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Scheme to evaluate the Green function by recursive polynomial expansion, and its applications

Research Project

Project/Area Number 12640376
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 物性一般(含基礎論)
Research InstitutionShimane University

Principal Investigator

TANAKA Hiroshi  Shimane Univ., Fac. Sci. & Eng., Assoc. Prof., 総合理工学部, 助教授 (10284019)

Co-Investigator(Kenkyū-buntansha) IITAKA Toshiaki  Riken, Comp. Sci., Researcher, 計算科学技術, 研究員 (60212700)
TOKIHIRO Tetsuji  Tokyo Univ., Mathematical Science, Prof., 大学院・数理科学研究科, 教授 (10163966)
ITOH Masakai  Shimane Univ., Fac. Sci. & Eng., Prof., 総合理工学部, 教授 (90184689)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2001: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2000: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordsrecursive polynomials / Green function / aperiodic system / ordered N / first principles calculation / error estimation / 直交多項式展開 / 数値解析
Research Abstract

We developed a new method to evaluate the Green function for the system described by a huge Hamiltonian matrix on the basis of recursive polynomial expansion. It has the following features ; (1) it is not necessary to diagonalize the Hamiltonian, (2) we can evaluate off-diagonal elements of the Green function as well as diagonal elements, (3) it is possible to evaluate products of the Green functions and other quantum operators. Then, several physical properties can be evaluated, (4) the method is applicable to the system with discrete nature, (5) we can also evaluate eigenvalues and eigenvectors on the same algorithm, (6) CPU time and memory size needed in the calculation is proportional to the system size N (i.e. ordered N method).
Combining this method with the first principles calculations, we can analyze several physical properties such as electronic conductivities from first principles in the large disordered systems. In practice, we applied this method to bcc and amorphous Fe system, and shows the usefulness of the method.
We also compared the method with time-dependent method such as particle source method and forced oscillator method, and showed that the time evolution operator can be derived from the Fourier transformation of the Green function obtained by our method.
Furthermore, we showed that the method can be extended to the evaluation of thermal Green function by analytical continuation to the imaginary time.
In the practical calculations, we have to terminate the expansion at finite order. We estimate the numerical error due to the termination, and showed that it decreases as proportional to the expansion order N.

Report

(3 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • Research Products

    (15 results)

All Other

All Publications (15 results)

  • [Publications] W.Kunishima, M.Itoh, H.Tanaka: "A new method to calculate the Green function by polynomial expansion"Prog. Theor. Phys supplement. 138. 149-151 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] W.Kunishima, M.Itoh, H.Tanaka: "Generalized polynomial expansion of Green's function with applications to electronic structure calculations"AIP Conference proceedings. 519. 350-351 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Tanaka, W.Kunishima, M.Itoh: "Efficient scheme to calculate the Green function by recursive polynomial expansion"Riken Review. 29. 20-24 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] W.Kunishima, T.Tokihiro, H.Tanaka: "Error estimation of recursive orthogonal polynomial expansion method for large Hamiltonian systems"Comput. Phys. Commun. (印刷中). (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] W. Kunishima, M. Itoh, and H. Tanaka: "A new method to calculate the Green function by polynomial expansion"Prog. Theor. Phys supplement. 138. 149-151 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] W. Kunishima, M. Itoh, and H. Tanaka: "Generalized polynomial expansion of Green's function with applications to electronic structure calculations"AIP Conference proceeding. 519, 350-519, 351 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Tanaka, W. Kunishima, and M. Itoh: "Efficient scheme to calculate the Green function by recursive polynomial expansion"Riken Review. 29. 20-24 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] W. Kunishima, T. Tokihiro, and H. Tanaka: "Error estimation of recursive orthogonal polynomial expansion method for large Hamiltonian systems"Comput. Phys. Commun.. (in press). (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] W.Kunishima, M.Itoh, H.Tanaka: "A new method to calculate the Green function by polynomial expansion"Prog.Theor.Phys supplement. 138. 149-151 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] W.Kunishima, M.Itoh, H.Tanaka: "Generalized polynomial expansion of Green's function with applications to electronic structure calculations"AIP Conference proceedings. 519. 350-351 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Tanaka, W.Kunishima, M.Itoh: "Efficient scheme to calculate the Green function by recursive polynomial expansion"Riken Review. 29. 20-24 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] W.Kunishima, T.Tokihiro, H.Tanaka: "Error estimation of recursive orthogonal polynomial expansion method for large Hamiltonian systems'"Comput.Phys.Commun.. (印刷中). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] W.Kunishima,M.Itoh,and H.Tanaka: "A new method to calculate the Green funciton by polynomial expansion"Prog.Theor.Phys.Supplement. 138. 149-150 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] W.Kunishima,M.Itoh,and H.Tanaka: "Generalized polynomial expansion of Green's function with applications to electronic structure calculations"AIP Conference proceedings. 519. 350-351 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Tanaka,W.Kunishima,and M.Itoh: "Efficient scheme to calculate Green functions by recursive polynomial expansion"Riken Review. No.29. 20-24 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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