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 The conventions for real spherical harmonics in OpenMX Date: 2011/10/06 12:06 Name: H. Dear Prof. T. Ozaki.The real spherical harmonics for l=1 are defined in Eqs. (157)-(159) in the technical note (http://www.openmx-square.org/tech_notes/tech2-1_0.pdf).Among these, the case for m=0 [Eq. (159), Y_{pz}=-Y_1^0] is defined with the "-" sign, which is different from the definition that I know [See (Cf.) below].I would like to know the definition for real spherical harmonics used in OpenMX for l=1 and 2 cases.[Though I check the subroutine, Set_Comp2Real() in SetPara_DFT.c, I still don't know the conventions for real spherical harmonics]I think the definition for even m (m=0, 2, 4, etc.) is different from the usual convention in that the "-1" is multiplied. Is it right?Regards.(Cf.)The definitions for real spherical harmonics, \bar{Y}_l^m, which I know are as follows:For positive m,\bar{Y}_l^m=\frac{1}{\sqrt{2}} [ Y_l^m+(-1)^m*Y_l^{-m} ]\bar{Y}_l^{-m}=\frac{1}{i\sqrt{2}} [ Y_l^m-(-1)^m*Y_l^{-m} ]For m=0,\bar{Y}_l^0=Y_l^0
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 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 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. 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. RegardsTO

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