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First-order semidefinite programming for the two-electron treatment ofmany-electron atoms and molecules

Published online by Cambridge University Press:  16 June 2007

David A. Mazziotti
David A. Mazziotti*
Affiliation:
Department of Chemistry and the James Franck Institute, The University of Chicago, Chicago, IL 60637, USA. damazz@uchicago.edu
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Abstract


The ground-state energy and properties of any many-electron atom ormolecule may be rigorously computed by variationally computing thetwo-electron reduced density matrix rather than the many-electronwavefunction. While early attempts fifty years ago to compute theground-state 2-RDM directly were stymied because the 2-RDM must beconstrained to represent an N-electron wavefunction, recentadvances in theory and optimization have made direct computation ofthe 2-RDM possible. The constraints in the variational calculationof the 2-RDM require a special optimization known as a semidefiniteprogramming. Development of first-order semidefinite programmingfor the 2-RDM method has reduced the computational costs of thecalculation by orders of magnitude [Mazziotti, Phys. Rev. Lett.93 (2004) 213001]. The variational 2-RDM approach is effective atcapturing multi-reference correlation effects that are especiallyimportant at non-equilibrium molecular geometries. Recent work on2-RDM methods will be reviewed and illustrated with particularemphasis on the importance of advances in large-scale semidefiniteprogramming.


Keywords

Semidefinite programmingelectron correlationreduced density matricesN-representability conditions.

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Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 2: Special issue on Molecular Modelling , March 2007, pp. 249 - 259
Copyright
© EDP Sciences, SMAI, 2007

