# Symmetry operation in reciprocal space

1. ### bsmile

27
For a given lattice, specifically a diamond, it has C3v symmetry and the symmetry operation in real space is easy to see, 2C3 around high symmetry axis and 3sigma_v. My question is, if the lattice is expressed in reciprocal space, (say wavefunction defined in momentum space), then how to define the symmetry operation?

2. ### Douasing

42
Hi,bsimle,in my opinion,the symmetry of the lattice depends only on the its geometry.That is to say,for the symmetry of the reciprocal lattice,it depends only on the geometry too.The essense of the symmetry balls down to the group space.

3. ### bsmile

27
Thanks for your response. (1) can the symmetry operation in the real space lattice be related to symmetry operation in the reciprocal lattice in some way, or they are not related at all? (2) For diamond, it seems the reciprocal lattice will be a BCC with specific K point defraction varied. This seems to me simplified the problem too much, as diamond in real space very plausibly has lower symmetry than the FCC structure.

4. ### Douasing

42
Hi,bsmile,you are welcome.(1)We can deduce the reciprocal lattice from the real space lattice.But it is not the case for their symmetry operations.(2) For diamond,we can understand and cope with it likewise.

4,243
6. ### M Quack

847
Diamond is FCC cubic with space group Fd-3m (#227). The crystal class is m-3m or O_h, which is of course also cubic.

The sites occupied by carbon in the diamond lattice have lower symmetry, -43m or T_d, because the space group is non-symmorphic, i.e. it contains glide planes and screw axes.

The reciprocal lattice is completely determined by the real space lattice, in each and every aspect, property, and parameter - including all its symmetries. In particular, the Brillouin zone has that same symmetry as the crystal class, for diamond that is m-3m.

This is a good place to play around and explore:

http://www.cryst.ehu.es/

The "translational symmetry" of the reciprocal lattice is a different story, but it is also determined by the real space lattice.

7. ### bsmile

27
These general information on space group really helps. To help me more precisely, I might give you the issue I am faced with. I am dealing with the NV center of a diamond, basically two nearest neighbor carbon atoms are removed from the lattice, and one vacancy is replaced with N atom. Thus I know it has the overall C3v symmetry. It will have C3 rotation symmetry along [1,1,1] direction, and equally the 3 Sigma_v reflection planes can be determined in real space, of course. Now I have a wavefunction defined in reciprocal space (in DFT-VASP language, CPTWFP(G,K=0)) and I need to determine what its irreducible representation is under the C3v symmetry. One thing to do is to transform this wavefunction back to real space and then determine its property, or I would like to work on it directly in reciprocal space. Then, the issue is how to define the symmetry operations in the reciprocal space. A related interesting (I think) question is how to relate these symmetry operations in reciprocal space to real space operations. Thanks for your further help. If you need more information, please let me know.

8. ### M Quack

847
OK, that makes a lot more sense now.

You are dealing with non-periodic point defects. There break the translational symmetry of the lattice. They also lower the local point symmetry as you correctly point out - C_3v is a subgroup of T_d. You are now dealing with a point group only, not a space group that also includes translations, screw axes etc.

To determine the IR of the wave function, you do not need to transform back to real space. You can do the symmetry analysis directly in reciprocal space, using the exact same symmetry operators. I don't recall off the top of my head how to prove this, but it is probably enough to note that the Fourier transform is linear.

Let R(D) be the reciprocal lattice of the direct lattice D. Let G be a symmetry operation of D, G(D)=D.

Then R(D) = G^-1(R(G(D))) = G^-1(R(D)), i.e. G^-1 is a symmetry operation of the reciprocal lattice.

(The first step is intuitively obvious, but I can't think of a formal argument as to why. It is kind of looking at the same problem from a different, symmetry-related point of view).

9. ### Douasing

42
That is very nice because your problem is now discussed deeper and deeper!
Though my research is very different from yours,I still decide to spend some time to discuss it.
(1) For your first problem,i.e."Now I have a wavefunction defined in reciprocal space (in DFT-VASP language, CPTWFP(G,K=0)) and I need to determine what its irreducible representation is under the C3v symmetry. One thing to do is to transform this wavefunction back to real space and then determine its property, or I would like to work on it directly in reciprocal space."
I think M.Quack is right,i.e.,"To determine the IR of the wave function, you do not need to transform back to real space. You can do the symmetry analysis directly in reciprocal space, using the exact same symmetry operators."
Here,I may supply a suitable formula for the symmetrized planewaves as follows
$$\Phi_{s}=\frac{1}{N_{op}}\sum_{R}exp[iRG(r-t_{R})]=\frac{1}{m_{s}}\sum_{m}\varphi_{m}exp[iR_{m}Gr]$$
From its mathematics,we know that it is fit for pseudo potential (or your software mentioned above,DFT-VASP) or interstitial repersentations in so-called APW method.

(2)For your second problem,i.e.,"Then, the issue is how to define the symmetry operations in the reciprocal space."
Of course,you can use the exact same symmetry operators in reciprocal space as in real space.
But, "how to relate these symmetry operations in reciprocal space to real space operations.",obviously,you can use Fourier transform (or more efficiently,FFT) to make it.However,(I think) those symmetry operations in reciprocal space should not be transformed into the real space,or you can't find the precise mathematical relation about their oprations between these two space (as mentioned above by me).

10. ### bsmile

27
Thanks for all the discussions contributed to the post. With insights acquired from the discussion, I finally figured out how to it. And, only after I figured out how to work it out did I understand the comments better.

Here I would like to contribute a simpler argument (I hope) towards my own questions.

First, of course, one can transform from G space to r space, and then apply symmetry operation on psi(r)

Second, if want to work in G space, then the following might be helpful

suppose the definition: Psi(G)=intg(exp(-i G r) psi(r)), apply the symmetry operation P on both sides,

P Psi(G) = intg(exp(-i G r) P psi(r) )=intg(exp(-i G r) Ppsi(Pr))=intg(exp(- PG Pr) Ppsi(Pr))=PPsi(PG)

where the fact G.r is a scalar, and G, r are both vectors are used under symmetry operation P.

Compare it against the wavefunction in real space,

P psi(r) = Ppsi(Pr)

one can see that the same symmetry operation can be applied similarly to Psi(G) to determine its irreducible representation (at least for A1,A2 type rank 1 irreducible representation).