David A. Mazziotti

Approximate solution for electron correlation through the use of Schwinger probes

Abstract Quantum energies and two-particle reduced density matrices (2-rdms) may be determined without using the N -particle wavefunction by combining the reconstruction of higher rdms from lower rdms and the contracted Schrodinger equation. In this letter we derive a systematic procedure for obtaining reconstruction functionals through the use of Schwinger probes. Previous functionals for the 3 and 4-rdms are generated as well as new functionals for higher rdms. Through a quasi-spin model we demonstrate that the 5-rdm, normalized to unity, may be reconstructed from lower rdms with the accuracy of its elements ranging from 10 −4 for five fermions to 10 −8 for fifty fermions.

Reduced-density-matrix mechanics with application to many-electron atoms and molecules

Part I. CHAPTER 1: N-REPRESENTABILITY (A. John Coleman). CHAPTER 2: HISTORICAL INTRODUCTION (Mitja Rosina). Part II. CHAPTER 3: VARIATIONAL TWO-ELECTRON REDUCED-DENSITY-MATRIX THEORY (David A. Mazziotti). CHAPTER 4: THE LOWER BOUND METHOD FOR DENSITY MATRICES AND SEMIDEFINITE PROGRAMMING (Robert M. Erdahl). CHAPTER 5: THE T1 AND T2 REPRESENTABILITY CONDITIONS (Bastiaan J. Braams, Jerome K. Percus, and Zhengji Zhao). CHAPTER 6: SEMIDEFINITE PROGRAMMING: FORMULATIONS AND PRIMAL-DUAL INTERIOR-POINT METHODS (Mituhiro Fukuda, Maho Nakata, and Makoto Yamashita). Part III. CHAPTER 7: THEORY AND METHODOLOGY OF THE CONTRACTED SCHRO DINGER EQUATION (By C. Valdemoro) CHAPTER 8: CONTRACTED SCHRO DINGER EQUATION (By David A. Mazziotti). CHAPTER 9: PURIFICATION OF CORRELATED REDUCED DENSITY MATRICES: REVIEW AND APPLICATIONS (By D. R. Alcoba). CHAPTER 10: CUMULANTS, EXTENSIVITY, AND THE CONNECTED FORMULATION OF THE CONTRACTED SCHRO DINGER EQUATION (By John M. Herbert and John E. Harriman). CHAPTER 11: GENERALIZED NORMAL ORDERING, IRREDUCIBLE BRILLOUIN CONDITIONS, AND CONTRACTED SCHRO DINGER EQUATIONS (By Werner Kutzelnigg and Debashis Mukherjee). CHAPTER 12: ANTI-HERMITIAN FORMULATION OF THE CONTRACTED SCHRO DINGER THEORY (By David A. Mazziotti). CHAPTER 13: CANONICAL TRANSFORMATION THEORY FOR DYNAMIC CORRELATIONS IN MULTIREFERENCE PROBLEMS (By Garnet Kin-Lic Chan and Takeshi Yanai). Part IV. CHAPTER 14: NATURAL ORBITAL FUNCTIONAL THEORY (By Mario Piris). CHAPTER 15: GEMINAL FUNCTIONAL THEORY (By B. C. Rinderspacher). CHAPTER 16: LINEAR INEQUALITIES FOR DIAGONAL ELEMENTS OF DENSITY MATRICES (By Paul W. Ayers and Ernest R. Davidson). Part V. CHAPTER 17: PARAMETERIZATION OF THE 2-RDM (A. John Coleman). CHAPTER 18: ENTANGLEMENT, ELECTRON CORRELATION, AND DENSITY MATRICES (By Sabre Kais). AUTHOR INDEX. SUBJECT INDEX.

Active-space two-electron reduced-density-matrix method: Complete active-space calculations without diagonalization of the N-electron Hamiltonian

Molecular systems in chemistry often have wave functions with substantial contributions from two-or-more electronic configurations. Because traditional complete-active-space self-consistent-field (CASSCF) methods scale exponentially with the number N of active electrons, their applicability is limited to small active spaces. In this paper we develop an active-space variational two-electron reduced-density-matrix (2-RDM) method in which the expensive diagonalization is replaced by a variational 2-RDM calculation where the 2-RDM is constrained by approximate N-representability conditions. Optimization of the constrained 2-RDM is accomplished by large-scale semidefinite programming [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Because the computational cost of the active-space 2-RDM method scales polynomially as r(a)(6) where r(a) is the number of active orbitals, the method can be applied to treat active spaces that are too large for conventional CASSCF. The active-space 2-RDM method performs two steps: (i) variational calculation of the 2-RDM in the active space and (ii) optimization of the active orbitals by Jacobi rotations. For large basis sets this two-step 2-RDM method is more efficient than the one-step, low-rank variational 2-RDM method [Gidofalvi and Mazziotti, J. Chem. Phys. 127, 244105 (2007)]. Applications are made to HF, H(2)O, and N(2) as well as n-acene chains for n=2-8. When n>4, the acenes cannot be treated by conventional CASSCF methods; for example, when n=8, CASSCF requires optimization over approximately 1.47x10(17) configuration state functions. The natural occupation numbers of the n-acenes show the emergence of bi- and polyradical character with increasing chain length.

