First-order semidefinite programming for the two-electron treatment of many-electron atoms and molecules
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 2, p. 249-259

The ground-state energy and properties of any many-electron atom or molecule may be rigorously computed by variationally computing the two-electron reduced density matrix rather than the many-electron wavefunction. While early attempts fifty years ago to compute the ground-state 2-RDM directly were stymied because the 2-RDM must be constrained to represent an $N$-electron wavefunction, recent advances in theory and optimization have made direct computation of the 2-RDM possible. The constraints in the variational calculation of the 2-RDM require a special optimization known as a semidefinite programming. Development of first-order semidefinite programming for the 2-RDM method has reduced the computational costs of the calculation by orders of magnitude [Mazziotti, Phys. Rev. Lett. 93 (2004) 213001]. The variational 2-RDM approach is effective at capturing multi-reference correlation effects that are especially important at non-equilibrium molecular geometries. Recent work on 2-RDM methods will be reviewed and illustrated with particular emphasis on the importance of advances in large-scale semidefinite programming.

DOI : https://doi.org/10.1051/m2an:2007021
Classification:  90C22,  81Q05,  52A40
Keywords: semidefinite programming, electron correlation, reduced density matrices, $N$-representability conditions
@article{M2AN_2007__41_2_249_0,
author = {Mazziotti, David A.},
title = {First-order semidefinite programming for the two-electron treatment of many-electron atoms and molecules},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {41},
number = {2},
year = {2007},
pages = {249-259},
doi = {10.1051/m2an:2007021},
zbl = {1135.81378},
mrnumber = {2339627},
language = {en},
url = {http://www.numdam.org/item/M2AN_2007__41_2_249_0}
}

Mazziotti, David A. First-order semidefinite programming for the two-electron treatment of many-electron atoms and molecules. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 2, pp. 249-259. doi : 10.1051/m2an:2007021. http://www.numdam.org/item/M2AN_2007__41_2_249_0/

[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.

[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.

[3] D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982). | MR 690767 | Zbl 0572.90067

[4] S. Burer and C. Choi, Computational enhancements in low-rank semidefinite programming. Optim. Methods Soft. 21 (2006) 493-512. | Zbl 1136.90429

[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. | Zbl 1030.90077

[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. | Zbl 1099.90040

[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. | MR 155637

[9] A.J. Coleman and V.I. Yukalov, Reduced Density Matrices: Coulson's Challenge. Springer-Verlag, New York (2000). | Zbl 0998.81506

[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.

[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.

[12] A.R. Conn, I.M. Gould and P.L. Toint, Trust-Region Methods. SIAM: Philadelphia (2000). | MR 1774899 | Zbl 0958.65071

[13] C.A. Coulson, Present state of molecular structure calculations. Rev. Mod. Phys. 32 (1960) 170-177.

[14] R.M. Erdahl, Representability. Int. J. Quantum Chem. 13 (1978) 697-718.

[15] R.M. Erdahl, Two algorithms for the lower bound method of reduced density matrix theory. Reports Math. Phys. 15 (1979) 147-162. | Zbl 0441.49056

[16] R.M. Erdahl and B. Jin, The lower bound method for reduced density matrices. J. Mol. Struc. (Theochem) 527 (2000) 207-220.

[17] R. Fletcher, Practical Methods of Optimization. John Wiley and Sons, New York (1987). | MR 955799 | Zbl 0905.65002

[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. | MR 2296564 | Zbl 1278.90495 | Zbl pre05131069

[19] C. Garrod and J. Percus, Reduction of N-particle variational problem. J. Math. Phys. 5 (1964) 1756-1776. | Zbl 0129.44401

[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.

[21] G. Gidofalvi and D.A. Mazziotti, Application of variational reduced-density-matrix theory to organic molecules. J. Chem. Phys. 122 (2005) 094107.

[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.

[23] G. Gidofalvi and D.A. Mazziotti, Spin- and symmetry-adapted two-electron reduced-density-matrix theory. Phys. Rev. A 72 (2005) 052505.

[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.

[25] G. Gidofalvi and D.A. Mazziotti, Computation of quantum phase transitions by reduced-density-matrix mechanics. Phys. Rev. A 74 (2006) 012501.

[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.

[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.

[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.

[29] J.E. Harriman, Geometry of density matrices 17 (1978) 1257-1268.

[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.

[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. | MR 69061 | Zbl 0065.44907

[33] J.E. Mayer, Electron correlation. Phys. Rev. 100 (1955) 1579-1586. | Zbl 0066.44602

[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.

[35] D.A. Mazziotti, Approximate solution for electron correlation through the use of Schwinger probes. Chem. Phys. Lett. 289 (1998) 419-427.

[36] D.A. Mazziotti, Pursuit of N-representability for the contracted Schrödinger equation through density-matrix reconstruction. Phys. Rev. A 60 (1999) 3618-3626.

[37] D.A. Mazziotti, Comparison of contracted Schrödinger and coupled-cluster theories. Phys. Rev. A 60 (1999) 4396-4408.

[38] D.A. Mazziotti, Correlated purification of reduced density matrices. Phys. Rev. E 65 (2002) 026704.

[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.

[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.

[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.

[42] D.A. Mazziotti, Realization of quantum chemistry without wavefunctions through first-order semidefinite programming. Phys. Rev. Lett. 93 (2004) 213001.

[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.

[44] D.A. Mazziotti, Variational two-electron reduced-density-matrix theory for many-electron atoms and molecules: Implementation of the spin- and symmetry-adapted T${}_{2}$ condition through first-order semidefinite programming. Phys. Rev. A 72 (2005) 032510.

[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.

[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.

[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.

[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.

[51] H. Nakatsuji, Equation for the direct determination of the density matrix. Phys. Rev. A 14 (1976) 41-50.

[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.

[53] M. Nayakkankuppam, Solving large-scale semidefinite programs in parallel. Math. Program., Ser. B 109 (2007) 477-504. | MR 2295152 | Zbl 1278.90301 | Zbl pre05131074

[54] Y. Nesterov and A.S. Nemirovskii, Interior Point Polynomial Method in Convex Programming: Theory and Applications. SIAM: Philadelphia (1993). | MR 1258086

[55] E. Polak, Optimization: Algorithms and Consistent Approximations. Springer-Verlag, New York (1997). | MR 1454128 | Zbl 0899.90148

[56] J.H. Sebold and J.K. Percus, Model derived reduced density matrix restrictions for correlated fermions. J. Chem. Phys. 104 (1996) 6606-6612.

[57] R.H. Tredgold, Density matrix and the many-body problem. Phys. Rev. 105 (1957) 1421-1423. | MR 91143

[58] L. Vandenberghe and S. Boyd, Semidefinite programming. SIAM Rev. 38 (1996) 49-95. | MR 1379041 | Zbl 0845.65023

[59] S. Wright, Primal-Dual Interior-Point Methods. SIAM, Philadelphia (1997). | MR 1422257 | Zbl 0863.65031

[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.

[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.