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.

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] Convergence enhancement in the iterative solution of the second-order contracted Schrödinger equation. Int. J. Quantum Chem. 102 (2005) 620-628.

, , , and ,[2] Invariance of the cumulant expansion under 1-particle unitary transformations in reduced density matrix theory. Chem. Phys. Lett. 387 (2004) 485-489.

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

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

and ,[5] Nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Program. Ser. B 95 (2003) 329-357. | Zbl 1030.90077

and ,[6] Local minima and convergence in low-rank semidefinite programming. Math. Program. Ser. A 103 (2005) 427-444. | Zbl 1099.90040

and ,[7] Hierarchy equations for reduced density matrices, Phys. Rev. A 13 (1976) 927-930.

and ,[8] Structure of fermion density matrices. Rev. Mod. Phys. 35 (1963) 668. | MR 155637

,[9] Reduced Density Matrices: Coulson's Challenge. Springer-Verlag, New York (2000). | Zbl 0998.81506

and ,[10] Approximating q-order reduced density-matrices in terms of the lower-order ones. 2. Applications. Phys. Rev. A 47 (1993) 979-985.

and ,[11] Self-consistent approximate solution of the 2nd-order contracted Schrödinger equation. Int. J. Quantum Chem. 51 (1994) 369-388.

and ,[12] Trust-Region Methods. SIAM: Philadelphia (2000). | MR 1774899 | Zbl 0958.65071

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

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

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

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

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

,[18] Large-scale semidefinite programs in electronic structure calculation. Math. Program., Ser. B 109 (2007) 553. | MR 2296564 | Zbl 1278.90495 | Zbl pre05131069

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

and ,[20] 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.

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

and ,[22] Application of variational reduced-density-matrix theory to the potential energy surfaces of the nitrogen and carbon dimers. J. Chem. Phys. 122 (2005) 194104.

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

and ,[24] 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.

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

and ,[26] Variational two-electron reduced-density-matrix theory: Partial 3-positivity conditions for $N$-representability. Phys. Rev. A 71 (2005) 062503.

and ,[27] Variational reduced-density-matrix calculations on radicals: a new approach to open-shell ab initio quantum chemistry. Phys. Rev. A 73 (2006) 012509.

and ,[28] Variational reduced-density-matrix calculation of the one-dimensional Hubbard model. Phys. Rev. A 73 (2006) 062505.

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

,[30] Perturbation theory corrections to the two-particle reduced density matrix variational method. J. Chem. Phys. 121 (2004) 1201-1205.

and ,[31] Irreducible Brillouin conditions and contracted Schrödinger equations for $n$-electron systems. IV. Perturbative analysis. J. Chem. Phys. (2004) 120 7350-7368.

and ,[32] 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] Electron correlation. Phys. Rev. 100 (1955) 1579-1586. | Zbl 0066.44602

,[34] Contracted Schrödinger equation: Determining quantum energies and two-particle density matrices without wave functions. Phys. Rev. A 57 (1998) 4219-4234.

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

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

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

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

,[39] 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] Variational minimization of atomic and molecular ground-state energies via the two-particle reduced density matrix. Phys. Rev. A 65 (2002) 062511.

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

,[43] 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] 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] Variational reduced-density-matrix method using three-particle $N$-representability conditions with application to many-electron molecules. Phys. Rev. A 74 (2006) 032501.

,[46] 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] Uncertainty relations and reduced density matrices: Mapping many-body quantum mechanics onto four particles. Phys. Rev. A 63 (2001) 042113.

and ,[48] Excitations as ground-state variational parameters. Nucl. Phys. A130 (1969) 386.

and ,[49] Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm. J. Chem. Phys. 114 (2001) 8282-8292.

, , , , and ,[50] Density matrix variational theory: Application to the potential energy surfaces and strongly correlated systems. J. Chem. Phys. 116 (2002) 5432-5439.

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

,[52] Direct determination of the quantum-mechanical density matrix using the density equation. Phys. Rev. Lett. 76 (1996) 1039-1042.

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

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

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

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

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

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

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

,[60] Direct determination of the quantum-mechanical density matrix using the density equation II. Phys. Rev. A 56 (1997) 2648-2657.

, and ,[61] The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions. J. Chem. Phys. 120 (2004) 2095-2104.

, , , and ,