A Recovering Wavefunction in Periodic Ab Initio Calculations

[bsmile]

In ab initio calculations for periodic systems, only an irreducible K grid is used for calculation, and consequently only those K points have their wavefunction calculated. My question is, how to recover wavefunction at other K points not included in the irreducible K grid? Similar questions to the density matrix.

 

[DrDu]

I hear the term "irreducible K grid" for the first time.

 

[bsmile]

I hear the term "irreducible K grid" for the first time.

sorry, I should be more careful towards my terms, it should be something like irreducible K wedge in the first Brillouin Zone.

 

[DrDu]

Ah, I see.
I suppose you get the other wedges applying the elements of the point group of your crystal to the wavefunctions.
In fact, there is a very general theorem from group theory, the orbit-stabilizer theorem. If G is the full point group of your crystal and H is the little group of the point in K space, then a point in your K wedge will mapped to #G/#H points in total K space. "#" means here the number of elements of the group. E.g. the little group of the Gamma point is G, i.e. H=G, so the gamma point is invariant, while the little group of a general point, not coinciding with a special point is H=1, so there will be #G points formed from it.

 

[bsmile]

Yes, I understand there is point group symmetry operation on how to related one K point to the other equivalent ones, but I don't know how to transform the eigen-wavefunction for (n,K) where n is band indices? I believe this could be done. A further question is, would the transformation depend on the choice of basis set, say planewave basis or atomic orbital basis set (say contracted Gaussian basis set)?

 

[DrDu]

You can use the general expression for transformation for this. So if a point transforms as $r'=Rr$, a function transforms as $\psi'(r)=\psi(R^{-1}r)$. Use the Bloch theorem and the form of your wavefunctions to work out how they transform. E.g. $exp(ik^TR^{-1} r)$ can be written as $exp(ik'^Tr)$ with $k'=(k^TR^{-1})^T=Rk$. Be careful here when direct and reciprocal lattice vectors are used.

 

[bsmile]

You can use the general expression for transformation for this. So if a point transforms as $r'=Rr$, a function transforms as $\psi'(r)=\psi(R^{-1}r)$. Use the Bloch theorem and the form of your wavefunctions to work out how they transform. E.g. $exp(ik^TR^{-1} r)$ can be written as $exp(ik'^Tr)$ with $k'=(k^TR^{-1})^T=Rk$. Be careful here when direct and reciprocal lattice vectors are used.

Thanks, I understand here $\psi$ is a scalar, thus a geometric transformation does not change it, while dot product being a scalar is also not affected. What if $\psi$ is a spinor with its axis along discrete z direction while the target K has a direction arbitrary in space, or another quantity differing from wavefunction but carrying indices in angular momentum, say a local orbital density matrix $\rho_{px,py}(K)$, to be transformed into arbitrary target K direction?

 

[DrDu]

Then you have to transform the spinor/vector too: $v'(r)=D_Rv(R^{-1}r)$. Where $D_R$ is a transformation matrix for the spinor/vector.