Some questions for Monkhorst-Pack's paper

[jackychenp]

Hi All,

I am trying to figure out how Monkhorst-Pack grits work, but I cannot repeat some results in their paper. (H. J. Monkhorst and J. D. Pack, Special points for Brillouin-zone integrations, 13, 5188, 1976)

1. In equation (9), when $|R^b_j - R^a_j|=p$, which is not multiples of q, $W^{ab}_j(q) = e^{-ip\pi/q}(-1)^p$ according to my calculation. Only when r = 0, 1 ,2 ...,q, $W^{ab}_j(q)=0$.
2. In equation (19), I got $\epsilon_{BZ} =\frac{8\pi^3}{v} \sum_{m>1}f_mS_{m1}(q)$
3. In equation A7, if q=4, P(q)=2, but some software gave a value of 10 when q=4 for FCC point group.

Please correct me if I missed something. Thanks ahead for your help!

 

[eljose79]

Thank you for bringing up this topic. The Monkhorst-Pack method for Brillouin zone integration is a widely used approach in computational materials science, and it is important to understand the details of its implementation. I have looked into your concerns and here are my thoughts:

1. In equation (9), the expression for W^{ab}_j(q) is correct. However, it is important to note that the expression is only valid when |R^b_j - R^a_j| is a multiple of q. In your case, when |R^b_j - R^a_j|=p, where p is not a multiple of q, the expression for W^{ab}_j(q) will be different. This is because the Monkhorst-Pack method uses a special set of k-points that are evenly spaced in the reciprocal space, and the expression for W^{ab}_j(q) is derived based on this assumption. Therefore, it is not valid for all values of p. I suggest you look into the details of the Monkhorst-Pack method and its assumptions to better understand this.

2. Your expression for \epsilon_{BZ} is correct. However, it is important to note that this expression is only valid for the case of a single atom basis. If you have multiple atoms in the basis, you will need to modify the expression accordingly. I suggest you refer to the original paper by Monkhorst and Pack for more details on this.

3. The value of P(q) in equation A7 depends on the point group of the crystal structure. For the FCC point group, P(q) is indeed 10 when q=4. This value is derived based on the symmetry of the FCC crystal structure and the Monkhorst-Pack method. I suggest you double check your software or consult the documentation to ensure that it is using the correct value for P(q).

I hope this helps clarify your concerns. If you have any further questions, please do not hesitate to ask. Best of luck with your research!