[Wien] inversion vs time reversal

[Peter Blaha]

> However, for example, the case : GaAs, the system is complex, which I 
> can understand, caused by the absence of inversion symmetry. but the 
> system don’t  have the (-1, -1, -1) diagonal matrix,  Is this mean GaAs 
> also without the time reversal symmetry?

As I said: WITHOUT magnetism and neglecting Spin-orbit interactions, 
time reversal is always present.
Therefore, when we calculate the IBZ k-mesh, we use not only the 24 
symops of this space group, but multiply every operation also with 
inversion and use 48 sym.ops.

> Is the (-1 -1 -1) diagonal both shows if the system has the inversion 
> symmetry or the time reversal symmetry?  But the (-1 -1 -1) diagonal 
> matrix absent cause the system has not only inversion symmetry, but also 
> the time reversal symmetry?

The (-1,-1,-1) diagonal matrix (in case.struct) shows, that we have 
INVERSION symmetry.  Time reversal has nothing to do with the local 
symmetry, but depends on ... (see above).

> By the way, why the system is complex when the system without the 
> inversion symmetry? As I know ,the result of inversion symmetry is : <r| 
> -k, j> = <-r | k, j> , Only the time reversal symmetry has some 
> relationship with the complex conjugate:  <r|- k, j> = <r |k, j >*.  Do 
> you have some relevant references of this ?

Now it becomes quite basic:
We use plane waves, right ? These are complex functions  psi= exp**(i k r).

However, this can be expanded into   cos (k r) + i sin (k r)

With inversion we know that   psi (r) = psi (-r).
Since sin (k r) is "odd";  (sin (k r) .ne. sin (k -r)), we know that 
this term must vanish and we can replace    exp**(i k r)  by   cos (k r) 
  and this is purely "real".

> 
> Looking forward to your reply.
> 
> Sincerely,
> 
> Jasmine.
> 
> *From: *Peter Blaha <mailto:pblaha at theochem.tuwien.ac.at>
> *Sent: *Thursday, February 6, 2020 7:31 PM
> *To: *wien at zeus.theochem.tuwien.ac.at 
> <mailto:wien at zeus.theochem.tuwien.ac.at>
> *Subject: *Re: [Wien] inversion vs time reversal
> 
> time inversion symmetry is used in WIEN2k only for the generation of the
> 
> k-mesh when "inversion" is not present (and it is not a magnetic case
> 
> with spin-orbit). In that case we use the fact that
> 
> epsilon(k)=epsilon(-k) and "add" the (-1,-1,-1) diagonal matrix to the
> 
> symmetry operations reducing the full BZ mesh to the IBZ.
> 
> Everywhere else, only "inversion" is used (if present), which makes
> 
> wavefunctions "real" and one can use lapw1 instead of lapw1c.
> 
> If you try to run Si with (0,0,0) and (1/4,1/4,1/4), you must "trick"
> 
> the initialization procedure, because WIEN2k will normally not allow it,
> 
> because it requires an 4 times larger computational effort.
> 
> On 2/6/20 6:26 AM, 姜若诗 wrote:
> 
>  > Dear professors,
> 
>  >
> 
>  > Do you know how to deal with the inversion symmetry and time reversal
> 
>  > symmetry in wien2k, how do you make them different ?
> 
>  >
> 
>  > For example, if the positions of two atoms Si is in (0.125 0.125 0.125)
> 
>  > and (0.875 0.875 0.875), the system has inversion and time reversal
> 
>  > symmetry, then the point group symmetry matrix has number 48.
> 
>  >
> 
>  > But if the positions are (0 0 0) and (0.25 0.25 0.25), the system
> 
>  > doesn’t has the inversion symmetry, with time reversal symmetry left,
> 
>  > will the symmetry matrix reduce to number 24, or also number 48?
> 
>  >
> 
>  > In a word, what my question is :  Does the symmetry matrix  (-1 0 0, 0
> 
>  > -1 0, 0 0 -1)  show not only the inversion symmetry , but also time
> 
>  > reversal symmetry in a system?
> 
>  >
> 
>  > Looking forward to your reply
> 
>  >
> 
>  > Regards,
> 
>  >
> 
>  > Jasmine.
> 
>  >
> 
>  >
> 
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> 
> -- 
> 
>                                         P.Blaha
> 
> --------------------------------------------------------------------------
> 
> Peter BLAHA, Inst.f. Materials Chemistry, TU Vienna, A-1060 Vienna
> 
> Phone: +43-1-58801-165300             FAX: +43-1-58801-165982
> 
> Email: blaha at theochem.tuwien.ac.at    WIEN2k: http://www.wien2k.at
> 
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-- 

                                       P.Blaha
--------------------------------------------------------------------------
Peter BLAHA, Inst.f. Materials Chemistry, TU Vienna, A-1060 Vienna
Phone: +43-1-58801-165300             FAX: +43-1-58801-165982
Email: blaha at theochem.tuwien.ac.at    WIEN2k: http://www.wien2k.at
WWW:   http://www.imc.tuwien.ac.at/TC_Blaha
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