Electron‐pair density decomposition for core–valence separable systems

The electron pair density of a core‐valence separable system can be decomposed into three parts: core‐core, core‐valence, and valence‐valence. The core‐core part has a Hartree‐Fock like structure. The core‐valence part can be written as Γcv (1,2) = γc (1,1)γv (2,2) − γc (1,2)γv (2,1) + γc (2,2)γv (1,1) − γc (2,1)γv (1,2), where only the 1‐matrices from the core and valence orbitals contribute. The valence‐valence part is left to be determined from the reduced frozen‐core type wave function, which often contains the essential information on the electron correlation and the chemical bond. We demonstrate the analysis to the ground state of negative ion Li− and 21Σu+ excited state of the Li2 molecule. © 2012 Wiley Periodicals, Inc.

[1]  David A Mazziotti,et al.  Quantum chemistry without wave functions: two-electron reduced density matrices. , 2006, Accounts of chemical research.

[2]  Paul Adrien Maurice Dirac,et al.  Note on the Interpretation of the Density Matrix in the Many-Electron Problem , 1931, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  E J Baerends,et al.  Charge transfer, double and bond-breaking excitations with time-dependent density matrix functional theory. , 2008, Physical review letters.

[4]  Yu Wang,et al.  Electron correlation in the GK state of the hydrogen molecule. , 2007, The Journal of chemical physics.

[5]  M Nekovee,et al.  Quantum Monte Carlo analysis of exchange and correlation in the strongly inhomogeneous electron gas. , 2000, Physical review letters.

[6]  R. Needs,et al.  Monte Carlo Investigation of Exchange and Correlation in Silicon , 1996 .

[7]  E J Baerends,et al.  Response calculations with an independent particle system with an exact one-particle density matrix. , 2010, Physical review letters.

[8]  John C. Slater,et al.  The Theory of Complex Spectra , 1929 .

[9]  A. J. Coleman,et al.  Reduced Density Matrices , 2000 .

[10]  Kenichi Fukui,et al.  A Molecular Orbital Theory of Reactivity in Aromatic Hydrocarbons , 1952 .

[11]  M. Y. Chou,et al.  Comparative study of density-functional theories of the exchange-correlation hole and energy in silicon , 2001 .

[12]  Wilfried Meyer,et al.  Ground- and excited-state properties of Li2 and Li2+ from ab initio calculations with effective core polarization potentials , 1985 .

[13]  A. Becke.  Correlation energy of an inhomogeneous electron gas: A coordinate‐space model , 1988 .

[14]  Jerzy Cioslowski,et al.  Electron intracule densities and Coulomb holes from energy-derivative two-electron reduced density matrices , 1998 .

[15]  R. Mcweeny,et al.  On the basis of orbital theories , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[16]  J. E. Sienkiewicz,et al.  Calculation of adiabatic potentials of Li2 , 2006 .

[17]  Yu Wang,et al.  Electron correlation in the 3Σg+1 and 2Σu+1 excited state lithium molecule , 2006 .

[18]  Manolo C Per,et al.  Anisotropic intracule densities and electron correlation in H2: a quantum Monte Carlo study. , 2009, The Journal of chemical physics.

[19]  Jesus M. Ugalde,et al.  The separation of the dynamical and non-dynamical electron correlation effects , 2001 .

[20]  J. Perdew,et al.  Accurate density functional for the energy: Real-space cutoff of the gradient expansion for the exchange hole. , 1985, Physical review letters.

[21]  Chen Ning Yang,et al.  Concept of Off-Diagonal Long-Range Order and the Quantum Phases of Liquid He and of Superconductors , 1962 .

[22]  X. López,et al.  A natural orbital functional for multiconfigurational states. , 2011, The Journal of chemical physics.

[23]  R. P. Hosteny,et al.  Ab initio study of the π‐electron states of trans‐butadiene , 1975 .

[24]  Harrison Shull,et al.  NATURAL ORBITALS IN THE QUANTUM THEORY OF TWO-ELECTRON SYSTEMS , 1956 .

[25]  Werner Kutzelnigg,et al.  Direct Determination of Natural Orbitals and Natural Expansion Coefficients of Many‐Electron Wavefunctions. I. Natural Orbitals in the Geminal Product Approximation , 1964 .

[26]  R. Mcweeny.  Some Recent Advances in Density Matrix Theory , 1960 .

[27]  Andreas Savin,et al.  Simple model for the spherically and system-averaged pair density: Results for two-electron atoms , 2005 .

[28]  Gustavo E. Scuseria,et al.  Optimization of density matrix functionals by the Hartree–Fock–Bogoliubov method , 2002 .

[29]  David L. Cooper,et al.  Core‐valence separation in the spin‐coupled wave function: A fully variational treatment based on a second‐order constrained optimization procedure , 1992 .

[30]  Michael W. Schmidt,et al.  The construction and interpretation of MCSCF wavefunctions. , 1998, Annual review of physical chemistry.

[31]  Jesus M. Ugalde,et al.  The evaluation of electronic extracule and intracule densities and related probability functions in terms of Gaussian basis functions , 1991 .

