Mathematical Programming

, Volume 109, Issue 2, pp 553-580

First online:

Large-scale semidefinite programs in electronic structure calculation

  • Mituhiro FukudaAffiliated withDepartment of Mathematical and Computing Sciences, Tokyo Institute of Technology Email author 
  • , Bastiaan J. BraamsAffiliated withDepartment of Mathematics and Computer Science, Emory University
  • , Maho NakataAffiliated withDepartment of Applied Chemistry, The University of Tokyo
  • , Michael L. OvertonAffiliated withDepartment of Computer Science, Courant Institute of Mathematical Sciences, New York University
  • , Jerome K. PercusAffiliated withCourant Institute of Mathematical Sciences and Department of Physics, New York University
  • , Makoto YamashitaAffiliated withDepartment of Information Systems Creation, Kanagawa University
  • , Zhengji ZhaoAffiliated withHigh Performance Computing Research Department, Lawrence Berkeley National Laboratory

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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