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JRCAT 244p and JAIST 2122p
Date: 2018/10/18 19:59
Name: J. Chapman   <jacob.chapman@ukaea.uk>

Dear Prof Ozaki and respected Openmx community.

I am looking through the source code of the openmx modules and regularly see notes for JRCAT-244p and JAIST-2122p in regards to the transformation of the C matrix. However, I have been unable to determine the references to fully appreciate mapping back onto the original eigenvectors. Could you please provide the details of these references so I may understand the process fully?

Best wishes,
Jacob
ƒƒ“ƒe
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Re: JRCAT 244p and JAIST 2122p ( No.1 )
Date: 2019/01/30 16:34
Name: Naoya Yamaguchi

Dear Jacob,

I guess that your question is about "C = U * lambda^{-1/2} * D" in some source codes to solve eigenvalue problems, and I have just learned it by reading the codes and so I leave some comments even though you may have solved it.
I believe that D is a matrix including eigenvectors obtained from eigenvalue problems transformed from generalized eigenvalue problems of eq. (27) in the following technical note: http://www.openmx-square.org/tech_notes/tech1-1_2.pdf. And, a overlap matrix S is expressed by S=Uƒ©^{1/2}ƒ©^{1/2}U^õ where ƒ© is a diagonal matrix including eigenvalues of S and U is a unitary matrix including the corresponding eigenvectors.

From the eq. (27), HC=ƒÃSC.
-> HC=ƒÃ(Uƒ©^{1/2}ƒ©^{1/2}U^õ)C.
-> (ƒ©^{-1/2}U^õHUƒ©^{-1/2})(ƒ©^{1/2}U^õC)=ƒÃ(ƒ©^{1/2}U^õC).
-> H'D=ƒÃD, where H'=ƒ©^{-1/2}U^õHUƒ©^{-1/2} and D=ƒ©^{1/2}U^õC.
Then, C=Uĩ^{-1/2}D.

Regards,
Naoya Yamaguchi
ƒƒ“ƒe

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