Matrix‐covariant representation of high‐order configuration interaction and coupled cluster theories

We present the closed form of the reduced density matrices (RDMs) of arbitrary order for configuration interaction (CI) wave functions at any excitation level, up to the full CI. A special operator technique due to Bogoliubov is applied and extended. It focuses on constructions of matrix‐covariant expressions independent of the basis set used. The corresponding variational CI equations are given in an explicit form containing the matrices related to conventional excitation operators. A subsequent transformation of the latter to an irreducible form makes it possible to generate the matrix‐covariant representation for coupled cluster (CC) models. Here this transformation is performed for a simplified high‐order CC scheme somewhat reminiscent of the quadratic CI model. A generalized spin‐flip approximation closely related to high‐order CI and CC models is presented, stressing on a possible inclusion of nondynamical and dynamical correlation effects for multiple bond breaking. A derivation of the full CI and simple CC models for systems involving effective three‐electron interactions is also given, thereby demonstrating the capability of the proposed method to deal with complicated many‐body problem. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Structure of Fukui matrices

Fukui matrices considered as the generalization of the concept of Fukui densities are decomposed into their pairing and unpairing contributions within the theory of the reduced density matrices. Their algebraic structure become clear from this decomposition providing their relationships with the spin density matrices and the irreducible part of the second‐order reduced density matrix cumulant, that is, the explicit contributions of the many‐body or correlation effects. The uncorrelated state function approximation is a simple way to emphasize the physical meaning of these matrices and represents the appropriate starting point for the treatment of a quasi‐analytical model to denote the occurrence of correlation effects.     

Correlated holes and their relationships with reduced density matrices and cumulants

This paper describes a matrix formulation for the correlated hole theory within the framework of the domain-averaged model in many electron systems (atoms, molecules, condensed matter, etc.). General relationships between this quantity and one-particle reduced density matrices for any independent particle or correlated state functions are presented. This formulation turns out to be suitable for computational purposes due to the straightforward introduction of cumulants of two-particle reduced density matrices within the quantum field structure. Numerical calculations in selected simple molecular systems have been performed in order to determine preliminary correlated values for such a quantity.     

Density matrix expansions and their application in the theory of the many-electron systems

A method of calculation of the correlation energy is proposed, which includes the superposition of configurations and the two particle approach. This method is based on the density matrix formalism. The approximate, but N-representable expressions for the reduced density matrices are used. The correlation energy of the beryllium atom is calculated as an example.     

Exact expressions for ensemble functionals from particle number dependence

Some properties of exact ensemble density functionals can be determined by examining the particle number dependence of ground state ensemble density matrices for systems where the integer ground state energies satisfy a convexity condition. The results include the observation that the integral of the product of the functional derivative and Fukui function of functionals that can be expressed as the trace of an operator is particle number independent for particle numbers between successive integers and the integral itself is equal to the difference between functionals evaluated at successive integer particle numbers. Expressions that must be satisfied by 2nd and higher order functional derivatives are formulated and equations that must be satisfied point by point in space are derived. Using the analytic Hooke's atom model, it is shown that commonly used correlation functional approximations do not bear any resemblance to a spatially dependent expression derived from the exact second order functional derivative of the correlation functional. It is also shown that two expressions for the mutual Coulomb energy are not equal when approximate exchange and correlation functionals are used.     

Correlation strength and information entropy

The correlation present in the nondegenerate ground state of an interacting Fermi system is discussed in terms of reduced density matrices and their cumulant expansion. By generalizing a result obtained for the interacting uniform electron gas (correlation induced exchange-hole narrowing), possible measures of the correlation strength in terms of natural occupation numbers (the eigenvalues of the true one-particle density matrix) are introduced. These quantities-the v-order nonidempotency and the information entropy of the natural occupation numbers-result from the correlated many-body wave function and characterize the ground-state correlation in addition to the usual correlation energy. The uniform electron gas serves as a first illustrative example. © 1995 John Wiley & Sons, Inc.     

