Free access
Issue
ESAIM: M2AN
Volume 41, Number 2, March-April 2007
Special issue on Molecular Modelling
Page(s) 249 - 259
DOI http://dx.doi.org/10.1051/m2an:2007021
Published online 16 June 2007
  1. D.R. Alcoba, F.J. Casquero, L.M. Tel, E. Perez-Romero and C. Valdemoro, Convergence enhancement in the iterative solution of the second-order contracted Schrödinger equation. Int. J. Quantum Chem. 102 (2005) 620–628. [CrossRef]
  2. M.D. Benayoun, A.Y. Lu and D.A. Mazziotti, Invariance of the cumulant expansion under 1-particle unitary transformations in reduced density matrix theory. Chem. Phys. Lett. 387 (2004) 485–489. [CrossRef]
  3. D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982).
  4. S. Burer and C. Choi, Computational enhancements in low-rank semidefinite programming. Optim. Methods Soft. 21 (2006) 493–512. [CrossRef]
  5. S. Burer and R.D.C. Monteiro, Nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Program. Ser. B 95 (2003) 329–357. [CrossRef]
  6. S. Burer and R.D.C. Monteiro, Local minima and convergence in low-rank semidefinite programming. Math. Program. Ser. A 103 (2005) 427–444. [CrossRef]
  7. L. Cohen and C. Frishberg, Hierarchy equations for reduced density matrices, Phys. Rev. A 13 (1976) 927–930.
  8. A.J. Coleman, Structure of fermion density matrices. Rev. Mod. Phys. 35 (1963) 668. [CrossRef]
  9. A.J. Coleman and V.I. Yukalov, Reduced Density Matrices: Coulson's Challenge. Springer-Verlag, New York (2000).
  10. F. Colmenero and C. Valdemoro, Approximating q-order reduced density-matrices in terms of the lower-order ones. 2. Applications. Phys. Rev. A 47 (1993) 979–985. [CrossRef] [PubMed]
  11. F. Colmenero and C. Valdemoro, Self-consistent approximate solution of the 2nd-order contracted Schrödinger equation. Int. J. Quantum Chem. 51 (1994) 369–388. [CrossRef]
  12. A.R. Conn, I.M. Gould and P.L. Toint, Trust-Region Methods. SIAM: Philadelphia (2000).
  13. C.A. Coulson, Present state of molecular structure calculations. Rev. Mod. Phys. 32 (1960) 170–177. [CrossRef] [MathSciNet]
  14. R.M. Erdahl, Representability. Int. J. Quantum Chem. 13 (1978) 697–718. [CrossRef]
  15. R.M. Erdahl, Two algorithms for the lower bound method of reduced density matrix theory. Reports Math. Phys. 15 (1979) 147–162. [CrossRef]
  16. R.M. Erdahl and B. Jin, The lower bound method for reduced density matrices. J. Mol. Struc. (Theochem) 527 (2000) 207–220. [CrossRef]
  17. R. Fletcher, Practical Methods of Optimization. John Wiley and Sons, New York (1987).
  18. M. Fukuda, B.J. Braams, M. Nakata, M.L. Overton, J.K. Percus, M. Yamashita and Z. Zhao, Large-scale semidefinite programs in electronic structure calculation. Math. Program., Ser. B 109 (2007) 553.
