Mathematical Programming

, Volume 109, Issue 2–3, pp 553–580 | Cite as

Large-scale semidefinite programs in electronic structure calculation

  • Mituhiro FukudaEmail author
  • Bastiaan J. Braams
  • Maho Nakata
  • Michael L. Overton
  • Jerome K. Percus
  • Makoto Yamashita
  • Zhengji Zhao


It has been a long-time dream in electronic structure theory in physical chemistry/chemical physics to compute ground state energies of atomic and molecular systems by employing a variational approach in which the two-body reduced density matrix (RDM) is the unknown variable. Realization of the RDM approach has benefited greatly from recent developments in semidefinite programming (SDP). We present the actual state of this new application of SDP as well as the formulation of these SDPs, which can be arbitrarily large. Numerical results using parallel computation on high performance computers are given. The RDM method has several advantages including robustness and provision of high accuracy compared to traditional electronic structure methods, although its computational time and memory consumption are still extremely large.


Large-scale optimization Computational chemistry Semidefinite programming relaxation Reduced density Matrix N-representability Parallel computation 


90C06 81Q05 90C22 68W10 


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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Mituhiro Fukuda
    • 1
    Email author
  • Bastiaan J. Braams
    • 2
  • Maho Nakata
    • 3
  • Michael L. Overton
    • 4
  • Jerome K. Percus
    • 5
  • Makoto Yamashita
    • 6
  • Zhengji Zhao
    • 7
  1. 1.Department of Mathematical and Computing SciencesTokyo Institute of TechnologyMeguro-kuJapan
  2. 2.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA
  3. 3.Department of Applied ChemistryThe University of TokyoBunkyo-kuJapan
  4. 4.Department of Computer Science, Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  5. 5.Courant Institute of Mathematical Sciences and Department of PhysicsNew York UniversityNew YorkUSA
  6. 6.Department of Information Systems CreationKanagawa UniversityKanagawa-ku, Yokohama-shiJapan
  7. 7.High Performance Computing Research DepartmentLawrence Berkeley National LaboratoryBerkeleyUSA

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