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Large-scale semidefinite programs in electronic structure calculation

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Abstract

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.

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References

  1. Blackford, L.S., Choi, J., Cleary, A., D’Azevedo E., Demmel, J., Dhillon, I., Dongarra, J., Hammarling, S., Henry, G., Petitet, A., Stanley, K., Walker, D., Whaley, R.C.: ScaLAPACK User’s Guide. SIAM, Philadelphia (1997)

  2. Burer, S., Monteiro, R.D.C., Choi, C.: SDPLR version 1.01 user’s guide (short). Department of Management Sciences, University of Iowa, Iowa City, IA 52242-1000 (2004). http://dollar.biz.uiowa.edu/~burer/software/SDPLR/

  3. Cioslowski J. (2000): Many-Electron Densities and Reduced Density Matrices. Kluwer Academic/Plenum Publishers, New York

    Google Scholar 

  4. Cohen L., Frishberg C. (1976): Hierarchy equations for reduced density matrices. Phys. Rev. A13, 927–930

    Article  Google Scholar 

  5. Coleman A.J. (1963): Structure of fermion density matrices. Rev. Mod. Phys. 35, 668–687

    Article  Google Scholar 

  6. Coleman, A.J.: RDMs: How did we get here? In: [3], pp. 1–17

  7. Colmenero F., Valdemoro C. (1994): Self-consistent approximate solution of the 2nd-order contracted Schrödinger-equation. Int. J. Quantum Chem. 51, 369–388

    Article  Google Scholar 

  8. Davidson E.R. (1969): Linear inequalities for density matrices. J. Math. Phys. 10, 725–734

    Article  Google Scholar 

  9. Deza M.M., Laurent M. (1997): Geometry of Cuts and Metrics. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  10. Dunning Jr T.H. (1970): Gaussian basis functions for use in molecular calculations I Contraction of (9s5p) atomic basis sets for the first-row atoms. J. Chem. Phys. 53: 2823–2833

    Article  Google Scholar 

  11. Dunning, Jr., T.H., Hay, P.J.: Gaussian basis sets for molecular calculations. In: Schaefer III, H.F. (eds) Modern Theoretical Chemistry, vol. 3: Methods of Electronic Structure Theory. Plenum, New York, pp. 1–28 (1977)

  12. Ehara M., Nakata M., Kou H., Yasuda K., Nakatsuji H. (1999): Direct determination of the density matrix using the density equation: Potential energy curves of HF, CH4, BH3, NH3, and H2O. Chem. Phys. Lett. 305, 483–488

    Article  Google Scholar 

  13. Erdahl R.M. (1978): Representability. Int. J. Quantum Chem. 13, 697–718

    Article  Google Scholar 

  14. Erdahl R.M., Jin B. (2000): The lower bound method for reduced density matrices. Journal of Molecular Structure: THEOCHEM 527, 207–220

    Google Scholar 

  15. Erdahl, R.M., Jin, B.: On calculating approximate and exact density matrices. In: [3], pp. 57–84

  16. Frisch, M.J., Trucks, G.W., Schlegel, H.B., et al.: Gaussian 98, Revision A.11.3, Gaussian, Inc., Pittsburgh PA (2002)

  17. Garrod C., Fusco M.A. (1976): A density matrix variational calculation for atomic Be. Int. J. Quantum Chem. 10, 495–510

    Article  Google Scholar 

  18. Garrod C., Percus J.K. (1964): Reduction of the N-particle variational problem. J. Math. Phys. 5, 1756–1776

    Article  MathSciNet  MATH  Google Scholar 

  19. Garrod C., Mihailović M.V., Rosina M. (1975): The variational approach to the two-body density matrix. J. Math. Phys. 16, 868–874

    Article  Google Scholar 

  20. Graner G., Hirota E., Iijima T., Kuchitsu K., Ramsay D.A., Vogt J., Vogt N. (1998): Landolt-Börnstein – Group II Molecules and Radicals, vol. 25, subvolume A. Springer, Berlin Heidelberg New York

    Google Scholar 

  21. Graner G., Hirota E., Iijima T., Kuchitsu K., Ramsay D.A., Vogt J., Vogt N. (1999): Landolt-Börnstein – Group II Molecules and Radicals, vol. 25, subvolume B. Springer, Berlin Heidelberg New York

    Google Scholar 

  22. Hehre W.J., Stewart R.F., Pople J.A. (1969): Self-consistent molecular-orbital methods. I. Use of Gaussian expansions of slater-type atomic orbitals. J. Chem. Phys. 51, 2657–2664