Correlated one-particle method: numerical results

In a previous paper a correlated one-particle method was formulated, where the effective Hamiltonian was composed of the Fock operator and a correlation potential. The objective was to define a correlated one-particle theory that would give all properties that can be obtained from a one-particle theory. The Fock-space coupled-cluster method was used to construct the infinite-order correlation potential, which yields correct ionization potentials (IP's) and electron affinities (EA's) as the negative of the eigenvalues. The model, however, was largely independent of orbital choice. To exploit the degree of freedom of improving the orbitals, the Brillouin-Brueckner condition is imposed, which leads to an effective Brueckner Hamiltonian. To assess its numerical properties, the effective Brueckner Hamiltonian is approximated through second order in perturbation. Its eigenvalues are the negative of IP's and EA's correct through second order, and its eigenfunctions are second-order Brueckner orbitals. We also give expressions for its energy and density matrix. Different partitioning schemes of the Hamiltonian are used and the intruder state problem is discussed. The results for ionization potentials, electron affinities, dipole moments, energies, and potential curves are given for some sample molecules.     

Study of correlation holes. II. CI calculations on model polyatomic systems

The shape of correlation holes in many-electron systems is at present scarcely known, even where correlated wave functions are available. We investigate here the kind of electron correlation brought about by configuration interaction (CI ), within a given basis set, in the wavefunction of a polyatomic system. The model ring system H₆ (in two different bonding circumstances) and H₁₄ have been chosen for a detailed study, because of their paradigmatic importance. We set out the equal-spin and different-spin correlation holes as obtained from complete CI calculations in H6 and partial ct in H₁₄, both within a minimal basis set. We basically find the spinless correlation as being short range, while the spin-dependent correlation holes show long-range oscillations of antiferromagnetic character. We also present a natural spin-geminal analysis of the two-body reduced density matrices in these systems; we find a peculiarity possibly related to the long-range correlation discussed above. Finally, we compare the electron correlation as given from our CI wavefunction to other pictures of electron correlation, as obtained essentially from alternant molecular orbital wave functions and from the electron–gas literature.     

Influence of electron correlation and degeneracy on the Fukui matrix and extension of frontier molecular orbital theory to correlated quantum chemical methods

The Fukui function is considered as the diagonal element of the Fukui matrix in position space, where the Fukui matrix is the derivative of the one particle density matrix (1DM) with respect to the number of electrons. Diagonalization of the Fukui matrix, expressed in an orthogonal orbital basis, explains why regions in space with negative Fukui functions exist. Using a test set of molecules, electron correlation is found to have a remarkable effect on the eigenvalues of the Fukui matrix. The Fukui matrices at the independent electron model level are mathematically proven to always have an eigenvalue equal to exactly unity while the rest of the eigenvalues possibly differ from zero but sum to zero. The loss of idempotency of the 1DM at correlated levels of theory causes the loss of these properties. The influence of electron correlation is examined in detail and the frontier molecular orbital concept is extended to correlated levels of theory by defining it as the eigenvector of the Fukui matrix with the largest eigenvalue. The effect of degeneracy on the Fukui matrix is examined in detail, revealing that this is another way by which the unity eigenvalue and perfect pairing of eigenvalues can disappear.     

Nonlocal energy density functionals: Models plus some exact general results

Holas and March (Phys Rev 1995 A51, 2040) gave a formally exact expression for the force ??V_xc( r )/? r associated with the exchange‐correlation potential V_xc( r ) of density functional theory. This forged a precise link between first‐ and second‐order density matrices and V_xc( r ). Here models are presented in which these low‐order matrices can be related to the ground‐state electron density. This allows nonlocal energy density functionals to be constructed within the framework of such models. Finally, results emerging from these models have led to the derivation of some exact “nuclear cusp” relations for exchange and correlation energy densities in molecules, clusters, and condensed phases. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Frequency dependent nonlinear optical properties of molecules: Formulation and implementation in the HONDO program

This article summarizes the detailed equations for the time-dependent Hartree–Fock treatment of nonlinear properties for perturbations made up of a static electric field and an oscillating field. Explicit expressions for all nonlinear processes up to third order are obtained in terms of the density matrices at the same order. For processes at second and third order in perturbation, expressions in terms of lower order quantities are also obtained by applying the (2n + 1) theorem of perturbation theory. The corresponding computer implementation in the HONDO program is described.     

