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Advances in chemical physics

Reduced‐Density‐Matrix Mechanics: With Application to Many‐Electron Atoms and Molecules

David A. Mazziotti

Published Mar 9, 2007• 250 citations
Publisher

Abstract

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.

DOI
10.1002/0470106603

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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.
Preface. I: Properties of Reduced Density Matrices. 1. RDMs: How Did We Get Here? A.J. Coleman. 2. Some Theorems on Uniqueness and Reconstruction of Higher-Order Density Matrices M. Rosina. 3. Cumulant Expansions of Reduced Densities, Reduced Density Matrices, and Green's Functions P. Ziesche. 4. On Calculating Approximate and Exact Density Matrices R. Erdahl, B. Jin. II: The Contracted Schroedinger Equation. 5. Density Equation Theory in Chemical Physics H. Nakatsuji. 6. Critical Questions Concerning Iterative Solution of the Contracted Schroedinger Equation C. Valdemoro, L.M. Tel, E. Perez-Romero 7. Cumulants and the Contracted Schroedinger Equation D.A. Mazziotti.. III: Density Matrix Functional Theory. 8. Natural Orbital Functional Theory S. Goedecker, C.J. Umrigar. 9. The Pair Density in Approximate Density Functional Theory: The Hidden Agent N.T. Maitra, K. Burke. 10. Functional N-representability in Density Matrix and Density Functional Theory: An Illustration for Hooke's Atom E.V. Ludena, V. Karasiev, A. Artemiev, D. Gomez.. IV: Electron Intracule and Extracule Densities. 11. Intracule and Extracule Densities: Historical Perspectives and Future Prospects E. Valderrama, J.M. Ugalde, R.J. Boyd. 12. Topology of Electron Correlation J. Cioslowski. 13. Electron-Pair Densities of Atoms T. Koga. Index.
For 50 years, progress toward the direct calculation of the ground-state two-electron reduced density matrix (2-RDM) was stymied from an inability to constrain the 2-RDM to represent an N-electron wave function. Recent advances in theory and optimization have realized the direct calculation of the 2-RDM. A variational 2-RDM procedure, using first-order semidefinite programming, has been shown to capture multireference correlation effects important at nonequilibrium geometries [Mazziotti, Phys. Rev. Lett. 2004, 93, No. 213001]. This method emerged from research on a nonvariational calculation of the 2-RDM by the contracted Schrödinger equation. Both approaches will be discussed and illustrated.
The energy and properties of a many-electron atom or molecule may be directly computed from a variational optimization of a two-electron reduced density matrix (2RDM) that is constrained to represent many-electron quantum systems. In this paper we implement a variational 2RDM method with a representability constraint, known as the T2{T}_{2}T2​ condition. The optimization of the 2RDM is performed with a first-order algorithm for semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] which, because of its lower computational cost in comparison to second-order methods, allows the treatment of larger basis sets. We also derive and implement a spin- and symmetry-adapted formulation of the T2{T}_{2}T2​ condition that significantly decreases the size of the largest block in the T2{T}_{2}T2​ matrix. The T2{T}_{2}T2​ condition, originally derived by Erdahl [Int. J. Quantum Chem. 13, 697 (1978)], was recently applied via a second-order algorithm to atoms and molecules [Z. Zhao et al., J. Chem. Phys. 120, 2095 (2004)]. While these calculations were restricted to molecules at equilibrium geometries in minimal basis sets, we apply the 2RDM method with the T2{T}_{2}T2​ condition to compute the electronic energies of molecules in both minimal and nonminimal basis sets at equilibrium as well as nonequilibrium geometries. Accurate potential energies curves are produced for BH, HF, and N2{\mathrm{N}}_{2}N2​. Results are compared with the 2RDM method without the T2{T}_{2}T2​ condition as well as several wave-function methods.
A Tribute to A. John Coleman - The Tame Mathematician.- Reduced Density Matrices: 1929-1989.- Some Aspects on the Development of the Theory of Reduced Density Matrices and the Representability Problem.- Representability Conditions.- On the Diagonal N-Representability Problem.- Fermion N-Representability Conditions Generated by a Decomposition of the 1-Particle Identity Operator onto Mutually Orthogonal Projection Operators.- The Unitarily Invariant Decomposition of Hermitian Operators.- Building Up N-Electron States with Symplectic Symmetry.- Time Dependent Antisymmetrized Geminal Power Theory Using a Coherent State Formulation.- Griffiths Inequalities for Fermion Systems.- Entropy of Reduced Density Matrices.- A Lower Bound to the Ground State Energy of a Boson System With Fermion Source.- Reduced Density Operators, Their Related von Neumann Density Operators, Close Cousins of These, and Their Physical Interpretation.- Theory and Practice of the Spin-Adapted Reduced Hamiltonians (SRH).- Variational Principle with Built-In Pure State N-Representability Conditions. The N-Electron Case.- Wigner Distributions as Representations of the Density Matrix.- Inter-Relationships Between Various Representations of One-Matrices and Related Densities: A Road Map and an Example ..- Current Problems in Density Functional Theory.- The Interface Between Reduced Density Matrices and Density Functional Theory.- The Physics Underlying the Langreth-Mehl Scheme for Non-Uniform Systems.- Understanding Energy Differences in Density Functional Theory.- Density Functional Calculations of Molecular Bond Energies.- Non-Local Effects on Atomic and Molecular Correlation Energies Studied with a Gradient-Corrected Density Functional.- An Evaluation of Local Electron Correlation Corrections and Non-Local Exchange Corrections to the Hartree-Fock-Slater Method from Calculations on Bond Energies and Electronic Spectra of Molecular Systems.- Correlation Energy Functionals of One-Matrices and Hartree-Fock Densities.- Some Remarks on Scaling Relations in Density Functional Theory.- Deduction of Semiempirical MO Methods from Density Functional Theory.- Charge and Spin Densities in Molecular Solids: Local Density Functional Calculations Versus Experiment.- A Functional of the Two-Particle Density Matrix for the Approximate Calculation of the Electronic Correlation Energy.- Extracules, Intracules, Correlation Holes, Potentials, Coefficients and All That.- The Exact Schroedinger Equation for the Electron Density.- Adiabatic Separation, Broken Symmetries and Geometry Optimization.- Asymptotic Results for Density Matrices and Electron Density in Atoms and Nearly Spherical Molecules.- An Algorithm for Calculating Isoelectronic Changes in Energies, Densities, and One-Matrices.- Atoms and Ions in the Limit of Large Nuclear Charge.- Improved Thomas-Fermi Theory for Atoms.- A Bond Energy from Quantum Mechanics.- Measured Electron Densities and Band Structure Calculations.- X-ray Orthonormal Orbital Model for Crystallography.
The commutation-anticommutation relations of q-electron operators imply a set of N representability conditions [A. J. Coleman, Rev. Mod. Phys. 31, 668 (1963)] for the corresponding q-order reduced density matrices (q-RDM) [C. Valdemoro, An. Fis. 79, 95 (1983); in Structure, Interaction and Reactivity, edited by S. Fraga (Elsevier, Amsterdam, 1992)]. From these conditions, a general and closed-form relation is obtained here. In this equation the part involving RDM's has the same structure as that involving hole reduced density matrices. This relation is the basis of a method for approximating a q-RDM in terms of the r-RDM's [C. Valdemoro, Phys. Rev. A 45, 4462 (1992)] with rq. The derivation of this relation can be simplified significantly by employing a graph method which is described here. These graphs are in a one-to-one correspondence with the elements of the symmetric group of permutations.