[Pw_forum] generating k point weights

[Gabriele Sclauzero]

xirainbow wrote:
> Dear Gabriele Sclauzero:
> 
>  >Just one additional note of caution: kpoints.x reduces the number of 
> k-points (and compute
>  >weights) according to the symmetry of the bravais lattice only, while 
> the subroutine
>  >kpoint_grid in PW/ used by pw.x takes into account the crystal 
> symmetry (which can be
>  >lower than the lattice symmetry if you have more than one atom per 
> cell or non-collinear
>  >magnetism)
>  
> I am confused with your statement:"crystal symmetry can be lower than 
> the lattice symmetry if you have more than one atom per cell".

Maybe I misused the standard terminology.

> Could you explain it more clearly?

Every periodic system can be described by one among the 14 Bravais lattices plus a 
so-called "basis", i.e. a set of atomic position within the unit cell of that lattice. The 
symmetry point group of the lattice is the set of all rotations which leave the lattice 
invariant (which means that it can be mapped to itself with the application of a 
translation of a l.c. of the basis vectors, if needed).
This is the symmetry group given by kpoints.x (which in fact does not ask anything about 
the basis of atoms).

To determine the symmetry of a crystallographic system you have to check which symmetry 
operations leave the system invariant, which means that, after a rotation, each atom 
overlaps with itself or with an equivalent atom (again, modulus a translation by a lattice 
vector).
If you have more than one atom, or your atom is not in the origin an operation with 
transform the lattice into itself may not send all the atoms on top of equivalent atoms, 
hence it is not a symmetry operation.
For some systems (e.g. diamond), you may recover some rotations by applying a translation 
by a fraction of a lattice vector after the rotation.


> Or could you give me a simple example?

Sorry, I don't have a crystallographic example in mind right now, but I can give you a 
practical example of what I've got my hands on at the moment.
In order to simulate monatomic nanowires I use a tetragonal cell (ibrav=6) with one atom 
per cell (along the z axis). The lattice symmetry (point group D_4d) is maintained also 
after checking the atomic basis (although this system in reality as D_{\inf h} point 
group, but this is another story...).
If you adsorb an impurity aside of the wire, e.g. a molecule lying in the xz plane, the 
symmetry of the system will be lowered (C_2v in this case). In fact you will lose the 
inversion symmetry (this halves the number of symmetry operations) and the rotations along z.

I hope I've not confused you further...

GS

> Thank you ：）
> 
> ____________________________________
> Hui Wang
> School of physics, Nankai University, Tianjin, China
> 
> 
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