Natural bond orbital‐based energy density analysis for correlated methods: Second‐order Møller–Plesset perturbation and coupled‐cluster singles and doubles

Natural bond orbital‐based energy density analysis (NBO‐EDA), which split energies into atomic and bonding contributions, is proposed for correlated methods such as coupled‐cluster singles and doubles (CCSD) and second‐order Møller–Plesset (MP2) perturbation. Applying NBO‐EDA for CCSD and MP2 to ethylene and the Diels–Alder reaction, we are successful in obtaining useful knowledge regarding electron correlation of π‐ and σ‐type orbitals, and clarifying the difference of the reaction barriers and heat of reaction calculated by CCSD and MP2. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Partitioning schemes for the 2-matrix and the pair density

Several schemes are discussed for partitioning the second-order reduced density matrix Γ into two parts, Γ⁰ and Γ′. The Γ⁰s are based on the independent particle model and the Γ′s are corrections due to electron correlation. The difficulties of choosing a Γ⁰ that will serve as a suitable reference point for studying electron correlation are discussed. In order to compare alternative partitioning schemes, an atomic wave function for the ¹S ground state of the Be atom in the configuration-interaction approximation was selected. A fifty-two configuration wave function was computed and contour graphs were made of the total pair density Γ(1 2) and of the “correlation pair density” Γ′(1 2) for several choices of the reference Γ⁰.     

Nuclear magnetic resonance J coupling constant polarizabilities of hydrogen peroxide: A basis set and correlation study

In this article, we present the so far most extended investigation of the calculation of the coupling constant polarizability of a molecule. The components of the coupling constant polarizability are derivatives of the nuclear magnetic resonance (NMR) indirect nuclear spin–spin coupling constant with respect to an external electric field and play an important role for both chiral discrimination and solvation effects on NMR coupling constants. In this study, we illustrate the effects of one‐electron basis sets and electron correlation both at the level of density functional theory as well as second‐order polarization propagator approximation for the small molecule hydrogen peroxide, which allowed us to perform calculations with the largest available basis sets optimized for the calculation of NMR coupling constants. We find a systematic but rather slow convergence with the one‐electron basis set and that augmentation functions are required. We observe also large and nonsystematic correlation effects with significant differences between the density functional and wave function theory methods. © 2012 Wiley Periodicals, Inc.     

Direct selected multireference configuration interaction calculations for large systems using localized orbitals

A selected multireference configuration interaction (CI) method and the corresponding code are presented. It is based on a procedure of localization that permits to obtain well localized occupied and virtual orbitals. Due to the local character of the electron correlation, using local orbitals allows one to neglect long range interactions. In a first step, three topological matrices are constructed, which determine whether two orbitals must be considered as interacting or not. Two of them concern the truncation of the determinant basis, one for occupied/virtual, the second one for dispersive interactions. The third one concerns the truncation of the list of two electron integrals. This approach permits a fine analysis of each kind of approximation and induces a huge reduction of the CI size and of the computational time. The procedure is tested on linear polyene aldehyde chains, dissociation potential energy curve, and reaction energy of a pesticide-Ca(2+) complex and finally on transition energies of a large iron system presenting a light-induced excited spin-state trapping effect.     

Second-order exchange-induction energy of intermolecular interactions from coupled cluster density matrices and their cumulants

A new formulation of the second-order exchange-induction energy of symmetry-adapted perturbation theory is presented. In the proposed formalism the exchange-induction energy is expressed through one- and two-particle reduced density matrices of monomers, which are of zeroth and first order with respect to the effective electrostatic potential of another monomer. The resulting expression is further modified by using the partition of two-particle density matrices into the antisymmetrized product of one-particle density matrices and the remaining cumulant part. The proposed formalism has been applied to the case of closed-shell monomers and for density matrices obtained from the expectation-value expression with coupled cluster singles and doubles wave functions. The performance of the new approach has been demonstrated on several benchmark van der Waals systems, including dimers of argon, water, and ethyne.     

Representable reference density matrices induced by information theory

A class of reference full density matrices and their reduced density matrices is presented. These density matrices are designed to be of value as references from which to describe and measure the effects of electron correlation in atoms, molecules, and solids. A given reference full density matrix is constructed to contain the least possible information consistent with having the (recognized) symmetry properties of—and reducing to the 1-matrix of—a given “true” full density matrix (which in a typical application is constructed from a correlated variational wave function). Therefore, the reduced density matrices derived from are representable and depend only upon the 1-matrix of and the (recognized) symmetry properties of for their construction. Furthermore, the property of containing the least possible information consistent with the given constraints makes these reference density matrices ideally suitable as references from which to describe the electron correlation contained in the “true” full density matrix .