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References

Alcoba, D.R., Casquero, F.J., Tel, L.M., Perez-Romero, E. and Valdemoro, C., Convergence enhancement in the iterative solution of the second-order contracted Schrödinger equation. Int. J. Quantum Chem. 102 (2005) 620–628. CrossRef
Benayoun, M.D., Lu, A.Y. and Mazziotti, D.A., Invariance of the cumulant expansion under 1-particle unitary transformations in reduced density matrix theory. Chem. Phys. Lett. 387 (2004) 485–489. CrossRef
D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982).
Burer, S. and Choi, C., Computational enhancements in low-rank semidefinite programming. Optim. Methods Soft. 21 (2006) 493–512. CrossRef
Burer, S. and Monteiro, R.D.C., Nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Program. Ser. B 95 (2003) 329–357. CrossRef
Burer, S. and Monteiro, R.D.C., Local minima and convergence in low-rank semidefinite programming. Math. Program. Ser. A 103 (2005) 427–444. CrossRef
L. Cohen and C. Frishberg, Hierarchy equations for reduced density matrices, Phys. Rev. A 13 (1976) 927–930.
Coleman, A.J., Structure of fermion density matrices. Rev. Mod. Phys. 35 (1963) 668. CrossRef
A.J. Coleman and V.I. Yukalov, Reduced Density Matrices: Coulson's Challenge. Springer-Verlag, New York (2000).
Colmenero, F. and Valdemoro, C., Approximating q-order reduced density-matrices in terms of the lower-order ones. 2. Applications. Phys. Rev. A 47 (1993) 979–985. CrossRef
Colmenero, F. and Valdemoro, C., Self-consistent approximate solution of the 2nd-order contracted Schrödinger equation. Int. J. Quantum Chem. 51 (1994) 369–388. CrossRef
A.R. Conn, I.M. Gould and P.L. Toint, Trust-Region Methods. SIAM: Philadelphia (2000).
Coulson, C.A., Present state of molecular structure calculations. Rev. Mod. Phys. 32 (1960) 170–177. CrossRef
Erdahl, R.M., Representability. Int. J. Quantum Chem. 13 (1978) 697–718. CrossRef
Erdahl, R.M., Two algorithms for the lower bound method of reduced density matrix theory. Reports Math. Phys. 15 (1979) 147–162. CrossRef
Erdahl, R.M. and Jin, B., The lower bound method for reduced density matrices. J. Mol. Struc. (Theochem) 527 (2000) 207–220. CrossRef
R. Fletcher, Practical Methods of Optimization. John Wiley and Sons, New York (1987).
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.
Garrod, C. and Percus, J., Reduction of N-particle variational problem. J. Math. Phys. 5 (1964) 1756–1776. CrossRef
Gidofalvi, G. and Mazziotti, D.A., 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
Gidofalvi, G. and Mazziotti, D.A., Application of variational reduced-density-matrix theory to organic molecules. J. Chem. Phys. 122 (2005) 094107. CrossRef
Gidofalvi, G. and Mazziotti, D.A., 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
Gidofalvi, G. and Mazziotti, D.A., Spin- and symmetry-adapted two-electron reduced-density-matrix theory. Phys. Rev. A 72 (2005) 052505. CrossRef
Gidofalvi, G. and Mazziotti, D.A., 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
Gidofalvi, G. and Mazziotti, D.A., Computation of quantum phase transitions by reduced-density-matrix mechanics. Phys. Rev. A 74 (2006) 012501. CrossRef
Hammond, J.R. and Mazziotti, D.A., Variational two-electron reduced-density-matrix theory: Partial 3-positivity conditions for N-representability. Phys. Rev. A 71 (2005) 062503. CrossRef
Hammond, J.R. and Mazziotti, D.A., Variational reduced-density-matrix calculations on radicals: a new approach to open-shell ab initio quantum chemistry. Phys. Rev. A 73 (2006) 012509. CrossRef
Hammond, J.R. and Mazziotti, D.A., Variational reduced-density-matrix calculation of the one-dimensional Hubbard model. Phys. Rev. A 73 (2006) 062505. CrossRef
Harriman, J.E., Geometry of density matrices. II. Reduced density matrices and N-representability. Phys. Rev. A 17 (1978) 1257–1268. CrossRef
Juhász, T. and Mazziotti, D.A., Perturbation theory corrections to the two-particle reduced density matrix variational method. J. Chem. Phys. 121 (2004) 1201–1205. CrossRef
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.
Löwdin, P.O., 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
Mayer, J.E., Electron correlation. Phys. Rev. 100 (1955) 1579–1586. CrossRef
Mazziotti, D.A., Contracted Schrödinger equation: Determining quantum energies and two-particle density matrices without wave functions. Phys. Rev. A 57 (1998) 4219–4234. CrossRef
Mazziotti, D.A., Approximate solution for electron correlation through the use of Schwinger probes. Chem. Phys. Lett. 289 (1998) 419–427. CrossRef
Mazziotti, D.A., Pursuit of N-representability for the contracted Schrödinger equation through density-matrix reconstruction. Phys. Rev. A 60 (1999) 3618–3626. CrossRef
Mazziotti, D.A., Comparison of contracted Schrödinger and coupled-cluster theories. Phys. Rev. A 60 (1999) 4396–4408. CrossRef
Mazziotti, D.A., Correlated purification of reduced density matrices. Phys. Rev. E 65 (2002) 026704. CrossRef
Mazziotti, D.A., 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
Mazziotti, D.A., Variational minimization of atomic and molecular ground-state energies via the two-particle reduced density matrix. Phys. Rev. A 65 (2002) 062511. CrossRef
Mazziotti, D.A., 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
Mazziotti, D.A., Realization of quantum chemistry without wavefunctions through first-order semidefinite programming. Phys. Rev. Lett. 93 (2004) 213001. CrossRef
Mazziotti, D.A., 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
Mazziotti, D.A., 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
Mazziotti, D.A., Variational reduced-density-matrix method using three-particle N-representability conditions with application to many-electron molecules. Phys. Rev. A 74 (2006) 032501. CrossRef
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).
Mazziotti, D.A. and Erdahl, R.M., Uncertainty relations and reduced density matrices: Mapping many-body quantum mechanics onto four particles. Phys. Rev. A 63 (2001) 042113. CrossRef
Mihailović, M.V. and Rosina, M., Excitations as ground-state variational parameters. Nucl. Phys. A130 (1969) 386. CrossRef
Nakata, M., Nakatsuji, H., Ehara, M., Fukuda, M., Nakata, K. and Fujisawa, K., Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm. J. Chem. Phys. 114 (2001) 8282–8292. CrossRef
Nakata, M., Ehara, M. and Nakatsuji, H., Density matrix variational theory: Application to the potential energy surfaces and strongly correlated systems. J. Chem. Phys. 116 (2002) 5432–5439. CrossRef
Nakatsuji, H., Equation for the direct determination of the density matrix. Phys. Rev. A 14 (1976) 41–50. CrossRef
Nakatsuji, H. and Yasuda, K., Direct determination of the quantum-mechanical density matrix using the density equation. Phys. Rev. Lett. 76 (1996) 1039–1042. CrossRef
M. Nayakkankuppam, Solving large-scale semidefinite programs in parallel. Math. Program., Ser. B 109 (2007) 477–504.
Y. Nesterov and A.S. Nemirovskii, Interior Point Polynomial Method in Convex Programming: Theory and Applications. SIAM: Philadelphia (1993).
E. Polak, Optimization: Algorithms and Consistent Approximations. Springer-Verlag, New York (1997).
Sebold, J.H. and Percus, J.K., Model derived reduced density matrix restrictions for correlated fermions. J. Chem. Phys. 104 (1996) 6606–6612. CrossRef
Tredgold, R.H., Density matrix and the many-body problem. Phys. Rev. 105 (1957) 1421–1423. CrossRef
Vandenberghe, L. and Boyd, S., Semidefinite programming. SIAM Rev. 38 (1996) 49–95. CrossRef
S. Wright, Primal-Dual Interior-Point Methods. SIAM, Philadelphia (1997).
Yasuda, K., and Nakatsuji, H., Direct determination of the quantum-mechanical density matrix using the density equation II. Phys. Rev. A 56 (1997) 2648–2657. CrossRef
Zhao, Z., Braams, B.J., Fukuda, H., Overton, M.L. and Percus, J.K., 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