    Article  Google Scholar 

  23. Hehre W.J., Ditchfield R., Stewart R.F., Pople J.A. (1970): Self-consistent molecular orbital methods. IV. Use of Gaussian expansions of slater-type orbitals. Extension to second-row molecules. J. Chem. Phys. 52, 2769–2773

    Article  Google Scholar 

  24. Huber K.P., Herzberg G. (1979): Molecular Spectra and Molecular Structure IV, Electronic Constants of Diatomic Molecules. Van Nostrand Reinhold, New York

    Google Scholar 

  25. Husimi K. (1940): Some formal properties of the density matrix. Proc. Phys. Math. Soc. Jpn. 22, 264–314

    MATH  Google Scholar 

  26. Karp R.M., Papadimitriou C.H. (1982): On linear characterization of combinatorial optimization problems. SIAM J. Comput. 11, 620–632

    Article  MathSciNet  MATH  Google Scholar 

  27. Kijewski L.J. (1972): Effectiveness of symmetry and the Pauli condition on the 1 matrix in the reduced-density-matrix variational principle. Phys. Rev. A6, 31–35

    Article  Google Scholar 

  28. Kijewski L.J. (1974): Strength of the G-matrix condition in the reduced-density-matrix variational principle. Phys. Rev. A9, 2263–2266

    Article  Google Scholar 

  29. Löwdin P.-O. (1955): Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals, and convergence problems in the method of configurational iteration. Phys. Rev. 97, 1474–1489

    Article  MathSciNet  Google Scholar 

  30. Mayer J.E. (1955): Electron correlation. Phys. Rev. 100, 1579–1586

    Article  MATH  Google Scholar 

  31. Mazziotti D.A. (1998): Contracted Schrödinger equation: Determining quantum energies and two-particle density matrices without wave functions. Phys. Rev. A57, 4219–4234

    Article  Google Scholar 

  32. Mazziotti, D.A.: Cumulants and the contracted Schrödinger equation. In: [3], pp. 139–163

  33. Mazziotti D.A. (2002): Variational minimization of atomic and molecular ground-state energies via the two-particle reduced density matrix. Phys. Rev. A65: 062511

    Article  Google Scholar 

  34. Mazziotti D.A. (2002): Solution of the 1,3-contracted Schrödinger equation through positivity conditions on the two-particle reduced density matrix. Phys. Rev. A66, 062503

    Article  Google Scholar 

  35. Mazziotti D.A. (2004): Realization of quantum chemistry without wave functions through first-order semidefinite programming. Phys. Rev. Lett. 93, 213001

    Article  Google Scholar 

  36. Mazziotti D.A. (2004): First-order semidefinite programming for the direct determination of two-electron reduced density matrices with application to many-electron atoms and molecules. J. Chem. Phys. 121, 10957–10966

    Article  Google Scholar 

  37. Mazziotti D.A., Erdahl R.M. (2001): Uncertainty relations and reduced density matrices: Mapping many-body quantum mechanics onto four particles. Phys. Rev. A63, 042113

    Article  Google Scholar 

  38. McRae W.B., Davidson E.R. (1972): Linear inequalities for density matrices II. J. Math. Phys. 13, 1527–1538

    Article  Google Scholar 

  39. Mihailović M.V., Rosina M. (1975): The variational approach to the density matrix for light nuclei. Nucl. Phys. A237: 221–228

    Article  Google Scholar 

  40. Monteiro R.D.C. (2003): First- and second-order methods for semidefinite programming. Math. Prog. B97, 209–244

    MathSciNet  MATH  Google Scholar 

  41. Nakata M., Nakatsuji H., Ehara M., Fukuda M., Nakata K., Fujisawa K. (2001): Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm. J. Chem. Phys. 114, 8282–8292

    Article  Google Scholar 

  42. Nakata M., Ehara M., Nakatsuji H. (2002): Density matrix variational theory: Application to the potential energy surfaces and strongly correlated systems. J. Chem. Phys. 116, 5432–5439

    Article  Google Scholar 

  43. Nakata M., Ehara M., Nakatsuji H. (2003): Density matrix variational theory: Strength of Weinhold–Wilson inequalities. In: Löwdin P.-O., Kryachko E.S (eds) Fundamental World of Quantum Chemistry. Kluwer Academic, Boston, pp. 543–557

    Google Scholar 

  44. Nakatsuji H. (1976): Equation for direct determination of density matrix. Phys. Rev. A14, 41–50

    Article  Google Scholar 

  45. Nakatsuji, H.: Density equation theory in chemical physics. In: [3], pp. 85–116

  46. Nayakkankuppam, M.V.: Optimization over symmetric cones. Ph.D. Thesis, Department of Computer Science, New York University, New York (1999)