Strong Correlation in Acene Sheets from the Active-Space Variational Two-Electron Reduced Density Matrix Method: Effects of Symmetry and Size

Polyaromatic hydrocarbons (PAHs) are a class of organic molecules with importance in several branches of science, including medicine, combustion chemistry, and materials science. The delocalized π-orbital systems in PAHs require highly accurate electronic structure methods to capture strong electron correlation. Treating correlation in PAHs has been challenging because (i) traditional wave function methods for strong correlation have not been applicable since they scale exponentially in the number of strongly correlated orbitals, and (ii) alternative methods such as the density-matrix renormalization group and variational two-electron reduced density matrix (2-RDM) methods have not been applied beyond linear acene chains. In this paper we extend the earlier results from active-space variational 2-RDM theory [Gidofalvi, G.; Mazziotti, D. A. J. Chem. Phys. 2008, 129, 134108] to the more general two-dimensional arrangement of rings--acene sheets--to study the relationship between geometry and electron correlation in PAHs. The acene-sheet calculations, if performed with conventional wave function methods, would require wave function expansions with as many as 1.5 × 10(17) configuration state functions. To measure electron correlation, we employ several RDM-based metrics: (i) natural-orbital occupation numbers, (ii) the 1-RDM von Neumann entropy, (iii) the correlation energy per carbon atom, and (iv) the squared Frobenius norm of the cumulant 2-RDM. The results confirm a trend of increasing polyradical character with increasing molecular size previously observed in linear PAHs and reveal a corresponding trend in two-dimensional (arch-shaped) PAHs. Furthermore, in PAHs of similar size they show significant variations in correlation with geometry. PAHs with the strictly linear geometry (chains) exhibit more electron correlation than PAHs with nonlinear geometries (sheets).

First-order semidefinite programming for the direct determination of two-electron reduced density matrices with application to many-electron atoms and molecules

Direct variational calculation of two-electron reduced density matrices (2-RDMs) for many-electron atoms and molecules in nonminimal basis sets has recently been achieved through the use of first-order semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. (in press)]. With semidefinite programming, the electronic ground-state energy of a molecule is minimized with respect to the 2-RDM subject to N-representability constraints known as positivity conditions. Here we present a detailed account of the first-order algorithm for semidefinite programming and its comparison with the primal-dual interior-point algorithms employed in earlier variational 2-RDM calculations. The first-order semidefinite-programming algorithm, computations show, offers an orders-of-magnitude reduction in floating-point operations and storage in comparison with previous implementations. We also examine the ability of the positivity conditions to treat strong correlation and multireference effects through an analysis of the Hamiltonians for which the conditions are exact. Calculations are performed in nonminimal basis sets for a variety of atoms and molecules and the potential-energy curves for CO and H(2)O.

Variational method for solving the contracted Schrödinger equation through a projection of the N-particle power method onto the two-particle space

The power method for solving N-particle eigenvalue equations is contracted onto the two-particle space to produce a reduced “variational” method for solving the contracted Schrodinger equation (CSE), also known as the density equation. In contrast to the methods which solve a system of approximate nonlinear equations to determine the two-particle reduced density matrix (2-RDM) nonvariationally, the contracted power method updates the 2-RDM iteratively through a “gradient” of the N-particle energy. After each power iteration we modify the 2-RDM to satisfy certain N-representability conditions through an extension of purification to correlated RDMs. The contracted power method is illustrated with a variety of molecules. Significant features of the present calculations include (i) accurate results for both first- and second-order functionals for building the 3- and the 4-RDM’s from the 2-RDM’s; (ii) the first molecular implementation of the Mazziotti correction within the CSE [Mazziotti, Phys. Rev. A 60, 3618 (1999)]; (iii) a spin–orbital formulation; (iv) the treatment of both core and valence orbitals as active; and; (v) a reduction of the CSE computational scaling through fast summation and the natural-orbital transformation.