  19. C. Garrod and J. Percus, Reduction of N-particle variational problem. J. Math. Phys. 5 (1964) 1756–1776. [CrossRef]
  20. G. Gidofalvi and D.A. Mazziotti, Boson correlation energies via variational minimization with the two-particle reduced density matrix: Exact N-representability conditions for harmonic interactions. Phys. Rev. A 69 (2004) 042511. [CrossRef]
  21. G. Gidofalvi and D.A. Mazziotti, Application of variational reduced-density-matrix theory to organic molecules. J. Chem. Phys. 122 (2005) 094107. [CrossRef] [PubMed]
  22. G. Gidofalvi and D.A. Mazziotti, Application of variational reduced-density-matrix theory to the potential energy surfaces of the nitrogen and carbon dimers. J. Chem. Phys. 122 (2005) 194104. [CrossRef] [PubMed]
  23. G. Gidofalvi and D.A. Mazziotti, Spin- and symmetry-adapted two-electron reduced-density-matrix theory. Phys. Rev. A 72 (2005) 052505. [CrossRef]
  24. G. Gidofalvi and D.A. Mazziotti, Potential energy surface of carbon monoxide in the presence and absence of an electric field using the two-electron reduced-density-matrix method. J. Phys. Chem. A 110 (2006) 5481–5486. [CrossRef] [PubMed]
  25. G. Gidofalvi and D.A. Mazziotti, Computation of quantum phase transitions by reduced-density-matrix mechanics. Phys. Rev. A 74 (2006) 012501. [CrossRef]
  26. J.R. Hammond and D.A. Mazziotti, Variational two-electron reduced-density-matrix theory: Partial 3-positivity conditions for N-representability. Phys. Rev. A 71 (2005) 062503. [CrossRef]
  27. J.R. Hammond and D.A. Mazziotti, Variational reduced-density-matrix calculations on radicals: a new approach to open-shell ab initio quantum chemistry. Phys. Rev. A 73 (2006) 012509. [CrossRef]
  28. J.R. Hammond and D.A. Mazziotti, Variational reduced-density-matrix calculation of the one-dimensional Hubbard model. Phys. Rev. A 73 (2006) 062505. [CrossRef]
  29. J.E. Harriman, Geometry of density matrices. II. Reduced density matrices and N-representability. Phys. Rev. A 17 (1978) 1257–1268. [CrossRef]
  30. T. Juhász and D.A. Mazziotti, Perturbation theory corrections to the two-particle reduced density matrix variational method. J. Chem. Phys. 121 (2004) 1201–1205. [CrossRef] [PubMed]
  31. W. Kutzelnigg and D. Mukherjee, Irreducible Brillouin conditions and contracted Schrödinger equations for n-electron systems. IV. Perturbative analysis. J. Chem. Phys. (2004) 120 7350–7368.
  32. P.O. Löwdin, Quantum theory of many-particle systems. 1. Physical interpretations by means of density matrices, natural spin-orbitals, and convergence problems in the method of configuration interaction. Phys. Rev. 97 (1955) 1474–1489. [CrossRef] [MathSciNet]
  33. J.E. Mayer, Electron correlation. Phys. Rev. 100 (1955) 1579–1586. [CrossRef]
  34. D.A. Mazziotti, Contracted Schrödinger equation: Determining quantum energies and two-particle density matrices without wave functions. Phys. Rev. A 57 (1998) 4219–4234. [CrossRef]
  35. D.A. Mazziotti, Approximate solution for electron correlation through the use of Schwinger probes. Chem. Phys. Lett. 289 (1998) 419–427. [CrossRef]
  36. D.A. Mazziotti, Pursuit of N-representability for the contracted Schrödinger equation through density-matrix reconstruction. Phys. Rev. A 60 (1999) 3618–3626. [CrossRef]
  37. D.A. Mazziotti, Comparison of contracted Schrödinger and coupled-cluster theories. Phys. Rev. A 60 (1999) 4396–4408. [CrossRef]
  38. D.A. Mazziotti, Correlated purification of reduced density matrices. Phys. Rev. E 65 (2002) 026704. [CrossRef]
  39. D.A. Mazziotti, A variational method for solving the contracted Schrödinger equation through a projection of the N-particle power method onto the two-particle space. J. Chem. Phys. 116 (2002) 1239–1249. [CrossRef]
