Re: The conventions for real spherical harmonics in OpenMX ( No.1 ) |
- Date: 2011/10/12 18:08
- Name: T.Ozaki
- Hi,
Sorry for causing your confusion about the convention of real spherical harmonics.
In the actual implementation, the convention follows the definition in Set_Comp2Real() as:
/* p */
Comp2Real[1][0][0] = Complex( 1.0/sqrt(2.0), 0.0); Comp2Real[1][0][1] = Complex( 0.0, 0.0); Comp2Real[1][0][2] = Complex(-1.0/sqrt(2.0), 0.0);
Comp2Real[1][1][0] = Complex( 0.0, 1.0/sqrt(2.0)); Comp2Real[1][1][1] = Complex( 0.0, 0.0); Comp2Real[1][1][2] = Complex( 0.0, 1.0/sqrt(2.0));
Comp2Real[1][2][0] = Complex( 0.0, 0.0); Comp2Real[1][2][1] = Complex( 1.0, 0.0); Comp2Real[1][2][2] = Complex( 0.0, 0.0);
/* d */
Comp2Real[2][0][0] = Complex( 0.0, 0.0); Comp2Real[2][0][1] = Complex( 0.0, 0.0); Comp2Real[2][0][2] = Complex( 1.0, 0.0); Comp2Real[2][0][3] = Complex( 0.0, 0.0); Comp2Real[2][0][4] = Complex( 0.0, 0.0);
Comp2Real[2][1][0] = Complex( 1.0/sqrt(2.0), 0.0); Comp2Real[2][1][1] = Complex( 0.0, 0.0); Comp2Real[2][1][2] = Complex( 0.0, 0.0); Comp2Real[2][1][3] = Complex( 0.0, 0.0); Comp2Real[2][1][4] = Complex( 1.0/sqrt(2.0), 0.0);
Comp2Real[2][2][0] = Complex( 0.0, 1.0/sqrt(2.0)); Comp2Real[2][2][1] = Complex( 0.0, 0.0); Comp2Real[2][2][2] = Complex( 0.0, 0.0); Comp2Real[2][2][3] = Complex( 0.0, 0.0); Comp2Real[2][2][4] = Complex( 0.0,-1.0/sqrt(2.0));
Comp2Real[2][3][0] = Complex( 0.0, 0.0); Comp2Real[2][3][1] = Complex( 1.0/sqrt(2.0), 0.0); Comp2Real[2][3][2] = Complex( 0.0, 0.0); Comp2Real[2][3][3] = Complex(-1.0/sqrt(2.0), 0.0); Comp2Real[2][3][4] = Complex( 0.0, 0.0);
So, the definition in Eq. (159) in the note is not consistent with that in the code.
Regards,
TO
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Re: The conventions for real spherical harmonics in OpenMX ( No.2 ) |
- Date: 2011/10/13 16:18
- Name: H.
- Dear Prof. T. Ozaki.
Given the conventions in Set_Comp2Real(),
is Eq. (158), i.e., Y_{py}=\frac{1}{i\sqrt{2}}(Y_1^{-1}+Y_1^1} right?
The convention below in Set_Comp2Real() differs from Eq. (158) in the overall "-" sign.
... Comp2Real[1][1][0] = Complex( 0.0, 1.0/sqrt(2.0)); Comp2Real[1][1][1] = Complex( 0.0, 0.0); Comp2Real[1][1][2] = Complex( 0.0, 1.0/sqrt(2.0)); ...
Also, is the imaginary part of Eq. (76) (spin up-spin down component) right?
I think that Eq. (76) in the technical note (http://www.openmx-square.org/tech_notes/tech2-1_0.pdf) is obtained with the conventions, Eq. (157)-(159), not those in Set_Comp2Real().
Thanks for your kindly reply.
Regards.
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Re: The conventions for real spherical harmonics in OpenMX ( No.3 ) |
- Date: 2011/11/11 21:55
- Name: T.Ozaki
- Hi,
In Set_Comp2Real(), the unitary matrices for p-states are defined by
Comp2Real[1][0][0] = Complex( 1.0/sqrt(2.0), 0.0); Comp2Real[1][0][1] = Complex( 0.0, 0.0); Comp2Real[1][0][2] = Complex(-1.0/sqrt(2.0), 0.0);
Comp2Real[1][1][0] = Complex( 0.0, 1.0/sqrt(2.0)); Comp2Real[1][1][1] = Complex( 0.0, 0.0); Comp2Real[1][1][2] = Complex( 0.0, 1.0/sqrt(2.0));
Comp2Real[1][2][0] = Complex( 0.0, 0.0); Comp2Real[1][2][1] = Complex( 1.0, 0.0); Comp2Real[1][2][2] = Complex( 0.0, 0.0);
They are all opposite compared to Eqs.(157)-(159) in the notes in terms of sign. Then, the difference regarding sign does not matter when the matrices F and G are calculated, since they are cancelled out.
Regards
TO
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