[32]  B. Judd,et al.  Reduced Density Matrices: Coulson's Challenge , 2000 .

[33]  David E. Woon,et al.  Gaussian basis sets for use in correlated molecular calculations. IV. Calculation of static electrical response properties , 1994 .

[34]  S. Goedecker,et al.  Natural Orbital Functional for the Many-Electron Problem , 1998 .

[35]  P. Löwdin.  Quantum Theory of Many-Particle Systems. I. Physical Interpretations by Means of Density Matrices, Natural Spin-Orbitals, and Convergence Problems in the Method of Configurational Interaction , 1955 .

[36]  K. E. Banyard,et al.  Coulomb holes and correlation coefficients for electronic shells: A comparative analysis of several wave functions for Be , 1981 .

[37]  Jian Wang,et al.  Counterintuitive effects of the electron correlation in the first-excited 1Σ+g state of the hydrogen molecule , 2006 .

[38]  Mario Piris,et al.  Iterative diagonalization for orbital optimization in natural orbital functional theory , 2009, J. Comput. Chem..

[39]  A. J. Williamson,et al.  Exchange and correlation in silicon , 1998 .

[40]  A. Becke,et al.  Exchange holes in inhomogeneous systems: A coordinate-space model. , 1989, Physical review. A, General physics.

[41]  R J Boyd,et al.  The Fermi hole in atoms , 1974 .

[42]  E. Gross,et al.  One-body reduced density matrix functionals for the homogeneous electron gas , 2006, cond-mat/0605531.

[43]  Peter G. Lykos,et al.  On the Pi‐Electron Approximation and Its Possible Refinement , 1956 .

[44]  J. Lennard-jones,et al.  The Spatial Correlation of Electrons in Molecules , 1952 .

[45]  V W Maslen,et al.  The Fermi, or Exchange, Hole in Atoms , 1956 .

[46]  Peter M. W. Gill,et al.  Distributions of r1·r2 and p1·p2 in Atoms. , 2012, Journal of chemical theory and computation.

[47]  Jian Wang,et al.  Counterintuitive Coulomb hole around the bond midplane. , 2010, The Journal of chemical physics.

[48]  Mark S. Gordon,et al.  General atomic and molecular electronic structure system , 1993, J. Comput. Chem..

[49]  Harrison Shull,et al.  Reduced-Density-Matrix Theory: The 2-Matrix of Four-Electron Systems , 1967 .

[50]  John C. Slater,et al.  Molecular Energy Levels and Valence Bonds , 1931 .

[51]  P. Dirac.  Note on Exchange Phenomena in the Thomas Atom , 1930, Mathematical Proceedings of the Cambridge Philosophical Society.

[52]  Wlodzimierz Jastrzebski,et al.  The C1Πu and 21Σu+ states in Li2: Experiment and comparison with theory , 2007 .

[53]  Wang,et al.  Generalized gradient approximation for the exchange-correlation hole of a many-electron system. , 1996, Physical review. B, Condensed matter.

[54]  K. E. Banyard,et al.  Coulomb holes and correlation coefficients for electronic shells: The Be‐like ions , 1977 .

[55]  Katarzyna Pernal,et al.  Effective potential for natural spin orbitals. , 2005, Physical review letters.

[56]  Jesus M. Ugalde,et al.  Electron Correlation: Quantum Chemistry’s Holy Grail , 2003 .

[57]  Md. Humayun Kabir,et al.  Doppler-free UV-visible optical–optical double resonance polarization spectroscopy of the 2 1Σu+ double minimum state and the C 1Πu state of Li2 , 2000 .

[58]  Jesus M. Ugalde,et al.  Can correlation bring electrons closer together? , 2009 .

[59]  E. Wigner.  On the Interaction of Electrons in Metals , 1934 .

[60]  John P. Perdew,et al.  Generalized gradient approximation to the angle- and system-averaged exchange hole , 1998 .

[61]  Evert Jan Baerends,et al.  An approximate exchange-correlation hole density as a functional of the natural orbitals , 2002 .

[62]  Peter M W Gill,et al.  A family of intracules, a conjecture and the electron correlation problem. , 2006, Physical chemistry chemical physics : PCCP.

[63]  Robert Erdahl,et al.  Density Matrices and Density Functionals , 1987 .

[64]  Peter M W Gill,et al.  Intracule functional models. II. Analytically integrable kernels. , 2007, The Journal of chemical physics.

[65]  K. E. Banyard,et al.  Coulomb holes and expectation values. I. Explicitly correlated wavefunctions , 1973 .

[66]  C. Coulson,et al.  Electron Correlation in the Ground State of Helium , 1961 .

[67]  Laura K McKemmish,et al.  The nature of electron correlation in a dissociating bond. , 2011, The Journal of chemical physics.

[68]  Jian Wang,et al.  Electron pair density in the lowest 1Σ(u)(+) and 1Σ(g)(+) states of H2. , 2011, The Journal of chemical physics.

[69]  J. C. Slater.  A Simplification of the Hartree-Fock Method , 1951 .

[70]  Ajit J. Thakkar,et al.  Extracules, Intracules, Correlation Holes, Potentials, Coefficients and All That , 1987 .