How to use the momentum matrix of the reduced k-points

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SUMMARY

The discussion focuses on calculating the momentum matrix elements at a specific k-point (k1) and using symmetry to derive optical properties from equivalent k-points (k1, k2, k3, k4). The key conclusion is that one can utilize the symmetry of the k-points to simplify calculations by multiplying the momentum matrix element at k1 by the weight of k1, given that the k-points are equivalent. Additionally, the discussion emphasizes the importance of understanding irreducible representations in the context of point group symmetry to ensure the total matrix element remains symmetric.

PREREQUISITES
  • Understanding of momentum matrix elements in quantum mechanics
  • Familiarity with k-point sampling in solid-state physics
  • Knowledge of symmetry operations and point group theory
  • Experience with optical property calculations in condensed matter physics
NEXT STEPS
  • Study the concept of irreducible representations in group theory
  • Learn about k-point integration techniques in density functional theory (DFT)
  • Research the application of symmetry in quantum mechanics and solid-state physics
  • Explore advanced optical property calculations using software like Quantum ESPRESSO or VASP
USEFUL FOR

This discussion is beneficial for physicists, materials scientists, and computational chemists involved in quantum mechanics, particularly those working on optical properties and symmetry in solid-state systems.

sandf
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I have a set of k-points, e.g. k1,k2,k3,k4. and they are equivalent by symmetry.

Now I have calculated the momentum matrix element <i|p|j> at k1 point ONLY,

and then calculate the optical properties which, for example, depend on <i|p|j><j|p|i>

I have to make a summation on four k-points (k1,k2,k3,k4) for dk integration.

So, how to make this summation only on the basis of <i|p|j> of k1 point?

Can I simply do by multiplying <i|p|j> by the weight of k1 point? because these four k-points are equivalent.

If not, could you give me some references?Any help will be appreciated.

Youzhao Lan
 
Last edited:
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I am no specialist on this, but I think that the p and |i> span a representation of the point group which is also responsible for the equivalence of the k points. So you have to find the irreducible representations spanned by p and |i> and find those combinations of irreducible representations formed from <j|, p and |i> that the total matrix element <j|p|i> is totally symmetric.
 
Dr.Du, thanks for your help.

Lan
 

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