Complete reconstruction of reduced density matrices

Abstract Different from traditional electronic structure methods, the contracted Schrodinger equation with reduced-density-matrix (RDM) reconstruction may be exact when only the 2-particle RDM is employed as the fundamental parameter. Although Rosinas theorem indicates that the 3 and the 4-RDMs are functionals of the 2-RDM, cumulant theory generates only those terms expressible as antisymmetrized products of lower RDMs. We present a formal solution for reconstruction where the approximate cumulant formulas are systematically corrected through contraction conditions. Using a part of the formal 3-RDM reconstruction, the CSE is compared with other methods through a quasi-spin model containing as many as eight-hundred fermions.

Structure of fermionic density matrices: complete N-representability conditions.

We present a constructive solution to the N-representability problem: a full characterization of the conditions for constraining the two-electron reduced density matrix to represent an N-electron density matrix. Previously known conditions, while rigorous, were incomplete. Here, we derive a hierarchy of constraints built upon (i) the bipolar theorem and (ii) tensor decompositions of model Hamiltonians. Existing conditions D, Q, G, T1, and T2, known classical conditions, and new conditions appear naturally. Subsets of the conditions are amenable to polynomial-time computations of strongly correlated systems.

Two-electron reduced density matrices from the anti-Hermitian contracted Schrodinger equation: enhanced energies and properties with larger basis sets.

Two-electron reduced density matrices (2-RDMs) have recently been directly determined from the solution of the anti-Hermitian contracted Schrodinger equation (ACSE) to obtain 95%-100% of the ground-state correlation energy of atoms and molecules, which significantly improves upon the accuracy of the contracted Schrodinger equation (CSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. Two subsets of the CSE, the ACSE and the contraction of the CSE onto the one-particle space, known as the 1,3-CSE, have two important properties: (i) dependence upon only the 3-RDM and (ii) inclusion of all second-order terms when the 3-RDM is reconstructed as only a first-order functional of the 2-RDM. The error in the 1,3-CSE has an important role as a stopping criterion in solving the ACSE for the 2-RDM. Using a computationally more efficient implementation of the ACSE, the author treats a variety of molecules, including H2O, NH3, HCN, and HO3-, in larger basis sets such as correlation-consistent polarized double- and triple-zeta. The ground-state energy of neon is also calculated in a polarized quadruple-zeta basis set with extrapolation to the complete basis-set limit, and the equilibrium bond length and harmonic frequency of N2 are computed with comparison to experimental values. The author observes that increasing the basis set enhances the ability of the ACSE to capture correlation effects in ground-state energies and properties. In the triple-zeta basis set, for example, the ACSE yields energies and properties that are closer in accuracy to coupled cluster with single, double, and triple excitations than to coupled cluster with single and double excitations. In all basis sets, the computed 2-RDMs very closely satisfy known N-representability conditions.

Strong electron correlation in the decomposition reaction of dioxetanone with implications for firefly bioluminescence

Dioxetanone, a key component of the bioluminescence of firefly luciferin, is itself a chemiluminescent molecule due to two conical intersections on its decomposition reaction surface. While recent calculations of firefly luciferin have employed four electrons in four active orbitals [(4,4)] for the dioxetanone moiety, a study of dioxetanone [F. Liu et al., J. Am. Chem. Soc. 131, 6181 (2009)] indicates that a much larger active space is required. Using a variational calculation of the two-electron reduced-density-matrix (2-RDM) [D. A. Mazziotti, Acc. Chem. Res. 39, 207 (2006)], we present the ground-state potential energy surface as a function of active spaces from (4,4) to (20,17) to determine the number of molecular orbitals required for a correct treatment of the strong electron correlation near the conical intersections. Because the 2-RDM method replaces exponentially scaling diagonalizations with polynomially scaling semidefinite optimizations, we readily computed large (18,15) and (20,17) active spaces that are inaccessible to traditional wave function methods. Convergence of the electron correlation with active-space size was measured with complementary RDM-based metrics, the von Neumann entropy of the one-electron RDM as well as the Frobenius and infinity norms of the cumulant 2-RDM. Results show that the electron correlation is not correctly described until the (14,12) active space with small variations present through the (20,17) space. Specifically, for active spaces smaller than (14,12), we demonstrate that at the first conical intersection, the electron in the σ(∗) orbital of the oxygen-oxygen bond is substantially undercorrelated with the electron of the σ orbital and overcorrelated with the electron of the carbonyl oxygens p orbital. Based on these results, we estimate that in contrast to previous treatments, an accurate calculation of the strong electron correlation in firefly luciferin requires an active space of 28 electrons in 25 orbitals, beyond the capacity of traditional multireference wave function methods.