  47. Rosina M., Garrod C. (1975): The variational calculation of reduced density matrices. J. Comput. Phys. 18, 300-310

    Article  MathSciNet  Google Scholar 

  48. Schmidt M.W., Baldridge K.K., Boatz J.A., Elbert S.T., Gordon M.S., Jensen J.H., Koseki S., Matsunaga N., Nguyen K.A., Su S.J., Windus T.L., Dupuis M., Montgomery J.A. (1993): General atomic and molecular electronic structure system. J. Comput. Chem. 14, 1347-1363

    Article  Google Scholar 

  49. Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653 (1999). http://sedumi.mcmaster.ca/

  50. Szabo A., Ostlund N.S. (1996): Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory. Dover Publications Inc., Mineola, New York

    Google Scholar 

  51. Todd M.J. (2001): Semidefinite optimization. Acta Numer. 10, 515–560

    Article  MathSciNet  MATH  Google Scholar 

  52. Toh K.-C. (2003): Solving large scale semidefinite programs via an iterative solver on the augmented systems. SIAM J. Optim. 14, 670–698

    Article  MathSciNet  MATH  Google Scholar 

  53. Toh, K.-C., Tütüncü, R.H., Todd, M.J.: On the implementation of SDPT3 (version 3.1) – a Matlab software package for semidefinite-quadratic-linear programming. In: IEEE Conference on Computer-Aided Control System Design (2004)

  54. Valdemoro C. (1992): Approximating the 2nd-order reduced density-matrix in terms of the 1st-order one. Phys. Rev. A45, 4462–4467

    Article  Google Scholar 

  55. Valdemoro, C., Tel, L.M., Pérez-Romero, E.: Critical questions concerning iterative solution of the contracted Schrödinger equation. In: [3], pp. 117–138

  56. Wolkowicz H., Saigal R., Vandenberghe L. (2000): Handbook of Semidefinite Programming: Theory, Algorithms, and Applications. Kluwer Academic, Boston

    Google Scholar 

  57. Yamashita, M., Fujisawa, K., Kojima, M.: SDPARA : SemiDefiniteProgramming Algorithm paRAllel version. Parallel Comput. 29, 1053–1067 (2003). http://grid.r.dendai.ac.jp/sdpa/

  58. Yasuda K. (1999) Direct determination of the quantum-mechanical density matrix: Parquet theory. Phys. Rev. A59, 4133–4149

    Article  Google Scholar 

  59. Yasuda K., Nakatsuji H. (1997): Direct determination of the quantum-mechanical density matrix using the density equation II. Phys. Rev. A56, 2648–2657

    Article  Google Scholar 

  60. Zhao, Z.: The reduced density matrix method for electronic structure calculation – application of semidefinite programming to N-fermion systems. Ph.D. Thesis, Department of Physics, New York University, New York, (2004)

  61. Zhao Z., Braams B.J., Fukuda M., Overton M.L., Percus J.K. (2004): The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions. J. Chem. Phys. 120, 2095–2104

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-29 Oh-okayama, Meguro-ku, Tokyo, 152-8552, Japan

    Mituhiro Fukuda

  2. Department of Mathematics and Computer Science, Emory University, Atlanta, GA, 30322, USA

    Bastiaan J. Braams

  3. Department of Applied Chemistry, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan

    Maho Nakata

  4. Department of Computer Science, Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Michael L. Overton

  5. Courant Institute of Mathematical Sciences and Department of Physics, New York University, New York, NY, 10012, USA

    Jerome K. Percus

  6. Department of Information Systems Creation, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama-shi, Kanagawa, 221-8686, Japan

    Makoto Yamashita

  7. High Performance Computing Research Department, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Mail Stop 50F1650, Berkeley, CA, 94720, USA

    Zhengji Zhao

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  1. Mituhiro Fukuda
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  2. Bastiaan J. Braams
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  3. Maho Nakata
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  4. Michael L. Overton
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  5. Jerome K. Percus
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  6. Makoto Yamashita
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  7. Zhengji Zhao
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Correspondence to Mituhiro Fukuda.

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In memory of Jos Sturm who made many contributions to the theory and practice of semidefinite programming, including the widely used SeDuMi software package, and whose tragic early death is a great loss to our community.

The work of Mituhiro Fukuda was primarily conducted at the Courant Institute of Mathematical Sciences, New York University.

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Fukuda, M., Braams, B.J., Nakata, M. et al. Large-scale semidefinite programs in electronic structure calculation. Math. Program. 109, 553–580 (2007). https://doi.org/10.1007/s10107-006-0027-y

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