  40. D.A. Mazziotti, Variational minimization of atomic and molecular ground-state energies via the two-particle reduced density matrix. Phys. Rev. A 65 (2002) 062511. [CrossRef]
  41. D.A. Mazziotti, Solution of the 1,3-contracted Schrödinger equation through positivity conditions on the 2-particle reduced density matrix. Phys. Rev. A 66 (2002) 062503. [CrossRef]
  42. D.A. Mazziotti, Realization of quantum chemistry without wavefunctions through first-order semidefinite programming. Phys. Rev. Lett. 93 (2004) 213001. [CrossRef] [PubMed]
  43. D.A. Mazziotti, 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 (2004) 10957–10966. [CrossRef] [PubMed]
  44. D.A. Mazziotti, Variational two-electron reduced-density-matrix theory for many-electron atoms and molecules: Implementation of the spin- and symmetry-adapted T2 condition through first-order semidefinite programming. Phys. Rev. A 72 (2005) 032510. [CrossRef]
  45. D.A. Mazziotti, Variational reduced-density-matrix method using three-particle N-representability conditions with application to many-electron molecules. Phys. Rev. A 74 (2006) 032501. [CrossRef]
  46. D.A. Mazziotti, Reduced-Density-Matrix with Application to Many-electron Atoms and Molecules, Advances in Chemical Physics 134, D.A. Mazziotti Ed., John Wiley and Sons, New York (2007).
  47. D.A. Mazziotti and R.M. Erdahl, Uncertainty relations and reduced density matrices: Mapping many-body quantum mechanics onto four particles. Phys. Rev. A 63 (2001) 042113. [CrossRef]
  48. M.V. Mihailović and M. Rosina, Excitations as ground-state variational parameters. Nucl. Phys. A130 (1969) 386.
  49. M. Nakata, H. Nakatsuji, M. Ehara, M. Fukuda, K. Nakata and K. Fujisawa, Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm. J. Chem. Phys. 114 (2001) 8282–8292. [CrossRef]
  50. M. Nakata, M. Ehara and H. Nakatsuji, Density matrix variational theory: Application to the potential energy surfaces and strongly correlated systems. J. Chem. Phys. 116 (2002) 5432–5439. [CrossRef]
  51. H. Nakatsuji, Equation for the direct determination of the density matrix. Phys. Rev. A 14 (1976) 41–50. [CrossRef]
  52. H. Nakatsuji and K. Yasuda, Direct determination of the quantum-mechanical density matrix using the density equation. Phys. Rev. Lett. 76 (1996) 1039–1042. [CrossRef] [PubMed]
  53. M. Nayakkankuppam, Solving large-scale semidefinite programs in parallel. Math. Program., Ser. B 109 (2007) 477–504.
  54. Y. Nesterov and A.S. Nemirovskii, Interior Point Polynomial Method in Convex Programming: Theory and Applications. SIAM: Philadelphia (1993).
  55. E. Polak, Optimization: Algorithms and Consistent Approximations. Springer-Verlag, New York (1997).
  56. J.H. Sebold and J.K. Percus, Model derived reduced density matrix restrictions for correlated fermions. J. Chem. Phys. 104 (1996) 6606–6612. [CrossRef]
  57. R.H. Tredgold, Density matrix and the many-body problem. Phys. Rev. 105 (1957) 1421–1423. [CrossRef] [MathSciNet]
  58. L. Vandenberghe and S. Boyd, Semidefinite programming. SIAM Rev. 38 (1996) 49–95. [CrossRef] [MathSciNet]
  59. S. Wright, Primal-Dual Interior-Point Methods. SIAM, Philadelphia (1997).
  60. K. Yasuda, and H. Nakatsuji, Direct determination of the quantum-mechanical density matrix using the density equation II. Phys. Rev. A 56 (1997) 2648–2657. [CrossRef]
  61. Z. Zhao, B.J. Braams, H. Fukuda, M.L. Overton and J.K. Percus, The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions. J. Chem. Phys. 120 (2004) 2095–2104. [CrossRef] [PubMed]

Recommended for you