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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

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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=UΛ^{1/2}Λ^{1/2}U^† where Λ is a diagonal matrix including eigenvalues of S and U is a unitary matrix including the corresponding eigenvectors. From the eq. (27), HC=εSC. -> HC=ε(UΛ^{1/2}Λ^{1/2}U^†)C. -> (Λ^{-1/2}U^†HUΛ^{-1/2})(Λ^{1/2}U^†C)=ε(Λ^{1/2}U^†C). -> H'D=εD, where H'=Λ^{-1/2}U^†HUΛ^{-1/2} and D=Λ^{1/2}U^†C. Then, C=UΛ^{-1/2}D. Regards, Naoya Yamaguchi

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