
True and the use_time_reversal keyword is set to True and LOCK
SPECIES_ATOMIC_SET is used to specify different initial magnetic moments for
different atoms of the same species (for more, see
Initial Guess Density: Setting initial charges and spins). If the initial
magnetic moments do change the symmetry, ONETEP will report the symmetry
operations written:
Number of symmetry operations = NUMBER
(reduced by initial magmoms)
Compiling ONETEP with spglib
ONETEP now ships with spglib and can be enabled by BUILD_SPGLIB=yes, i.e.,
make onetep ARCH=YOUR_ARCH BUILD_SPGLIB=yes
You can also change the C-compiler by setting the CC environment variable in
your ARCH file, e.g.,
CC=gcc-11
Running ONETEP with symmetry
To run ONETEP with symmetry, the user only needs to set the use_symmetry
keyword to True. To leverage the time-reversal symmetry to reduce the number
of k-points, the use_time_reversal keyword can also be set to True.
One thing to note is that only MP grids set up by kpoint_grid_size will get
symmetrised. User-supplied k-point list (via block kpoints_list) will not be
modified.
Once these tags are set, ONETEP will report the symmetry operations and the
reduced k-points in the output file (if output_detail is set to
verbose), and the symmetry-related k-points will be used in the calculation.
Keywords
use_symmetry: [bool] turn symmetry on or off (only works if compiled with spglib) | default: offuse_time_reversal: [bool] use TRS to reduce the number of k-points or not | default: onsymmetry_tol: [float] Precision in determining the symmetry. | default: 0.001889726 Bohr (0.001 Å)